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Mathmatics ofSound Analysis

ContentsArticles

Fast Fourier transform 1Discrete Hartley transform 8Discrete Fourier transform 11Fourier analysis 25Sine 32Trigonometric functions 46Complex number 66Microphone practice 83Wave 86

ReferencesArticle Sources and Contributors 100Image Sources, Licenses and Contributors 102

Article LicensesLicense 104

Fast Fourier transform 1

Fast Fourier transformA fast Fourier transform (FFT) is an efficient algorithm to compute the discrete Fourier transform (DFT) and itsinverse. "The FFT has been called the most important numerical algorithm of our lifetime (Strang, 1994)." (Kent &Read 2002, 61) There are many distinct FFT algorithms involving a wide range of mathematics, from simplecomplex-number arithmetic to group theory and number theory; this article gives an overview of the availabletechniques and some of their general properties, while the specific algorithms are described in subsidiary articleslinked below.A DFT decomposes a sequence of values into components of different frequencies. This operation is useful in manyfields (see discrete Fourier transform for properties and applications of the transform) but computing it directly fromthe definition is often too slow to be practical. An FFT is a way to compute the same result more quickly: computinga DFT of N points in the naive way, using the definition, takes O(N2) arithmetical operations, while an FFT cancompute the same result in only O(N log N) operations. The difference in speed can be substantial, especially forlong data sets where N may be in the thousands or millions—in practice, the computation time can be reduced byseveral orders of magnitude in such cases, and the improvement is roughly proportional to N / log(N). This hugeimprovement made many DFT-based algorithms practical; FFTs are of great importance to a wide variety ofapplications, from digital signal processing and solving partial differential equations to algorithms for quickmultiplication of large integers.The most well known FFT algorithms depend upon the factorization of N, but (contrary to popular misconception)there are FFTs with O(N log N) complexity for all N, even for prime N. Many FFT algorithms only depend on thefact that is an th primitive root of unity, and thus can be applied to analogous transforms over any finitefield, such as number-theoretic transforms.Since the inverse DFT is the same as the DFT, but with the opposite sign in the exponent and a 1/N factor, any FFTalgorithm can easily be adapted for it.

Definition and speedAn FFT computes the DFT and produces exactly the same result as evaluating the DFT definition directly; the onlydifference is that an FFT is much faster. (In the presence of round-off error, many FFT algorithms are also muchmore accurate than evaluating the DFT definition directly, as discussed below.)Let x0, ...., xN-1 be complex numbers. The DFT is defined by the formula

Evaluating this definition directly requires O(N2) operations: there are N outputs Xk, and each output requires a sumof N terms. An FFT is any method to compute the same results in O(N log N) operations. More precisely, all knownFFT algorithms require Θ(N log N) operations (technically, O only denotes an upper bound), although there is noknown proof that better complexity is impossible.To illustrate the savings of an FFT, consider the count of complex multiplications and additions. Evaluating theDFT's sums directly involves N2 complex multiplications and N(N − 1) complex additions [of which O(N) operationscan be saved by eliminating trivial operations such as multiplications by 1]. The well-known radix-2 Cooley–Tukeyalgorithm, for N a power of 2, can compute the same result with only (N/2) log2 N complex multiplies (again,ignoring simplifications of multiplications by 1 and similar) and N log2N complex additions. In practice, actualperformance on modern computers is usually dominated by factors other than arithmetic and is a complicated subject(see, e.g., Frigo & Johnson, 2005), but the overall improvement from O(N2) to O(N log N) remains.

Fast Fourier transform 2

Algorithms

Cooley–Tukey algorithmBy far the most common FFT is the Cooley–Tukey algorithm. This is a divide and conquer algorithm thatrecursively breaks down a DFT of any composite size N = N1N2 into many smaller DFTs of sizes N1 and N2, alongwith O(N) multiplications by complex roots of unity traditionally called twiddle factors (after Gentleman and Sande,1966).This method (and the general idea of an FFT) was popularized by a publication of J. W. Cooley and J. W. Tukey in1965, but it was later discovered (Heideman & Burrus, 1984) that those two authors had independently re-inventedan algorithm known to Carl Friedrich Gauss around 1805 (and subsequently rediscovered several times in limitedforms).

The most well-known use of the Cooley–Tukey algorithm is to divide the transform into two pieces of size ateach step, and is therefore limited to power-of-two sizes, but any factorization can be used in general (as was knownto both Gauss and Cooley/Tukey). These are called the radix-2 and mixed-radix cases, respectively (and othervariants such as the split-radix FFT have their own names as well). Although the basic idea is recursive, mosttraditional implementations rearrange the algorithm to avoid explicit recursion. Also, because the Cooley–Tukeyalgorithm breaks the DFT into smaller DFTs, it can be combined arbitrarily with any other algorithm for the DFT,such as those described below.

Other FFT algorithms

There are other FFT algorithms distinct from Cooley–Tukey. For with coprime and , one canuse the Prime-Factor (Good-Thomas) algorithm (PFA), based on the Chinese Remainder Theorem, to factorize theDFT similarly to Cooley–Tukey but without the twiddle factors. The Rader-Brenner algorithm (1976) is aCooley–Tukey-like factorization but with purely imaginary twiddle factors, reducing multiplications at the cost ofincreased additions and reduced numerical stability; it was later superseded by the split-radix variant ofCooley–Tukey (which achieves the same multiplication count but with fewer additions and without sacrificingaccuracy). Algorithms that recursively factorize the DFT into smaller operations other than DFTs include the Bruunand QFT algorithms. (The Rader-Brenner and QFT algorithms were proposed for power-of-two sizes, but it ispossible that they could be adapted to general composite . Bruun's algorithm applies to arbitrary even compositesizes.) Bruun's algorithm, in particular, is based on interpreting the FFT as a recursive factorization of thepolynomial , here into real-coefficient polynomials of the form and .Another polynomial viewpoint is exploited by the Winograd algorithm, which factorizes into cyclotomicpolynomials—these often have coefficients of 1, 0, or −1, and therefore require few (if any) multiplications, soWinograd can be used to obtain minimal-multiplication FFTs and is often used to find efficient algorithms for smallfactors. Indeed, Winograd showed that the DFT can be computed with only irrational multiplications,leading to a proven achievable lower bound on the number of multiplications for power-of-two sizes; unfortunately,this comes at the cost of many more additions, a tradeoff no longer favorable on modern processors with hardwaremultipliers. In particular, Winograd also makes use of the PFA as well as an algorithm by Rader for FFTs of primesizes.Rader's algorithm, exploiting the existence of a generator for the multiplicative group modulo prime , expresses aDFT of prime size as a cyclic convolution of (composite) size , which can then be computed by a pair ofordinary FFTs via the convolution theorem (although Winograd uses other convolution methods). Anotherprime-size FFT is due to L. I. Bluestein, and is sometimes called the chirp-z algorithm; it also re-expresses a DFT asa convolution, but this time of the same size (which can be zero-padded to a power of two and evaluated by radix-2Cooley–Tukey FFTs, for example), via the identity .

Fast Fourier transform 3

FFT algorithms specialized for real and/or symmetric dataIn many applications, the input data for the DFT are purely real, in which case the outputs satisfy the symmetry

and efficient FFT algorithms have been designed for this situation (see e.g. Sorensen, 1987). One approach consistsof taking an ordinary algorithm (e.g. Cooley–Tukey) and removing the redundant parts of the computation, savingroughly a factor of two in time and memory. Alternatively, it is possible to express an even-length real-input DFT asa complex DFT of half the length (whose real and imaginary parts are the even/odd elements of the original realdata), followed by O(N) post-processing operations.It was once believed that real-input DFTs could be more efficiently computed by means of the discrete Hartleytransform (DHT), but it was subsequently argued that a specialized real-input DFT algorithm (FFT) can typically befound that requires fewer operations than the corresponding DHT algorithm (FHT) for the same number of inputs.Bruun's algorithm (above) is another method that was initially proposed to take advantage of real inputs, but it hasnot proved popular.There are further FFT specializations for the cases of real data that have even/odd symmetry, in which case one cangain another factor of (roughly) two in time and memory and the DFT becomes the discrete cosine/sine transform(s)(DCT/DST). Instead of directly modifying an FFT algorithm for these cases, DCTs/DSTs can also be computed viaFFTs of real data combined with O(N) pre/post processing.

Computational issues

Bounds on complexity and operation countsA fundamental question of longstanding theoretical interest is to prove lower bounds on the complexity and exactoperation counts of fast Fourier transforms, and many open problems remain. It is not even rigorously provedwhether DFTs truly require (i.e., order or greater) operations, even for the simple case ofpower of two sizes, although no algorithms with lower complexity are known. In particular, the count of arithmeticoperations is usually the focus of such questions, although actual performance on modern-day computers isdetermined by many other factors such as cache or CPU pipeline optimization.Following pioneering work by Winograd (1978), a tight lower bound is known for the number of realmultiplications required by an FFT. It can be shown that only irrational realmultiplications are required to compute a DFT of power-of-two length . Moreover, explicit algorithmsthat achieve this count are known (Heideman & Burrus, 1986; Duhamel, 1990). Unfortunately, these algorithmsrequire too many additions to be practical, at least on modern computers with hardware multipliers.A tight lower bound is not known on the number of required additions, although lower bounds have been provedunder some restrictive assumptions on the algorithms. In 1973, Morgenstern proved an lower boundon the addition count for algorithms where the multiplicative constants have bounded magnitudes (which is true formost but not all FFT algorithms). Pan (1986) proved an lower bound assuming a bound on a measureof the FFT algorithm's "asynchronicity", but the generality of this assumption is unclear. For the case ofpower-of-two , Papadimitriou (1979) argued that the number of complex-number additions achievedby Cooley–Tukey algorithms is optimal under certain assumptions on the graph of the algorithm (his assumptionsimply, among other things, that no additive identities in the roots of unity are exploited). (This argument wouldimply that at least real additions are required, although this is not a tight bound because extra additionsare required as part of complex-number multiplications.) Thus far, no published FFT algorithm has achieved fewerthan complex-number additions (or their equivalent) for power-of-two .A third problem is to minimize the total number of real multiplications and additions, sometimes called the "arithmetic complexity" (although in this context it is the exact count and not the asymptotic complexity that is being

Fast Fourier transform 4

considered). Again, no tight lower bound has been proven. Since 1968, however, the lowest published count forpower-of-two was long achieved by the split-radix FFT algorithm, which requires real multiplications and additions

for . This was recently reduced to (Johnson and Frigo, 2007; Lundy and Van Buskirk, 2007).

Most of the attempts to lower or prove the complexity of FFT algorithms have focused on the ordinary complex-datacase, because it is the simplest. However, complex-data FFTs are so closely related to algorithms for relatedproblems such as real-data FFTs, discrete cosine transforms, discrete Hartley transforms, and so on, that anyimprovement in one of these would immediately lead to improvements in the others (Duhamel & Vetterli, 1990).

Accuracy and approximationsAll of the FFT algorithms discussed below compute the DFT exactly (in exact arithmetic, i.e. neglectingfloating-point errors). A few "FFT" algorithms have been proposed, however, that compute the DFT approximately,with an error that can be made arbitrarily small at the expense of increased computations. Such algorithms trade theapproximation error for increased speed or other properties. For example, an approximate FFT algorithm byEdelman et al. (1999) achieves lower communication requirements for parallel computing with the help of a fastmultipole method. A wavelet-based approximate FFT by Guo and Burrus (1996) takes sparse inputs/outputs(time/frequency localization) into account more efficiently than is possible with an exact FFT. Another algorithm forapproximate computation of a subset of the DFT outputs is due to Shentov et al. (1995). Only the Edelman algorithmworks equally well for sparse and non-sparse data, however, since it is based on the compressibility (rank deficiency)of the Fourier matrix itself rather than the compressibility (sparsity) of the data.Even the "exact" FFT algorithms have errors when finite-precision floating-point arithmetic is used, but these errorsare typically quite small; most FFT algorithms, e.g. Cooley–Tukey, have excellent numerical properties as aconsequence of the pairwise summation structure of the algorithms. The upper bound on the relative error for theCooley–Tukey algorithm is O(ε log N), compared to O(εN3/2) for the naïve DFT formula (Gentleman and Sande,1966), where ε is the machine floating-point relative precision. In fact, the root mean square (rms) errors are muchbetter than these upper bounds, being only O(ε √log N) for Cooley–Tukey and O(ε √N) for the naïve DFT(Schatzman, 1996). These results, however, are very sensitive to the accuracy of the twiddle factors used in the FFT(i.e. the trigonometric function values), and it is not unusual for incautious FFT implementations to have much worseaccuracy, e.g. if they use inaccurate trigonometric recurrence formulas. Some FFTs other than Cooley–Tukey, suchas the Rader-Brenner algorithm, are intrinsically less stable.In fixed-point arithmetic, the finite-precision errors accumulated by FFT algorithms are worse, with rms errorsgrowing as O(√N) for the Cooley–Tukey algorithm (Welch, 1969). Moreover, even achieving this accuracy requirescareful attention to scaling in order to minimize the loss of precision, and fixed-point FFT algorithms involverescaling at each intermediate stage of decompositions like Cooley–Tukey.To verify the correctness of an FFT implementation, rigorous guarantees can be obtained in O(N log N) time by asimple procedure checking the linearity, impulse-response, and time-shift properties of the transform on randominputs (Ergün, 1995).

Fast Fourier transform 5

Multidimensional FFTsAs defined in the multidimensional DFT article, the multidimensional DFT

transforms an array with a -dimensional vector of indices by a set of nestedsummations (over for each ), where the division , defined as

, is performed element-wise. Equivalently, it is simply the composition of asequence of sets of one-dimensional DFTs, performed along one dimension at a time (in any order).This compositional viewpoint immediately provides the simplest and most common multidimensional DFTalgorithm, known as the row-column algorithm (after the two-dimensional case, below). That is, one simplyperforms a sequence of one-dimensional FFTs (by any of the above algorithms): first you transform along the dimension, then along the dimension, and so on (or actually, any ordering will work). This method is easilyshown to have the usual complexity, where is the total number of data pointstransformed. In particular, there are transforms of size , etcetera, so the complexity of the sequence ofFFTs is:

In two dimensions, the can be viewed as an matrix, and this algorithm corresponds to first performingthe FFT of all the rows and then of all the columns (or vice versa), hence the name.In more than two dimensions, it is often advantageous for cache locality to group the dimensions recursively. Forexample, a three-dimensional FFT might first perform two-dimensional FFTs of each planar "slice" for each fixed

, and then perform the one-dimensional FFTs along the direction. More generally, an asymptotically optimalcache-oblivious algorithm consists of recursively dividing the dimensions into two groups and

that are transformed recursively (rounding if is not even) (see Frigo and Johnson, 2005). Still,this remains a straightforward variation of the row-column algorithm that ultimately requires only a one-dimensionalFFT algorithm as the base case, and still has complexity. Yet another variation is to perform matrixtranspositions in between transforming subsequent dimensions, so that the transforms operate on contiguous data;this is especially important for out-of-core and distributed memory situations where accessing non-contiguous data isextremely time-consuming.There are other multidimensional FFT algorithms that are distinct from the row-column algorithm, although all ofthem have complexity. Perhaps the simplest non-row-column FFT is the vector-radix FFT algorithm,which is a generalization of the ordinary Cooley–Tukey algorithm where one divides the transform dimensions by avector of radices at each step. (This may also have cache benefits.) The simplest case ofvector-radix is where all of the radices are equal (e.g. vector-radix-2 divides all of the dimensions by two), but this isnot necessary. Vector radix with only a single non-unit radix at a time, i.e. , isessentially a row-column algorithm. Other, more complicated, methods include polynomial transform algorithms dueto Nussbaumer (1977), which view the transform in terms of convolutions and polynomial products. See Duhameland Vetterli (1990) for more information and references.

Fast Fourier transform 6

Other generalizationsAn O(N5/2 log N) generalization to spherical harmonics on the sphere S2 with N2 nodes was described byMohlenkamp (1999), along with an algorithm conjectured (but not proven) to have O(N2 log2 N) complexity;Mohlenkamp also provides an implementation in the libftsh library [1]. A spherical-harmonic algorithm with O(N2

log N) complexity is described by Rokhlin and Tygert (2006).Various groups have also published "FFT" algorithms for non-equispaced data, as reviewed in Potts et al. (2001).Such algorithms do not strictly compute the DFT (which is only defined for equispaced data), but rather someapproximation thereof (a non-uniform discrete Fourier transform, or NDFT, which itself is often computed onlyapproximately).

References[1] http:/ / www. math. ohiou. edu/ ~mjm/ research/ libftsh. html

• Brenner, N.; Rader, C. (1976). "A New Principle for Fast Fourier Transformation". IEEE Acoustics, Speech &Signal Processing 24 (3): 264–266. doi:10.1109/TASSP.1976.1162805.

• Brigham, E. O. (2002). The Fast Fourier Transform. New York: Prentice-Hall• Cooley, James W.; Tukey, John W. (1965). "An algorithm for the machine calculation of complex Fourier series".

Math. Comput. 19 (90): 297–301. doi:10.1090/S0025-5718-1965-0178586-1.• Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein, 2001. Introduction to Algorithms,

2nd. ed. MIT Press and McGraw-Hill. ISBN 0-262-03293-7. Especially chapter 30, "Polynomials and the FFT."• Duhamel, Pierre (1990). "Algorithms meeting the lower bounds on the multiplicative complexity of length-

DFTs and their connection with practical algorithms". IEEE Trans. Acoust. Speech. Sig. Proc. 38 (9): 1504–151.doi:10.1109/29.60070.

• P. Duhamel and M. Vetterli, 1990, Fast Fourier transforms: a tutorial review and a state of the art(doi:10.1016/0165-1684(90)90158-U), Signal Processing 19: 259–299.

• A. Edelman, P. McCorquodale, and S. Toledo, 1999, The Future Fast Fourier Transform?(doi:10.1137/S1064827597316266), SIAM J. Sci. Computing 20: 1094–1114.

• D. F. Elliott, & K. R. Rao, 1982, Fast transforms: Algorithms, analyses, applications. New York: AcademicPress.

• Funda Ergün, 1995, Testing multivariate linear functions: Overcoming the generator bottleneck(doi:10.1145/225058.225167), Proc. 27th ACM Symposium on the Theory of Computing: 407–416.

• M. Frigo and S. G. Johnson, 2005, " The Design and Implementation of FFTW3 (http:/ / fftw. org/fftw-paper-ieee. pdf)," Proceedings of the IEEE 93: 216–231.

• Carl Friedrich Gauss, 1866. "Nachlass: Theoria interpolationis methodo nova tractata," Werke band 3, 265–327.Göttingen: Königliche Gesellschaft der Wissenschaften.

• W. M. Gentleman and G. Sande, 1966, "Fast Fourier transforms—for fun and profit," Proc. AFIPS 29: 563–578.doi:10.1145/1464291.1464352

• H. Guo and C. S. Burrus, 1996, Fast approximate Fourier transform via wavelets transform(doi:10.1117/12.255236), Proc. SPIE Intl. Soc. Opt. Eng. 2825: 250–259.

• H. Guo, G. A. Sitton, C. S. Burrus, 1994, The Quick Discrete Fourier Transform(doi:10.1109/ICASSP.1994.389994), Proc. IEEE Conf. Acoust. Speech and Sig. Processing (ICASSP) 3:445–448.

• Heideman, M. T.; Johnson, D. H.; Burrus, C. S. (1984). "Gauss and the history of the fast Fourier transform".IEEE ASSP Magazine 1 (4): 14–21. doi:10.1109/MASSP.1984.1162257.

• Heideman, Michael T.; Burrus, C. Sidney (1986). "On the number of multiplications necessary to compute alength- DFT". IEEE Trans. Acoust. Speech. Sig. Proc. 34 (1): 91–95. doi:10.1109/TASSP.1986.1164785.

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• S. G. Johnson and M. Frigo, 2007. " A modified split-radix FFT with fewer arithmetic operations (http:/ / www.fftw. org/ newsplit. pdf)," IEEE Trans. Signal Processing 55 (1): 111–119.

• T. Lundy and J. Van Buskirk, 2007. "A new matrix approach to real FFTs and convolutions of length 2k,"Computing 80 (1): 23-45.

• Kent, Ray D. and Read, Charles (2002). Acoustic Analysis of Speech. ISBN 0-7693-0112-6. Cites Strang, G.(1994)/May-June). Wavelets. American Scientist, 82, 250-255.

• Morgenstern, Jacques (1973). "Note on a lower bound of the linear complexity of the fast Fourier transform". J.ACM 20 (2): 305–306. doi:10.1145/321752.321761.

• Mohlenkamp, M. J. (1999). "A fast transform for spherical harmonics" (http:/ / www. math. ohiou. edu/ ~mjm/research/ MOHLEN1999P. pdf). J. Fourier Anal. Appl. 5 (2-3): 159–184. doi:10.1007/BF01261607.

• Nussbaumer, H. J. (1977). "Digital filtering using polynomial transforms". Electronics Lett. 13 (13): 386–387.doi:10.1049/el:19770280.

• V. Pan, 1986, The trade-off between the additive complexity and the asyncronicity of linear and bilinearalgorithms (doi:10.1016/0020-0190(86)90035-9), Information Proc. Lett. 22: 11-14.

• Christos H. Papadimitriou, 1979, Optimality of the fast Fourier transform (doi:10.1145/322108.322118), J. ACM26: 95-102.

• D. Potts, G. Steidl, and M. Tasche, 2001. " Fast Fourier transforms for nonequispaced data: A tutorial (http:/ /www. tu-chemnitz. de/ ~potts/ paper/ ndft. pdf)", in: J.J. Benedetto and P. Ferreira (Eds.), Modern SamplingTheory: Mathematics and Applications (Birkhauser).

• Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Chapter 12. Fast Fourier Transform" (http:/ /apps. nrbook. com/ empanel/ index. html#pg=600), Numerical Recipes: The Art of Scientific Computing (3rd ed.),New York: Cambridge University Press, ISBN 978-0-521-88068-8

• Rokhlin, Vladimir; Tygert, Mark (2006). "Fast algorithms for spherical harmonic expansions". SIAM J. Sci.Computing 27 (6): 1903–1928. doi:10.1137/050623073.

• James C. Schatzman, 1996, Accuracy of the discrete Fourier transform and the fast Fourier transform (http:/ /portal. acm. org/ citation. cfm?id=240432), SIAM J. Sci. Comput. 17: 1150–1166.

• Shentov, O. V.; Mitra, S. K.; Heute, U.; Hossen, A. N. (1995). "Subband DFT. I. Definition, interpretations andextensions". Signal Processing 41 (3): 261–277. doi:10.1016/0165-1684(94)00103-7.

• Sorensen, H. V.; Jones, D. L.; Heideman, M. T.; Burrus, C. S. (1987). "Real-valued fast Fourier transformalgorithms". IEEE Trans. Acoust. Speech Sig. Processing 35 (35): 849–863. doi:10.1109/TASSP.1987.1165220.See also Sorensen, H.; Jones, D.; Heideman, M.; Burrus, C. (1987). "Corrections to "Real-valued fast Fouriertransform algorithms"". IEEE Transactions on Acoustics, Speech, and Signal Processing 35 (9): 1353–1353.doi:10.1109/TASSP.1987.1165284.

• Welch, Peter D. (1969). "A fixed-point fast Fourier transform error analysis". IEEE Trans. Audio Electroacoustics17 (2): 151–157. doi:10.1109/TAU.1969.1162035.

• Winograd, S. (1978). "On computing the discrete Fourier transform". Math. Computation 32 (141): 175–199.doi:10.1090/S0025-5718-1978-0468306-4. JSTOR 2006266

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External links• Fast Fourier Algorithm (http:/ / www. cs. pitt. edu/ ~kirk/ cs1501/ animations/ FFT. html)• Fast Fourier Transforms (http:/ / cnx. org/ content/ col10550/ ), Connexions online book edited by C. Sidney

Burrus, with chapters by C. Sidney Burrus, Ivan Selesnick, Markus Pueschel, Matteo Frigo, and Steven G.Johnson (2008).

• Links to FFT code and information online. (http:/ / www. fftw. org/ links. html)• National Taiwan University – FFT (http:/ / www. cmlab. csie. ntu. edu. tw/ cml/ dsp/ training/ coding/ transform/

fft. html)• FFT programming in C++ — Cooley–Tukey algorithm. (http:/ / www. librow. com/ articles/ article-10)• Online documentation, links, book, and code. (http:/ / www. jjj. de/ fxt/ )• Using FFT to construct aggregate probability distributions (http:/ / www. vosesoftware. com/ ModelRiskHelp/

index. htm#Aggregate_distributions/ Aggregate_modeling_-_Fast_Fourier_Transform_FFT_method. htm)• Sri Welaratna, " 30 years of FFT Analyzers (http:/ / www. dataphysics. com/ support/ library/ downloads/ articles/

DP-30 Years of FFT. pdf)", Sound and Vibration (January 1997, 30th anniversary issue). A historical review ofhardware FFT devices.

• FFT Basics and Case Study Using Multi-Instrument (http:/ / www. multi-instrument. com/ doc/ D1002/FFT_Basics_and_Case_Study_using_Multi-Instrument_D1002. pdf)

• FFT Textbook notes, PPTs, Videos (http:/ / numericalmethods. eng. usf. edu/ topics/ fft. html) at HolisticNumerical Methods Institute.

• ALGLIB FFT Code (http:/ / www. alglib. net/ fasttransforms/ fft. php) GPL Licensed multilanguage (VBA, C++,Pascal, etc.) numerical analysis and data processing library.

Discrete Hartley transformA discrete Hartley transform (DHT) is a Fourier-related transform of discrete, periodic data similar to the discreteFourier transform (DFT), with analogous applications in signal processing and related fields. Its main distinctionfrom the DFT is that it transforms real inputs to real outputs, with no intrinsic involvement of complex numbers. Justas the DFT is the discrete analogue of the continuous Fourier transform, the DHT is the discrete analogue of thecontinuous Hartley transform, introduced by R. V. L. Hartley in 1942.Because there are fast algorithms for the DHT analogous to the fast Fourier transform (FFT), the DHT was originallyproposed by R. N. Bracewell in 1983 as a more efficient computational tool in the common case where the data arepurely real. It was subsequently argued, however, that specialized FFT algorithms for real inputs or outputs canordinarily be found with slightly fewer operations than any corresponding algorithm for the DHT (see below).

DefinitionFormally, the discrete Hartley transform is a linear, invertible function H : Rn -> Rn (where R denotes the set ofreal numbers). The N real numbers x0, ...., xN-1 are transformed into the N real numbers H0, ..., HN-1 according to theformula

.

The combination is sometimes denoted , and should be contrasted

with the that appears in the DFT definition (where i is the imaginary unit).As with the DFT, the overall scale factor in front of the transform and the sign of the sine term are a matter ofconvention. Although these conventions occasionally vary between authors, they do not affect the essential

Discrete Hartley transform 9

properties of the transform.

PropertiesThe transform can be interpreted as the multiplication of the vector (x0, ...., xN-1) by an N-by-N matrix; therefore, thediscrete Hartley transform is a linear operator. The matrix is invertible; the inverse transformation, which allows oneto recover the xn from the Hk, is simply the DHT of Hk multiplied by 1/N. That is, the DHT is its own inverse(involutary), up to an overall scale factor.The DHT can be used to compute the DFT, and vice versa. For real inputs xn, the DFT output Xk has a real part (Hk +HN-k)/2 and an imaginary part (HN-k - Hk)/2. Conversely, the DHT is equivalent to computing the DFT of xnmultiplied by 1+i, then taking the real part of the result.As with the DFT, a cyclic convolution z = x*y of two vectors x = (xn) and y = (yn) to produce a vector z = (zn), all oflength N, becomes a simple operation after the DHT. In particular, suppose that the vectors X, Y, and Z denote theDHT of x, y, and z respectively. Then the elements of Z are given by:

where we take all of the vectors to be periodic in N (XN = X0, etcetera). Thus, just as the DFT transforms aconvolution into a pointwise multiplication of complex numbers (pairs of real and imaginary parts), the DHTtransforms a convolution into a simple combination of pairs of real frequency components. The inverse DHT thenyields the desired vector z. In this way, a fast algorithm for the DHT (see below) yields a fast algorithm forconvolution. (Note that this is slightly more expensive than the corresponding procedure for the DFT, not includingthe costs of the transforms below, because the pairwise operation above requires 8 real-arithmetic operationscompared to the 6 of a complex multiplication. This count doesn't include the division by 2, which can be absorbede.g. into the 1/N normalization of the inverse DHT.)

Fast algorithmsJust as for the DFT, evaluating the DHT definition directly would require O(N2) arithmetical operations (see Big Onotation). There are fast algorithms similar to the FFT, however, that compute the same result in only O(N log N)operations. Nearly every FFT algorithm, from Cooley-Tukey to Prime-Factor to Winograd (Sorensen et al., 1985) toBruun's (Bini & Bozzo, 1993), has a direct analogue for the discrete Hartley transform. (However, a few of the moreexotic FFT algorithms, such as the QFT, have not yet been investigated in the context of the DHT.)In particular, the DHT analogue of the Cooley-Tukey algorithm is commonly known as the Fast HartleyTransform (FHT) algorithm, and was first described by Bracewell in 1984. This FHT algorithm, at least whenapplied to power-of-two sizes N, is the subject of the United States patent number 4,646,256, issued in 1987 toStanford University. Stanford placed this patent in the public domain in 1994 (Bracewell, 1995).As mentioned above, DHT algorithms are typically slightly less efficient (in terms of the number of floating-pointoperations) than the corresponding DFT algorithm (FFT) specialized for real inputs (or outputs). This was firstargued by Sorensen et al. (1987) and Duhamel & Vetterli (1987). The latter authors obtained what appears to be thelowest published operation count for the DHT of power-of-two sizes, employing a split-radix algorithm (similar tothe split-radix FFT) that breaks a DHT of length N into a DHT of length N/2 and two real-input DFTs (not DHTs) oflength N/4. In this way, they argued that a DHT of power-of-two length can be computed with, at best, 2 moreadditions than the corresponding number of arithmetic operations for the real-input DFT.On present-day computers, performance is determined more by cache and CPU pipeline considerations than by strict operation counts, and a slight difference in arithmetic cost is unlikely to be significant. Since FHT and real-input FFT algorithms have similar computational structures, neither appears to have a substantial a priori speed advantage (Popovic and Sevic, 1994). As a practical matter, highly optimized real-input FFT libraries are available from many

Discrete Hartley transform 10

sources (e.g. from CPU vendors such as Intel), whereas highly optimized DHT libraries are less common.On the other hand, the redundant computations in FFTs due to real inputs are more difficult to eliminate for largeprime N, despite the existence of O(N log N) complex-data algorithms for such cases, because the redundancies arehidden behind intricate permutations and/or phase rotations in those algorithms. In contrast, a standard prime-sizeFFT algorithm, Rader's algorithm, can be directly applied to the DHT of real data for roughly a factor of two lesscomputation than that of the equivalent complex FFT (Frigo and Johnson, 2005). On the other hand, anon-DHT-based adaptation of Rader's algorithm for real-input DFTs is also possible (Chu & Burrus, 1982).

References• R. N. Bracewell, "Discrete Hartley transform," J. Opt. Soc. Am. 73 (12), 1832–1835 (1983).• R. N. Bracewell, "The fast Hartley transform," Proc. IEEE 72 (8), 1010–1018 (1984).• R. N. Bracewell, The Hartley Transform (Oxford Univ. Press, New York, 1986).• R. N. Bracewell, "Computing with the Hartley Transform," Computers in Physics 9 (4), 373–379 (1995).• R. V. L. Hartley, "A more symmetrical Fourier analysis applied to transmission problems," Proc. IRE 30,

144–150 (1942).• H. V. Sorensen, D. L. Jones, C. S. Burrus, and M. T. Heideman, "On computing the discrete Hartley transform,"

IEEE Trans. Acoust. Speech Sig. Processing ASSP-33 (4), 1231–1238 (1985).• H. V. Sorensen, D. L. Jones, M. T. Heideman, and C. S. Burrus, "Real-valued fast Fourier transform algorithms,"

IEEE Trans. Acoust. Speech Sig. Processing ASSP-35 (6), 849–863 (1987).• Pierre Duhamel and Martin Vetterli, "Improved Fourier and Hartley transform algorithms: application to cyclic

convolution of real data," IEEE Trans. Acoust. Speech Sig. Processing ASSP-35, 818–824 (1987).• Mark A. O'Neill, "Faster than Fast Fourier", Byte 13(4):293-300, (1988).• J. Hong and M. Vetterli and P. Duhamel, "Basefield transforms with the convolution property," Proc. IEEE 82

(3), 400-412 (1994).• D. A. Bini and E. Bozzo, "Fast discrete transform by means of eigenpolynomials," Computers & Mathematics

(with Applications) 26 (9), 35–52 (1993).• Miodrag Popović and Dragutin Šević, "A new look at the comparison of the fast Hartley and Fourier transforms,"

IEEE Trans. Signal Processing 42 (8), 2178-2182 (1994).• Matteo Frigo and Steven G. Johnson, "The Design and Implementation of FFTW3 [1]," Proc. IEEE 93 (2),

216–231 (2005).• S. Chu and C. Burrus, "A prime factor FTT [sic] algorithm using distributed arithmetic," IEEE Transactions on

Acoustics, Speech, and Signal Processing 30 (2), 217–227 (1982).

References[1] http:/ / fftw. org/ fftw-paper-ieee. pdf

Discrete Fourier transform 11

Discrete Fourier transformIn mathematics, the discrete Fourier transform (DFT) is a specific kind of discrete transform, used in Fourieranalysis. It transforms one function into another, which is called the frequency domain representation, or simply theDFT, of the original function (which is often a function in the time domain). But the DFT requires an input functionthat is discrete and whose non-zero values have a limited (finite) duration. Such inputs are often created by samplinga continuous function, like a person's voice. Unlike the discrete-time Fourier transform (DTFT), it only evaluatesenough frequency components to reconstruct the finite segment that was analyzed. Using the DFT implies that thefinite segment that is analyzed is one period of an infinitely extended periodic signal; if this is not actually true, awindow function has to be used to reduce the artifacts in the spectrum. For the same reason, the inverse DFT cannotreproduce the entire time domain, unless the input happens to be periodic (forever). Therefore it is often said that theDFT is a transform for Fourier analysis of finite-domain discrete-time functions. The sinusoidal basis functions ofthe decomposition have the same properties.The input to the DFT is a finite sequence of real or complex numbers (with more abstract generalizations discussedbelow), making the DFT ideal for processing information stored in computers. In particular, the DFT is widelyemployed in signal processing and related fields to analyze the frequencies contained in a sampled signal, to solvepartial differential equations, and to perform other operations such as convolutions or multiplying large integers. Akey enabling factor for these applications is the fact that the DFT can be computed efficiently in practice using a fastFourier transform (FFT) algorithm.FFT algorithms are so commonly employed to compute DFTs that the term "FFT" is often used to mean "DFT" incolloquial settings. Formally, there is a clear distinction: "DFT" refers to a mathematical transformation or function,regardless of how it is computed, whereas "FFT" refers to a specific family of algorithms for computing DFTs. Theterminology is further blurred by the (now rare) synonym finite Fourier transform for the DFT, which apparentlypredates the term "fast Fourier transform" (Cooley et al., 1969) but has the same initialism.

DefinitionThe sequence of N complex numbers x0, ..., xN−1 is transformed into the sequence of N complex numbers X0, ...,XN−1 by the DFT according to the formula:

where i is the imaginary unit and is a primitive N'th root of unity. (This expression can also be written interms of a DFT matrix; when scaled appropriately it becomes a unitary matrix and the Xk can thus be viewed ascoefficients of x in an orthonormal basis.)The transform is sometimes denoted by the symbol , as in or or .The inverse discrete Fourier transform (IDFT) is given by

A simple description of these equations is that the complex numbers represent the amplitude and phase of thedifferent sinusoidal components of the input "signal" . The DFT computes the from the , while theIDFT shows how to compute the as a sum of sinusoidal components with frequency

cycles per sample. By writing the equations in this form, we are making extensive use of Euler's formula to expresssinusoids in terms of complex exponentials, which are much easier to manipulate. In the same way, by writing in polar form, we obtain the sinusoid amplitude and phase from the complex modulus and argument of

, respectively:

Discrete Fourier transform 12

where atan2 is the two-argument form of the arctan function. Note that the normalization factor multiplying the DFTand IDFT (here 1 and 1/N) and the signs of the exponents are merely conventions, and differ in some treatments. Theonly requirements of these conventions are that the DFT and IDFT have opposite-sign exponents and that theproduct of their normalization factors be 1/N. A normalization of for both the DFT and IDFT makes thetransforms unitary, which has some theoretical advantages, but it is often more practical in numerical computation toperform the scaling all at once as above (and a unit scaling can be convenient in other ways).(The convention of a negative sign in the exponent is often convenient because it means that is the amplitude ofa "positive frequency" . Equivalently, the DFT is often thought of as a matched filter: when looking for afrequency of +1, one correlates the incoming signal with a frequency of −1.)In the following discussion the terms "sequence" and "vector" will be considered interchangeable.

Properties

CompletenessThe discrete Fourier transform is an invertible, linear transformation

with C denoting the set of complex numbers. In other words, for any N > 0, an N-dimensional complex vector has aDFT and an IDFT which are in turn N-dimensional complex vectors.

Orthogonality

The vectors form an orthogonal basis over the set of N-dimensional complex vectors:

where is the Kronecker delta. This orthogonality condition can be used to derive the formula for the IDFT fromthe definition of the DFT, and is equivalent to the unitarity property below.

The Plancherel theorem and Parseval's theoremIf Xk and Yk are the DFTs of xn and yn respectively then the Plancherel theorem states:

where the star denotes complex conjugation. Parseval's theorem is a special case of the Plancherel theorem andstates:

These theorems are also equivalent to the unitary condition below.

Discrete Fourier transform 13

Periodicity

If the expression that defines the DFT is evaluated for all integers k instead of just for , thenthe resulting infinite sequence is a periodic extension of the DFT, periodic with period N.The periodicity can be shown directly from the definition:

Similarly, it can be shown that the IDFT formula leads to a periodic extension.

The shift theorem

Multiplying by a linear phase for some integer m corresponds to a circular shift of the output :is replaced by , where the subscript is interpreted modulo N (i.e., periodically). Similarly, a circular

shift of the input corresponds to multiplying the output by a linear phase. Mathematically, if represents the vector x then

if

then

and

Circular convolution theorem and cross-correlation theoremThe convolution theorem for the continuous and discrete time Fourier transforms indicates that a convolution of twoinfinite sequences can be obtained as the inverse transform of the product of the individual transforms. Withsequences and transforms of length N, a circularity arises:

The quantity in parentheses is 0 for all values of m except those of the form , where p is any integer.At those values, it is 1. It can therefore be replaced by an infinite sum of Kronecker delta functions, and we continueaccordingly. Note that we can also extend the limits of m to infinity, with the understanding that the x and ysequences are defined as 0 outside [0,N-1]:

which is the convolution of the sequence with a periodically extended sequence defined by:

Discrete Fourier transform 14

Similarly, it can be shown that:

which is the cross-correlation of     and  

A direct evaluation of the convolution or correlation summation (above) requires operations for an outputsequence of length N. An indirect method, using transforms, can take advantage of the efficiency ofthe fast Fourier transform (FFT) to achieve much better performance. Furthermore, convolutions can be used toefficiently compute DFTs via Rader's FFT algorithm and Bluestein's FFT algorithm.Methods have also been developed to use circular convolution as part of an efficient process that achieves normal(non-circular) convolution with an or sequence potentially much longer than the practical transform size (N).Two such methods are called overlap-save and overlap-add.[1]

Convolution theorem dualityIt can also be shown that:

  which is the circular convolution of and .

Trigonometric interpolation polynomialThe trigonometric interpolation polynomial

for

N even ,

for N

odd,where the coefficients Xk are given by the DFT of xn above, satisfies the interpolation property for .

For even N, notice that the Nyquist component is handled specially.

This interpolation is not unique: aliasing implies that one could add N to any of the complex-sinusoid frequencies(e.g. changing to ) without changing the interpolation property, but giving different values inbetween the points. The choice above, however, is typical because it has two useful properties. First, it consistsof sinusoids whose frequencies have the smallest possible magnitudes: the interpolation is bandlimited. Second, ifthe are real numbers, then is real as well.In contrast, the most obvious trigonometric interpolation polynomial is the one in which the frequencies range from0 to (instead of roughly to as above), similar to the inverse DFT formula. Thisinterpolation does not minimize the slope, and is not generally real-valued for real ; its use is a common mistake.

Discrete Fourier transform 15

The unitary DFTAnother way of looking at the DFT is to note that in the above discussion, the DFT can be expressed as aVandermonde matrix:

where

is a primitive Nth root of unity. The inverse transform is then given by the inverse of the above matrix:

With unitary normalization constants , the DFT becomes a unitary transformation, defined by a unitarymatrix:

where det()  is the determinant function. The determinant is the product of the eigenvalues, which are always oras described below. In a real vector space, a unitary transformation can be thought of as simply a rigid rotation

of the coordinate system, and all of the properties of a rigid rotation can be found in the unitary DFT.The orthogonality of the DFT is now expressed as an orthonormality condition (which arises in many areas ofmathematics as described in root of unity):

If is defined as the unitary DFT of the vector then

and the Plancherel theorem is expressed as:

If we view the DFT as just a coordinate transformation which simply specifies the components of a vector in a newcoordinate system, then the above is just the statement that the dot product of two vectors is preserved under aunitary DFT transformation. For the special case , this implies that the length of a vector is preserved aswell—this is just Parseval's theorem:

Discrete Fourier transform 16

Expressing the inverse DFT in terms of the DFTA useful property of the DFT is that the inverse DFT can be easily expressed in terms of the (forward) DFT, viaseveral well-known "tricks". (For example, in computations, it is often convenient to only implement a fast Fouriertransform corresponding to one transform direction and then to get the other transform direction from the first.)First, we can compute the inverse DFT by reversing the inputs:

(As usual, the subscripts are interpreted modulo N; thus, for , we have .)Second, one can also conjugate the inputs and outputs:

Third, a variant of this conjugation trick, which is sometimes preferable because it requires no modification of thedata values, involves swapping real and imaginary parts (which can be done on a computer simply by modifyingpointers). Define swap( ) as with its real and imaginary parts swapped—that is, if thenswap( ) is . Equivalently, swap( ) equals . Then

That is, the inverse transform is the same as the forward transform with the real and imaginary parts swapped forboth input and output, up to a normalization (Duhamel et al., 1988).The conjugation trick can also be used to define a new transform, closely related to the DFT, that is involutary—thatis, which is its own inverse. In particular, is clearly its own inverse: . Aclosely related involutary transformation (by a factor of (1+i) /√2) is , since the

factors in cancel the 2. For real inputs , the real part of is none other than thediscrete Hartley transform, which is also involutary.

Eigenvalues and eigenvectorsThe eigenvalues of the DFT matrix are simple and well-known, whereas the eigenvectors are complicated, notunique, and are the subject of ongoing research.Consider the unitary form defined above for the DFT of length N, where

This matrix satisfies the matrix polynomial equation:

This can be seen from the inverse properties above: operating twice gives the original data in reverse order, sooperating four times gives back the original data and is thus the identity matrix. This means that the eigenvalues

satisfy the equation:

Therefore, the eigenvalues of are the fourth roots of unity: is +1, −1, +i, or −i.Since there are only four distinct eigenvalues for this matrix, they have some multiplicity. The multiplicitygives the number of linearly independent eigenvectors corresponding to each eigenvalue. (Note that there are Nindependent eigenvectors; a unitary matrix is never defective.)The problem of their multiplicity was solved by McClellan and Parks (1972), although it was later shown to havebeen equivalent to a problem solved by Gauss (Dickinson and Steiglitz, 1982). The multiplicity depends on the valueof N modulo 4, and is given by the following table:

Discrete Fourier transform 17

Multiplicities of the eigenvalues λ of the unitary DFT matrix U as a function of thetransform size N (in terms of an integer m).

size N λ = +1 λ = −1 λ = -i λ = +i

4m m + 1 m m m − 1

4m + 1 m + 1 m m m

4m + 2 m + 1 m + 1 m m

4m + 3 m + 1 m + 1 m + 1 m

Otherwise stated, the characteristic polynomial of is:

No simple analytical formula for general eigenvectors is known. Moreover, the eigenvectors are not unique becauseany linear combination of eigenvectors for the same eigenvalue is also an eigenvector for that eigenvalue. Variousresearchers have proposed different choices of eigenvectors, selected to satisfy useful properties like orthogonalityand to have "simple" forms (e.g., McClellan and Parks, 1972; Dickinson and Steiglitz, 1982; Grünbaum, 1982;Atakishiyev and Wolf, 1997; Candan et al., 2000; Hanna et al., 2004; Gurevich and Hadani, 2008).A straightforward approach is to discretize the eigenfunction of the continuous Fourier transform, namely theGaussian function. Since periodic summation of the function means discretizing its frequency spectrum anddiscretization means periodic summation of the spectrum, the discretized and periodically summed Gaussianfunction yields an eigenvector of the discrete transform:

• .

A closed form expression for the series is not known, but it converges rapidly.Two other simple closed-form analytical eigenvectors for special DFT period N were found (Kong, 2008):For DFT period N = 2L + 1 = 4K +1, where K is an integer, the following is an eigenvector of DFT:

•

For DFT period N = 2L = 4K, where K is an integer, the following is an eigenvector of DFT:

•

The choice of eigenvectors of the DFT matrix has become important in recent years in order to define a discreteanalogue of the fractional Fourier transform—the DFT matrix can be taken to fractional powers by exponentiatingthe eigenvalues (e.g., Rubio and Santhanam, 2005). For the continuous Fourier transform, the natural orthogonaleigenfunctions are the Hermite functions, so various discrete analogues of these have been employed as theeigenvectors of the DFT, such as the Kravchuk polynomials (Atakishiyev and Wolf, 1997). The "best" choice ofeigenvectors to define a fractional discrete Fourier transform remains an open question, however.

Discrete Fourier transform 18

Uncertainty principle

If the random variable is constrained by:

then may be considered to represent a discrete probability mass function of n, with an associatedprobability mass function constructed from the transformed variable:

For the case of continuous functions P(x) and Q(k), the Heisenberg uncertainty principle states that:

where and are the variances of and respectively, with the equality attained in the caseof a suitably normalized Gaussian distribution. Although the variances may be analogously defined for the DFT, ananalogous uncertainty principle is not useful, because the uncertainty will not be shift-invariant.However, the Hirschman uncertainty will have a useful analog for the case of the DFT.[2] . The Hirschmanuncertainty principle is expressed in terms of the Shannon entropy of the two probability functions. In the discretecase, the Shannon entropies are defined as:

and

and the Hirschman uncertainty principle becomes[2] :

The equality is obtained for equal to translations and modulations of a suitably normalized Kronecker comb ofperiod A where A is any exact integer divisor of N. The probability mass function will then be proportional to asuitably translated Kronecker comb of period B=N/A.[2]

The real-input DFTIf are real numbers, as they often are in practical applications, then the DFT obeys the symmetry:

The star denotes complex conjugation. The subscripts are interpreted modulo N.Therefore, the DFT output for real inputs is half redundant, and one obtains the complete information by onlylooking at roughly half of the outputs . In this case, the "DC" element is purely real, and foreven N the "Nyquist" element is also real, so there are exactly N non-redundant real numbers in the first half+ Nyquist element of the complex output X.Using Euler's formula, the interpolating trigonometric polynomial can then be interpreted as a sum of sine and cosinefunctions.

Discrete Fourier transform 19

Generalized/shifted DFTIt is possible to shift the transform sampling in time and/or frequency domain by some real shifts a and b,respectively. This is sometimes known as a generalized DFT (or GDFT), also called the shifted DFT or offsetDFT, and has analogous properties to the ordinary DFT:

Most often, shifts of (half a sample) are used. While the ordinary DFT corresponds to a periodic signal in bothtime and frequency domains, produces a signal that is anti-periodic in frequency domain (

) and vice-versa for . Thus, the specific case of is known as an odd-time

odd-frequency discrete Fourier transform (or O2 DFT). Such shifted transforms are most often used for symmetricdata, to represent different boundary symmetries, and for real-symmetric data they correspond to different forms ofthe discrete cosine and sine transforms.Another interesting choice is , which is called the centered DFT (or CDFT). Thecentered DFT has the useful property that, when N is a multiple of four, all four of its eigenvalues (see above) haveequal multiplicities (Rubio and Santhanam, 2005)[3]

The discrete Fourier transform can be viewed as a special case of the z-transform, evaluated on the unit circle in thecomplex plane; more general z-transforms correspond to complex shifts a and b above.

Multidimensional DFTThe ordinary DFT transforms a one-dimensional sequence or array that is a function of exactly one discretevariable n. The multidimensional DFT of a multidimensional array that is a function of d discretevariables for in is defined by:

where as above and the d output indices run from . This ismore compactly expressed in vector notation, where we define and

as d-dimensional vectors of indices from 0 to , which we define as:

where the division is defined as to be performed element-wise, and thesum denotes the set of nested summations above.The inverse of the multi-dimensional DFT is, analogous to the one-dimensional case, given by:

As the one-dimensional DFT expresses the input as a superposition of sinusoids, the multidimensional DFTexpresses the input as a superposition of plane waves, or multidimensional sinusoids. The direction of oscillation inspace is . The amplitudes are . This decomposition is of great importance for everything from digitalimage processing (two-dimensional) to solving partial differential equations. The solution is broken up into planewaves.The multidimensional DFT can be computed by the composition of a sequence of one-dimensional DFTs along each dimension. In the two-dimensional case the independent DFTs of the rows (i.e., along ) are

Discrete Fourier transform 20

computed first to form a new array . Then the independent DFTs of y along the columns (along ) are computed toform the final result . Alternatively the columns can be computed first and then the rows. The order is immaterialbecause the nested summations above commute.An algorithm to compute a one-dimensional DFT is thus sufficient to efficiently compute a multidimensional DFT.This approach is known as the row-column algorithm. There are also intrinsically multidimensional FFT algorithms.

The real-input multidimensional DFTFor input data consisting of real numbers, the DFT outputs have a conjugate symmetry similar to theone-dimensional case above:

where the star again denotes complex conjugation and the -th subscript is again interpreted modulo (for).

ApplicationsThe DFT has seen wide usage across a large number of fields; we only sketch a few examples below (see also thereferences at the end). All applications of the DFT depend crucially on the availability of a fast algorithm to computediscrete Fourier transforms and their inverses, a fast Fourier transform.

Spectral analysis

When the DFT is used for spectral analysis, the sequence usually represents a finite set of uniformly-spacedtime-samples of some signal , where t represents time. The conversion from continuous time to samples(discrete-time) changes the underlying Fourier transform of x(t) into a discrete-time Fourier transform (DTFT),which generally entails a type of distortion called aliasing. Choice of an appropriate sample-rate (see Nyquistfrequency) is the key to minimizing that distortion. Similarly, the conversion from a very long (or infinite) sequenceto a manageable size entails a type of distortion called leakage, which is manifested as a loss of detail (akaresolution) in the DTFT. Choice of an appropriate sub-sequence length is the primary key to minimizing that effect.When the available data (and time to process it) is more than the amount needed to attain the desired frequencyresolution, a standard technique is to perform multiple DFTs, for example to create a spectrogram. If the desiredresult is a power spectrum and noise or randomness is present in the data, averaging the magnitude components ofthe multiple DFTs is a useful procedure to reduce the variance of the spectrum (also called a periodogram in thiscontext); two examples of such techniques are the Welch method and the Bartlett method; the general subject ofestimating the power spectrum of a noisy signal is called spectral estimation.A final source of distortion (or perhaps illusion) is the DFT itself, because it is just a discrete sampling of the DTFT,which is a function of a continuous frequency domain. That can be mitigated by increasing the resolution of theDFT. That procedure is illustrated in the discrete-time Fourier transform article.• The procedure is sometimes referred to as zero-padding, which is a particular implementation used in conjunction

with the fast Fourier transform (FFT) algorithm. The inefficiency of performing multiplications and additionswith zero-valued "samples" is more than offset by the inherent efficiency of the FFT.

• As already noted, leakage imposes a limit on the inherent resolution of the DTFT. So there is a practical limit tothe benefit that can be obtained from a fine-grained DFT.

Discrete Fourier transform 21

Data compressionThe field of digital signal processing relies heavily on operations in the frequency domain (i.e. on the Fouriertransform). For example, several lossy image and sound compression methods employ the discrete Fouriertransform: the signal is cut into short segments, each is transformed, and then the Fourier coefficients of highfrequencies, which are assumed to be unnoticeable, are discarded. The decompressor computes the inverse transformbased on this reduced number of Fourier coefficients. (Compression applications often use a specialized form of theDFT, the discrete cosine transform or sometimes the modified discrete cosine transform.)

Partial differential equationsDiscrete Fourier transforms are often used to solve partial differential equations, where again the DFT is used as anapproximation for the Fourier series (which is recovered in the limit of infinite N). The advantage of this approach isthat it expands the signal in complex exponentials einx, which are eigenfunctions of differentiation: d/dx einx = in einx.Thus, in the Fourier representation, differentiation is simple—we just multiply by i n. (Note, however, that thechoice of n is not unique due to aliasing; for the method to be convergent, a choice similar to that in thetrigonometric interpolation section above should be used.) A linear differential equation with constant coefficients istransformed into an easily solvable algebraic equation. One then uses the inverse DFT to transform the result backinto the ordinary spatial representation. Such an approach is called a spectral method.

Polynomial multiplicationSuppose we wish to compute the polynomial product c(x) = a(x) · b(x). The ordinary product expression for thecoefficients of c involves a linear (acyclic) convolution, where indices do not "wrap around." This can be rewrittenas a cyclic convolution by taking the coefficient vectors for a(x) and b(x) with constant term first, then appendingzeros so that the resultant coefficient vectors a and b have dimension d > deg(a(x)) + deg(b(x)). Then,

Where c is the vector of coefficients for c(x), and the convolution operator is defined so

But convolution becomes multiplication under the DFT:

Here the vector product is taken elementwise. Thus the coefficients of the product polynomial c(x) are just the terms0, ..., deg(a(x)) + deg(b(x)) of the coefficient vector

With a fast Fourier transform, the resulting algorithm takes O (N log N) arithmetic operations. Due to its simplicityand speed, the Cooley–Tukey FFT algorithm, which is limited to composite sizes, is often chosen for the transformoperation. In this case, d should be chosen as the smallest integer greater than the sum of the input polynomialdegrees that is factorizable into small prime factors (e.g. 2, 3, and 5, depending upon the FFT implementation).

Discrete Fourier transform 22

Multiplication of large integers

The fastest known algorithms for the multiplication of very large integers use the polynomial multiplication methodoutlined above. Integers can be treated as the value of a polynomial evaluated specifically at the number base, withthe coefficients of the polynomial corresponding to the digits in that base. After polynomial multiplication, arelatively low-complexity carry-propagation step completes the multiplication.

Some discrete Fourier transform pairs

Some DFT pairs

Note

Shift theorem

Real DFT

from the geometric progression formula

from the binomial theorem

is a rectangular window function of W pointscentered on n=0, where W is an odd integer, and is a

sinc-like function (specifically, is a Dirichletkernel)

Discretization and periodic summation of the scaledGaussian functions for . Since either or is

larger than one and thus warrants fast convergence ofone of the two series, for large you may choose to

compute the frequency spectrum and convert to the timedomain using the discrete Fourier transform.

Derivation as Fourier seriesThe DFT can be derived as a truncation of the Fourier series of a periodic sequence of impulses.

Generalizations

Representation theoryThe DFT can be interpreted as the complex-valued representation theory of the finite cyclic group. In other words, asequence of n complex numbers can be thought of as an element of n-dimensional complex space orequivalently a function from the finite cyclic group of order n to the complex numbers, This latter maybe suggestively written to emphasize that this is a complex vector space whose coordinates are indexed by then-element set From this point of view, one may generalize the DFT to representation theory generally, or more narrowly to therepresentation theory of finite groups.More narrowly still, one may generalize the DFT by either changing the target (taking values in a field other than thecomplex numbers), or the domain (a group other than a finite cyclic group), as detailed in the sequel.

Discrete Fourier transform 23

Other fields

Many of the properties of the DFT only depend on the fact that is a primitive root of unity, sometimesdenoted or (so that ). Such properties include the completeness, orthogonality,Plancherel/Parseval, periodicity, shift, convolution, and unitarity properties above, as well as many FFT algorithms.For this reason, the discrete Fourier transform can be defined by using roots of unity in fields other than the complexnumbers, and such generalizations are commonly called number-theoretic transforms (NTTs) in the case of finitefields. For more information, see number-theoretic transform and discrete Fourier transform (general).

Other finite groupsThe standard DFT acts on a sequence x0, x1, …, xN−1 of complex numbers, which can be viewed as a function {0, 1,…, N − 1} → C. The multidimensional DFT acts on multidimensional sequences, which can be viewed as functions

This suggests the generalization to Fourier transforms on arbitrary finite groups, which act on functions G → Cwhere G is a finite group. In this framework, the standard DFT is seen as the Fourier transform on a cyclic group,while the multidimensional DFT is a Fourier transform on a direct sum of cyclic groups.

AlternativesAs with other Fourier transforms, there are various alternatives to the DFT for various applications, prominentamong which are wavelets. The analog of the DFT is the discrete wavelet transform (DWT). From the point of viewof time–frequency analysis, a key limitation of the Fourier transform is that it does not include location information,only frequency information, and thus has difficulty in representing transients. As wavelets have location as well asfrequency, they are better able to represent location, at the expense of greater difficulty representing frequency. Fordetails, see comparison of the discrete wavelet transform with the discrete Fourier transform.

Notes[1] T. G. Stockham, Jr., "High-speed convolution and correlation," in 1966 Proc. AFIPS Spring Joint Computing Conf. Reprinted in Digital

Signal Processing, L. R. Rabiner and C. M. Rader, editors, New York: IEEE Press, 1972.[2] DeBrunner, Victor; Havlicek, Joseph P.; Przebinda, Tomasz; Özaydin, Murad (2005). "Entropy-Based Uncertainty Measures for

, and With a Hirschman Optimal Transform for " (http:/ / redwood. berkeley. edu/ w/

images/ 9/ 95/ 2002-26. pdf). IEEE Transactions on Signal Processing 53 (8): 2690. . Retrieved 2011-06-23.[3] Santhanam, Balu; Santhanam, Thalanayar S. "Discrete Gauss-Hermite functions and eigenvectors of the centered discrete Fourier transform"

(http:/ / thamakau. usc. edu/ Proceedings/ ICASSP 2007/ pdfs/ 0301385. pdf), Proceedings of the 32nd IEEE International Conference onAcoustics, Speech, and Signal Processing (ICASSP 2007, SPTM-P12.4), vol. III, pp. 1385-1388.

References• Brigham, E. Oran (1988). The fast Fourier transform and its applications. Englewood Cliffs, N.J.: Prentice Hall.

ISBN 0-13-307505-2.• Oppenheim, Alan V.; Schafer, R. W.; and Buck, J. R. (1999). Discrete-time signal processing. Upper Saddle

River, N.J.: Prentice Hall. ISBN 0-13-754920-2.• Smith, Steven W. (1999). "Chapter 8: The Discrete Fourier Transform" (http:/ / www. dspguide. com/ ch8/ 1.

htm). The Scientist and Engineer's Guide to Digital Signal Processing (Second ed.). San Diego, Calif.: CaliforniaTechnical Publishing. ISBN 0-9660176-3-3.

• Cormen, Thomas H.; Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein (2001). "Chapter 30:Polynomials and the FFT". Introduction to Algorithms (Second ed.). MIT Press and McGraw-Hill. pp. 822–848.ISBN 0-262-03293-7. esp. section 30.2: The DFT and FFT, pp. 830–838.

Discrete Fourier transform 24

• P. Duhamel, B. Piron, and J. M. Etcheto (1988). "On computing the inverse DFT". IEEE Trans. Acoust., Speechand Sig. Processing 36 (2): 285–286. doi:10.1109/29.1519.

• J. H. McClellan and T. W. Parks (1972). "Eigenvalues and eigenvectors of the discrete Fourier transformation".IEEE Trans. Audio Electroacoust. 20 (1): 66–74. doi:10.1109/TAU.1972.1162342.

• Bradley W. Dickinson and Kenneth Steiglitz (1982). "Eigenvectors and functions of the discrete Fouriertransform". IEEE Trans. Acoust., Speech and Sig. Processing 30 (1): 25–31. doi:10.1109/TASSP.1982.1163843.(Note that this paper has an apparent typo in its table of the eigenvalue multiplicities: the +i/−i columns areinterchanged. The correct table can be found in McClellan and Parks, 1972, and is easily confirmed numerically.)

• F. A. Grünbaum (1982). "The eigenvectors of the discrete Fourier transform". J. Math. Anal. Appl. 88 (2):355–363. doi:10.1016/0022-247X(82)90199-8.

• Natig M. Atakishiyev and Kurt Bernardo Wolf (1997). "Fractional Fourier-Kravchuk transform". J. Opt. Soc. Am.A 14 (7): 1467–1477. doi:10.1364/JOSAA.14.001467.

• C. Candan, M. A. Kutay and H. M.Ozaktas (2000). "The discrete fractional Fourier transform". IEEE Trans. onSignal Processing 48 (5): 1329–1337. doi:10.1109/78.839980.

• Magdy Tawfik Hanna, Nabila Philip Attalla Seif, and Waleed Abd El Maguid Ahmed (2004)."Hermite-Gaussian-like eigenvectors of the discrete Fourier transform matrix based on the singular-valuedecomposition of its orthogonal projection matrices". IEEE Trans. Circ. Syst. I 51 (11): 2245–2254.doi:10.1109/TCSI.2004.836850.

• Shamgar Gurevich and Ronny Hadani (2009). "On the diagonalization of the discrete Fourier transform". Appliedand Computational Harmonic Analysis 27 (1): 87–99. arXiv:0808.3281. doi:10.1016/j.acha.2008.11.003. preprintat.

• Shamgar Gurevich, Ronny Hadani, and Nir Sochen (2008). "The finite harmonic oscillator and its applications tosequences, communication and radar". IEEE Transactions on Information Theory 54 (9): 4239–4253.arXiv:0808.1495. doi:10.1109/TIT.2008.926440. preprint at.

• Juan G. Vargas-Rubio and Balu Santhanam (2005). "On the multiangle centered discrete fractional Fouriertransform". IEEE Sig. Proc. Lett. 12 (4): 273–276. doi:10.1109/LSP.2005.843762.

• J. Cooley, P. Lewis, and P. Welch (1969). "The finite Fourier transform". IEEE Trans. Audio Electroacoustics 17(2): 77–85. doi:10.1109/TAU.1969.1162036.

• F.N. Kong (2008). "Analytic Expressions of Two Discrete Hermite-Gaussian Signals". IEEE Trans. Circuits andSystems –II: Express Briefs. 55 (1): 56–60. doi:10.1109/TCSII.2007.909865.

External links• Interactive flash tutorial on the DFT (http:/ / www. fourier-series. com/ fourierseries2/ DFT_tutorial. html)• Mathematics of the Discrete Fourier Transform by Julius O. Smith III (http:/ / ccrma. stanford. edu/ ~jos/ mdft/

mdft. html)• Fast implementation of the DFT - coded in C and under General Public License (GPL) (http:/ / www. fftw. org)• The DFT “à Pied”: Mastering The Fourier Transform in One Day (http:/ / www. dspdimension. com/ admin/

dft-a-pied/ )

Fourier analysis 25

Fourier analysisIn mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject beganwith the study of the way general functions may be represented by sums of simpler trigonometric functions. Fourieranalysis is named after Joseph Fourier, who showed that representing a function by a trigonometric series greatlysimplifies the study of heat propagation.Today, the subject of Fourier analysis encompasses a vast spectrum of mathematics. In the sciences and engineering,the process of decomposing a function into simpler pieces is often called Fourier analysis, while the operation ofrebuilding the function from these pieces is known as Fourier synthesis. In mathematics, the term Fourier analysisoften refers to the study of both operations.The decomposition process itself is called a Fourier transform. The transform is often given a more specific namewhich depends upon the domain and other properties of the function being transformed. Moreover, the originalconcept of Fourier analysis has been extended over time to apply to more and more abstract and general situations,and the general field is often known as harmonic analysis. Each transform used for analysis (see list ofFourier-related transforms) has a corresponding inverse transform that can be used for synthesis.

ApplicationsFourier analysis has many scientific applications — in physics, partial differential equations, number theory,combinatorics, signal processing, imaging, probability theory, statistics, option pricing, cryptography, numericalanalysis, acoustics, oceanography, optics, diffraction, geometry, and other areas.This wide applicability stems from many useful properties of the transforms:

• The transforms are linear operators and, with proper normalization, are unitary as well (a property known asParseval's theorem or, more generally, as the Plancherel theorem, and most generally via Pontryaginduality)(Rudin 1990).

• The transforms are usually invertible.• The exponential functions are eigenfunctions of differentiation, which means that this representation transforms

linear differential equations with constant coefficients into ordinary algebraic ones (Evans 1998). Therefore, thebehavior of a linear time-invariant system can be analyzed at each frequency independently.

• By the convolution theorem, Fourier transforms turn the complicated convolution operation into simplemultiplication, which means that they provide an efficient way to compute convolution-based operations such aspolynomial multiplication and multiplying large numbers (Knuth 1997).

• The discrete version of the Fourier transform (see below) can be evaluated quickly on computers using fastFourier transform (FFT) algorithms. (Conte & de Boor 1980)

Fourier transformation is also useful as a compact representation of a signal. For example, JPEG compression uses avariant of the Fourier transformation (discrete cosine transform) of small square pieces of a digital image. TheFourier components of each square are rounded to lower arithmetic precision, and weak components are eliminatedentirely, so that the remaining components can be stored very compactly. In image reconstruction, eachFourier-transformed image square is reassembled from the preserved approximate components, and theninverse-transformed to produce an approximation of the original image.

Fourier analysis 26

Applications in signal processingWhen processing signals, such as audio, radio waves, light waves, seismic waves, and even images, Fourier analysiscan isolate individual components of a compound waveform, concentrating them for easier detection and/or removal.A large family of signal processing techniques consist of Fourier-transforming a signal, manipulating theFourier-transformed data in a simple way, and reversing the transformation.Some examples include:• Telephone dialing; the touch-tone signals for each telephone key, when pressed, are each a sum of two separate

tones (frequencies). Fourier analysis can be used to separate (or analyze) the telephone signal, to reveal the twocomponent tones and therefore which button was pressed.

• Removal of unwanted frequencies from an audio recording (used to eliminate hum from leakage of AC powerinto the signal, to eliminate the stereo subcarrier from FM radio recordings);

• Noise gating of audio recordings to remove quiet background noise by eliminating Fourier components that donot exceed a preset amplitude;

• Equalization of audio recordings with a series of bandpass filters;• Digital radio reception with no superheterodyne circuit, as in a modern cell phone or radio scanner;• Image processing to remove periodic or anisotropic artifacts such as jaggies from interlaced video, stripe artifacts

from strip aerial photography, or wave patterns from radio frequency interference in a digital camera;• Cross correlation of similar images for co-alignment;• X-ray crystallography to reconstruct a crystal structure from its diffraction pattern;• Fourier transform ion cyclotron resonance mass spectrometry to determine the mass of ions from the frequency of

cyclotron motion in a magnetic field.• Many other forms of spectroscopy also rely upon Fourier Transforms to determine the three-dimensional structure

and/or identity of the sample being analyzed, including Infrared and Nuclear Magnetic Resonance spectroscopies.• Generation of sound spectrograms used to analyze sounds.

Variants of Fourier analysisFourier analysis has different forms, some of which have different names. The more common variants are shownbelow. The different names usually reflect different properties of the function or data being analyzed. The resultanttransforms can be seen as special cases or generalizations of each other.

(Continuous) Fourier transformMost often, the unqualified term Fourier transform refers to the transform of functions of a continuous realargument, such as time (t). In this case the Fourier transform describes a function ƒ(t) in terms of basic complexexponentials of various frequencies. In terms of ordinary frequency, ν, the Fourier transform is given by the complexnumber:

Evaluating this quantity for all values of ν produces the frequency-domain function.See Fourier transform for even more information, including:

• the inverse transform, F(ν) → ƒ(t)• conventions for amplitude normalization and frequency scaling/units• transform properties• tabulated transforms of specific functions• an extension/generalization for functions of multiple dimensions, such as images

Fourier analysis 27

Fourier seriesA Fourier series is a representation of a function in terms of a summation of a potentially infinite number ofharmonically-related sinusoids or complex exponential functions with different amplitudes and phases. Theamplitude and phase of a sinusoid can be combined into a single complex number, called a Fourier coefficient. TheFourier series is a periodic function. So it cannot represent any arbitrary function. It can represent either:

(a) a periodic function, or(b) a function that is defined only over a finite-length interval (or compact support). Then the values producedby the Fourier series outside the finite interval are irrelevant.

The general form of a Fourier series is:

where τ is the period (case a) or the interval length (case b), and the S[n] sequence are the Fourier coefficients. Whenthe coefficients are derived from a function, ƒ(t) as follows:

then, aside from possible convergence issues, s(t) will equal ƒ(t) in the interval [u,u+τ ]. It follows that if ƒ(t) isτ-periodic (case a), s(t) and ƒ(t) are equal everywhere.The Fourier series is analogous to the inverse Fourier transform, in that it is the reconstruction of the originalfunction that was transformed into the Fourier series coefficients.See Fourier series for more information, including the historical development.

Discrete-time Fourier transform (DTFT)For functions of an integer index, the discrete-time Fourier transform (DTFT) provides a usefulfrequency-domain transform.A useful "discrete-time" function can be obtained by sampling a "continuous-time" function, s(t), which produces asequence, s(nT), for integer values of n and some time-interval T. If information is lost, then only an approximationto the original transform, S(f), can be obtained by looking at one period of the periodic function:

which is the DTFT. The identity above is a result of the Poisson summation formula. The DTFT is also equivalent tothe Fourier transform of a "continuous" function that is constructed by using the s[n] sequence to modulate a Diraccomb.Applications of the DTFT are not limited to sampled functions. It can be applied to any discrete sequence. SeeDiscrete-time Fourier transform for more information on this and other topics, including:

• the inverse transform• normalized frequency units• windowing (finite-length sequences)• transform properties• tabulated transforms of specific functions

Fourier analysis 28

Discrete Fourier transform (DFT)When s[n] is periodic, with period N,  S1/T(ƒ) is another Dirac comb function, modulated by the coefficients of aFourier series.  And the integral formula for the coefficients simplifies to:

    for all integer values of k.

This sequence is N-periodic, and so the entire sequence can be described by just N coefficients, known most often asthe DFT and sometimes as the discrete Fourier series (DFS).[1] The DFT also has an inverse transform thatreproduces the periodic s[n] sequence.When s[n] is not periodic, but its non-zero portion has finite duration (N),  S1/T(ƒ) is continuous and finite-valued.But a discrete subset of its values is sufficient to reconstruct/represent the (finite) portion of s[n] that was analyzed,analogous to case b (above) of the Fourier series. And that subset is again the DFT.• When N is larger than the non-zero portion of s[n], known as zero-padding, the DFT computes more

closely-spaced samples of one period of  S1/T(ƒ). That is frequently done to provide an interpolated view of theDTFT.

• The term DFT is ambiguous in the sense that it does not tell us whether the inverse transform is valid for all n orjust for a sequence of length N; i.e. whether the original s[n] sequence is periodic or finite. The term DFS,mentioned above, is sometimes used instead of DFT to convey that s[n] is periodic.

The DFT can be computed using a fast Fourier transform (FFT) algorithm, which makes it a practical and importanttransformation on computers.See Discrete Fourier transform for much more information, including:

• the inverse transform• transform properties• applications• tabulated transforms of specific functions

SummaryThis section consolidates the information above, using a consistent notation to emphasize the relationships betweenthe four transforms.Notation:

is a non-periodic function with Fourier transform   is a time-interval of N samples taken at sample-intervals of time T.

  is a periodic summation of

  is a periodic summation of

  is a periodic summation of discrete sequence

  is the integral of over any interval of length P.

  is the summation of sequence over any interval of length N.

In the columns labeled Finite duration or periodic the transforms are most useful when and the sequence are restricted to duration , but that is not a requirement for the inverse transforms to work as shown. The

Fourier analysis 29

periodic case (e.g. Fourier series) is seen by considering to be the function being transformed. Any periodic functioncan be represented as a periodic summation of another function, so there is no loss of generality. In either case, theperiodic function ( )  is always the inverse of these discrete-frequency transforms, a fact that is often misinterpreted.

Continuous-timetransforms

Any duration (continuousfrequency)

Finite duration or periodic (discrete frequencies)

Transform

Inverse

Discrete-time

transforms

Any duration (continuous frequency) Finite duration or periodic (discrete frequencies)

Transform

Inverse

Fourier transforms on arbitrary locally compact abelian topological groupsThe Fourier variants can also be generalized to Fourier transforms on arbitrary locally compact abelian topologicalgroups, which are studied in harmonic analysis; there, the Fourier transform takes functions on a group to functionson the dual group. This treatment also allows a general formulation of the convolution theorem, which relatesFourier transforms and convolutions. See also the Pontryagin duality for the generalized underpinnings of theFourier transform.

Time–frequency transformsIn signal processing terms, a function (of time) is a representation of a signal with perfect time resolution, but nofrequency information, while the Fourier transform has perfect frequency resolution, but no time information.As alternatives to the Fourier transform, in time–frequency analysis, one uses time–frequency transforms torepresent signals in a form that has some time information and some frequency information – by the uncertaintyprinciple, there is a trade-off between these. These can be generalizations of the Fourier transform, such as theshort-time Fourier transform, the Gabor transform or fractional Fourier transform, or can use different functions torepresent signals, as in wavelet transforms and chirplet transforms, with the wavelet analog of the (continuous)Fourier transform being the continuous wavelet transform.

Fourier analysis 30

HistoryA primitive form of harmonic series dates back to ancient Babylonian mathematics, where they were used tocompute ephemerides (tables of astronomical positions).[2]

In modern times, variants of the discrete Fourier transform were used by Alexis Clairaut in 1754 to compute anorbit,[3] which has been described as the first formula for the DFT,[4] and in 1759 by Joseph Louis Lagrange, incomputing the coefficients of a trigonometric series for a vibrating string.[5] Technically, Clairaut's work was acosine-only series (a form of discrete cosine transform), while Lagrange's work was a sine-only series (a form ofdiscrete sine transform); a true cosine+sine DFT was used by Gauss in 1805 for trigonometric interpolation ofasteroid orbits.[6] Euler and Lagrange both discretized the vibrating string problem, using what would today be calledsamples.[5]

An early modern development toward Fourier analysis was the 1770 paper Réflexions sur la résolution algébriquedes équations by Lagrange, which in the method of Lagrange resolvents used a complex Fourier decomposition tostudy the solution of a cubic:[7] Lagrange transformed the roots into the resolvents:

where ζ is a cubic root of unity, which is the DFT of order 3.A number of authors, notably Jean le Rond d'Alembert,, and Carl Friedrich Gauss used trigonometric series to studythe heat equation, but the breakthrough development was the 1807 paper Mémoire sur la propagation de la chaleurdans les corps solides by Joseph Fourier, whose crucial insight was to model all functions by trigonometric series,introducing the Fourier series.Historians are divided as to how much to credit Lagrange and others for the development of Fourier theory: DanielBernoulli and Leonhard Euler had introduced trigonometric representations of functions,[4] and Lagrange had giventhe Fourier series solution to the wave equation,[4] so Fourier's contribution was mainly the bold claim that anarbitrary function could be represented by a Fourier series.[4]

The subsequent development of the field is known as harmonic analysis, and is also an early instance ofrepresentation theory.The first fast Fourier transform (FFT) algorithm for the DFT was discovered around 1805 by Carl Friedrich Gausswhen interpolating measurements of the orbit of the asteroids Juno and Pallas, although that particular FFT algorithmis more often attributed to its modern rediscoverers Cooley and Tukey.[6] [8]

Interpretation in terms of time and frequencyIn signal processing, the Fourier transform often takes a time series or a function of continuous time, and maps it intoa frequency spectrum. That is, it takes a function from the time domain into the frequency domain; it is adecomposition of a function into sinusoids of different frequencies; in the case of a Fourier series or discrete Fouriertransform, the sinusoids are harmonics of the fundamental frequency of the function being analyzed.When the function ƒ is a function of time and represents a physical signal, the transform has a standard interpretationas the frequency spectrum of the signal. The magnitude of the resulting complex-valued function F at frequency ωrepresents the amplitude of a frequency component whose initial phase is given by the phase of F.Fourier transforms are not limited to functions of time, and temporal frequencies. They can equally be applied toanalyze spatial frequencies, and indeed for nearly any function domain. This justifies their use in branches suchdiverse as image processing, heat conduction and automatic control.

Fourier analysis 31

Notes[1] We note that DFS is actually a misnomer, since a Fourier series is a sum of sinusoids, not the sequence of coefficients.[2] Prestini, Elena (2004), The evolution of applied harmonic analysis: models of the real world (http:/ / books. google. com/

?id=fye--TBu4T0C), Birkhäuser, ISBN 978 0 81764125 2, , p. 62 (http:/ / books. google. com/ books?id=fye--TBu4T0C& pg=PA62)Rota, Gian-Carlo; Palombi, Fabrizio (1997), Indiscrete thoughts (http:/ / books. google. com/ ?id=H5smrEExNFUC), Birkhäuser, ISBN 978 081763866 5, , p. 11 (http:/ / books. google. com/ books?id=H5smrEExNFUC& pg=PA11)Neugebauer, Otto (1969) [1957], The Exact Sciences in Antiquity (http:/ / books. google. com/ ?id=JVhTtVA2zr8C) (2 ed.), DoverPublications, ISBN 978-048622332-2,Brack-Bernsen, Lis; Brack, Matthias, Analyzing shell structure from Babylonian and modern times, arXiv:physics/0310126

[3] Terras, Audrey (1999), Fourier analysis on finite groups and applications (http:/ / books. google. com/ ?id=-B2TA669dJMC), CambridgeUniversity Press, ISBN 978 0 52145718 7, , p. 30 (http:/ / books. google. com/ books?id=-B2TA669dJMC& pg=PA30#PPA30,M1)

[4] Briggs, William L.; Henson, Van Emden (1995), The DFT : an owner's manual for the discrete Fourier transform (http:/ / books. google.com/ ?id=coq49_LRURUC), SIAM, ISBN 978 0 89871342 8, , p. 4 (http:/ / books. google. com/ books?id=coq49_LRURUC&pg=PA2#PPA4,M1)

[5] Briggs, William L.; Henson, Van Emden (1995), The DFT: an owner's manual for the discrete Fourier transform (http:/ / books. google.com/ ?id=coq49_LRURUC), SIAM, ISBN 978 0 89871342 8, , p. 2 (http:/ / books. google. com/ books?id=coq49_LRURUC&pg=PA2#PPA2,M1)

[6] Heideman, M. T., D. H. Johnson, and C. S. Burrus, " Gauss and the history of the fast Fourier transform (http:/ / ieeexplore. ieee. org/ xpls/abs_all. jsp?arnumber=1162257)," IEEE ASSP Magazine, 1, (4), 14–21 (1984)

[7] Knapp, Anthony W. (2006), Basic algebra (http:/ / books. google. com/ ?id=KVeXG163BggC), Springer, ISBN 978 0 81763248 9, , p. 501(http:/ / books. google. com/ books?id=KVeXG163BggC& pg=PA501)

[8] Terras, Audrey (1999), Fourier analysis on finite groups and applications (http:/ / books. google. com/ ?id=-B2TA669dJMC), CambridgeUniversity Press, ISBN 978 0 52145718 7, , p. 31 (http:/ / books. google. com/ books?id=-B2TA669dJMC& pg=PA30#PPA31,M1)

References• Conte, S. D.; de Boor, Carl (1980), Elementary Numerical Analysis (Third ed.), New York: McGraw Hill, Inc.,

ISBN 0070662282• Evans, L. (1998), Partial Differential Equations, American Mathematical Society, ISBN 3540761241• Howell, Kenneth B. (2001). Principles of Fourier Analysis, CRC Press. ISBN 9780849382758• Kamen, E.W., and B.S. Heck. "Fundamentals of Signals and Systems Using the Web and Matlab". ISBN

0-13-017293-6• Knuth, Donald E. (1997), The Art of Computer Programming Volume 2: Seminumerical Algorithms (3rd ed.),

Section 4.3.3.C: Discrete Fourier transforms, pg.305: Addison-Wesley Professional, ISBN 0201896842• Polyanin, A.D., and A.V. Manzhirov (1998). Handbook of Integral Equations, CRC Press, Boca Raton. ISBN

0-8493-2876-4• Rudin, Walter (1990), Fourier Analysis on Groups, Wiley-Interscience, ISBN 047152364X• Smith, Steven W. (1999), The Scientist and Engineer's Guide to Digital Signal Processing (http:/ / www.

dspguide. com/ pdfbook. htm) (Second ed.), San Diego, Calif.: California Technical Publishing,ISBN 0-9660176-3-3

• Stein, E.M., and G. Weiss (1971). Introduction to Fourier Analysis on Euclidean Spaces. Princeton UniversityPress. ISBN 0-691-08078-X

Fourier analysis 32

External links• Tables of Integral Transforms (http:/ / eqworld. ipmnet. ru/ en/ auxiliary/ aux-inttrans. htm) at EqWorld: The

World of Mathematical Equations.• An Intuitive Explanation of Fourier Theory (http:/ / cns-alumni. bu. edu/ ~slehar/ fourier/ fourier. html) by Steven

Lehar.• Lectures on Image Processing: A collection of 18 lectures in pdf format from Vanderbilt University. Lecture 6 is

on the 1- and 2-D Fourier Transform. Lectures 7-15 make use of it. (http:/ / www. archive. org/ details/Lectures_on_Image_Processing), by Alan Peters

Sine

Sine

Basic features

Parity odd

Domain (-∞,∞)

Codomain [-1,1]

Period 2π

Specific values

At zero 0

Maxima ((2k+1/2)π,1)

Minima ((2k-1/2)π,-1)

Specific features

Root kπ

Critical point kπ-π/2

Inflection point kπ

Fixed point 0

Variable k is an integer.

Sine 33

For the angle α, the sine function gives the ratioof the length of the opposite side to the length of

the hypotenuse.

The sine function graphed on the Cartesian plane.In this graph, the angle x is given in radians (π =

180°).

The sine and cosine functions are related inmultiple ways. The derivative of is

. Also they are out of phase by 90°:

= . And for a

given angle, cos and sin give the respective x, ycoordinates on a unit circle.

In mathematics, the sine function is a function of an angle. In a righttriangle, sine gives the ratio of the length of the side opposite to anangle to the length of the hypotenuse.

Sine is usually listed first amongst the trigonometric functions.Trigonometric functions are commonly defined as ratios of two sidesof a right triangle containing the angle, and can equivalently be definedas the lengths of various line segments from a unit circle. More moderndefinitions express them as infinite series or as solutions of certaindifferential equations, allowing their extension to arbitrary positive andnegative values and even to complex numbers.

The sine function is commonly used to model periodic phenomenasuch as sound and light waves, the position and velocity of harmonicoscillators, sunlight intensity and day length, and average temperaturevariations throughout the year.

The function sine can be traced to the jyā and koṭi-jyā functions used inGupta period Indian astronomy (Aryabhatiya, Surya Siddhanta), viatranslation from Sanskrit to Arabic and then from Arabic to Latin.[1]

The word "sine" comes from a Latin mistranslation of the Arabic jiba,which is a transliteration of the Sanskrit word for half the chord,jya-ardha.[2]

Right-angled triangle definition

For any similar triangle the ratio of the length of the sides remains thesame. For example, if the hypotenuse is twice as long, so are the othersides. Therefore respective trigonometric functions, depending only onthe size of the angle, express those ratios: between the hypotenuse andthe "opposite" side to an angle A in question (see illustration) in thecase of sine function; or between the hypotenuse and the "adjacent"side (cosine) or between the "opposite" and the "adjacent" side(tangent), etc.

To define the trigonometric functions for an acute angle A, start withany right triangle that contains the angle A. The three sides of thetriangle are named as follows:

• The hypotenuse is the side opposite the right angle, in this caseside h. The hypotenuse is always the longest side of a right-angled triangle.

• The opposite side is the side opposite to the angle we are interested in (angle A), in this case side a.• The adjacent side is the side that is in contact with (adjacent to) both the angle we are interested in (angle A) and

the right angle, in this case side b.

In ordinary Euclidean geometry, according to the triangle postulate the inside angles of every triangle total 180° (πradians). Therefore, in a right-angled triangle, the two non-right angles total 90° (π/2 radians), so each of theseangles must be greater than 0° and less than 90°. The following definition applies to such angles.

The angle A (having measure α) is the angle between the hypotenuse and the adjacent line.The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. In our case

Sine 34

Note that this ratio does not depend on the size of the particular right triangle chosen, as long as it contains the angleA, since all such triangles are similar.

Relation to slopeThe trigonometric functions can be defined in terms of the rise, run, and slope of a line segment relative to somehorizontal line.• When the length of the line segment is 1, sine takes an angle and tells the rise• Sine takes an angle and tells the rise per unit length of the line segment.• Rise is equal to sin θ multiplied by the length of the line segmentIn contrast, cosine is used for the telling the run from the angle; and tangent is used for telling the slope from theangle. Arctan is used for telling the angle from the slope.The line segment is the equivalent of the hypotenuse in the right-triangle, and when it has a length of 1 it is alsoequivalent to the radius of the unit circle.

Relation to the unit circleIn trigonometry, a unit circle is the circle of radius one centered at the origin (0, 0) in the Cartesian coordinatesystem.Let a line through the origin, making an angle of θ with the positive half of the x-axis, intersect the unit circle. The x-and y-coordinates of this point of intersection are equal to cos θ and sin θ, respectively. The point's distance from theorigin is always 1.Unlike the definitions with the right triangle or slope, the angle can be extended to the full set of real arguments byusing the unit circle. This can also be achieved by requiring certain symmetries and that sine be a periodic function.

The unit circle. Illustration of a unit circle. The radius has alength of 1. The variable t is an angle measure.

Point P(x,y) on the circle of unit radius at anobtuse angle θ > π/2

Sine 35

Animation showing the graphing process of y = sin x (where x is the angle in radians)using a unit circle

Identities

Exact identities (using radians):These apply for all values of .

ReciprocalThe reciprocal of sine is cosecant. i.e. the reciprocal of sin(A) is csc(A), or cosec(A). Cosecant gives the ratio of thelength of the hypotenuse to the length of the opposite side:

Inverse

The usual principal values of the arcsin(x)function graphed on the cartesian plane. Arcsin is

the inverse of sin.

The inverse function of sine is arcsine (arcsin or asin) or inverse sine(sin−1). As sine is multivalued, it is not an exact inverse function but apartial inverse function. For example, sin(0) = 0, but also sin(π) = 0,sin(2π) = 0 etc. It follows that the arcsine function is also multivalued:arcsin(0) = 0, but also arcsin(0) = π, arcsin(0) = 2π, etc. When onlyone value is desired, the function may be restricted to its principalbranch. With this restriction, for each x in the domain the expressionarcsin(x) will evaluate only to a single value, called its principal value.

k is some integer:

Sine 36

Arcsin satisfies:

and

CalculusFor the sine function:

The derivative is:

The antiderivative is:

C denotes the constant of integration.

Other trigonomic functions

The four quadrants of a Cartesiancoordinate system.

It is possible to express any trigonometric function in terms of any other (upto a plus or minus sign, or using the sign function).

Sine in terms of the other common trigonometric functions:

f θ Using plus/minus (±) Using sign function (sgn)

f θ = ± per Quadrant f θ =

I II III IV

cos + + - -

+ - - +

cot + + - -

+ - - +

tan + - - +

+ - - +

Sine 37

sec + - + -

+ - - +

Note that for all equations which use plus/minus (±), it is positive in the first quadrant.The basic relationship between the sine and the cosine can also be expressed as the Pythagorean trigonometricidentity:

where sin2x means (sin(x))2.

Properties relating to the quadrantsOver the four quadrants of the sine function is as follows.

Quadrant Degrees Radians Value Sign Monotony Convexity

1st Quadrant increasing concave

2nd Quadrant decreasing concave

3rd Quadrant decreasing convex

4th Quadrant increasing convex

Points between the quadrants. k is an integer.

The quadrants of the unit circle and of sin x, using the Cartesian coordinate system.

Degrees Radians 0 ≤ x < 2π Radians sin x Point type

0 0 Root, Inflection

1 Maxima

0 Root, Inflection

-1 Minima

For arguments outside of those in the table, get the value using the fact the sine function has a period of 360° (or 2πrad): , or use .

Sine 38

Series definition

The sine function (blue) is closely approximatedby its Taylor polynomial of degree 7 (pink) for a

full cycle centered on the origin.

Using only geometry and properties of limits, it can be shown that thederivative of sine is cosine and the derivative of cosine is the negativeof sine.

Using the reflection from the calculated geometric derivation of thesine is with the 4n + k-th derivative at the point 0:

This gives the following Taylor series expansion at x = 0. One can then use the theory of Taylor series to show thatthe following identities hold for all real numbers x (where x is the angle in radians) :[3]

If x were expressed in degrees then the series would contain messy factors involving powers of π/180: if x is thenumber of degrees, the number of radians is y = πx /180, so

Mathematically important relationships between the sine and cosine functions and the exponential function (see, forexample, Euler's formula) are, again, elegant when the functions' arguments are in radians and messy otherwise. Inmost branches of mathematics beyond practical geometry, angles are generally measured in radians.A similar series is Gregory's series for arctan, which is obtained by omitting the factorials in the denominator.

Sine 39

Continued fraction

The fixed point iteration xn+1 = sin xn with initialvalue x0 = 2 converges to 0.

The sine function can also be represented as a generalized continuedfraction:

The continued fraction representation expresses the real number values, both rational and irrational, of the sinefunction.

Law of sinesThe law of sines states that for an arbitrary triangle with sides a, b, and c and angles opposite those sides A, B and C:

This is equivalent to the equality of the first three expressions below:

where R is the triangle's circumradius.It can be proven by dividing the triangle into two right ones and using the above definition of sine. The law of sinesis useful for computing the lengths of the unknown sides in a triangle if two angles and one side are known. This is acommon situation occurring in triangulation, a technique to determine unknown distances by measuring two anglesand an accessible enclosed distance.

Sine 40

Values

sin(x)

Some common angles (θ) shown on the unit circle. The angles aregiven in degrees and radians, together with the corresponding

intersection point on the unit circle, (cos θ, sin θ).

x (angle) sin x

Degrees Radians Grads Exact Decimal

0° 0 0g 0 0

180° 200g

15° 162⁄3g 0.258819045102521

165° 1831⁄3g

30° 331⁄3g 0.5

150° 1662⁄3g

45° 50g 0.707106781186548

135° 150g

60° 662⁄3g 0.866025403784439

120° 1331⁄3g

75° 831⁄3g 0.965925826289068

105° 1162⁄3g

90° 100g 1 1

A memory aid (note it does not include 15° and 75°):

Sine 41

x in degrees 0° 30° 45° 60° 90°

x in radians 0 π/6 π/4 π/3 π/2

90 degree increments:

x in degrees 0° 90° 180° 270° 360°

x in radians 0 π/2 π 3π/2 2π

0 1 0 -1 0

Other values not listed above:

A019812 [4]

A019815 [5]

A019818 [6]

A019821 [7]

A019827 [8]

A019830 [9]

A019833 [10]

A019836 [11]

A019842 [12]

A019845 [13]

A019848 [14]

A019851 [15]

For angles greater than 2π or less than −2π, simply continue to rotate around the circle; sine periodic function withperiod 2π:

for any angle θ and any integer k.The primitive period (the smallest positive period) of sine is a full circle, i.e. 2π radians or 360 degrees.

Sine 42

Relationship to complex numbers

An illustration of the complex plane. Theimaginary numbers are on the vertical coordinate

axis.

Sine is used to determine the imaginary part of a complex numbergiven in polar coordinates (r,φ):

the imaginary part is:

r and φ represent the magnitude and angle of the complex number respectively. i is the imaginary unit. z is acomplex number.Although dealing with complex numbers, sine's parameter in this usage is still a real number. Sine can also take acomplex number as an argument.

Sine with a complex argument

Domain coloring of sin(z) over (-π,π) on x and yaxes. Brightness indicates absolute magnitude,

saturation represents imaginary and realmagnitude.

The definition of the sine function for complex arguments z:

Sine 43

sin(z) as a vector field

where i 2 = −1. This is an entire function. Also, for purely real x,

For purely imaginary numbers:

It is also sometimes useful to express the complex sine function in terms of the real and imaginary parts of itsargument:

Usage of complex sine

sin z is found in the functional equation for the Gamma function,

.

Which in turn is found in the functional equation for the Riemann zeta-function,

As a holomorphic function, sin z is a 2D solution of Laplace's equation:

Sine 44

Complex graphs

Sine function in the complex plane

real component imaginary component magnitude

Sin-1 in the complex plane

real component imaginary component magnitude

HistoryWhile the early study of trigonometry can be traced to antiquity, the trigonometric functions as they are in use todaywere developed in the medieval period. The chord function was discovered by Hipparchus of Nicaea (180–125 BC)and Ptolemy of Roman Egypt (90–165 AD).The function sine (and cosine) can be traced to the jyā and koṭi-jyā functions used in Gupta period Indian astronomy(Aryabhatiya, Surya Siddhanta), via translation from Sanskrit to Arabic and then from Arabic to Latin.[1]

The first published use of the abbreviations 'sin', 'cos', and 'tan' is by the 16th century French mathematician AlbertGirard; these were further promulgated by Euler (see below). The Opus palatinum de triangulis of Georg JoachimRheticus, a student of Copernicus, was probably the first in Europe to define trigonometric functions directly interms of right triangles instead of circles, with tables for all six trigonometric functions; this work was finished byRheticus' student Valentin Otho in 1596.In a paper published in 1682, Leibniz proved that sin x is not an algebraic function of x.[16] Roger Cotes computedthe derivative of sine in his Harmonia Mensurarum (1722).[17] Leonhard Euler's Introductio in analysin infinitorum(1748) was mostly responsible for establishing the analytic treatment of trigonometric functions in Europe, alsodefining them as infinite series and presenting "Euler's formula", as well as the near-modern abbreviations sin., cos.,tang., cot., sec., and cosec.[18]

Sine 45

EtymologyEtymologically, the word sine derives from the Sanskrit word for chord, jiva*(jya being its more popular synonym).This was transliterated in Arabic as jiba جــيــب , abbreviated jb جــــب . Since Arabic is written without short vowels,"jb" was interpreted as the word jaib جــيــب , which means "bosom", when the Arabic text was translated in the 12thcentury into Latin by Gerard of Cremona. The translator used the Latin equivalent for "bosom", sinus (which means"bosom" or "bay" or "fold") [19] [20] The English form sine was introduced in the 1590s.

Software implementationsThe sin function, along with other trigonometric functions, is widely available across programming languages andplatforms. Some CPU architectures have a built-in instruction for sin, including the Intel x86 FPU. In programminglanguages, sin is usually either a built in function or found within the language's standard math library. There is nostandard algorithm for calculating sin. IEEE 754-2008, the most widely-used standard for floating-pointcomputation, says nothing on the topic of calculating trigonometric functions such as sin.[21] Algorithms forcalculating sin may be balanced for such constraints as speed, accuracy, portability, or range of input valuesaccepted. This can lead to very different results for different algorithms in special circumstances, such as for verylarge inputs, e.g. sin(1022).A once common programming optimization, used especially in 3D graphics, was to pre-calculate a table of sinvalues, for example a value per degree. This allowed results to be looked up from a table rather than being calculatedin real time. With modern CPU architectures this method typically offers no advantage.

Notes[1] Boyer, Carl B. (1991). A History of Mathematics (Second ed.). John Wiley & Sons, Inc.. ISBN 0-471-54397-7, p. 210.[2] Victor J Katx, A history of mathematics, p210, sidebar 6.1.[3] See Ahlfors, pages 43–44.[4] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa019812[5] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa019815[6] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa019818[7] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa019821[8] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa019827[9] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa019830[10] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa019833[11] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa019836[12] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa019842[13] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa019845[14] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa019848[15] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa019851[16] Nicolás Bourbaki (1994). Elements of the History of Mathematics. Springer.[17] " Why the sine has a simple derivative (http:/ / www. math. usma. edu/ people/ rickey/ hm/ CalcNotes/ Sine-Deriv. pdf)", in Historical Notes

for Calculus Teachers (http:/ / www. math. usma. edu/ people/ rickey/ hm/ CalcNotes/ default. htm) by V. Frederick Rickey (http:/ / www.math. usma. edu/ people/ rickey/ )

[18] See Boyer (1991).[19] See Maor (1998), chapter 3, regarding the etymology.[20] Victor J Katx, A history of mathematics, p210, sidebar 6.1.[21] Grand Challenges of Informatics, Paul Zimmermann. September 20, 2006 – p. 14/31 (http:/ / www. jaist. ac. jp/ ~bjorner/ ae-is-budapest/

talks/ Sept20pm2_Zimmermann. pdf)

Trigonometric functions 46

Trigonometric functionsIn mathematics, the trigonometric functions (also called circular functions) are functions of an angle. They areused to relate the angles of a triangle to the lengths of the sides of a triangle. Trigonometric functions are importantin the study of triangles and modeling periodic phenomena, among many other applications.The most familiar trigonometric functions are the sine, cosine, and tangent. In the context of the standard unit circlewith radius 1, where a triangle is formed by a ray originating at the origin and making some angle with the x-axis,the sine of the angle gives the length of the y-component (rise) of the triangle, the cosine gives the length of thex-component (run), and the tangent function gives the slope (y-component divided by the x-component). Moreprecise definitions are detailed below. Trigonometric functions are commonly defined as ratios of two sides of aright triangle containing the angle, and can equivalently be defined as the lengths of various line segments from aunit circle. More modern definitions express them as infinite series or as solutions of certain differential equations,allowing their extension to arbitrary positive and negative values and even to complex numbers.Trigonometric functions have a wide range of uses including computing unknown lengths and angles in triangles(often right triangles). In this use, trigonometric functions are used, for instance, in navigation, engineering, andphysics. A common use in elementary physics is resolving a vector into Cartesian coordinates. The sine and cosinefunctions are also commonly used to model periodic function phenomena such as sound and light waves, the positionand velocity of harmonic oscillators, sunlight intensity and day length, and average temperature variations throughthe year.In modern usage, there are six basic trigonometric functions, tabulated here with equations that relate them to oneanother. Especially with the last four, these relations are often taken as the definitions of those functions, but one candefine them equally well geometrically, or by other means, and then derive these relations.

Right-angled triangle definitions

The notion that there should be some standard correspondence betweenthe lengths of the sides of a triangle and the angles of the trianglecomes as soon as one recognizes that similar triangles maintain thesame ratios between their sides. That is, for any similar triangle theratio of the hypotenuse (for example) and another of the sides remainsthe same. If the hypotenuse is twice as long, so are the sides. It is theseratios that the trigonometric functions express.

To define the trigonometric functions for the angle A, start with anyright triangle that contains the angle A. The three sides of the triangleare named as follows:

• The hypotenuse is the side opposite the right angle, in this caseside h. The hypotenuse is always the longest side of a right-angledtriangle.

• The opposite side is the side opposite to the angle we are interested in (angle A), in this case side a.• The adjacent side is the side having both the angles of interest (angle A and right-angle C), in this case side b.

Trigonometric functions 47

(Top): Trigonometric function sinθ for selectedangles θ, π − θ, π + θ, and 2π − θ in the fourquadrants. (Bottom) Graph of sine functionversus angle. Angles from the top panel are

identified.

In ordinary Euclidean geometry, according to the triangle postulate theinside angles of every triangle total 180° (π radians). Therefore, in aright-angled triangle, the two non-right angles total 90° (π/2 radians),so each of these angles must be in the range of (0°,90°) as expressed ininterval notation. The following definitions apply to angles in this0° – 90° range. They can be extended to the full set of real argumentsby using the unit circle, or by requiring certain symmetries and thatthey be periodic functions. For example, the figure shows sin θ forangles θ, π − θ, π + θ, and 2π − θ depicted on the unit circle (top) andas a graph (bottom). The value of the sine repeats itself apart from signin all four quadrants, and if the range of θ is extended to additionalrotations, this behavior repeats periodically with a period 2π.

The trigonometric functions are summarized in the following table anddescribed in more detail below. The angle θ is the angle between thehypotenuse and the adjacent line – the angle at A in the accompanyingdiagram.

Function Abbreviation Description Identities (using radians)

Sine sin opposite / hypotenuse

Cosine cos adjacent / hypotenuse

Tangent tan (or tg) opposite / adjacent

Cotangent cot (or ctg or ctn) adjacent / opposite

Secant sec hypotenuse / adjacent

Cosecant csc (or cosec) hypotenuse / opposite

Trigonometric functions 48

The sine, tangent, and secant functions of an angle constructedgeometrically in terms of a unit circle. The number θ is the length ofthe curve; thus angles are being measured in radians. The secant andtangent functions rely on a fixed vertical line and the sine function ona moving vertical line. ("Fixed" in this context means not moving asθ changes; "moving" means depending on θ.) Thus, as θ goes from 0

up to a right angle, sin θ goes from 0 to 1, tan θ goes from 0 to ∞,and sec θ goes from 1 to ∞.

Sine, cosine, and tangent

The sine of an angle is the ratio of the length of theopposite side to the length of the hypotenuse. In ourcase

Trigonometric functions 49

The cosine, cotangent, and cosecant functions of an angle θconstructed geometrically in terms of a unit circle. The functionswhose names have the prefix co- use horizontal lines where the

others use vertical lines.

Note that this ratio does not depend on size of the particular right triangle chosen, as long as it contains the angle A,since all such triangles are similar.The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse. In our case

The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side (called sobecause it can be represented as a line segment tangent to the circle).[1] In our case

The acronyms "SOHCAHTOA" and "OHSAHCOAT" are commonly used mnemonics for these ratios.

Reciprocal functionsThe remaining three functions are best defined using the above three functions.The cosecant csc(A), or cosec(A), is the reciprocal of sin(A), i.e. the ratio of the length of the hypotenuse to thelength of the opposite side:

The secant sec(A) is the reciprocal of cos(A), i.e. the ratio of the length of the hypotenuse to the length of theadjacent side:

The cotangent cot(A) is the reciprocal of tan(A), i.e. the ratio of the length of the adjacent side to the length of theopposite side:

Trigonometric functions 50

Slope definitionsEquivalent to the right-triangle definitions the trigonometric functions can be defined in terms of the rise, run, andslope of a line segment relative to some horizontal line. The slope is commonly taught as "rise over run" or rise/run.The three main trigonometric functions are commonly taught in the order sine, cosine, tangent. With a segmentlength of 1 (as in a unit circle) the following correspondence of definitions exists:1. Sine is first, rise is first. Sine takes an angle and tells the rise when the length of the line is 1.2. Cosine is second, run is second. Cosine takes an angle and tells the run when the length of the line is 1.3. Tangent is the slope formula that combines the rise and run. Tangent takes an angle and tells the slope, and tells

the rise when the run is 1.This shows the main use of tangent and arctangent: converting between the two ways of telling the slant of a line,i.e., angles and slopes. (Note that the arctangent or "inverse tangent" is not to be confused with the cotangent, whichis cosine divided by sine.)While the length of the line segment makes no difference for the slope (the slope does not depend on the length ofthe slanted line), it does affect rise and run. To adjust and find the actual rise and run when the line does not have alength of 1, just multiply the sine and cosine by the line length. For instance, if the line segment has length 5, the runat an angle of 7° is 5 cos(7°)

Unit-circle definitions

The unit circle

The six trigonometric functions can also bedefined in terms of the unit circle, the circleof radius one centered at the origin. The unitcircle definition provides little in the way ofpractical calculation; indeed it relies on righttriangles for most angles.

The unit circle definition does, however,permit the definition of the trigonometricfunctions for all positive and negativearguments, not just for angles between 0 andπ/2 radians.

It also provides a single visual picture thatencapsulates at once all the importanttriangles. From the Pythagorean theorem theequation for the unit circle is:

In the picture, some common angles, measured in radians, are given. Measurements in the counterclockwisedirection are positive angles and measurements in the clockwise direction are negative angles.

Trigonometric functions 51

Let a line through the origin, making an angle of θ with the positive half of the x-axis, intersect the unit circle. The x-and y-coordinates of this point of intersection are equal to cos θ and sin θ, respectively.The triangle in the graphic enforces the formula; the radius is equal to the hypotenuse and has length 1, so we havesin θ = y/1 and cos θ = x/1. The unit circle can be thought of as a way of looking at an infinite number of triangles byvarying the lengths of their legs but keeping the lengths of their hypotenuses equal to 1.Note that these values can easily be memorized in the form

but the angles are not equally spaced.The values for 15°, 54° and 75° are slightly more complicated.

The sine and cosine functions graphed on the Cartesian plane.

For angles greater than 2π or less than −2π,simply continue to rotate around the circle;sine and cosine are periodic functions withperiod 2π:

for any angle θ and any integer k.The smallest positive period of a periodic function is called the primitive period of the function.The primitive period of the sine or cosine is a full circle, i.e. 2π radians or 360 degrees.Above, only sine and cosine were defined directly by the unit circle, but other trigonometric functions can be definedby:

So :• The primitive period of the secant, or cosecant is also a full circle, i.e. 2π radians or 360 degrees.• The primitive period of the tangent or cotangent is only a half-circle, i.e. π radians or 180 degrees.

Trigonometric functions 52

Trigonometric functions: Sine, Cosine, Tangent, Cosecant (dotted), Secant(dotted), Cotangent (dotted)

The image at right includes a graph of thetangent function.• Its θ-intercepts correspond to those of

sin(θ) while its undefined valuescorrespond to the θ-intercepts of cos(θ).

• The function changes slowly aroundangles of kπ, but changes rapidly atangles close to (k + 1/2)π.

• The graph of the tangent function alsohas a vertical asymptote at θ = (k + 1/2)π,the θ-intercepts of the cosine function,because the function approaches infinityas θ approaches (k + 1/2)π from the leftand minus infinity as it approaches(k + 1/2)π from the right.

All of the trigonometric functions of the angle θ can be constructed geometricallyin terms of a unit circle centered at O.

Alternatively, all of the basic trigonometricfunctions can be defined in terms of a unitcircle centered at O (as shown in the pictureto the right), and similar such geometricdefinitions were used historically.

• In particular, for a chord AB of the circle,where θ is half of the subtended angle,sin(θ) is AC (half of the chord), adefinition introduced in India[2] (seehistory).

• cos(θ) is the horizontal distance OC, andversin(θ) = 1 − cos(θ) is CD.

• tan(θ) is the length of the segment AE ofthe tangent line through A, hence theword tangent for this function. cot(θ) isanother tangent segment, AF.

• sec(θ) = OE and csc(θ) = OF are segments of secant lines (intersecting the circle at two points), and can also beviewed as projections of OA along the tangent at A to the horizontal and vertical axes, respectively.

• DE is exsec(θ) = sec(θ) − 1 (the portion of the secant outside, or ex, the circle).• From these constructions, it is easy to see that the secant and tangent functions diverge as θ approaches π/2 (90

degrees) and that the cosecant and cotangent diverge as θ approaches zero. (Many similar constructions arepossible, and the basic trigonometric identities can also be proven graphically.[3] )

Trigonometric functions 53

Series definitions

The sine function (blue) is closely approximated by its Taylor polynomial ofdegree 7 (pink) for a full cycle centered on the origin.

Using only geometry and properties oflimits, it can be shown that the derivative ofsine is cosine and the derivative of cosine isthe negative of sine. (Here, and generally incalculus, all angles are measured in radians;see also the significance of radians below.)One can then use the theory of Taylor seriesto show that the following identities hold forall real numbers x:[4]

These identities are sometimes taken as the definitions of the sine and cosine function. They are often used as thestarting point in a rigorous treatment of trigonometric functions and their applications (e.g., in Fourier series), sincethe theory of infinite series can be developed, independent of any geometric considerations, from the foundations ofthe real number system. The differentiability and continuity of these functions are then established from the seriesdefinitions alone.Combining these two series gives Euler's formula: cos x + i sin x = eix.Other series can be found.[5] For the following trigonometric functions:

Un is the nth up/down number,Bn is the nth Bernoulli number, andEn (below) is the nth Euler number.

Tangent

Trigonometric functions 54

When this series for the tangent function is expressed in a form in which the denominators are the correspondingfactorials, the numerators, called the "tangent numbers", have a combinatorial interpretation: they enumeratealternating permutations of finite sets of odd cardinality.[6]

Cosecant

Secant

When this series for the secant function is expressed in a form in which the denominators are the correspondingfactorials, the numerators, called the "secant numbers", have a combinatorial interpretation: they enumeratealternating permutations of finite sets of even cardinality.[7]

Cotangent

From a theorem in complex analysis, there is a unique analytic continuation of this real function to the domain ofcomplex numbers. They have the same Taylor series, and so the trigonometric functions are defined on the complexnumbers using the Taylor series above.There is a series representation as partial fraction expansion where just translated reciprocal functions are summedup, such that the poles of the cotangent function and the reciprocal functions match:[8]

This identity can be proven with the Herglotz trick.[9] By combining the -th with the -th term, it can beexpressed as an absolutely convergent series:

Trigonometric functions 55

Relationship to exponential function and complex numbers

Euler's formula illustrated with the three dimensional helix, starting with the 2-Dorthogonal components of the unit circle, sine and cosine (using θ = t ).

It can be shown from the seriesdefinitions[10] that the sine and cosinefunctions are the imaginary and realparts, respectively, of the complexexponential function when itsargument is purely imaginary:

This identity is called Euler's formula. In this way, trigonometric functions become essential in the geometricinterpretation of complex analysis. For example, with the above identity, if one considers the unit circle in thecomplex plane, parametrized by e ix, and as above, we can parametrize this circle in terms of cosines and sines, therelationship between the complex exponential and the trigonometric functions becomes more apparent.Furthermore, this allows for the definition of the trigonometric functions for complex arguments z:

where i 2 = −1. The sine and cosine defined by this are entire functions. Also, for purely real x,

It is also sometimes useful to express the complex sine and cosine functions in terms of the real and imaginary partsof their arguments.

This exhibits a deep relationship between the complex sine and cosine functions and their real (sin, cos) andhyperbolic real (sinh, cosh) counterparts.

Trigonometric functions 56

Complex graphs

In the following graphs, the domain is the complex plane pictured, and the range values are indicated at each pointby color. Brightness indicates the size (absolute value) of the range value, with black being zero. Hue varies withargument, or angle, measured from the positive real axis. (more)

Trigonometric functions in the complex plane

Definitions via differential equationsBoth the sine and cosine functions satisfy the differential equation

That is to say, each is the additive inverse of its own second derivative. Within the 2-dimensional function space Vconsisting of all solutions of this equation,• the sine function is the unique solution satisfying the initial condition and• the cosine function is the unique solution satisfying the initial condition .Since the sine and cosine functions are linearly independent, together they form a basis of V. This method of definingthe sine and cosine functions is essentially equivalent to using Euler's formula. (See linear differential equation.) Itturns out that this differential equation can be used not only to define the sine and cosine functions but also to provethe trigonometric identities for the sine and cosine functions.Further, the observation that sine and cosine satisfies y′′ = −y means that they are eigenfunctions of thesecond-derivative operator.The tangent function is the unique solution of the nonlinear differential equation

satisfying the initial condition y(0) = 0. There is a very interesting visual proof that the tangent function satisfies thisdifferential equation.[11]

Trigonometric functions 57

The significance of radiansRadians specify an angle by measuring the length around the path of the unit circle and constitute a special argumentto the sine and cosine functions. In particular, only sines and cosines that map radians to ratios satisfy the differentialequations that classically describe them. If an argument to sine or cosine in radians is scaled by frequency,

then the derivatives will scale by amplitude.

Here, k is a constant that represents a mapping between units. If x is in degrees, then

This means that the second derivative of a sine in degrees does not satisfy the differential equation

but rather

The cosine's second derivative behaves similarly.This means that these sines and cosines are different functions, and that the fourth derivative of sine will be sineagain only if the argument is in radians.

IdentitiesMany identities interrelate the trigonometric functions. Among the most frequently used is the Pythagoreanidentity, which states that for any angle, the square of the sine plus the square of the cosine is 1. This is easy to seeby studying a right triangle of hypotenuse 1 and applying the Pythagorean theorem. In symbolic form, thePythagorean identity is written

where sin2 x + cos2 x is standard notation for (sin x)2 + (cos x)2.Other key relationships are the sum and difference formulas, which give the sine and cosine of the sum anddifference of two angles in terms of sines and cosines of the angles themselves. These can be derived geometrically,using arguments that date to Ptolemy. One can also produce them algebraically using Euler's formula.

When the two angles are equal, the sum formulas reduce to simpler equations known as the double-angle formulae.

These identities can also be used to derive the product-to-sum identities that were used in antiquity to transform theproduct of two numbers into a sum of numbers and greatly speed operations, much like the logarithm function.

Trigonometric functions 58

CalculusFor integrals and derivatives of trigonometric functions, see the relevant sections of Differentiation of trigonometricfunctions, Lists of integrals and List of integrals of trigonometric functions. Below is the list of the derivatives andintegrals of the six basic trigonometric functions. The number C is a constant of integration.

Definitions using functional equationsIn mathematical analysis, one can define the trigonometric functions using functional equations based on propertieslike the sum and difference formulas. Taking as given these formulas and the Pythagorean identity, for example, onecan prove that only two real functions satisfy those conditions. Symbolically, we say that there exists exactly onepair of real functions — and  — such that for all real numbers and , the following equations hold:

with the added condition that.

Other derivations, starting from other functional equations, are also possible, and such derivations can be extended tothe complex numbers. As an example, this derivation can be used to define trigonometry in Galois fields.

ComputationThe computation of trigonometric functions is a complicated subject, which can today be avoided by most peoplebecause of the widespread availability of computers and scientific calculators that provide built-in trigonometricfunctions for any angle. This section, however, describes details of their computation in three important contexts: thehistorical use of trigonometric tables, the modern techniques used by computers, and a few "important" angles wheresimple exact values are easily found.The first step in computing any trigonometric function is range reduction—reducing the given angle to a "reducedangle" inside a small range of angles, say 0 to π/2, using the periodicity and symmetries of the trigonometricfunctions.Prior to computers, people typically evaluated trigonometric functions by interpolating from a detailed table of theirvalues, calculated to many significant figures. Such tables have been available for as long as trigonometric functionshave been described (see History below), and were typically generated by repeated application of the half-angle andangle-addition identities starting from a known value (such as sin(π/2) = 1).Modern computers use a variety of techniques.[12] One common method, especially on higher-end processors with floating point units, is to combine a polynomial or rational approximation (such as Chebyshev approximation, best uniform approximation, and Padé approximation, and typically for higher or variable precisions, Taylor and Laurent series) with range reduction and a table lookup—they first look up the closest angle in a small table, and then use the

Trigonometric functions 59

polynomial to compute the correction.[13] Devices that lack hardware multipliers often use an algorithm calledCORDIC (as well as related techniques), which uses only addition, subtraction, bitshift, and table lookup. Thesemethods are commonly implemented in hardware floating-point units for performance reasons.For very high precision calculations, when series expansion convergence becomes too slow, trigonometric functionscan be approximated by the arithmetic-geometric mean, which itself approximates the trigonometric function by the(complex) elliptic integral.[14]

Finally, for some simple angles, the values can be easily computed by hand using the Pythagorean theorem, as in thefollowing examples. For example, the sine, cosine and tangent of any integer multiple of radians (3°) can befound exactly by hand.Consider a right triangle where the two other angles are equal, and therefore are both radians (45°). Then thelength of side b and the length of side a are equal; we can choose . The values of sine, cosine andtangent of an angle of radians (45°) can then be found using the Pythagorean theorem:

Therefore:

To determine the trigonometric functions for angles of π/3 radians (60 degrees) and π/6 radians (30 degrees), westart with an equilateral triangle of side length 1. All its angles are π/3 radians (60 degrees). By dividing it into two,we obtain a right triangle with π/6 radians (30 degrees) and π/3 radians (60 degrees) angles. For this triangle, theshortest side = 1/2, the next largest side =(√3)/2 and the hypotenuse = 1. This yields:

Special values in trigonometric functionsThere are some commonly used special values in trigonometric functions, as shown in the following table.

Function

sin

cos

tan

cot

sec

csc

Trigonometric functions 60

Inverse functionsThe trigonometric functions are periodic, and hence not injective, so strictly they do not have an inverse function.Therefore to define an inverse function we must restrict their domains so that the trigonometric function is bijective.In the following, the functions on the left are defined by the equation on the right; these are not proved identities. Theprincipal inverses are usually defined as:

Function Definition Value Field

For inverse trigonometric functions, the notations sin−1 and cos−1 are often used for arcsin and arccos, etc. When thisnotation is used, the inverse functions could be confused with the multiplicative inverses of the functions. Thenotation using the "arc-" prefix avoids such confusion, though "arcsec" can be confused with "arcsecond".Just like the sine and cosine, the inverse trigonometric functions can also be defined in terms of infinite series. Forexample,

These functions may also be defined by proving that they are antiderivatives of other functions. The arcsine, forexample, can be written as the following integral:

Analogous formulas for the other functions can be found at Inverse trigonometric functions. Using the complexlogarithm, one can generalize all these functions to complex arguments:

Trigonometric functions 61

Properties and applicationsThe trigonometric functions, as the name suggests, are of crucial importance in trigonometry, mainly because of thefollowing two results.

Law of sinesThe law of sines states that for an arbitrary triangle with sides a, b, and c and angles opposite those sides A, B and C:

or, equivalently,

where R is the triangle's circumradius.

A Lissajous curve, a figure formed with atrigonometry-based function.

It can be proven by dividing the triangle into two right ones and usingthe above definition of sine. The law of sines is useful for computingthe lengths of the unknown sides in a triangle if two angles and oneside are known. This is a common situation occurring in triangulation,a technique to determine unknown distances by measuring two anglesand an accessible enclosed distance.

Law of cosines

The law of cosines (also known as the cosine formula) is an extensionof the Pythagorean theorem:

or equivalently,

In this formula the angle at C is opposite to the side c. This theorem can be proven by dividing the triangle into tworight ones and using the Pythagorean theorem.The law of cosines can be used to determine a side of a triangle if two sides and the angle between them are known.It can also be used to find the cosines of an angle (and consequently the angles themselves) if the lengths of all thesides are known.

Trigonometric functions 62

Law of tangentsThe following all form the law of tangents[15]

The explanation of the formulae in words would be cumbersome, but the patterns of sums and differences; for thelengths and corresponding opposite angles, are apparent in the theorem.

Law of cotangentsIf

(the radius of the inscribed circle for the triangle) and

(the semi-perimeter for the triangle), then the following all form the law of cotangents[16]

It follows that

In words the theorem is: the cotangent of a half-angle equals the ratio of the semi-perimeter minus the opposite sideto the said angle, to the inradius for the triangle.

Trigonometric functions 63

Other useful properties

Periodic functions

Click on the image to see an animation of the additive synthesis of a square wave with anincreasing number of harmonics

Superimposed sinusoidal wave basis functions (bottom) form a sawtoothwave (top) when added; the basis functions have wavelengths λ/k

(k = integer) shorter than the wavelength λ of the sawtooth itself (except fork = 1). All basis functions have nodes at the nodes of the sawtooth, but all butthe fundamental have additional nodes. The oscillation about the sawtooth is

called the Gibbs phenomenon

The trigonometric functions are alsoimportant in physics. The sine and thecosine functions, for example, are usedto describe simple harmonic motion,which models many naturalphenomena, such as the movement of amass attached to a spring and, forsmall angles, the pendular motion of amass hanging by a string. The sine andcosine functions are one-dimensionalprojections of uniform circular motion.

Trigonometric functions also prove tobe useful in the study of generalperiodic functions. The characteristicwave patterns of periodic functions areuseful for modeling recurringphenomena such as sound or lightwaves.[17]

Under rather general conditions, aperiodic function ƒ(x) can be expressedas a sum of sine waves or cosine wavesin a Fourier series.[18] Denoting thesine or cosine basis functions by φk,the expansion of the periodic functionƒ(t) takes the form:

For example, the square wave can be written as the Fourier series

In the animation of a square wave at top right it can be seen that just a few terms already produce a fairly goodapproximation. The superposition of several terms in the expansion of a sawtooth wave are shown underneath.

Trigonometric functions 64

HistoryWhile the early study of trigonometry can be traced to antiquity, the trigonometric functions as they are in use todaywere developed in the medieval period. The chord function was discovered by Hipparchus of Nicaea (180–125 BC)and Ptolemy of Roman Egypt (90–165 AD).The functions sine and cosine can be traced to the jyā and koti-jyā functions used in Gupta period Indian astronomy(Aryabhatiya, Surya Siddhanta), via translation from Sanskrit to Arabic and then from Arabic to Latin.[19]

All six trigonometric functions in current use were known in Islamic mathematics by the 9th century, as was the lawof sines, used in solving triangles.[20] al-Khwārizmī produced tables of sines, cosines and tangents. They werestudied by authors including Omar Khayyám, Bhāskara II, Nasir al-Din al-Tusi, Jamshīd al-Kāshī (14th century),Ulugh Beg (14th century), Regiomontanus (1464), Rheticus, and Rheticus' student Valentinus OthoMadhava of Sangamagrama (c. 1400) made early strides in the analysis of trigonometric functions in terms ofinfinite series.[21]

The first published use of the abbreviations 'sin', 'cos', and 'tan' is by the 16th century French mathematician AlbertGirard.In a paper published in 1682, Leibniz proved that sin x is not an algebraic function of x.[22]

Leonhard Euler's Introductio in analysin infinitorum (1748) was mostly responsible for establishing the analytictreatment of trigonometric functions in Europe, also defining them as infinite series and presenting "Euler's formula",as well as the near-modern abbreviations sin., cos., tang., cot., sec., and cosec.[2]

A few functions were common historically, but are now seldom used, such as the chord (crd(θ) = 2 sin(θ/2)), theversine (versin(θ) = 1 − cos(θ) = 2 sin2(θ/2)) (which appeared in the earliest tables [2] ), the haversine (haversin(θ) =versin(θ) / 2 = sin2(θ/2)), the exsecant (exsec(θ) = sec(θ) − 1) and the excosecant (excsc(θ) = exsec(π/2 − θ) =csc(θ) − 1). Many more relations between these functions are listed in the article about trigonometric identities.Etymologically, the word sine derives from the Sanskrit word for half the chord, jya-ardha, abbreviated to jiva. Thiswas transliterated in Arabic as jiba, written jb, vowels not being written in Arabic. Next, this transliteration wasmis-translated in the 12th century into Latin as sinus, under the mistaken impression that jb stood for the word jaib,which means "bosom" or "bay" or "fold" in Arabic, as does sinus in Latin.[23] Finally, English usage converted theLatin word sinus to sine.[24] The word tangent comes from Latin tangens meaning "touching", since the line touchesthe circle of unit radius, whereas secant stems from Latin secans — "cutting" — since the line cuts the circle.

Notes[1] Dictionary.com: Tangent def 5b, 5c (http:/ / dictionary. reference. com/ browse/ tangent)[2] See Boyer (1991).[3] See Maor (1998)[4] See Ahlfors, pages 43–44.[5] Abramowitz; Weisstein.[6] Stanley, Enumerative Combinatorics, Vol I., page 149[7] Stanley, Enumerative Combinatorics, Vol I[8] Aigner, Martin; Ziegler, Günter M. (2000). Proofs from THE BOOK (http:/ / www. springer. com/ mathematics/ book/ 978-3-642-00855-9)

(Second ed.). Springer-Verlag. p. 149. ISBN 9773540678657. .[9] Remmert, Reinhold (1991). Theory of complex functions (http:/ / books. google. com/ books?id=CC0dQxtYb6kC). Springer. p. 327.

ISBN 0-387-97195-5. ., Extract of page 327 (http:/ / books. google. com/ books?id=CC0dQxtYb6kC& pg=PA327)[10] For a demonstration, see Euler's_formula#Using_power_series[11] Needham, p. Visual Complex Analysis. ISBN 0198534469.[12] Kantabutra.[13] However, doing that while maintaining precision is nontrivial, and methods like Gal's accurate tables, Cody and Waite reduction, and Payne

and Hanek reduction algorithms can be used.[14] "R. P. Brent, "Fast Multiple-Precision Evaluation of Elementary Functions", J. ACM 23, 242 (1976)." (http:/ / doi. acm. org/ 10. 1145/

321941. 321944). .

Trigonometric functions 65

[15] The Universal Encyclopaedia of Mathematics, Pan Reference Books, 1976, page 529. English version George Allen and Unwin, 1964.Translated from the German version Meyers Rechenduden, 1960.

[16] The Universal Encyclopaedia of Mathematics, Pan Reference Books, 1976, page 530. English version George Allen and Unwin, 1964.Translated from the German version Meyers Rechenduden, 1960.

[17] Stanley J Farlow (1993). Partial differential equations for scientists and engineers (http:/ / books. google. com/books?id=DLUYeSb49eAC& pg=PA82) (Reprint of Wiley 1982 ed.). Courier Dover Publications. p. 82. ISBN 048667620X. .

[18] See for example, Gerald B Folland (2009). "Convergence and completeness" (http:/ / books. google. com/ books?id=idAomhpwI8MC&pg=PA77). Fourier Analysis and its Applications (Reprint of Wadsworth & Brooks/Cole 1992 ed.). American Mathematical Society. pp. 77 ff.ISBN 0821847902. .

[19] Boyer, Carl B. (1991). A History of Mathematics (Second ed.). John Wiley & Sons, Inc.. ISBN 0471543977, p. 210.[20] Owen Gingerich (1986). Islamic Astronomy (http:/ / faculty. kfupm. edu. sa/ PHYS/ alshukri/ PHYS215/ Islamic_astronomy. htm). 254.

Scientific American. p. 74. . Retrieved 2010-37-13.[21] J J O'Connor and E F Robertson. "Madhava of Sangamagrama" (http:/ / www-gap. dcs. st-and. ac. uk/ ~history/ Biographies/ Madhava.

html). School of Mathematics and Statistics University of St Andrews, Scotland. . Retrieved 2007-09-08.[22] Nicolás Bourbaki (1994). Elements of the History of Mathematics. Springer.[23] See Maor (1998), chapter 3, regarding the etymology.[24] "Clark University" (http:/ / www. clarku. edu/ ~djoyce/ trig/ ). .

References• Abramowitz, Milton and Irene A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and

Mathematical Tables, Dover, New York. (1964). ISBN 0-486-61272-4.• Lars Ahlfors, Complex Analysis: an introduction to the theory of analytic functions of one complex variable,

second edition, McGraw-Hill Book Company, New York, 1966.• Boyer, Carl B., A History of Mathematics, John Wiley & Sons, Inc., 2nd edition. (1991). ISBN 0-471-54397-7.• Gal, Shmuel and Bachelis, Boris. An accurate elementary mathematical library for the IEEE floating point

standard, ACM Transaction on Mathematical Software (1991).• Joseph, George G., The Crest of the Peacock: Non-European Roots of Mathematics, 2nd ed. Penguin Books,

London. (2000). ISBN 0-691-00659-8.• Kantabutra, Vitit, "On hardware for computing exponential and trigonometric functions," IEEE Trans. Computers

45 (3), 328–339 (1996).• Maor, Eli, Trigonometric Delights (http:/ / www. pupress. princeton. edu/ books/ maor/ ), Princeton Univ. Press.

(1998). Reprint edition (February 25, 2002): ISBN 0-691-09541-8.• Needham, Tristan, "Preface" (http:/ / www. usfca. edu/ vca/ PDF/ vca-preface. pdf)" to Visual Complex Analysis

(http:/ / www. usfca. edu/ vca/ ). Oxford University Press, (1999). ISBN 0-19-853446-9.• O'Connor, J.J., and E.F. Robertson, "Trigonometric functions" (http:/ / www-gap. dcs. st-and. ac. uk/ ~history/

HistTopics/ Trigonometric_functions. html), MacTutor History of Mathematics archive. (1996).• O'Connor, J.J., and E.F. Robertson, "Madhava of Sangamagramma" (http:/ / www-groups. dcs. st-and. ac. uk/

~history/ Mathematicians/ Madhava. html), MacTutor History of Mathematics archive. (2000).• Pearce, Ian G., "Madhava of Sangamagramma" (http:/ / www-history. mcs. st-andrews. ac. uk/ history/ Projects/

Pearce/ Chapters/ Ch9_3. html), MacTutor History of Mathematics archive. (2002).• Weisstein, Eric W., "Tangent" (http:/ / mathworld. wolfram. com/ Tangent. html) from MathWorld, accessed 21

January 2006.

Trigonometric functions 66

External links• Visionlearning Module on Wave Mathematics (http:/ / www. visionlearning. com/ library/ module_viewer.

php?mid=131& l=& c3=)• GonioLab (http:/ / glab. trixon. se/ ): Visualization of the unit circle, trigonometric and hyperbolic functions• Dave's draggable diagram. (http:/ / www. clarku. edu/ ~djoyce/ trig/ ) (Requires java browser plugin)

Complex number

A complex number can be visually represented asa pair of numbers forming a vector on a diagram

called an Argand diagram, representing thecomplex plane. Re is the real axis, Im is the

imaginary axis, and i is the square root of –1.

A complex number is a number consisting of a real part and animaginary part. Complex numbers extend the idea of theone-dimensional number line to the two-dimensional complex plane byusing the number line for the real part and adding a vertical axis to plotthe imaginary part. In this way the complex numbers contain theordinary real numbers while extending them in order to solve problemsthat would be impossible with only real numbers.

Complex numbers are used in many scientific fields, includingengineering, electromagnetism, quantum physics, applied mathematics,and chaos theory. Italian mathematician Gerolamo Cardano is the firstknown to have conceived complex numbers; he called them"fictitious", during his attempts to find solutions to cubic equations inthe 16th century.[1]

Introduction and definition

Complex numbers have been introduced to allow for solutions ofcertain equations that have no real solution: the equation

has no real solution x, since the square of x is 0 or positive, so x2 + 1 cannot be zero. Complex numbers are a solutionto this problem. The idea is to enhance the real numbers by introducing a non-real number i whose square is −1, sothat x = i and x = −i are the two solutions to the preceding equation.

DefinitionA complex number is an expression of the form

where a and b are real numbers and i is a mathematical symbol which is called the imaginary unit. For example, −3.5+ 2i is a complex number.The real number a of the complex number z = a + bi is called the real part of z and the real number b is theimaginary part.[2] They are denoted Re(z) or ℜ(z) and Im(z) or ℑ(z), respectively. For example,

Some authors write a+ib instead of a+bi. In some disciplines (in particular, electrical engineering, where i is asymbol for current), the imaginary unit i is instead written as j, so complex numbers are written as a + bj or a + jb.

Complex number 67

A real number a can usually be regarded as a complex number with an imaginary part of zero, that is to say, a + 0i.However the sets are defined differently and have slightly different operations defined, for instance comparisonoperations are not defined for complex numbers. Complex numbers whose real part is zero, that is to say, those ofthe form 0 + bi, are called imaginary numbers. It is common to write a for a + 0i and bi for 0 + bi. Moreover, when bis negative, it is common to write a − bi instead of a + (−b)i, for example 3 − 4i instead of 3 + (−4)i.The set of all complex numbers is denoted by C or .

The complex plane

Figure 1: A complex number plotted as a point(red) and position vector (blue) on an Argand

diagram; is the rectangular expressionof the point.

A complex number can be viewed as a point or position vector in atwo-dimensional Cartesian coordinate system called the complex planeor Argand diagram (see Pedoe 1988 and Solomentsev 2001), namedafter Jean-Robert Argand. The numbers are conventionally plottedusing the real part as the horizontal component, and imaginary part asvertical (see Figure 1). These two values used to identify a givencomplex number are therefore called its Cartesian, rectangular, oralgebraic form.

The defining characteristic of a position vector is that it has magnitudeand direction. These are emphasised in a complex number's polar formand it turns out notably that the operations of addition andmultiplication take on a very natural geometric character whencomplex numbers are viewed as position vectors: addition correspondsto vector addition while multiplication corresponds to multiplying their magnitudes and adding their arguments (i.e.the angles they make with the x axis). Viewed in this way the multiplication of a complex number by i correspondsto rotating a complex number anticlockwise through 90° about the origin.

History in briefMain section: History

The solution of a general cubic equation in radicals (without trigonometric functions) may require intermediatecalculations containing the square roots of negative numbers, even when the final solutions are real numbers, asituation known as casus irreducibilis. This conundrum led Italian mathematician Gerolamo Cardano to conceive ofcomplex numbers in around 1545, though his understanding was rudimentary.Work on the problem of general polynomials ultimately led to the fundamental theorem of algebra, which shows thatwith complex numbers, a solution exists to every polynomial equation of degree one or higher. Complex numbersthus form an algebraically closed field, where any polynomial equation has a root.Many mathematicians contributed to the full development of complex numbers. The rules for addition, subtraction,multiplication, and division of complex numbers were developed by the Italian mathematician Rafael Bombelli.[3] Amore abstract formalism for the complex numbers was further developed by the Irish mathematician William RowanHamilton, who extended this abstraction to the theory of quaternions.

Complex number 68

Elementary operations

Conjugation

Geometric representation of and its conjugatein the complex plane

The complex conjugate of the complex number z = x + yi is defined tobe x − yi. It is denoted or . Geometrically, is the "reflection"of z about the real axis. In particular, conjugating twice gives theoriginal complex number: .

The real and imaginary parts of a complex number can be extractedusing the conjugate:

Moreover, a complex number is real if and only if it equals its conjugate.Conjugation distributes over the standard arithmetic operations:

The reciprocal of a nonzero complex number z = x + yi is given by

This formula can be used to compute the multiplicative inverse of a complex number if it is given in rectangularcoordinates. Inversive geometry, a branch of geometry studying more general reflections than ones about a line, canalso be expressed in terms of complex numbers.

Complex number 69

Addition and subtraction

Addition of two complex numbers can be donegeometrically by constructing a parallelogram.

Complex numbers are added by adding the real and imaginaryparts of the summands. That is to say:

Similarly, subtraction is defined by

Using the visualization of complex numbers in the complex plane, the addition has the following geometricinterpretation: the sum of two complex numbers A and B, interpreted as points of the complex plane, is the point Xobtained by building a parallelogram three of whose vertices are 0, A and B. Equivalently, X is the point such that thetriangles with vertices 0, A, B, and X, B, A, are congruent.

Multiplication and divisionThe multiplication of two complex numbers is defined by the following formula:

In particular, the square of the imaginary unit is −1:

The preceding definition of multiplication of general complex numbers is the natural way of extending thisfundamental property of the imaginary unit. Indeed, treating i as a variable, the formula follows from this

(distributive law)(commutative law of addition—the order of the summands can be

changed)

(commutative law of multiplication—the order of the factors canbe changed)

(fundamental property of the imaginary unit).The division of two complex numbers is defined in terms of complex multiplication, which is described above, andreal division:

Division can be defined in this way because of the following observation:

Complex number 70

As shown earlier, is the complex conjugate of the denominator . The real part c and the imaginarypart d of the denominator must not both be zero for division to be defined.

Square root

The square roots of a + bi (with b ≠ 0) are , where

and

where sgn is the signum function. This can be seen by squaring to obtain a + bi.[4] [5] Here is called the modulus of a + bi, and the square root with non-negative real part is called the principal square root.

Polar form

Figure 2: The argument φ and modulus r locate apoint on an Argand diagram;

or are polar

expressions of the point.

Absolute value and argument

Another way of encoding points in the complex plane other than usingthe x- and y-coordinates is to use the distance of a point P to O, thepoint whose coordinates are (0, 0) (origin), and the angle of the linethrough P and O. This idea leads to the polar form of complexnumbers.

The absolute value (or modulus or magnitude) of a complex number z= x+yi is

If z is a real number (i.e., y = 0), then r = |x|. In general, by Pythagoras' theorem, r is the distance of the point Prepresenting the complex number z to the origin.

The argument or phase of z is the angle of the radius OP with the positive real axis, and is written as . Aswith the modulus, the argument can be found from the rectangular form :[6]

Complex number 71

The value of φ must always be expressed in radians. It can change by any multiple of 2π and still give the sameangle. Hence, the arg function is sometimes considered as multivalued. Normally, as given above, the principal valuein the interval is chosen. Values in the range are obtained by adding if the value is negative.The polar angle for the complex number 0 is undefined, but arbitrary choice of the angle 0 is common.The value of φ equals the result of atan2: .Together, r and φ give another way of representing complex numbers, the polar form, as the combination of modulusand argument fully specify the position of a point on the plane. Recovering the original rectangular co-ordinatesfrom the polar form is done by the formula called trigonometric form

Using Euler's formula this can be written as

Using the cis function, this is sometimes abbreviated to

In angle notation, often used in electronics to represent a phasor with amplitude r and phase φ it is written as[7]

Multiplication, division and exponentiation in polar form

Multiplication of 2+i (blue triangle) and 3+i (redtriangle). The red triangle is rotated to match thevertex of the blue one and stretched by √5, thelength of the hypotenuse of the blue triangle.

The relevance of representing complex numbers in polar form stemsfrom the fact that the formulas for multiplication, division andexponentiation are simpler than the ones using Cartesian coordinates.Given two complex numbers z1 = r1(cos φ1 + isin φ1) and z2 =r2(cosφ2 + isin φ2) the formula for multiplication is

In other words, the absolute values are multiplied and the arguments are added to yield the polar form of the product.For example, multiplying by i corresponds to a quarter-rotation counter-clockwise, which gives back i 2 = −1. Thepicture at the right illustrates the multiplication of

Complex number 72

Since the real and imaginary part of 5+5i are equal, the argument of that number is 45 degrees, or π/4 (in radian). Onthe other hand, it is also the sum of the angles at the origin of the red and blue triangle are arctan(1/3) andarctan(1/2), respectively. Thus, the formula

holds. As the arctan function can be approximated highly efficiently, formulas like this—known as Machin-likeformulas—are used for high-precision approximations of π.Similarly, division is given by

This also implies de Moivre's formula for exponentiation of complex numbers with integer exponents:

The n-th roots of z are given by

for any integer k satisfying 0 ≤ k ≤ n − 1. Here is the usual (positive) nth root of the positive real number r.While the nth root of a positive real number r is chosen to be the positive real number c satisfying cn = x there is nonatural way of distinguishing one particular complex nth root of a complex number. Therefore, the nth root of z isconsidered as a multivalued function (in z), as opposed to a usual function f, for which f(z) is a uniquely definednumber. Formulas such as

(which holds for positive real numbers), do in general not hold for complex numbers.

Properties

Field structureThe set C of complex numbers is a field. Briefly, this means that the following facts hold: first, any two complexnumbers can be added and multiplied to yield another complex number. Second, for any complex number a, itsnegative −a is also a complex number; and third, every nonzero complex number has a reciprocal complex number.Moreover, these operations satisfy a number of laws, for example the law of commutativity of addition andmultiplication for any two complex numbers z1 and z2:

These two laws and the other requirements on a field can be proven by the formulas given above, using the fact thatthe real numbers themselves form a field.Unlike the reals, C is not an ordered field, that is to say, it is not possible to define a relation z1 < z2 that iscompatible with the addition and multiplication. In fact, in any ordered field, the square of any element is necessarilypositive, so i2 = −1 precludes the existence of an ordering on C.When the underlying field for a mathematical topic or construct is the field of complex numbers, the thing's name isusually modified to reflect that fact. For example: complex analysis, complex matrix, complex polynomial, andcomplex Lie algebra.

Complex number 73

Solutions of polynomial equationsGiven any complex numbers (called coefficients) a0, ..., an, the equation

has at least one complex solution z, provided that at least one of the higher coefficients, a1, ..., an, is nonzero. This isthe statement of the fundamental theorem of algebra. Because of this fact, C is called an algebraically closed field.This property does not hold for the field of rational numbers Q (the polynomial x2 − 2 does not have a rational root,since √2 is not a rational number) nor the real numbers R (the polynomial x2 + a does not have a real solution for a >0, since the square of x is positive for any real number x).There are various proofs of this theorem, either by analytic methods such as Liouville's theorem, or topological onessuch as the winding number, or a proof combining Galois theory and the fact that any real polynomial of odd degreehas at least one root.Because of this fact, theorems that hold "for any algebraically closed field", apply to C. For example, any complexmatrix has at least one (complex) eigenvalue.

Algebraic characterizationThe field C has the following three properties: first, it has characteristic 0. This means that 1 + 1 + ... + 1 ≠ 0 for anynumber of summands (all of which equal one). Second, its transcendence degree over Q, the prime field of C is thecardinality of the continuum. Third, it is algebraically closed (see above). It can be shown that any field having theseproperties is isomorphic (as a field) to C. For example, the algebraic closure of Qp also satisfies these threeproperties, so these two fields are isomorphic. Also, C is isomorphic to the field of complex Puiseux series.However, specifying an isomorphism requires the axiom of choice. Another consequence of this algebraiccharacterization is that C contains many proper subfields which are isomorphic to C (the same is true of R, whichcontains many sub fields isomorphic to itself).

Characterization as a topological fieldThe preceding characterization of C describes the algebraic aspects of C, only. That is to say, the properties ofnearness and continuity, which matter in areas such as analysis and topology, are not dealt with. The followingdescription of C as a topological field (that is, a field that is equipped with a topology, which allows to specifynotions such as convergence) does take into account the topological properties. C contains a subset P (namely the setof positive real numbers) of nonzero elements satisfying the following three conditions:• P is closed under addition, multiplication and taking inverses.• If x and y are distinct elements of P, then either x−y or y−x is in P.• If S is any nonempty subset of P, then S + P = x + P for some x in C.Moreover, C has a nontrivial involutive automorphism (namely the complex conjugation), fixing P andsuch that xx∗ is in P for any nonzero x in C.Any field F with these properties can be endowed with a topology by taking the sets B(x, p) = {y | P − (y − x)(y −x) ∈P} as a base, where x ranges over the field and p ranges over P. With this topology F is isomorphic as a topologicalfield to C.The only connected locally compact topological fields are R and C. This gives another characterization of C as atopological field, since C can be distinguished from R because the nonzero complex numbers are connected, whilethe nonzero real numbers are not.

Complex number 74

Formal construction

Formal developmentAbove, complex numbers have been defined by introducing i, the imaginary unit, as a symbol. More rigorously, theset C of complex numbers can be defined as the set R2 of ordered pairs (a, b) of real numbers. In this notation, theabove formulas for addition and multiplication read

It is then just a matter of notation to express (a, b) as a + ib.Though this low-level construction does accurately describe the structure of the complex numbers, the followingequivalent definition reveals the algebraic nature of C more immediately. This characterization relies on the notionof fields and polynomials. A field is a set endowed with an addition, subtraction, multiplication and divisionoperations which behave as is familiar from, say, rational numbers. For example, the distributive law

is required to hold for any three elements x, y and z of a field. The set R of real numbers does form a field. Apolynomial p(X) with real coefficients is an expression of the form

where the a0, ..., an are real numbers. The usual addition and multiplication of polynomials endows the set R[X] ofall such polynomials with a ring structure. This ring is called polynomial ring. The quotient ring R[X]/(X2+1) can beshown to be a field. This extension field contains two square roots of −1, namely (the cosets of) X and −X,respectively. (The cosets of) 1 and X form a basis of R[X]/(X2+1) as a real vector space, which means that eachelement of the extension field can be uniquely written as a linear combination in these two elements. Equivalently,elements of the extension field can be written as ordered pairs (a,b) of real numbers. Moreover, the above formulasfor addition etc. correspond to the ones yielded by this abstract algebraic approach—the two definitions of the fieldC are said to be isomorphic (as fields). Together with the above-mentioned fact that C is algebraically closed, thisalso shows that C is an algebraic closure of R.

Matrix representation of complex numbersComplex numbers can also be represented by 2×2 matrices that have the following form:

Here the entries a and b are real numbers. The sum and product of two such matrices is again of this form, and thesum and product of complex numbers corresponds to the sum and product of such matrices. The geometricdescription of the multiplication of complex numbers can also be phrased in terms of rotation matrices by using thiscorrespondence between complex numbers and such matrices. Moreover, the square of the absolute value of acomplex number expressed as a matrix is equal to the determinant of that matrix:

The conjugate corresponds to the transpose of the matrix.Though this representation of complex numbers with matricies is the most common, many other representations arise

from matrices other than that square to the negative of the identity matrix. See the article on 2 × 2 real

matrices for other representations of complex numbers.

Complex number 75

Complex analysis

Color wheel graph of sin(1/z). Black parts inside refer to numbers havinglarge absolute values.

The study of functions of a complex variable isknown as complex analysis and has enormouspractical use in applied mathematics as well as inother branches of mathematics. Often, the mostnatural proofs for statements in real analysis oreven number theory employ techniques fromcomplex analysis (see prime number theorem foran example). Unlike real functions which arecommonly represented as two-dimensional graphs,complex functions have four-dimensional graphsand may usefully be illustrated by color coding athree-dimensional graph to suggest fourdimensions, or by animating the complexfunction's dynamic transformation of the complexplane.

Complex exponential and relatedfunctions

The notions of convergent series and continuousfunctions in (real) analysis have natural analogs in complex analysis. A sequence of complex numbers is said toconverge if and only if its real and imaginary parts do. This is equivalent to the (ε, δ)-definition of limits, where theabsolute value of real numbers is replaced by the one of complex numbers. From a more abstract point of view, C,endowed with the metric

is a complete metric space, which notably includes the triangle inequality

for any two complex numbers z1 and z2.Like in real analysis, this notion of convergence is used to construct a number of elementary functions: theexponential function exp(z), also written ez, is defined as the infinite series

and the series defining the real trigonmetric functions sine and cosine, as well as hyperbolic functions such as sinhalso carry over to complex arguments without change. Euler's identity states:

for any real number φ, in particular

Unlike in the situation of real numbers, there is an infinitude of complex solutions z of the equation

for any complex number w ≠ 0. It can be shown that any such solution z—called complex logarithm of a—satisfies

where arg is the argument defined above, and ln the (real) natural logarithm. As arg is a multivalued function, unique only up to a multiple of 2π, log is also multivalued. The principal value of log is often taken by restricting the

Complex number 76

imaginary part to the interval (−π,π].Complex exponentiation zω is defined as

Consequently, they are in general multi-valued. For ω = 1 / n, for some natural number n, this recovers thenon-unicity of n-th roots mentioned above.

Holomorphic functionsA function f : C → C is called holomorphic if it satisfies the Cauchy-Riemann equations. For example, any R-linearmap C → C can be written in the form

with complex coefficients a and b. This map is holomorphic if and only if b = 0. The second summand isreal-differentiable, but does not satisfy the Cauchy-Riemann equations.Complex analysis shows some features not apparent in real analysis. For example, any two holomorphic functions fand g that agree on an arbitrarily small open subset of C necessarily agree everywhere. Meromorphic functions,functions that can locally be written as f(z)/(z − z0)n with a holomorphic function f(z), still share some of the featuresof holomorphic functions. Other functions have essential singularities, such as sin(1/z) at z = 0.

ApplicationsSome applications of complex numbers are:

Control theoryIn control theory, systems are often transformed from the time domain to the frequency domain using the Laplacetransform. The system's poles and zeros are then analyzed in the complex plane. The root locus, Nyquist plot, andNichols plot techniques all make use of the complex plane.In the root locus method, it is especially important whether the poles and zeros are in the left or right half planes, i.e.have real part greater than or less than zero. If a system has poles that are• in the right half plane, it will be unstable,• all in the left half plane, it will be stable,• on the imaginary axis, it will have marginal stability.If a system has zeros in the right half plane, it is a nonminimum phase system.

Signal analysisComplex numbers are used in signal analysis and other fields for a convenient description for periodically varyingsignals. For given real functions representing actual physical quantities, often in terms of sines and cosines,corresponding complex functions are considered of which the real parts are the original quantities. For a sine wave ofa given frequency, the absolute value |z| of the corresponding z is the amplitude and the argument arg(z) the phase.If Fourier analysis is employed to write a given real-valued signal as a sum of periodic functions, these periodicfunctions are often written as complex valued functions of the form

where ω represents the angular frequency and the complex number z encodes the phase and amplitude as explainedabove.In electrical engineering, the Fourier transform is used to analyze varying voltages and currents. The treatment of resistors, capacitors, and inductors can then be unified by introducing imaginary, frequency-dependent resistances

Complex number 77

for the latter two and combining all three in a single complex number called the impedance. (Electrical engineers andsome physicists use the letter j for the imaginary unit since i is typically reserved for varying currents and may comeinto conflict with i.) This approach is called phasor calculus. This use is also extended into digital signal processingand digital image processing, which utilize digital versions of Fourier analysis (and wavelet analysis) to transmit,compress, restore, and otherwise process digital audio signals, still images, and video signals.

Improper integralsIn applied fields, complex numbers are often used to compute certain real-valued improper integrals, by means ofcomplex-valued functions. Several methods exist to do this; see methods of contour integration.

Quantum mechanicsThe complex number field is relevant in the mathematical formulations of quantum mechanics, where complexHilbert spaces provide the context for one such formulation that is convenient and perhaps most standard. Theoriginal foundation formulas of quantum mechanics – the Schrödinger equation and Heisenberg's matrix mechanics– make use of complex numbers.

RelativityIn special and general relativity, some formulas for the metric on spacetime become simpler if one takes the timevariable to be imaginary. (This is no longer standard in classical relativity, but is used in an essential way in quantumfield theory.) Complex numbers are essential to spinors, which are a generalization of the tensors used in relativity.

Dynamic equationsIn differential equations, it is common to first find all complex roots r of the characteristic equation of a lineardifferential equation or equation system and then attempt to solve the system in terms of base functions of the formf(t) = ert. Likewise, in difference equations, the complex roots r of the characteristic equation of the differenceequation system are used, to attempt to solve the system in terms of base functions of the form f(t) = r t.

Fluid dynamicsIn fluid dynamics, complex functions are used to describe potential flow in two dimensions.

FractalsCertain fractals are plotted in the complex plane, e.g. the Mandelbrot set and Julia sets.

Algebraic number theory

Complex number 78

Construction of a regular polygon usingstraightedge and compass.

As mentioned above, any nonconstant polynomial equation (incomplex coefficients) has a solution in C. A fortiori, the same is true ifthe equation has rational coefficients. The roots of such equations arecalled algebraic numbers–they are a principal object of study inalgebraic number theory. Compared to Q, the algebraic closure of Q,which also contains all algebraic numbers, C has the advantage ofbeing easily understandable in geometric terms. In this way, algebraicmethods can be used to study geometric questions and vice versa. Withalgebraic methods, more specifically applying the machinery of fieldtheory to the number field containing roots of unity, it can be shownthat it is not possible to construct a regular nonagon using onlycompass and straightedge–a purely geometric problem.

Another example are Pythagorean triples (a, b, c), that is to sayintegers satisfying

(which implies that the triangle having sidelengths a, b, and c is a right triangle). They can be studied by consideringGaussian integers, that is, numbers of the form x + iy, where x and y are integers.

Analytic number theoryAnalytic number theory studies numbers, often integers or rationals, by taking advantage of the fact that they can beregarded as complex numbers, in which analytic methods can be used. This is done by encoding number-theoreticinformation in complex-valued functions. For example, the Riemann zeta-function ζ(s) is related to the distributionof prime numbers.

HistoryThe earliest fleeting reference to square roots of negative numbers can perhaps be said to occur in the work of theGreek mathematician Heron of Alexandria in the 1st century AD, where in his Stereometrica he considers,apparently in error, the volume of an impossible frustum of a pyramid to arrive at the term in hiscalculations, although negative quantities were not conceived of in Hellenistic mathematics and Heron merelyreplaced it by its positive.[8]

The impetus to study complex numbers proper first arose in the 16th century when algebraic solutions for the rootsof cubic and quartic polynomials were discovered by Italian mathematicians (see Niccolo Fontana Tartaglia,Gerolamo Cardano). It was soon realized that these formulas, even if one was only interested in real solutions,sometimes required the manipulation of square roots of negative numbers. As an example, Tartaglia's cubic formulagives the solution to the equation x3 − x = 0 as

Complex number 79

The three cube roots of −1, two of which arecomplex

and when the three cube roots of -1

of which two are complex, are substituted into this expression the three real roots, 0, 1 and −1, result. Of course thisparticular equation can be solved at sight but it does illustrate that when general formulas are used to solve cubicequations with real roots then, as later mathematicians showed rigorously, the use of complex numbers isunavoidable. Rafael Bombelli was the first to explicitly address these seemingly paradoxical solutions of cubicequations and developed the rules for complex arithmetic trying to resolve these issues.The term "imaginary" for these quantities was coined by René Descartes in 1637, although he was at pains to stresstheir imaginary nature [9]

[...] quelquefois seulement imaginaires c’est-à-dire que l’on peut toujours en imaginer autant que j'ai dit enchaque équation, mais qu’il n’y a quelquefois aucune quantité qui corresponde à celle qu’on imagine. ([...]sometimes only imaginary, that is one can imagine as many as I said in each equation, but sometimes thereexists no quantity that matches that which we imagine.)

A further source of confusion was that the equation seemed to be capriciously

inconsistent with the algebraic identity , which is valid for non-negative real numbers a and b,and which was also used in complex number calculations with one of a, b positive and the other negative. Theincorrect use of this identity (and the related identity ) in the case when both a and b are negative evenbedeviled Euler. This difficulty eventually led to the convention of using the special symbol i in place of toguard against this mistake . Even so Euler considered it natural to introduce students to complex numbers muchearlier than we do today. In his elementary algebra text book, Elements of Algebra [10], he introduces these numbersalmost at once and then uses them in a natural way throughout.In the 18th century complex numbers gained wider use, as it was noticed that formal manipulation of complexexpressions could be used to simplify calculations involving trigonometric functions. For instance, in 1730 Abrahamde Moivre noted that the complicated identities relating trigonometric functions of an integer multiple of an angle topowers of trigonometric functions of that angle could be simply re-expressed by the following well-known formulawhich bears his name, de Moivre's formula:

Complex number 80

In 1748 Leonhard Euler went further and obtained Euler's formula of complex analysis:

by formally manipulating complex power series and observed that this formula could be used to reduce anytrigonometric identity to much simpler exponential identities.The idea of a complex number as a point in the complex plane (above) was first described by Caspar Wessel in 1799,although it had been anticipated as early as 1685 in Wallis's De Algebra tractatus.Wessel's memoir appeared in the Proceedings of the Copenhagen Academy but went largely unnoticed. In 1806Jean-Robert Argand independently issued a pamphlet on complex numbers and provided a rigorous proof of thefundamental theorem of algebra. Gauss had earlier published an essentially topological proof of the theorem in 1797but expressed his doubts at the time about "the true metaphysics of the square root of -1". It was not until 1831 thathe overcame these doubts and published his treatise on complex numbers as points in the plane, largely establishingmodern notation and terminology. The English mathematician G. H. Hardy remarked that Gauss was the firstmathematician to use complex numbers in 'a really confident and scientific way' although mathematicians such asNiels Henrik Abel and Carl Gustav Jacob Jacobi were necessarily using them routinely before Gauss published his1831 treatise.[11] Augustin Louis Cauchy and Bernhard Riemann together brought the fundamental ideas of complexanalysis to a high state of completion, commencing around 1825 in Cauchy's case.

The common terms used in the theory are chiefly due to the founders. Argand called the direction

factor, and the modulus; Cauchy (1828) called the reduced form (l'expressionréduite) and apparently introduced the term argument; Gauss used i for , introduced the term complex

number for a + bi, and called a2 + b2 the norm. The expression direction coefficient, often used for , is due to Hankel (1867), and absolute value, for modulus, is due to Weierstrass.Later classical writers on the general theory include Richard Dedekind, Otto Hölder, Felix Klein, Henri Poincaré,Hermann Schwarz, Karl Weierstrass and many others.

Generalizations and related notionsThe process of extending the field R of reals to C is known as Cayley-Dickson construction. It can be carried furtherto higher dimensions, yielding the quaternions H and octonions O which (as a real vector space) are of dimension 4and 8, respectively. However, with increasing dimension, the algebraic properties familiar from real and complexnumbers vanish: the quaternions are only a skew field, i.e. x·y ≠ y·x for two quaternions, the multiplication ofoctonions fails (in addition to not being commutative) to be associative: (x·y)·z ≠ x·(y·z). However, all of these arenormed division algebras over R. By Hurwitz's theorem they are the only ones. The next step in the Cayley-Dicksonconstruction, the sedenions fail to have this structure.The Cayley-Dickson construction is closely related to the regular representation of C, thought of as an R-algebra (anR-vector space with a multiplication), with respect to the basis 1, i. This means the following: the R-linear map

for some fixed complex number w can be represented by a 2×2 matrix (once a basis has been chosen). With respectto the basis 1, i, this matrix is

i.e., the one mentioned in the section on matrix representation of complex numbers above. While this is a linearrepresentation of C in the 2 × 2 real matrices, it is not the only one. Any matrix

has the property that its square is the negative of the identity matrix: J2 = -I. Then

Complex number 81

is also isomorphic to the field C, and gives an alternative complex structure on R2. This is generalized by the notionof a linear complex structure.Hypercomplex numbers also generalize R, C, H, and O. For example this notion contains the split-complexnumbers, which are elements of the ring R[x]/(x2 − 1) (as opposed to R[x]/(x2 + 1)). In this ring, the equation a2 = 1has four solutions.The field R is the completion of Q, the field of rational numbers, with respect to the usual absolute value metric.Other choices of metrics on Q lead to the fields Qp of p-adic numbers (for any prime number p), which are therebyanalogous to R. There are no other nontrivial ways of completing Q than R and Qp, by Ostrowski's theorem. Thealgebraic closure of Qp still carry a norm, but (unlike C) are not complete with respect to it. The completion

of turns out to be algebraically closed. This field is called p-adic complex numbers by analogy.The fields R and Qp and their finite field extensions, including C, are local fields.

Notes[1] Burton (1995, p. 294)[2] Aufmann, Richard N.; Barker, Vernon C.; Nation, Richard D. (2007), College Algebra and Trigonometry (http:/ / books. google. com/

?id=g5j-cT-vg_wC) (6 ed.), Cengage Learning, p. 66, ISBN 0618825150, , Chapter P, p. 66 (http:/ / books. google. com/books?id=g5j-cT-vg_wC& pg=PA66)

[3] Katz (2004, §9.1.4)[4] Abramowitz, Miltonn; Stegun, Irene A. (1964). Handbook of mathematical functions with formulas, graphs, and mathematical tables (http:/ /

books. google. com/ books?id=MtU8uP7XMvoC). Courier Dover Publications. p. 17. ISBN 0-486-61272-4. ., Section 3.7.26, p. 17 (http:/ /www. math. sfu. ca/ ~cbm/ aands/ page_17. htm)

[5] Cooke, Roger (2008). Classical algebra: its nature, origins, and uses (http:/ / books. google. com/ books?id=lUcTsYopfhkC). John Wileyand Sons. p. 59. ISBN 0-470-25952-3. ., Extract: page 59 (http:/ / books. google. com/ books?id=lUcTsYopfhkC& pg=PA59)

[6] Kasana, H.S. (2005). Complex Variables: Theory And Applications (http:/ / books. google. com/ books?id=rFhiJqkrALIC) (2nd ed.). PHILearning Pvt. Ltd. p. 14. ISBN 81-203-2641-5. ., Extract of chapter 1, page 14 (http:/ / books. google. com/ books?id=rFhiJqkrALIC&pg=PA14)

[7] Nilsson, James William; Riedel, Susan A. (2008). Electric circuits (http:/ / books. google. com/ books?id=sxmM8RFL99wC) (8th ed.).Prentice Hall. p. 338. ISBN 0-131-98925-1. ., Chapter 9, page 338 (http:/ / books. google. com/ books?id=sxmM8RFL99wC& pg=PA338)

[8] Nahin, Paul J. (2007). An Imaginary Tale: The Story of (http:/ / mathforum. org/ kb/ thread. jspa?forumID=149& threadID=383188&messageID=1181284). Princeton University Press. ISBN 9780691127989. . Retrieved 20 April 2011.

[9] Descartes, René (1954) [1637]. La Géométrie (http:/ / www. gutenberg. org/ ebooks/ 26400). Dover Publications. ISBN 0486600688. .Retrieved 20 April 2011.

[10] http:/ / web. mat. bham. ac. uk/ C. J. Sangwin/ euler/[11] Hardy, G. H.; Wright, E. M. (2000) [1938]. An Introduction to the Theory of Numbers. OUP Oxford. p. 189 (fourth edition).

ISBN 0199219869.

References

Mathematical references• Ahlfors, Lars (1979), Complex analysis (3rd ed.), McGraw-Hill, ISBN 978-0070006577• Conway, John B. (1986), Functions of One Complex Variable I, Springer, ISBN 0-387-90328-3• Joshi, Kapil D. (1989), Foundations of Discrete Mathematics, New York: John Wiley & Sons,

ISBN 978-0-470-21152-6• Pedoe, Dan (1988), Geometry: A comprehensive course, Dover, ISBN 0-486-65812-0• Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 5.5 Complex Arithmetic" (http:/ /

apps. nrbook. com/ empanel/ index. html?pg=225), Numerical Recipes: The Art of Scientific Computing (3rd ed.),New York: Cambridge University Press, ISBN 978-0-521-88068-8

• Solomentsev, E.D. (2001), "Complex number" (http:/ / eom. springer. de/ c/ c024140. htm), in Hazewinkel,Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104

Complex number 82

Historical references• Burton, David M. (1995), The History of Mathematics (3rd ed.), New York: McGraw-Hill,

ISBN 978-0-07-009465-9• Katz, Victor J. (2004), A History of Mathematics, Brief Version, Addison-Wesley, ISBN 978-0-321-16193-2• Nahin, Paul J. (1998), An Imaginary Tale: The Story of (hardcover ed.), Princeton University Press,

ISBN 0-691-02795-1A gentle introduction to the history of complex numbers and the beginnings of complex analysis.

• H.-D. Ebbinghaus ... (1991), Numbers (hardcover ed.), Springer, ISBN 0-387-97497-0An advanced perspective on the historical development of the concept of number.

Further reading• The Road to Reality: A Complete Guide to the Laws of the Universe, by Roger Penrose; Alfred A. Knopf, 2005;

ISBN 0-679-45443-8. Chapters 4-7 in particular deal extensively (and enthusiastically) with complex numbers.• Unknown Quantity: A Real and Imaginary History of Algebra, by John Derbyshire; Joseph Henry Press; ISBN

0-309-09657-X (hardcover 2006). A very readable history with emphasis on solving polynomial equations andthe structures of modern algebra.

• Visual Complex Analysis, by Tristan Needham; Clarendon Press; ISBN 0-19-853447-7 (hardcover, 1997). Historyof complex numbers and complex analysis with compelling and useful visual interpretations.

External links• Imaginary Numbers (http:/ / www. bbc. co. uk/ programmes/ b00tt6b2) on In Our Time at the BBC.• Euler's work on Complex Roots of Polynomials (http:/ / mathdl. maa. org/ convergence/ 1/ ?pa=content&

sa=viewDocument& nodeId=640& bodyId=1038) at Convergence. MAA Mathematical Sciences Digital Library.• John and Betty's Journey Through Complex Numbers (http:/ / mathforum. org/ johnandbetty/ )• Dimensions: a math film. (http:/ / www. dimensions-math. org/ Dim_regarder_E. htm) Chapter 5 presents an

introduction to complex arithmetic and stereographic projection. Chapter 6 discusses transformations of thecomplex plane, Julia sets, and the Mandelbrot set.

Microphone practice 83

Microphone practiceThere exist a number of well-developed microphone techniques used for miking musical, film, or voice sources.Choice of technique depends on a number of factors, including:• The collection of extraneous noise. This can be a concern, especially in amplified performances, where audio

feedback can be a significant problem. Alternatively, it can be a desired outcome, in situations where ambientnoise is useful (hall reverberation, audience reaction).

• Choice of a signal type: Mono, stereo or multi-channel.• Type of sound-source: Acoustic instruments produce a very different sound than electric instruments, which are

again different from the human voice.• Situational circumstances: Sometimes a microphone should not be visible, or having a microphone nearby is not

appropriate. In scenes for a movie the microphone may be held above the pictureframe, just out of sight. In thisway there is always a certain distance between the actor and the microphone.

• Processing: If the signal is destined to be heavily processed, or "mixed down", a different type of input may berequired.

• The use of a windshield as well as a pop shield, designed to reduce vocal plosives.

Basic techniquesThere are several classes of microphone placement for recording and amplification.• In close micing, a microphone is placed relatively close to an instrument or sound source. This serves to reduce

extraneous noise, including room reverberation, and is commonly used when attempting to record a number ofseparate instruments while keeping the signals separate, or when trying to avoid feedback in an amplifiedperformance. Close micing often affects the frequency response of the microphone, especially for directional micswhich exhibit bass boost from the proximity effect.

• In ambient or distant micing, a microphone — typically a sensitive one — is placed at some distance from thesound source. The goal of this technique is to get a broader, natural mix of the sound source or sources, alongwith ambient sound, including reverberation from the room or hall.

Multi-track recordingOften each instrument or vocalist is miked separately, with one or more microphones recording to separate channels(tracks). At a later stage, the channels are combined ('mixed-down') to two channels for stereo or more for surroundsound. The artists need not perform in the same place at the same time, and individual tracks (or sections of tracks)can be re-recorded to correct errors. Generally effects such as reverberation are added to each recorded channel, anddifferent levels sent to left and right final channels to position the artist in the stereo sound-stage. Microphones mayalso be used to record the overall effect, or just the effect of the performance room.This permits greater control over the final sound, but recording two channels (stereo recording) is simpler andcheaper, and can give a sound that is more natural.

Stereo recording techniquesThere are two features of sound that the human brain uses to place objects in the stereo sound-field between the loudspeakers. These are the relative level (or loudness) difference between the two channels Δ L, and the time delay difference in arrival times for the same sound in each channel Δ t. The "interaural" signals (binaural ILD and ITD) at the ears are not the stereo microphone signals which are coming from the loudspeakers, and are called "interchannel" signals (Δ L and Δ t). These signals are normally not mixed. Loudspeaker signals are different from the sound

Microphone practice 84

arriving at the ear. See the article "Binaural recording for earphones".

Various methods of stereo recording

X-Y technique: intensity stereophony

XY Stereo

Here there are two directional microphones at the same place, and typically placed at 90°or more to each other.[1] A stereo effect is achieved through differences in soundpressure level between two microphones. Due to the lack of differences in time-of-arrivaland phase ambiguities, the sonic characteristic of X-Y recordings is generally less"spacey" and has less depth compared to recordings employing an AB setup.

Blumlein Stereo

When the microphones are bidirectional and placed facing +-45° with respect to thesound source, the X-Y-setup is called a Blumlein Pair. The sonic image produced by thisconfiguration is considered by many authorities to create a realistic, almost holographicsoundstage.

A further refinement of the Blumlein Pair was developed by EMI in 1958, who called it"Stereosonic". They added a little in-phase crosstalk above 700 Hz to better align the midand treble phantom sources with the bass ones. [2]

A-B technique: time-of-arrival stereophonyThis uses two parallel omnidirectional microphones some distance apart, so capturing time-of-arrival stereoinformation as well as some level (amplitude) difference information, especially if employed close to the soundsource(s). At a distance of about 50 cm (0.5 m) the time delay for a signal reaching first one and then the othermicrophone from the side is approximately 1.5 ms (1 to 2 ms). If the distance is increased between the microphonesit effectively decreases the pickup angle. At 70 cm distance it is about equivalent to the pickup angle of thenear-coincident ORTF setup.

Microphone practice 85

M/S technique: Mid/Side stereophony

Mid-Side Stereo

This coincident technique employs a bidirectional microphone facingsideways and a cardioid (generally a variety of cardioid, although AlanBlumlein described the usage of an omnidirectional transducer in hisoriginal patent) at an angle of 90° facing the sound source. One mic isphysically inverted over the other, so they share the same distance. Theleft and right channels are produced through a simple matrix: Left =Mid + Side, Right = Mid − Side (the polarity-reversed side-signal).This configuration produces a completely mono-compatible signal and,if the Mid and Side signals are recorded (rather than the matrixed Leftand Right), the stereo width can be manipulated after the recording hastaken place. This makes it especially useful for film-based projects.

There is some controversy as to whether MS micing technique cancreate translation issues when used with matrix encoded cinemasurround formats such as Dolby SR LtRt, which relies on phaserelationships between left and right channels of the stereo recording inorder to decode surround information.

Choosing a technique

If a stereo signal is to be reproduced in mono, out-of-phase parts of thesignal will cancel, which may cause the unwanted reduction or loss ofsome parts of the signal. This can be an important factor in choosingwhich technique to use.• Since the A-B techniques use phase differences to give the stereo image, they are the least compatible with mono.• In the X-Y techniques, the microphones would ideally be in exactly the same place, which is not possible – if

they are slightly separated left to right, there may be some loss of treble when played back in mono, so they areoften separated vertically. This only causes problems with sound from above or below the height of themicrophones.

• The M/S technique is ideal for mono compatibility, since summing Left+Right just gives the Mid signal back.The equipment for the techniques also varies from the bulky to the small and convenient. A-B techniques generallyuse two separate microphone units, often mounted on a bar to define the separation. X-Y microphone capsules canbe mounted in one unit, or even on the top of a handheld digital recorder. Since M/S setups can give a variablesoundstage width, they are often used in small 'pencil microphones' to mount on video cameras, matching a zoomlens.

Microphone practice 86

References[1] Michael Williams. "The Stereophonic Zoom" (http:/ / www. rycote. com/ images/ uploads/ The_Stereophonic_Zoom. pdf) (PDF). Rycote

Microphone Windshields Ltd. .[2] Eargle, John (2004). The Microphone Book (http:/ / books. google. co. uk/ books?id=w8kXMVKOsY0C& pg=PA170) (2 ed.). Focal Press.

p. 170. ISBN 0240519612. .

External links• Michael Williams, The Stereophonic Zoom (http:/ / www. rycote. com/ images/ uploads/

The_Stereophonic_Zoom. pdf)• Visualization XY Stereo System - Blumlein Eight/Eight 90° - Intensity Stereo (http:/ / www. sengpielaudio. com/

Visualization-Blumlein-E. htm)

Wave

Surface waves in water

In physics, a wave is a disturbance thattravels through space and time, usuallyaccompanied by the transfer of energy.

Waves travel and the wave motion transfersenergy from one point to another, often withno permanent displacement of the particlesof the medium—that is, with little or noassociated mass transport. They consist,instead, of oscillations or vibrations aroundalmost fixed locations. For example, a corkon rippling water will bob up and down,staying in about the same place while thewave itself moves onwards.

One type of wave is a mechanical wave,which propagates through a medium in which the substance of this medium is deformed. The deformation reversesitself owing to restoring forces resulting from its deformation. For example, sound waves propagate via airmolecules bumping into their neighbors. This transfers some energy to these neighbors, which will cause a cascadeof collisions between neighbouring molecules. When air molecules collide with their neighbors, they also bounceaway from them (restoring force). This keeps the molecules from continuing to travel in the direction of the wave.

Another type of wave can travel through a vacuum, e.g. electromagnetic radiation (including visible light, ultravioletradiation, infrared radiation, gamma rays, X-rays, and radio waves). This type of wave consists of periodicoscillations in electrical and magnetic fields.A main distinction can be made between transverse and longitudinal waves. Transverse waves occur when adisturbance sends waves perpendicular (at right angles) to the original wave. Longitudinal waves occur when adisturbance sends waves in the same direction as the original wave.Waves are described by a wave equation which sets out how the disturbance proceeds over time. The mathematicalform of this equation varies depending on the type of wave.

Wave 87

General featuresA single, all-encompassing definition for the term wave is not straightforward. A vibration can be defined as aback-and-forth motion around a reference value. However, a vibration is not necessarily a wave. An attempt todefine the necessary and sufficient characteristics that qualify a phenomenon to be called a wave results in a fuzzyborder line.The term wave is often intuitively understood as referring to a transport of spatial disturbances that are generally notaccompanied by a motion of the medium occupying this space as a whole. In a wave, the energy of a vibration ismoving away from the source in the form of a disturbance within the surrounding medium (Hall 1980, p. 8).However, this notion is problematic for a standing wave (for example, a wave on a string), where energy is movingin both directions equally, or for electromagnetic / light waves in a vacuum, where the concept of medium does notapply. There are water waves on the ocean surface; light waves emitted by the Sun; microwaves used in microwaveovens; radio waves broadcast by radio stations; and sound waves generated by radio receivers, telephone handsetsand living creatures (as voices).It may appear that the description of waves is closely related to their physical origin for each specific instance of awave process. For example, acoustics is distinguished from optics in that sound waves are related to a mechanicalrather than an electromagnetic wave transfer caused by vibration. Concepts such as mass, momentum, inertia, orelasticity, become therefore crucial in describing acoustic (as distinct from optic) wave processes. This difference inorigin introduces certain wave characteristics particular to the properties of the medium involved. For example, inthe case of air: vortices, radiation pressure, shock waves etc.; in the case of solids: Rayleigh waves, dispersion etc.;and so on.Other properties, however, although they are usually described in an origin-specific manner, may be generalized toall waves. For such reasons, wave theory represents a particular branch of physics that is concerned with theproperties of wave processes independently from their physical origin.[1] For example, based on the mechanicalorigin of acoustic waves, a moving disturbance in space–time can exist if and only if the medium involved is neitherinfinitely stiff nor infinitely pliable. If all the parts making up a medium were rigidly bound, then they would allvibrate as one, with no delay in the transmission of the vibration and therefore no wave motion. This is impossiblebecause it would violate general relativity. On the other hand, if all the parts were independent, then there would notbe any transmission of the vibration and again, no wave motion. Although the above statements are meaningless inthe case of waves that do not require a medium, they reveal a characteristic that is relevant to all waves regardless oforigin: within a wave, the phase of a vibration (that is, its position within the vibration cycle) is different for adjacentpoints in space because the vibration reaches these points at different times.Similarly, wave processes revealed from the study of waves other than sound waves can be significant to theunderstanding of sound phenomena. A relevant example is Thomas Young's principle of interference (Young, 1802,in Hunt 1992, p. 132). This principle was first introduced in Young's study of light and, within some specificcontexts (for example, scattering of sound by sound), is still a researched area in the study of sound.

Wave 88

Mathematical description of one-dimensional waves

Wave equationConsider a traveling transverse wave (which may be a pulse) on a string (the medium). Consider the string to have asingle spatial dimension. Consider this wave as traveling

Wavelength λ, can be measured between any twocorresponding points on a waveform

• in the direction in space. E.g., let the positive direction beto the right, and the negative direction be to the left.

• with constant amplitude • with constant velocity , where is

• independent of wavelength (no dispersion)• independent of amplitude (linear media, not nonlinear).[2]

• with constant waveform, or shapeThis wave can then be described by the two-dimensional functions

(waveform traveling to the right)(waveform traveling to the left)

or, more generally, by d'Alembert's formula:[3]

representing two component waveforms and traveling through the medium in opposite directions. This wavecan also be represented by the partial differential equation

General solutions are based upon Duhamel's principle.[4]

Wave forms

Sine, square, triangle and sawtooth waveforms.

The form or shape of F in d'Alembert's formulainvolves the argument x − vt. Constant values ofthis argument correspond to constant values ofF, and these constant values occur if x increasesat the same rate that vt increases. That is, thewave shaped like the function F will move in thepositive x-direction at velocity v (and G willpropagate at the same speed in the negativex-direction).[5]

In the case of a periodic function F with periodλ, that is, F(x + λ − vt) = F(x − vt), theperiodicity of F in space means that a snapshotof the wave at a given time t finds the wavevarying periodically in space with period λ (thewavelength of the wave). In a similar fashion,this periodicity of F implies a periodicity in time as well: F(x − v(t + T)) = F(x − vt) provided vT = λ, so anobservation of the wave at a fixed location x finds the wave undulating periodically in time with period T = λ/v.[6]

Wave 89

Amplitude and modulation

Illustration of the envelope (the slowly varyingred curve) of an amplitude modulated wave. Thefast varying blue curve is the carrier wave, which

is being modulated.

The amplitude of a wave may be constant (in which case the wave is ac.w. or continuous wave), or may be modulated so as to vary with timeand/or position. The outline of the variation in amplitude is called theenvelope of the wave. Mathematically, the modulated wave can bewritten in the form:[7] [8] [9]

where is the amplitude envelope of the wave, is the wavenumber and is the phase. If the groupvelocity (see below) is wavelength-independent, this equation can be simplified as:[10]

showing that the envelope moves with the group velocity and retains its shape. Otherwise, in cases where the groupvelocity varies with wavelength, the pulse shape changes in a manner often described using an envelope equation.[10]

[11]

Phase velocity and group velocity

Frequency dispersion in groups of gravity waveson the surface of deep water. The red dot moves

with the phase velocity, and the green dotspropagate with the group velocity.

There are two velocities that are associated with waves, the phasevelocity and the group velocity. To understand them, one mustconsider several types of waveform. For simplification, examination isrestricted to one dimension.

This shows a wave with the Group velocity andPhase velocity going in different directions.

The most basic wave (a form of plane wave) may be expressed in theform:

which can be related to the usual sine and cosine forms using Euler's formula. Rewriting the argument,, makes clear that this expression describes a vibration of wavelength

traveling in the x-direction with a constant phase velocity .[12]

Wave 90

The other type of wave to be considered is one with localized structure described by an envelope, which may beexpressed mathematically as, for example:

where now A(k1) (the integral is the inverse fourier transform of A(k1)) is a function exhibiting a sharp peak in aregion of wave vectors Δk surrounding the point k1 = k. In exponential form:

with Ao the magnitude of A. For example, a common choice for Ao is a Gaussian wave packet:[13]

where σ determines the spread of k1-values about k, and N is the amplitude of the wave.The exponential function inside the integral for ψ oscillates rapidly with its argument, say φ(k1), and where it variesrapidly, the exponentials cancel each other out, interfere destructively, contributing little to ψ.[12] However, anexception occurs at the location where the argument φ of the exponential varies slowly. (This observation is the basisfor the method of stationary phase for evaluation of such integrals.[14] ) The condition for φ to vary slowly is that itsrate of change with k1 be small; this rate of variation is:[12]

where the evaluation is made at k1 = k because A(k1) is centered there. This result shows that the position x where thephase changes slowly, the position where ψ is appreciable, moves with time at a speed called the group velocity:

The group velocity therefore depends upon the dispersion relation connecting ω and k. For example, in quantummechanics the energy of a particle represented as a wave packet is E = ħω = (ħk)2/(2m). Consequently, for that wavesituation, the group velocity is

showing that the velocity of a localized particle in quantum mechanics is its group velocity.[12] Because the groupvelocity varies with k, the shape of the wave packet broadens with time, and the particle becomes less localized.[15]

In other words, the velocity of the constituent waves of the wave packet travel at a rate that varies with theirwavelength, so some move faster than others, and they cannot maintain the same interference pattern as the wavepropagates.

Sinusoidal waves

Sinusoidal waves correspond to simple harmonicmotion.

Mathematically, the most basic wave is the (spatially) one-dimensionalsine wave (or harmonic wave or sinusoid) with an amplitude described by the equation:

where

Wave 91

• is the maximum amplitude of the wave, maximum distance from the highest point of the disturbance in themedium (the crest) to the equilibrium point during one wave cycle. In the illustration to the right, this is themaximum vertical distance between the baseline and the wave.

• is the space coordinate• is the time coordinate• is the wavenumber• is the angular frequency• is the phase.The units of the amplitude depend on the type of wave. Transverse mechanical waves (e.g., a wave on a string) havean amplitude expressed as a distance (e.g., meters), longitudinal mechanical waves (e.g., sound waves) use units ofpressure (e.g., pascals), and electromagnetic waves (a form of transverse vacuum wave) express the amplitude interms of its electric field (e.g., volts/meter).The wavelength is the distance between two sequential crests or troughs (or other equivalent points), generally ismeasured in meters. A wavenumber , the spatial frequency of the wave in radians per unit distance (typically permeter), can be associated with the wavelength by the relation

The period is the time for one complete cycle of an oscillation of a wave. The frequency is the number ofperiods per unit time (per second) and is typically measured in hertz. These are related by:

In other words, the frequency and period of a wave are reciprocals.The angular frequency represents the frequency in radians per second. It is related to the frequency or period by

The wavelength of a sinusoidal waveform traveling at constant speed is given by:[16]

where is called the phase speed (magnitude of the phase velocity) of the wave and is the wave's frequency.Wavelength can be a useful concept even if the wave is not periodic in space. For example, in an ocean waveapproaching shore, the incoming wave undulates with a varying local wavelength that depends in part on the depthof the sea floor compared to the wave height. The analysis of the wave can be based upon comparison of the localwavelength with the local water depth.[17]

Although arbitrary wave shapes will propagate unchanged in lossless linear time-invariant systems, in the presenceof dispersion the sine wave is the unique shape that will propagate unchanged but for phase and amplitude, making iteasy to analyze.[18] Due to the Kramers–Kronig relations, a linear medium with dispersion also exhibits loss, so thesine wave propagating in a dispersive medium is attenuated in certain frequency ranges that depend upon themedium.[19] The sine function is periodic, so the sine wave or sinusoid has a wavelength in space and a period intime.[20] [21]

The sinusoid is defined for all times and distances, whereas in physical situations we usually deal with waves thatexist for a limited span in space and duration in time. Fortunately, an arbitrary wave shape can be decomposed intoan infinite set of sinusoidal waves by the use of Fourier analysis. As a result, the simple case of a single sinusoidalwave can be applied to more general cases.[22] [23] In particular, many media are linear, or nearly so, so thecalculation of arbitrary wave behavior can be found by adding up responses to individual sinusoidal waves using thesuperposition principle to find the solution for a general waveform.[24] When a medium is nonlinear, the response tocomplex waves cannot be determined from a sine-wave decomposition.

Wave 92

Standing waves

Standing wave in stationary medium. The reddots represent the wave nodes

A standing wave, also known as a stationary wave, is a wave thatremains in a constant position. This phenomenon can occur because themedium is moving in the opposite direction to the wave, or it can arisein a stationary medium as a result of interference between two wavestraveling in opposite directions.

The sum of two counter-propagating waves (of equal amplitude andfrequency) creates a standing wave. Standing waves commonly arise when a boundary blocks further propagation ofthe wave, thus causing wave reflection, and therefore introducing a counter-propagating wave. For example when aviolin string is displaced, transverse waves propagate out to where the string is held in place at the bridge and thenut, where the waves are reflected back. At the bridge and nut, the two opposed waves are in antiphase and canceleach other, producing a node. Halfway between two nodes there is an antinode, where the two counter-propagatingwaves enhance each other maximally. There is no net propagation of energy over time.

One-dimensionalstanding waves; the

fundamental mode andthe first 6 overtones.

A two-dimensional standing wave on a disk;this is the fundamental mode.

A standing wave on a disk with two nodallines crossing at the center; this is an

overtone.

Physical properties

Light beam exhibiting reflection,refraction, transmission and

dispersion when encountering aprism

Waves exhibit common behaviors under a number of standard situations, e.g.,

Transmission and media

Waves normally move in in a straight line (i.e. rectilinearly) through atransmission medium. Such media can be classified into one or more of thefollowing categories:

• A bounded medium if it is finite in extent, otherwise an unbounded medium• A linear medium if the amplitudes of different waves at any particular point in

the medium can be added• A uniform medium or homogeneous medium if its physical properties are

unchanged at different locations in space• An anisotropic medium if one or more of its physical properties differ in one

or more directions

• An isotropic medium if its physical properties are the same in all directions

Wave 93

ReflectionWhen a wave strikes a reflective surface, it changes direction, such that the angle made by the incident wave and linenormal to the surface equals the angle made by the reflected wave and the same normal line.

InterferenceWaves that encounter each other combine through superposition to create a new wave called an interference pattern.Important interference patterns occur for waves that are in phase.

Refraction

Sinusoidal traveling plane wave entering a region oflower wave velocity at an angle, illustrating thedecrease in wavelength and change of direction

(refraction) that results.

Refraction is the phenomenon of a wave changing its speed.Mathematically, this means that the size of the phase velocitychanges. Typically, refraction occurs when a wave passes fromone medium into another. The amount by which a wave isrefracted by a material is given by the refractive index of thematerial. The directions of incidence and refraction are related tothe refractive indices of the two materials by Snell's law.

Diffraction

A wave exhibits diffraction when it encounters an obstacle thatbends the wave or when it spreads after emerging from anopening. Diffraction effects are more pronounced when the size ofthe obstacle or opening is comparable to the wavelength of thewave.

Polarization

A wave is polarized if it oscillates in one direction or plane. A wavecan be polarized by the use of a polarizing filter. The polarization of atransverse wave describes the direction of oscillation in the planeperpendicular to the direction of travel.Longitudinal waves such as sound waves do not exhibit polarization.For these waves the direction of oscillation is along the direction of

travel.

Wave 94

DispersionA wave undergoes dispersion when either the phase velocity or the group velocity depends on the wave frequency.Dispersion is most easily seen by letting white light pass through a prism, the result of which is to produce thespectrum of colours of the rainbow. Isaac Newton performed experiments with light and prisms, presenting hisfindings in the Opticks (1704) that white light consists of several colours and that these colours cannot bedecomposed any further.[25]

Mechanical waves

Waves on stringsThe speed of a wave traveling along a vibrating string ( v ) is directly proportional to the square root of the tension ofthe string ( T ) over the linear mass density ( μ ):

where the linear density μ is the mass per unit length of the string.

Acoustic wavesAcoustic or sound waves travel at speed given by

or the square root of the adiabatic bulk modulus divided by the ambient fluid density (see speed of sound).

Water waves

• Ripples on the surface of a pond are actually a combination oftransverse and longitudinal waves; therefore, the points on thesurface follow orbital paths.

• Sound—a mechanical wave that propagates through gases, liquids, solids and plasmas;• Inertial waves, which occur in rotating fluids and are restored by the Coriolis effect;• Ocean surface waves, which are perturbations that propagate through water.

Shock waves

Wave 95

Other

• Waves of traffic, that is, propagation ofdifferent densities of motor vehicles, andso forth, which can be modeled askinematic waves[26]

• Metachronal wave refers to theappearance of a traveling wave producedby coordinated sequential actions.

Electromagnetic waves

(radio, micro, infrared, visible, uv)An electromagnetic wave consists of two waves that areoscillations of the electric and magnetic fields. An electromagneticwave travels in a direction that is at right angles to the oscillationdirection of both fields. In the 19th century, James Clerk Maxwellshowed that, in vacuum, the electric and magnetic fields satisfy thewave equation both with speed equal to that of the speed of light.From this emerged the idea that light is an electromagnetic wave.Electromagnetic waves can have different frequencies (and thuswavelengths), giving rise to various types of radiation such asradio waves, microwaves, infrared, visible light, ultraviolet and X-rays.

Quantum mechanical wavesThe Schrödinger equation describes the wave-like behavior of particles in quantum mechanics. Solutions of thisequation are wave functions which can be used to describe the probability density of a particle. Quantum mechanicsalso describes particle properties that other waves, such as light and sound, have on the atomic scale and below.

A propagating wave packet; in general, theenvelope of the wave packet moves at a different

speed than the constituent waves.[27]

de Broglie waves

Louis de Broglie postulated that all particles with momentum have awavelength

Wave 96

where h is Planck's constant, and p is the magnitude of the momentum of the particle. This hypothesis was at thebasis of quantum mechanics. Nowadays, this wavelength is called the de Broglie wavelength. For example, theelectrons in a CRT display have a de Broglie wavelength of about 10−13 m.A wave representing such a particle traveling in the k-direction is expressed by the wave function:

where the wavelength is determined by the wave vector k as:

and the momentum by:

However, a wave like this with definite wavelength is not localized in space, and so cannot represent a particlelocalized in space. To localize a particle, de Broglie proposed a superposition of different wavelengths rangingaround a central value in a wave packet,[28] a waveform often used in quantum mechanics to describe the wavefunction of a particle. In a wave packet, the wavelength of the particle is not precise, and the local wavelengthdeviates on either side of the main wavelength value.In representing the wave function of a localized particle, the wave packet is often taken to have a Gaussian shape andis called a Gaussian wave packet.[29] Gaussian wave packets also are used to analyze water waves.[30]

For example, a Gaussian wavefunction ψ might take the form:[31]

at some initial time t = 0, where the central wavelength is related to the central wave vector k0 as λ0 = 2π / k0. It iswell known from the theory of Fourier analysis,[32] or from the Heisenberg uncertainty principle (in the case ofquantum mechanics) that a narrow range of wavelengths is necessary to produce a localized wave packet, and themore localized the envelope, the larger the spread in required wavelengths. The Fourier transform of a Gaussian isitself a Gaussian.[33] Given the Gaussian:

the Fourier transform is:

The Gaussian in space therefore is made up of waves:

that is, a number of waves of wavelengths λ such that kλ = 2 π.The parameter σ decides the spatial spread of the Gaussian along the x-axis, while the Fourier transform shows aspread in wave vector k determined by 1/σ. That is, the smaller the extent in space, the larger the extent in k, andhence in λ = 2π/k.

Wave 97

Animation showing the effect of a cross-polarizedgravitational wave on a ring of test particles

Gravitational waves

Researchers believe that gravitational waves also travel through space,although gravitational waves have never been directly detected. Not tobe confused with gravity waves, gravitational waves are disturbancesin the curvature of spacetime, predicted by Einstein's theory of generalrelativity.

WKB method

In a nonuniform medium, in which the wavenumber k can depend onthe location as well as the frequency, the phase term kx is typicallyreplaced by the integral of k(x)dx, according to the WKB method. Suchnonuniform traveling waves are common in many physical problems,including the mechanics of the cochlea and waves on hanging ropes.

References[1] Lev A. Ostrovsky & Alexander I. Potapov (2002). Modulated waves: theory and application (http:/ / www. amazon. com/ gp/ product/

0801873258). Johns Hopkins University Press. ISBN 0801873258. .[2] Michael A. Slawinski (2003). "Wave equations" (http:/ / books. google. com/ ?id=s7bp6ezoRhcC& pg=PA134). Seismic waves and rays in

elastic media. Elsevier. pp. 131 ff. ISBN 0080439306. .[3] Karl F Graaf (1991). Wave motion in elastic solids (http:/ / books. google. com/ ?id=5cZFRwLuhdQC& printsec=frontcover) (Reprint of

Oxford 1975 ed.). Dover. pp. 13–14. ISBN 9780486667454. .[4] Jalal M. Ihsan Shatah, Michael Struwe (2000). "The linear wave equation" (http:/ / books. google. com/ ?id=zsasG2axbSoC& pg=PA37).

Geometric wave equations. American Mathematical Society Bookstore. pp. 37 ff. ISBN 0821827499. .[5] Louis Lyons (1998). All you wanted to know about mathematics but were afraid to ask (http:/ / books. google. com/ ?id=WdPGzHG3DN0C&

pg=PA128). Cambridge University Press. pp. 128 ff. ISBN 052143601X. .[6] Alexander McPherson (2009). "Waves and their properties" (http:/ / books. google. com/ ?id=o7sXm2GSr9IC& pg=PA77). Introduction to

Macromolecular Crystallography (2 ed.). Wiley. p. 77. ISBN 0470185902. .[7] Christian Jirauschek (2005). FEW-cycle Laser Dynamics and Carrier-envelope Phase Detection (http:/ / books. google. com/

?id=6kOoT_AX2CwC& pg=PA9). Cuvillier Verlag. p. 9. ISBN 3865374190. .[8] Fritz Kurt Kneubühl (1997). Oscillations and waves (http:/ / books. google. com/ ?id=geYKPFoLgoMC& pg=PA365). Springer. p. 365.

ISBN 354062001X. .[9] Mark Lundstrom (2000). Fundamentals of carrier transport (http:/ / books. google. com/ ?id=FTdDMtpkSkIC& pg=PA33). Cambridge

University Press. p. 33. ISBN 0521631343. .[10] Chin-Lin Chen (2006). "§13.7.3 Pulse envelope in nondispersive media" (http:/ / books. google. com/ ?id=LxzWPskhns0C& pg=PA363).

Foundations for guided-wave optics. Wiley. p. 363. ISBN 0471756873. .[11] Stefano Longhi, Davide Janner (2008). "Localization and Wannier wave packets in photonic crystals" (http:/ / books. google. com/

?id=xxbXgL967PwC& pg=PA329). In Hugo E. Hernández-Figueroa, Michel Zamboni-Rached, Erasmo Recami. Localized Waves.Wiley-Interscience. p. 329. ISBN 0470108851. .

[12] Albert Messiah (1999). Quantum Mechanics (http:/ / books. google. com/ ?id=mwssSDXzkNcC& pg=PA52& dq=intitle:quantum+inauthor:messiah+ "group+ velocity"+ "center+ of+ the+ wave+ packet") (Reprint of two-volume Wiley 1958 ed.). Courier Dover. pp. 50–52.ISBN 9780486409245. .

[13] See, for example, Eq. 2(a) in

Walter Greiner, D. Allan Bromley (2007). Quantum Mechanics: An introduction (http:/ / books. google. com/?id=7qCMUfwoQcAC& pg=PA61) (2nd ed.). Springer. pp. 60–61. ISBN 3540674586. .[14] John W. Negele, Henri Orland (1998). Quantum many-particle systems (http:/ / books. google. com/ ?id=mx5CfeeEkm0C& pg=PA121)

(Reprint in Advanced Book Classics ed.). Westview Press. p. 121. ISBN 0738200522. .[15] Donald D. Fitts (1999). Principles of quantum mechanics: as applied to chemistry and chemical physics (http:/ / books. google. com/

?id=8t4DiXKIvRgC& pg=PA15). Cambridge University Press. pp. 15 ff. ISBN 0521658411. .[16] David C. Cassidy, Gerald James Holton, Floyd James Rutherford (2002). Understanding physics (http:/ / books. google. com/

?id=rpQo7f9F1xUC& pg=PA340). Birkhäuser. pp. 339 ff. ISBN 0387987568. .[17] Paul R Pinet (2009). op. cit. (http:/ / books. google. com/ ?id=6TCm8Xy-sLUC& pg=PA242). p. 242. ISBN 0763759937. .

Wave 98

[18] Mischa Schwartz, William R. Bennett, and Seymour Stein (1995). Communication Systems and Techniques (http:/ / books. google. com/?id=oRSHWmaiZwUC& pg=PA208& dq=sine+ wave+ medium+ + linear+ time-invariant). John Wiley and Sons. p. 208.ISBN 9780780347151. .

[19] See Eq. 5.10 and discussion in A. G. G. M. Tielens (2005). The physics and chemistry of the interstellar medium (http:/ / books. google.com/ ?id=wMnvg681JXMC& pg=PA119). Cambridge University Press. pp. 119 ff. ISBN 0521826349. .; Eq. 6.36 and associated discussionin Otfried Madelung (1996). Introduction to solid-state theory (http:/ / books. google. com/ ?id=yK_J-3_p8_oC& pg=PA261) (3rd ed.).Springer. pp. 261 ff. ISBN 354060443X. .; and Eq. 3.5 in F Mainardi (1996). "Transient waves in linear viscoelastic media" (http:/ / books.google. com/ ?id=UfSk45nCVKMC& pg=PA134). In Ardéshir Guran, A. Bostrom, Herbert Überall, O. Leroy. Acoustic Interactions withSubmerged Elastic Structures: Nondestructive testing, acoustic wave propagation and scattering. World Scientific. p. 134. ISBN 9810242719..

[20] Aleksandr Tikhonovich Filippov (2000). The versatile soliton (http:/ / books. google. com/ ?id=TC4MCYBSJJcC& pg=PA106). Springer.p. 106. ISBN 0817636358. .

[21] Seth Stein, Michael E. Wysession (2003). An introduction to seismology, earthquakes, and earth structure (http:/ / books. google. com/?id=Kf8fyvRd280C& pg=PA31). Wiley-Blackwell. p. 31. ISBN 0865420785. .

[22] Seth Stein, Michael E. Wysession (2003). op. cit. (http:/ / books. google. com/ ?id=Kf8fyvRd280C& pg=PA32). p. 32. ISBN 0865420785. .[23] Kimball A. Milton, Julian Seymour Schwinger (2006). Electromagnetic Radiation: Variational Methods, Waveguides and Accelerators

(http:/ / books. google. com/ ?id=x_h2rai2pYwC& pg=PA16). Springer. p. 16. ISBN 3540293043. . "Thus, an arbitrary function f(r, t) can besynthesized by a proper superposition of the functions exp[i (k·r−ωt)]…"

[24] Raymond A. Serway and John W. Jewett (2005). "§14.1 The Principle of Superposition" (http:/ / books. google. com/ ?id=1DZz341Pp50C&pg=PA433). Principles of physics (4th ed.). Cengage Learning. p. 433. ISBN 053449143X. .

[25] Newton, Isaac (1704). "Prop VII Theor V" (http:/ / gallica. bnf. fr/ ark:/ 12148/ bpt6k3362k. image. f128. pagination). Opticks: Or, Atreatise of the Reflections, Refractions, Inflexions and Colours of Light. Also Two treatises of the Species and Magnitude of CurvilinearFigures. 1. London. p. 118. . "All the Colours in the Universe which are made by Light... are either the Colours of homogeneal Lights, orcompounded of these..."

[26] M. J. Lighthill; G. B. Whitham (1955). "On kinematic waves. II. A theory of traffic flow on long crowded roads". Proceedings of the RoyalSociety of London. Series A 229: 281–345. Bibcode 1955RSPSA.229..281L. doi:10.1098/rspa.1955.0088. And: P. I. Richards (1956)."Shockwaves on the highway". Operations Research 4 (1): 42–51. doi:10.1287/opre.4.1.42.

[27] A. T. Fromhold (1991). "Wave packet solutions" (http:/ / books. google. com/ ?id=3SOwc6npkIwC& pg=PA59). Quantum Mechanics forApplied Physics and Engineering (Reprint of Academic Press 1981 ed.). Courier Dover Publications. pp. 59 ff. ISBN 0486667413. . "(p. 61)…the individual waves move more slowly than the packet and therefore pass back through the packet as it advances"

[28] Ming Chiang Li (1980). "Electron Interference" (http:/ / books. google. com/ ?id=g5q6tZRwUu4C& pg=PA271). In L. Marton & ClaireMarton. Advances in Electronics and Electron Physics. 53. Academic Press. p. 271. ISBN 0120146533. .

[29] See for example Walter Greiner, D. Allan Bromley (2007). Quantum Mechanics (http:/ / books. google. com/ ?id=7qCMUfwoQcAC&pg=PA60) (2 ed.). Springer. p. 60. ISBN 3540674586. . and John Joseph Gilman (2003). Electronic basis of the strength of materials (http:/ /books. google. com/ ?id=YWd7zHU0U7UC& pg=PA57). Cambridge University Press. p. 57. ISBN 0521620058. .,Donald D. Fitts (1999).Principles of quantum mechanics (http:/ / books. google. com/ ?id=8t4DiXKIvRgC& pg=PA17). Cambridge University Press. p. 17.ISBN 0521658411. ..

[30] Chiang C. Mei (1989). The applied dynamics of ocean surface waves (http:/ / books. google. com/ ?id=WHMNEL-9lqkC& pg=PA47) (2nded.). World Scientific. p. 47. ISBN 9971507897. .

[31] Walter Greiner, D. Allan Bromley (2007). Quantum Mechanics (http:/ / books. google. com/ ?id=7qCMUfwoQcAC& pg=PA60) (2nd ed.).Springer. p. 60. ISBN 3540674586. .

[32] Siegmund Brandt, Hans Dieter Dahmen (2001). The picture book of quantum mechanics (http:/ / books. google. com/?id=VM4GFlzHg34C& pg=PA23) (3rd ed.). Springer. p. 23. ISBN 0387951415. .

[33] Cyrus D. Cantrell (2000). Modern mathematical methods for physicists and engineers (http:/ / books. google. com/ ?id=QKsiFdOvcwsC&pg=PA677). Cambridge University Press. p. 677. ISBN 0521598273. .

Sources• Campbell, M. and Greated, C. (1987). The Musician’s Guide to Acoustics. New York: Schirmer Books.• French, A.P. (1971). Vibrations and Waves (M.I.T. Introductory physics series). Nelson Thornes.

ISBN 0-393-09936-9. OCLC 163810889.• Hall, D. E. (1980). Musical Acoustics: An Introduction. Belmont, California: Wadsworth Publishing Company.

ISBN 0534007589..• Hunt, F. V. (1992). Origins in Acoustics (http:/ / asa. aip. org/ publications. html#pub17). New York: Acoustical

Society of America Press..• Ostrovsky, L. A.; Potapov, A. S. (1999). Modulated Waves, Theory and Applications. Baltimore: The Johns

Hopkins University Press. ISBN 0801858704..

Wave 99

• Vassilakis, P.N. (2001) (http:/ / www. acousticslab. org/ papers/ diss. htm). Perceptual and Physical Properties ofAmplitude Fluctuation and their Musical Significance. Doctoral Dissertation. University of California, LosAngeles.

External links• Interactive Visual Representation of Waves (http:/ / resonanceswavesandfields. blogspot. com/ 2007/ 08/

true-waves. html)• Science Aid: Wave properties—Concise guide aimed at teens (http:/ / www. scienceaid. co. uk/ physics/ waves/

properties. html)• Simulation of diffraction of water wave passing through a gap (http:/ / www. phy. hk/ wiki/ englishhtm/

Diffraction. htm)• Simulation of interference of water waves (http:/ / www. phy. hk/ wiki/ englishhtm/ Interference. htm)• Simulation of longitudinal traveling wave (http:/ / www. phy. hk/ wiki/ englishhtm/ Lwave. htm)• Simulation of stationary wave on a string (http:/ / www. phy. hk/ wiki/ englishhtm/ StatWave. htm)• Simulation of transverse traveling wave (http:/ / www. phy. hk/ wiki/ englishhtm/ TwaveA. htm)• Sounds Amazing—AS and A-Level learning resource for sound and waves (http:/ / www. acoustics. salford. ac.

uk/ feschools/ )• chapter from an online textbook (http:/ / www. lightandmatter. com/ html_books/ lm/ ch19/ ch19. html)• Simulation of waves on a string (http:/ / www. physics-lab. net/ applets/ mechanical-waves)• of longitudinal and transverse mechanical wave (http:/ / www. cbu. edu/ ~jvarrian/ applets/ waves1/ lontra_g.

htm-simulation)• MIT OpenCourseWare 8.03: Vibrations and Waves (http:/ / ocw. mit. edu/ courses/ physics/

8-03-physics-iii-vibrations-and-waves-fall-2004/ ) Free, independent study course with video lectures,assignments, lecture notes and exams.

Velocities of waves

Phase velocity • Group velocity • Front velocity • Signal velocity

Article Sources and Contributors 100

Article Sources and ContributorsFast Fourier transform  Source: http://en.wikipedia.org/w/index.php?oldid=447035554  Contributors: 16@r, 2ganesh, Adam Zivner, Adashiel, Akoesters, Amitparikh, Apexfreak, Artur Nowak,Audriusa, Avalcarce, AxelBoldt, Bartosz, BehnamFarid, Bemoeial, Bender235, Bender2k14, Blablahblablah, Bmitov, Boxplot, Cameronc, Captain Disdain, Coneslayer, Conversion script,Crossz, Cxxl, Daniel Brockman, David spector, Davidmeo, Dcoetzee, Dcwill1285, DeadTotoro, Decrease789, Dekart, Delirium, Djg2006, Dmsar, Domitori, Donarreiskoffer, DrBob, Efryevt,Eras-mus, Excirial, Eyreland, Faestning, Fredrik, Fresheneesz, Furrykef, Gareth Owen, Gene93k, Geoffeg, Giftlite, Glutamin, Gopimanne, GreenSpigot, Grendelkhan, Gunnar Larsson, H2g2bob,HalfShadow, Hashar, Helwr, HenningThielemann, Herry41341, Herve661, Hess88, Hmo, Hyacinth, Ixfd64, Jaredwf, Jeltz, Jitse Niesen, Johnbibby, JustUser, Kellybundybrain, Kkmurray,Klokbaske, Kuashio, LMB, Lavaka, LeoTrottier, LiDaobing, Lorem Ip, LouScheffer, Lupo, MarylandArtLover, Materialscientist, MaxSem, Maxim, Michael Hardy, MrOllie, Mschlindwein,Mwilde, Nagesh Anupindi, Nbarth, Nixdorf, Norm mit, Ntsimp, Oleg Alexandrov, Oli Filth, Omkar lon, Palfrey, Pankajp, Pit, Pt, QueenCake, Quibik, Quintote, Qwertyus, Qwfp, R. J. Mathar,R.e.b., RTV User 545631548625, Requestion, Riemann'sZeta, Rjwilmsi, Roadrunner, Robertvan1, Rogerbrent, Rubin joseph 10, Sam Hocevar, Sangwine, SciCompTeacher, SebastianHelm,Smallman12q, Solongmarriane, Spectrogram, Squell, Steina4, Stevenj, TakuyaMurata, Tarquin, Teorth, The Anome, The Yowser, Thenub314, Tim32, TimBentley, Timendum, Tuhinbhakta,Twexcom, Ulterior19802005, Unyoyega, Vincent kraeutler, Wik, Wikichicheng, Yacht, Ylai, ZeroOne, Zven, Zxcvbnm, 191 anonymous edits

Discrete Hartley transform  Source: http://en.wikipedia.org/w/index.php?oldid=431133573  Contributors: Charles Matthews, Cojoco, Cyp, Edgar181, Eras-mus, Evil saltine, Fred Bradstadt,Fredrik, Heinzelmann, Keenan Pepper, Lockeownzj00, Lorem Ip, MCBracewell, Maximus Rex, Michael Hardy, Miym, Nixdorf, Oleg Alexandrov, PV=nRT, Robin S, Robsavoie, Rolfyu,Stevenj, Woohookitty, 30 anonymous edits

Discrete Fourier transform  Source: http://en.wikipedia.org/w/index.php?oldid=447093571  Contributors: 1exec1, Alberto Orlandini, Alexjs, Arialblack, Arthur Rubin, Astronautics, AxelBoldt,BD2412, Bill411432, Bmitov, Bo Jacoby, Bob K, Bob.v.R, Bongomatic, Booyabazooka, Borching, Bryan Derksen, CRGreathouse, Cburnett, Centrx, Charles Matthews, ChrisRuvolo, Connelly,Conversion script, CoolBlue1234, Crazyvas, Cybedu, Cyp, Davidmeo, Dcoetzee, Dicklyon, DopefishJustin, DrBob, Dysprosia, Dzordzm, EconoPhysicist, Ed g2s, Edward, Emvee, Epolk,EverettColdwell, Evil saltine, Foobaz, Fru1tbat, Furrykef, Galaxiaad, Gareth Owen, Gauge, Gene Nygaard, Geoeg, Giftlite, Glogger, Graham87, GregRM, Hanspi, HappyCamper, Helwr,HenningThielemann, Herr Satz, Hesam7, Hobit, Humatronic, Iyerkri, Java410, JeromeJerome, Jitse Niesen, Jwkuehne, Kohtala, Kompik, Kramer, Kvng, LMB, Lockeownzj00, Loisel, LokiClock,Lorem Ip, Madmardigan53, Man It's So Loud In Here, Martynas Patasius, MathMartin, Matithyahu, Mcclurmc, Mckee, Mdebets, Metacomet, Michael Hardy, MightyWarrior, MikeJ9919,Minesweeper, Minghong, Mjb, Msuzen, Mulligatawny, Nbarth, Nikai, Nitin.mehta, NormDor, Oleg Alexandrov, Oli Filth, Omegatron, OrgasGirl, Ospalh, Oyz, PAR, Pak21, Paul August, PaulMatthews, Phil Boswell, Poslfit, Pseudomonas, Qz, Rabbanis, Rdrk, Recognizance, Robin S, Ronhjones, Sam Hocevar, Sampletalk, SebastianHelm, Shakeel3442, Sharpoo7, SheeEttin, Srleffler,Ssd, Stevenj, Sverdrup, Svick, Swagato Barman Roy, Tbackstr, The wub, TheMightyOrb, Thenub314, Thorwald, Tobias Bergemann, Uncia, Uncle Dick, User A1, Verdy p, Wikid77, Wile E.Heresiarch, Woohookitty, Zundark, Zvika, Zxcvbnm, Пика Пика, 216 anonymous edits

Fourier analysis  Source: http://en.wikipedia.org/w/index.php?oldid=446009665  Contributors: 212.153.190.xxx, Ahoerstemeier, Andreas Kaufmann, Andrei Polyanin, AndrewKepert,Andykass, Anonymous Dissident, Ashok567, AxelBoldt, Ayda D, Ayonbd2000, Baa, BenBreen2003, BenFrantzDale, Bo Jacoby, Bob K, Bobo192, Borgx, Boxplot, CRGreathouse, CSTAR,CanadianLinuxUser, Cburnett, Cdnc, ChangChienFu, Charles Matthews, Chubby Chicken, Cogburnd02, Conversion script, Cyrius, Damian Yerrick, Davidmeo, Dicklyon, Dingar, Don4of4,Dysprosia, Ejrh, Eliyak, Enochlau, EugeneZelenko, Evil saltine, Fintor, Frade, Fred Bradstadt, Freestyle, French Tourist, Fresheneesz, Furrykef, Futurebird, GT5162, Gareth Owen, GcSwRhIc,Gene Ward Smith, Geoeg, Giftlite, Giuscarl, Grubber, Hankwang, Headbomb, Helptry, Helwr, HenningThielemann, Ht686rg90, JAIG, Jclaer, Jitse Niesen, JohnOwens, Jumbuck, JustUser,Jyossarian, Jyril, KYN, Kat, Kevin Baas, Kman543210, Lamoidfl, Lenthe, LiDaobing, Linas, LokiClock, Lupin, Mahlerite, Marcosaedro, Math.geek3.1415926, MauriceJFox3, Mazi, Melcombe,Michael Hardy, Miguelgoldstein, Mormegil, Morwan, Mwilde, Mütze, Nbarth, Nixdorf, Ojigiri, Oleg Alexandrov, Oli Filth, Omegatron, OwenX, Oxymoron83, PAR, Parsecboy, Pdenapo, Planb89, Qz, RFightmaster, Radagast83, Rade Kutil, Rbj, Reddi, RexNL, Rmcguire, Ross Burgess, Rs2, Samsara, SebastianHelm, Sepia tone, Silly rabbit, Slehar, Smack, Ste4k, Stevenj, Strangealibi,Supten, Sverdrup, Sylvestersteele, Syncategoremata, Szidomingo, Sławomir Biały, Tbackstr, Tbsmith, The Anome, Theanthrope, Thenub314, Thrapper, Tobias Bergemann, Unaiaia, Uncia,Viriditas, Voyajer, Wikid77, Wile E. Heresiarch, Yardgnome, Yurivict, Zeimusu, Zowie, Zundark, Zvika, 163 anonymous edits

Sine  Source: http://en.wikipedia.org/w/index.php?oldid=446147979  Contributors: Anonymous Dissident, Artyom, Auddhav, Ben Ben, Chowbok, Dan131m, Darkwind, Elauminri,Ewlyahoocom, FHannes, Farisori, Glenn L, Jim.belk, KuduIO, Michael Hardy, Muslim-Researcher, Nbarth, Ninthabout, Noq, Norm mit, Ntsimp, Pengo, Prenn, Sankalpdravid, Seijihyouronka,SuperCow, Supernaturalist, Tamsier, Teapeat, Termininja, The Evil IP address, Tryptographer, Txus.aparicio, Uuor, William M. Connolley, 31 ,שדדשכ anonymous edits

Trigonometric functions  Source: http://en.wikipedia.org/w/index.php?oldid=446848723  Contributors: 15trik, 216.7.146.xxx, 24.176.164.xxx, 313.kohler, A bit iffy, ATMACI, AbcXyz, Acdx,AdjustShift, AdnanSa, Ahsirakh, Aitias, Ajmas, Akriasas, Al-khowarizmi, Alan Liefting, Alansohn, Alejo2083, Alerante, Aminiz, Andrewlp1991, Andrewmc123, Angelo De La Paz, Animum,Anonymous Dissident, Anthony Appleyard, Anville, Arabic Pilot, ArglebargleIV, Armend, Asyndeton, Atrian, Avoided, Avono, Awaterl, AxelBoldt, Aymatth2, Baccala@freesoft.org, Baxtrom,Bay77, Bcherkas, Beantwo, Beland, BenFrantzDale, Bevo, Bfwalach, Bhadani, BlackGriffen, Bobblewik, Bobo192, Bonadea, BoomerAB, Brews ohare, Brian Huffman, Brighterorange,Bromskloss, Bryan Derksen, Buba14, CBM, CTZMSC3, Canjth, Capricorn42, Captain-tucker, Carifio24, Carolus m, CaseyPenk, Catslash, Cdibuduo, Cenarium, Centroyd, CesarB, Ceyockey,Charles Matthews, Charvest, Chewie, Chicocvenancio, Chipuni, ChongDae, Chris55, Christopher Parham, 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Complex number  Source: http://en.wikipedia.org/w/index.php?oldid=446136248  Contributors: 144.118.193.xxx, 160.94.28.xxx, 165.123.179.xxx, 2.718281828459..., 203.111.1.xxx, 7severn7, Aaron Kauppi, Abtract, AdamSmithee, Adhanali, Aka042, Akashkumarrath, Alanius, Ale2006, Alison, AllTheThings, AnakngAraw, Andres, Andymc, Angus Lepper, Anonymous Dissident, Ap, Argentino, Armend, Arundhati bakshi, Asyndeton, Ato, Atongmy, AugPi, AxelBoldt, Barno uk, Barticus88, Basispoeter, Ben Ben, BenBaker, Bender2k14, Betacommand, Bevo, Bidabadi, BigJohnHenry, BitterMilton, Biógrafo, Bkell, Blogeswar, Bo Jacoby, Bob A, Bob K, Bobrayner, Boulaur, Br77rino, Brandon Brunner, Brazmyth, Brianga, Brion VIBBER, Bromskloss, Brona, Bruno Simões, Bryan Derksen, Bwe203, C S, CBM, CRGreathouse, Calréfa Wéná, Capricorn42, Card, Catherineyronwode, CeNobiteElf, Cenarium, Centrx, Charles Matthews, ChongDae, Chowbok, Chris-gore, ChrisHodgesUK, Chrisandtaund, Christian List, ChristopherWillis, Christopherkennedy1994, Chun-hian, Cometstyles, Commander, Conversion script, Crowsnest, Crust, Crywalt, Cybercobra, Cypa, DE logics, DVD R W, DVdm, Da nuke, DanBishop, Darth Panda, David R. Ingham, DavidCBryant, Deeptrivia, Denelson83, Dert34, Dgies, Diannaa, Dima373, Dino, Dissimul, Dlo1986, Dmcq, Dod1, Dominus, Dragon Dave, Dual Freq, Duoduoduo, Dysprosia, Eastlaw, Ehrenkater, Elb2000, Elf, Emmelie, Enchanter, Epbr123, Eric Shalov, Evil saltine, Ewlyahoocom, Falcon8765, Favonian, Felizdenovo, Fibonacci, Fiedorow, Fieldday-sunday, FightingMac, FilipeS, Finlay McWalter, Fireaxe888, Flamurai, Fleminra, Flex, FocalPoint, Freakofnurture, Fredrik, Fresheneesz, Fropuff, Furrykef, GTBacchus, Gandalf61, Gareth Owen, Gauss, Georgesawyer, Gesslein, Giftlite, GlenShennan, Goldencako, Googl, GraemeMcRae, Graemeb1967, Graham87, Gunark, Guru6969, Haein45, Haham hanuka, Hairy Dude, HalfShadow, Hannes Eder, Happy-melon, HappyCamper, Hard Sin, Harley peters, Haroldco,

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HenningThielemann, HenryLi, Henrygb, Herbee, HolIgor, Huppybanny, Hut 8.5, Iapetusuk, Igiffin, Igoldste, Inkington, Intangir, Iridescent, Isaac Dupree, Isheden, Itsmejudith, Ivan Štambuk,Ivysaur, Ixfd64, JDSperling, JForget, JRSpriggs, JRocketeer, Jackol, Jagged 85, JaimeLesMaths, Jakob.scholbach, JamesBWatson, Jamesg, Janisozaur, Jarek Duda, Jashaszun, Jasondet,JensMueller, Jesin, Jfredrickson, Jimbryho, Jitse Niesen, Jmvicent, Jni, JohnBlackburne, Jol123, Joriki, Josh Grosse, Jujutacular, Juliancolton, Kaldari, Kan8eDie, KarlJacobi, Kbdank71, KeenanPepper, Kenneth Brookes, Kenyon, Kerotan, Keta, Ketiltrout, Khalid Mahmood, Kiensvay, Kine, Koeplinger, Krischik, Kudret abi, KyraVixen, L Kensington, L33tminion, LHOON,Lagrange123, Lambiam, Lambyte, Laurens-af, Laurent MAYER, Leafyplant, LeaveSleaves, Lessbread, Lifeonahilltop, LilHelpa, Linas, Livius3, Loadmaster, Loom91, Looxix, Lovibond,M4gnum0n, Madhero88, Madmath789, Majorly, Mange01, Manuel Trujillo Berges, MarSch, Marc van Leeuwen, Marek69, Maria Renee Jenkins, MarkSweep, Marozols, MarsRover,MathMan64, MathMartin, Matqkks, Matthew Stannard, Matty j, Matusz, Mct mht, Mdd, MeltBanana, Mentifisto, Mets501, Mh, Michael Hardy, Michael Kinyon, Micro01, Miguel, Mild BillHiccup, Mindmatrix, Mlewis000, Mobower, Modelpanicer, Mogmiester, Momo kiki, Monroetransfer, Moralshearer, Mr Stephen, Ms2ger, Msh210, MuZemike, Nanshu, Nbarth, Ncrfgs, Nejko,Netrapt, Newone, Nic bor, Nicoli nicolivich, Nijdam, Niofis, Nivaca, Nixdorf, Nmulder, Obradovic Goran, ObserverFromAbove, Octahedron80, OdedSchramm, Oleg Alexandrov, Oli Filth,Oliphaunt, Omnieiunium, Onewhohelps, Onkel Tuca, Opticyclic, Oxinabox, P0807670, P0lyglut, Pablogrb, Pablothegreat85, Paolo.dL, Papa November, Patrick, Patrick.ar, Paul August, Paul D.Anderson, Pegship, Petecrawford, Phlembowper99, PierreAbbat, Pitel, PizzaMargherita, Pj.de.bruin, Pjacobi, Pleasantville, Plugwash, Pmanderson, Pne, Polymath618, Poor Yorick, Profjohn,Pstanton, Pt, Qe2eqe, Quaeler, Quantum Burrito, Qwfp, RDBury, RG2, RJChapman, Rabkid15, RainbowOfLight, Raise exception, Raja Hussain, Randomblue, Raphaelsss, Rasmus Faber,Ravisingh57547, Rcog, Readysoaper, ReallyNiceGuy, Recentchanges, Red Winged Duck, Remyrem1, RepublicanJacobite, Revolver, RexNL, Rgdboer, Rhanekom, Ricardo sandoval,Roadrunner, Robinh, Rohan Ghatak, Romanm, Rookkey, Rossami, Rs2, Rsmuruga, RyanJones, Saga City, Salix alba, Santosga, Saric, Sbharris, Scentoni, Scottie 000, SeanMack, Sergiacid,ServiceAT, Seth Ilys, SeventyThree, Shizhao, Shoessss, Shoujun, Sidonath, Silly rabbit, SimonTrew, SirFozzie, Sligocki, Snoyes, Soap, Soliloquial, Som subhra, Spellcast, Spireguy, Steorra,Stevenj, Storkk, StradivariusTV, Strebe, Streyeder, SuperMidget, Sverdrup, Symane, Szidomingo, Taggart Transcontinental, Takatodigi, TakuyaMurata, Tariqhada, Tarquin, Taw, Taxman,Tcha216, TechnoFaye, Template namespace initialisation script, Tentoes, Terry2012, Tesseran, Thalesfernandes, The Anome, The Thing That Should Not Be, Thenub314, Tiddly Tom, Tikiwont,Timir2, Tommy2010, Tosha, Treisijs, Trevor MacInnis, Trovatore, Truthnlove, Ttimespan, Twigletmac, Twooars, Tyco.skinner, Urdutext, User27091, VKokielov, VTBushyTail, Vipinhari,Virga, Virginia-American, Voorlandt, Wafulz, Waltpohl, Watcharakorn, Wayp123, Wesaq, Whommighter, Wiki alf, WikiDao, Wikiborg, Wolfkeeper, Wolfrock, Wshun, Wurzel, Wwoods,Wyatt915, X42bn6, Xantharius, Xxglennxx, Yellowstone6, Yjwong, Zachlipton, Zirland, Zoffdino, Zoicon5, Zundark, Zzuuzz, Ævar Arnfjörð Bjarmason, Александър, 700 anonymous edits

Microphone practice  Source: http://en.wikipedia.org/w/index.php?oldid=446098192  Contributors: (aeropagitica), Altaphon, Annunciation, Binksternet, Decaren, December21st2012Freak,Gabelstaplerfahrer, Ggegan, HairyWombat, II MusLiM HyBRiD II, ILike2BeAnonymous, Iain, John of Reading, Lhanneus, Redheylin, Rjwilmsi, Tohd8BohaithuGh1, UkiahBass, Wg0867, 35anonymous edits

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Image Sources, Licenses and ContributorsImage:Sinus.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Sinus.svg  License: Creative Commons Attribution-Sharealike 2.5  Contributors: Juiced lemon, Quark67File:Trigono sine en2.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Trigono_sine_en2.svg  License: Creative Commons Attribution-Sharealike 3.0,2.5,2.0,1.0  Contributors:Trigono_a10.svg: Dnu72 derivative work: Pengo (talk)Image:Sine.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Sine.svg  License: Creative Commons Attribution-Share Alike  Contributors: Geek3File:Sine cosine one period.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Sine_cosine_one_period.svg  License: Creative Commons Attribution 3.0  Contributors: Geek3File:Unit circle3.png  Source: http://en.wikipedia.org/w/index.php?title=File:Unit_circle3.png  License: Public Domain  Contributors: JonasMeinertzFile:Unit circle.svg  Source: 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in en:Sine: Pengo (talk) First version: http://ja.wikipedia.org/wiki/%E7%94%BB%E5%83%8F:Complex.png by jp:user:たまなるたみ Vectorized and tweaked:Complex_conjugate_picture.svg by Oleg AlexandrovFile:ComplexPlot-Sin-z-,1024-.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:ComplexPlot-Sin-z-,1024-.jpg  License: Creative Commons Attribution-Share Alike  Contributors:Athanase E-OFile:Sin z vector field 02 Pengo.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Sin_z_vector_field_02_Pengo.svg  License: Public Domain  Contributors: PengoFile:Complex sin real 01 Pengo.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Complex_sin_real_01_Pengo.svg  License: Public Domain  Contributors: PengoFile:Complex sin imag 01 Pengo.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Complex_sin_imag_01_Pengo.svg  License: Public Domain  Contributors: PengoFile:Complex sin abs 01 Pengo.svg  Source: 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Based on en:Image:Circle-trig6.png, which was donated to Wikipedia underGFDL by Steven G. Johnson.Image:Sine and Cosine fundamental relationship to Circle (and Helix).gif  Source:http://en.wikipedia.org/w/index.php?title=File:Sine_and_Cosine_fundamental_relationship_to_Circle_(and_Helix).gif  License: Creative Commons Attribution-Sharealike 3.0  Contributors:TdadamemdImage:Complex sin.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Complex_sin.jpg  License: Public Domain  Contributors: Jan HomannImage:Complex cos.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Complex_cos.jpg  License: Public Domain  Contributors: Jan HomannImage:Complex tan.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Complex_tan.jpg  License: Public Domain  Contributors: Jan HomannImage:Complex Cot.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Complex_Cot.jpg  License: Public Domain  Contributors: Jan HomannImage:Complex Sec.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Complex_Sec.jpg  License: Public Domain  Contributors: Jan HomannImage:Complex Csc.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Complex_Csc.jpg  License: Public Domain  Contributors: Jan HomannImage:Lissajous curve 5by4.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Lissajous_curve_5by4.svg  License: Creative Commons Attribution-ShareAlike 3.0 Unported Contributors: Alessio DamatoImage:Synthesis square.gif  Source: http://en.wikipedia.org/w/index.php?title=File:Synthesis_square.gif  License: GNU Free Documentation License  Contributors: Alejo2083, Cuddlyable3,Kieff, Omegatron, Pieter Kuiper, 3 anonymous editsFile:Sawtooth Fourier Analysis.JPG  Source: http://en.wikipedia.org/w/index.php?title=File:Sawtooth_Fourier_Analysis.JPG  License: Creative Commons Attribution-Sharealike 3.0 Contributors: Brews ohareImage:Complex number illustration.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Complex_number_illustration.svg  License: Creative Commons Attribution-Sharealike3.0,2.5,2.0,1.0  Contributors: Original uploader was Wolfkeeper at en.wikipediaImage:Complex number illustration.png  Source: http://en.wikipedia.org/w/index.php?title=File:Complex_number_illustration.png  License: Creative Commons Attribution-Sharealike 3.0 Contributors: Kan8eDieFile:Complex conjugate picture.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Complex_conjugate_picture.svg  License: GNU Free Documentation License  Contributors: OlegAlexandrovImage:Vector Addition.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Vector_Addition.svg  License: Public Domain  Contributors: Booyabazooka, Kilom691Image:Complex_number_illustration_modarg.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Complex_number_illustration_modarg.svg  License: GNU Free DocumentationLicense  Contributors: Complex_number_illustration.svg: Original uploader was Wolfkeeper at en.wikipedia derivative work: Kan8eDie (talk)File:ComplexMultiplication.png  Source: http://en.wikipedia.org/w/index.php?title=File:ComplexMultiplication.png  License: Creative Commons Attribution-Sharealike 3.0  Contributors:Jakob.scholbach

Image Sources, Licenses and Contributors 103

Image:Sin1perz.png  Source: http://en.wikipedia.org/w/index.php?title=File:Sin1perz.png  License: Creative Commons Attribution-Sharealike 3.0  Contributors: KovzolFile:Pentagon construct.gif  Source: http://en.wikipedia.org/w/index.php?title=File:Pentagon_construct.gif  License: Public domain  Contributors: TokyoJunkie at the English WikipediaImage:NegativeOne3Root.svg  Source: http://en.wikipedia.org/w/index.php?title=File:NegativeOne3Root.svg  License: Creative Commons Attribution-Sharealike 3.0  Contributors: Loadmaster(David R. Tribble)Image:XY stereo.svg  Source: http://en.wikipedia.org/w/index.php?title=File:XY_stereo.svg  License: GNU Free Documentation License  Contributors: Iainf 23:51, 21 September 2007 (UTC)Image:Blumlein Stereo.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Blumlein_Stereo.svg  License: GNU Free Documentation License  Contributors: Iainf 23:51, 21 September2007 (UTC)Image:MS stereo.svg  Source: http://en.wikipedia.org/w/index.php?title=File:MS_stereo.svg  License: GNU Free Documentation License  Contributors: Iainf 23:51, 21 September 2007 (UTC)File:2006-01-14 Surface waves.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:2006-01-14_Surface_waves.jpg  License: GNU Free Documentation License  Contributors: RogerMcLassusFile:Nonsinusoidal wavelength.JPG  Source: http://en.wikipedia.org/w/index.php?title=File:Nonsinusoidal_wavelength.JPG  License: Creative Commons Attribution-Sharealike 3.0 Contributors: Brews ohareFile:Waveforms.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Waveforms.svg  License: unknown  Contributors: Jafeluv, Omegatron, Pieter Kuiper, 5 anonymous editsFile:Wave packet.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Wave_packet.svg  License: Public Domain  Contributors: Oleg AlexandrovImage:Wave group.gif  Source: http://en.wikipedia.org/w/index.php?title=File:Wave_group.gif  License: GNU Free Documentation License  Contributors: KraaiennestImage:Wave opposite-group-phase-velocity.gif  Source: http://en.wikipedia.org/w/index.php?title=File:Wave_opposite-group-phase-velocity.gif  License: Creative Commons Attribution 3.0 Contributors: Geek3File:Simple harmonic motion animation.gif  Source: http://en.wikipedia.org/w/index.php?title=File:Simple_harmonic_motion_animation.gif  License: Public Domain  Contributors:User:Evil_saltineFile:Standing wave.gif  Source: http://en.wikipedia.org/w/index.php?title=File:Standing_wave.gif  License: Public Domain  Contributors: BrokenSegue, Cdang, Joolz, Kersti Nebelsiek, Kieff,Mike.lifeguard, Pieter Kuiper, PtjImage:Harmonic partials on strings.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Harmonic_partials_on_strings.svg  License: Public Domain  Contributors: QefImage:Drum vibration mode01.gif  Source: http://en.wikipedia.org/w/index.php?title=File:Drum_vibration_mode01.gif  License: Public Domain  Contributors: Oleg AlexandrovImage:Drum vibration mode21.gif  Source: http://en.wikipedia.org/w/index.php?title=File:Drum_vibration_mode21.gif  License: Public Domain  Contributors: Oleg AlexandrovFile:Light dispersion of a mercury-vapor lamp with a flint glass prism IPNr°0125.jpg  Source:http://en.wikipedia.org/w/index.php?title=File:Light_dispersion_of_a_mercury-vapor_lamp_with_a_flint_glass_prism_IPNr°0125.jpg  License: unknown  Contributors: D-KuruFile:Wave refraction.gif  Source: http://en.wikipedia.org/w/index.php?title=File:Wave_refraction.gif  License: Creative Commons Attribution-Sharealike 3.0  Contributors: Dicklyon (Richard F.Lyon)File:Circular.Polarization.Circularly.Polarized.Light Circular.Polarizer Creating.Left.Handed.Helix.View.svg  Source:http://en.wikipedia.org/w/index.php?title=File:Circular.Polarization.Circularly.Polarized.Light_Circular.Polarizer_Creating.Left.Handed.Helix.View.svg  License: Public Domain  Contributors:Dave3457File:Shallow water wave.gif  Source: http://en.wikipedia.org/w/index.php?title=File:Shallow_water_wave.gif  License: GNU Free Documentation License  Contributors: KraaiennestFile:Transonico-en.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Transonico-en.svg  License: Creative Commons Attribution-ShareAlike 3.0 Unported  Contributors: Cmprince,Cobatfor, Ignacio Icke, Pieter Kuiper, Rocket000File:Onde electromagnétique.png  Source: http://en.wikipedia.org/w/index.php?title=File:Onde_electromagnétique.png  License: Creative Commons Attribution-ShareAlike 3.0 Unported Contributors: ploufandsplashFile:Wave packet (dispersion).gif  Source: http://en.wikipedia.org/w/index.php?title=File:Wave_packet_(dispersion).gif  License: Public Domain  Contributors: Cdang, Fffred, Kersti NebelsiekFile:GravitationalWave CrossPolarization.gif  Source: http://en.wikipedia.org/w/index.php?title=File:GravitationalWave_CrossPolarization.gif  License: Public Domain  Contributors: Originaluploader was MOBle at en.wikipedia

License 104

LicenseCreative Commons Attribution-Share Alike 3.0 Unportedhttp:/ / creativecommons. org/ licenses/ by-sa/ 3. 0/