Nyquist–Shannon sampling theorem

X ( ƒ)

ƒB−B

Example of magnitude of the Fourier transform of a bandlimitedfunction

In the field of digital signal processing, the sampling the-orem is a fundamental bridge between continuous-timesignals (often called “analog signals”) and discrete-timesignals (often called “digital signals”). It establishes a suf-ficient condition for a sample rate that permits a discretesequence of samples to capture all the information froma continuous-time signal of finite bandwidth.Strictly speaking, the theorem only applies to a class ofmathematical functions having a Fourier transform that iszero outside of a finite region of frequencies. Intuitivelywe expect that when one reduces a continuous function toa discrete sequence and interpolates back to a continuousfunction, the fidelity of the result depends on the density(or sample rate) of the original samples. The samplingtheorem introduces the concept of a sample rate that issufficient for perfect fidelity for the class of functions thatare bandlimited to a given bandwidth, such that no actualinformation is lost in the sampling process. It expressesthe sufficient sample rate in terms of the bandwidth for theclass of functions. The theorem also leads to a formulafor perfectly reconstructing the original continuous-timefunction from the samples.Perfect reconstruction may still be possible when thesample-rate criterion is not satisfied, provided other con-straints on the signal are known. (See § Sampling ofnon-baseband signals below, and compressed sensing.) Insome cases (when the sample-rate criterion is not satis-fied), utilizing additional constraints allows for approx-imate reconstructions. The fidelity of these reconstruc-tions can be verified and quantified utilizing Bochner’stheorem.[1]

The name Nyquist–Shannon sampling theorem honorsHarry Nyquist and Claude Shannon. The theoremwas also discovered independently by E. T. Whit-taker, by Vladimir Kotelnikov, and by others. Itis thus also known by the names Nyquist–Shannon–

Kotelnikov, Whittaker–Shannon–Kotelnikov, Whittaker–Nyquist–Kotelnikov–Shannon, and cardinal theorem ofinterpolation.

1 Introduction

Sampling is a process of converting a signal (for exam-ple, a function of continuous time and/or space) into a nu-meric sequence (a function of discrete time and/or space).Shannon’s version of the theorem states:[2]

If a function x(t) contains no frequencies higher than Bhertz, it is completely determined by giving its ordinatesat a series of points spaced 1/(2B) seconds apart.A sufficient sample-rate is therefore 2B samples/second,or anything larger. Equivalently, for a given sample ratef , perfect reconstruction is guaranteed possible for abandlimit B < f /2.When the bandlimit is too high (or there is no ban-dlimit), the reconstruction exhibits imperfections knownas aliasing. Modern statements of the theorem are some-times careful to explicitly state that x(t) must containno sinusoidal component at exactly frequency B, or thatB must be strictly less than ½ the sample rate. Thetwo thresholds, 2B and f /2 are respectively called theNyquist rate and Nyquist frequency. And respectively,they are attributes of x(t) and of the sampling equipment.The condition described by these inequalities is calledthe Nyquist criterion, or sometimes the Raabe condi-tion. The theorem is also applicable to functions of otherdomains, such as space, in the case of a digitized image.The only change, in the case of other domains, is the unitsof measure applied to t, f , and B.The symbol T = 1/f is customarily used to represent theinterval between samples and is called the sample pe-riod or sampling interval. And the samples of functionx(t) are commonly denoted by x[n] = x(nT) (alternatively"xn" in older signal processing literature), for all integervalues of n. A mathematically ideal way to interpolate thesequence involves the use of sinc functions. Each samplein the sequence is replaced by a sinc function, centeredon the time axis at the original location of the sample,nT, with the amplitude of the sinc function scaled to thesample value, x[n]. Subsequently, the sinc functions aresummed into a continuous function. A mathematicallyequivalent method is to convolve one sinc function with aseries of Dirac delta pulses, weighted by the sample val-ues. Neither method is numerically practical. Instead,

1

2 3 DERIVATION AS A SPECIAL CASE OF POISSON SUMMATION

1.0

0.8

0.6

0.4

0.2

-0.2

-6 -4 -2 2 4 6

sin(πx)πx

x

The normalized sinc function: sin(πx) / (πx) ... showing the cen-tral peak at x= 0, and zero-crossings at the other integer valuesof x.

some type of approximation of the sinc functions, finitein length, is used. The imperfections attributable to theapproximation are known as interpolation error.Practical digital-to-analog converters produce neitherscaled and delayed sinc functions, nor ideal Dirac pulses.Instead they produce a piecewise-constant sequence ofscaled and delayed rectangular pulses (the zero-orderhold), usually followed by an “anti-imaging filter” to cleanup spurious high-frequency content.

2 Aliasing

Main article: AliasingWhen x(t) is a function with a Fourier transform, X(f):

The samples of two sine waves can be identical, when at leastone of them is at a frequency above half the sample rate.

X(f)def=

∫ ∞

−∞x(t) e−i2πft dt,

the Poisson summation formula indicates that the sam-ples, x(nT), of x(t) are sufficient to create a periodic sum-mation of X(f). The result is:

which is a periodic function and its equivalent represen-tation as a Fourier series, whose coefficients are T•x(nT).This function is also known as the discrete-time Fourier

X(f) (top blue) and XA(f) (bottom blue) are continuous Fouriertransforms of two different functions, x(t) and xA(t) (notshown). When the functions are sampled at rate fs, the images(green) are added to the original transforms (blue) when one ex-amines the discrete-time Fourier transforms (DTFT) of the se-quences. In this hypothetical example, the DTFTs are identical,which means the sampled sequences are identical, even thoughthe original continuous pre-sampled functions are not. If thesewere audio signals, x(t) and xA(t) might not sound the same.But their samples (taken at rate fs) are identical and would leadto identical reproduced sounds; thus xA(t) is an alias of x(t) atthis sample rate.

transform (DTFT) of the sequence T•x(nT), for integersn.As depicted, copies of X(f) are shifted by multiples offs and combined by addition. For a band-limited func-tion (X(f) = 0 for all |f | ≥ B), and sufficiently large fs,it is possible for the copies to remain distinct from eachother. But if the Nyquist criterion is not satisfied, ad-jacent copies overlap, and it is not possible in general todiscern an unambiguous X(f). Any frequency componentabove fs/2 is indistinguishable from a lower-frequencycomponent, called an alias, associated with one of thecopies. In such cases, the customary interpolation tech-niques produce the alias, rather than the original compo-nent. When the sample-rate is pre-determined by otherconsiderations (such as an industry standard), x(t) is usu-ally filtered to reduce its high frequencies to acceptablelevels before it is sampled. The type of filter requiredis a lowpass filter, and in this application it is called ananti-aliasing filter.

3 Derivation as a special case ofPoisson summation

When there is no overlap of the copies (aka “images”) ofX(f), the k = 0 term of Xs(f) can be recovered by theproduct:

X(f) = H(f) ·Xs(f), where:

3

Spectrum, X (f), of a properly sampled bandlimited signal (blue)and the adjacent DTFT images (green) that do not overlap. Abrick-wall low-pass filter, H(f), removes the images, leaves theoriginal spectrum, X(f), and recovers the original signal from itssamples.

H(f)def=

{1 |f | < B

0 |f | > fs −B.

At this point, the sampling theorem is proved, since X(f)uniquely determines x(t).All that remains is to derive the formula for reconstruc-tion. H(f) need not be precisely defined in the region [B,fs − B] because X (f) is zero in that region. However, theworst case is when B = fs/2, the Nyquist frequency. Afunction that is sufficient for that and all less severe casesis:

H(f) = rect(

f

fs

)=

{1 |f | < fs

2

0 |f | > fs2 ,

where rect(•) is the rectangular function. Therefore:

X(f) = rect(

ffs

)·Xs(f)

= rect(Tf) ·∑∞

n=−∞ T ·x(nT ) e−i2πnTf (fromEq.1, above).=

∑∞n=−∞ x(nT ) ·

T · rect(Tf) · e−i2πnTf︸ ︷︷ ︸F{sinc( t−nT

T )}

.

[3]

The inverse transform of both sides produces theWhittaker–Shannon interpolation formula:

x(t) =∞∑

n=−∞x(nT ) · sinc

(t− nT

T

),

which shows how the samples, x(nT), can be combinedto reconstruct x(t).

• Larger-than-necessary values of fs (smaller valuesof T), called oversampling, have no effect on the

outcome of the reconstruction and have the benefitof leaving room for a transition band in which H(f)is free to take intermediate values. Undersampling,which causes aliasing, is not in general a reversibleoperation.

• Theoretically, the interpolation formula can beimplemented as a low pass filter, whose im-pulse response is sinc(t/T) and whose input is∑∞

n=−∞ x(nT ) ·δ(t−nT ), which is a Dirac combfunction modulated by the signal samples. Practi-cal digital-to-analog converters (DAC) implementan approximation like the zero-order hold. In thatcase, oversampling can reduce the approximationerror.

4 Shannon’s original proof

Poisson shows that the Fourier series in Eq.1 producesthe periodic summation of X(f), regardless of fs and B.Shannon, however, only derives the series coefficients forthe case fs = 2B. Virtually quoting Shannon’s original pa-per:

Let X(ω) be the spectrum of x(t). Then

since X(ω) is assumed to be zero outside theband | ω

2π |<B . If we let

t =n

2B

where n is any positive or negative integer, weobtain

x(

n2B

)=

1

2π

∫ 2πB

−2πB

X(ω)eiωn2B dω.

On the left are values of x(t) at the samplingpoints. The integral on the right will be recog-nized as essentially[note 1] the nth coefficient in aFourier-series expansion of the function X(ω),

taking the interval –B to B as a fundamentalperiod. This means that the values of the sam-ples x(n/2B) determine the Fourier coefficientsin the series expansion of X(ω). Thus they de-termine X(ω), since X(ω) is zero for frequen-cies greater than B, and for lower frequenciesX(ω) is determined if its Fourier coefficientsare determined. But X(ω) determines the orig-inal function x(t) completely, since a functionis determined if its spectrum is known. There-fore the original samples determine the func-tion x(t) completely.

4 5 APPLICATION TO MULTIVARIABLE SIGNALS AND IMAGES

Shannon’s proof of the theorem is complete at that point,but he goes on to discuss reconstruction via sinc func-tions, what we now call the Whittaker–Shannon interpo-lation formula as discussed above. He does not derive orprove the properties of the sinc function, but these wouldhave been familiar to engineers reading his works at thetime, since the Fourier pair relationship between rect (therectangular function) and sinc was well known.

Let xn be the nth sample. Then the function x(t)

is represented by:

x(t) =∞∑

n=−∞xn

sinπ(2Bt− n)

π(2Bt− n).

As in the other proof, the existence of the Fourier trans-form of the original signal is assumed, so the proof doesnot say whether the sampling theorem extends to ban-dlimited stationary random processes.

4.1 Notes

[1] The actual coefficient formula contains an additional fac-tor of 1/2B=T. So Shannon’s coefficients are T ·x(nT ),

which agrees with Eq.1.

5 Application to multivariable sig-nals and images

Main article: Multidimensional samplingThe sampling theorem is usually formulated for functionsof a single variable. Consequently, the theorem is directlyapplicable to time-dependent signals and is normally for-mulated in that context. However, the sampling theoremcan be extended in a straightforward way to functions ofarbitrarily many variables. Grayscale images, for exam-ple, are often represented as two-dimensional arrays (ormatrices) of real numbers representing the relative inten-sities of pixels (picture elements) located at the intersec-tions of row and column sample locations. As a result,images require two independent variables, or indices, tospecify each pixel uniquely — one for the row, and onefor the column.Color images typically consist of a composite of threeseparate grayscale images, one to represent each of thethree primary colors — red, green, and blue, or RGB forshort. Other colorspaces using 3-vectors for colors in-clude HSV, CIELAB, XYZ, etc. Some colorspaces suchas cyan, magenta, yellow, and black (CMYK) may repre-sent color by four dimensions. All of these are treated asvector-valued functions over a two-dimensional sampleddomain.

Subsampled image showing a Moiré pattern

Properly sampled image

Similar to one-dimensional discrete-time signals, imagescan also suffer from aliasing if the sampling resolution,or pixel density, is inadequate. For example, a digitalphotograph of a striped shirt with high frequencies (inother words, the distance between the stripes is small),can cause aliasing of the shirt when it is sampled by thecamera’s image sensor. The aliasing appears as a moirépattern. The “solution” to higher sampling in the spatialdomain for this case would be to move closer to the shirt,

5

use a higher resolution sensor, or to optically blur the im-age before acquiring it with the sensor.Another example is shown to the right in the brick pat-terns. The top image shows the effects when the sam-pling theorem’s condition is not satisfied. When soft-ware rescales an image (the same process that createsthe thumbnail shown in the lower image) it, in effect,runs the image through a low-pass filter first and thendownsamples the image to result in a smaller image thatdoes not exhibit the moiré pattern. The top image is whathappens when the image is downsampled without low-pass filtering: aliasing results.The application of the sampling theorem to images shouldbe made with care. For example, the sampling process inany standard image sensor (CCD or CMOS camera) isrelatively far from the ideal sampling which would mea-sure the image intensity at a single point. Instead thesedevices have a relatively large sensor area at each sam-ple point in order to obtain sufficient amount of light. Inother words, any detector has a finite-width point spreadfunction. The analog optical image intensity functionwhich is sampled by the sensor device is not in generalbandlimited, and the non-ideal sampling is itself a use-ful type of low-pass filter, though not always sufficientto remove enough high frequencies to sufficiently reducealiasing. When the area of the sampling spot (the size ofthe pixel sensor) is not large enough to provide sufficientspatial anti-aliasing, a separate anti-aliasing filter (opticallow-pass filter) is typically included in a camera systemto further blur the optical image. Despite images havingthese problems in relation to the sampling theorem, thetheorem can be used to describe the basics of down andup sampling of images.

6 Critical frequency

To illustrate the necessity of fs > 2B, consider the fam-ily of sinusoids generated by different values of θ in thisformula:

x(t) =cos(2πBt+ θ)

cos(θ) = cos(2πBt)−sin(2πBt) tan(θ), −π/2 < θ < π/2.

With fs = 2B or equivalently T = 1/(2B), the samples aregiven by:

x(nT ) = cos(πn)− sin(πn)︸ ︷︷ ︸0

tan(θ) = (−1)n

regardless of the value of θ. That sort of ambiguity is thereason for the strict inequality of the sampling theorem’scondition.

−3

−2

−1

0

1

2

3

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

−3 −2 −1 0 1 2 3

A family of sinusoids at the critical frequency, all having thesame sample sequences of alternating +1 and –1. That is, theyall are aliases of each other, even though their frequency is notabove half the sample rate.

7 Sampling of non-baseband sig-nals

As discussed by Shannon:[2]

A similar result is true if theband does not start at zero fre-quency but at some higher value,and can be proved by a linear trans-lation (corresponding physically tosingle-sideband modulation) of thezero-frequency case. In this casethe elementary pulse is obtainedfrom sin(x)/x by single-side-bandmodulation.

That is, a sufficient no-loss condition for sampling signalsthat do not have baseband components exists that involvesthe width of the non-zero frequency interval as opposedto its highest frequency component. See Sampling (signalprocessing) for more details and examples.For example, in order to sample the FM radio signals inthe frequency range of 100-102 MHz, it is not necessaryto sample at 204 MHz (twice the upper frequency), butrather it is sufficient to sample at 4 MHz (twice the widthof the frequency interval).A bandpass condition is that X(f) = 0, for all nonnegativef outside the open band of frequencies:

(N

2fs,

N + 1

2fs

),

for some nonnegative integer N. This formulation in-cludes the normal baseband condition as the case N=0.

6 10 HISTORICAL BACKGROUND

The corresponding interpolation function is the impulseresponse of an ideal brick-wall bandpass filter (as op-posed to the ideal brick-wall lowpass filter used above)with cutoffs at the upper and lower edges of the specifiedband, which is the difference between a pair of lowpassimpulse responses:

(N+1) sinc((N + 1)t

T

)−N sinc

(Nt

T

).

Other generalizations, for example to signals occupyingmultiple non-contiguous bands, are possible as well. Eventhe most generalized form of the sampling theorem doesnot have a provably true converse. That is, one cannotconclude that information is necessarily lost just becausethe conditions of the sampling theorem are not satisfied;from an engineering perspective, however, it is generallysafe to assume that if the sampling theorem is not satisfiedthen information will most likely be lost.

8 Nonuniform sampling

The sampling theory of Shannon can be generalizedfor the case of nonuniform sampling, that is, samplesnot taken equally spaced in time. The Shannon sam-pling theory for non-uniform sampling states that a band-limited signal can be perfectly reconstructed from its sam-ples if the average sampling rate satisfies the Nyquistcondition.[4] Therefore, although uniformly spaced sam-ples may result in easier reconstruction algorithms, it isnot a necessary condition for perfect reconstruction.The general theory for non-baseband and nonuniformsamples was developed in 1967 by Landau.[5] He provedthat, to paraphrase roughly, the average sampling rate(uniform or otherwise) must be twice the occupied band-width of the signal, assuming it is a priori known whatportion of the spectrum was occupied. In the late 1990s,this work was partially extended to cover signals of whenthe amount of occupied bandwidth was known, but theactual occupied portion of the spectrum was unknown.[6]

In the 2000s, a complete theory was developed (see thesection Beyond Nyquist below) using compressed sens-ing. In particular, the theory, using signal processinglanguage, is described in this 2009 paper.[7] They show,among other things, that if the frequency locations are un-known, then it is necessary to sample at least at twice theNyquist criteria; in other words, you must pay at least afactor of 2 for not knowing the location of the spectrum.Note that minimum sampling requirements do not neces-sarily guarantee stability.

9 Sampling below the Nyquist rateunder additional restrictions

Main article: Undersampling

The Nyquist–Shannon sampling theorem provides asufficient condition for the sampling and reconstructionof a band-limited signal. When reconstruction is donevia the Whittaker–Shannon interpolation formula, theNyquist criterion is also a necessary condition to avoidaliasing, in the sense that if samples are taken at a slowerrate than twice the band limit, then there are some sig-nals that will not be correctly reconstructed. However,if further restrictions are imposed on the signal, then theNyquist criterion may no longer be a necessary condition.A non-trivial example of exploiting extra assumptionsabout the signal is given by the recent field of compressedsensing, which allows for full reconstruction with a sub-Nyquist sampling rate. Specifically, this applies to sig-nals that are sparse (or compressible) in some domain.As an example, compressed sensing deals with signalsthat may have a low over-all bandwidth (say, the effec-tive bandwidth EB), but the frequency locations are un-known, rather than all together in a single band, so that thepassband technique doesn't apply. In other words, the fre-quency spectrum is sparse. Traditionally, the necessarysampling rate is thus 2B. Using compressed sensing tech-niques, the signal could be perfectly reconstructed if it issampled at a rate slightly lower than 2EB. The downsideof this approach is that reconstruction is no longer givenby a formula, but instead by the solution to a convex op-timization program which requires well-studied but non-linear methods.

10 Historical background

The sampling theorem was implied by the work of HarryNyquist in 1928 (“Certain topics in telegraph transmis-sion theory”), in which he showed that up to 2B inde-pendent pulse samples could be sent through a system ofbandwidth B; but he did not explicitly consider the prob-lem of sampling and reconstruction of continuous signals.About the same time, Karl Küpfmüller showed a simi-lar result,[8] and discussed the sinc-function impulse re-sponse of a band-limiting filter, via its integral, the stepresponse Integralsinus; this bandlimiting and reconstruc-tion filter that is so central to the sampling theorem issometimes referred to as a Küpfmüller filter (but seldomso in English).The sampling theorem, essentially a dual of Nyquist’s re-sult, was proved by Claude E. Shannon in 1949 (“Com-munication in the presence of noise”). V. A. Kotelnikovpublished similar results in 1933 (“On the transmissioncapacity of the 'ether' and of cables in electrical com-munications”, translation from the Russian), as did the

10.2 Why Nyquist? 7

mathematician E. T. Whittaker in 1915 (“Expansions ofthe Interpolation-Theory”, “Theorie der Kardinalfunktio-nen”), J. M. Whittaker in 1935 (“Interpolatory functiontheory”), and Gabor in 1946 (“Theory of communica-tion”). In 1999, the Eduard Rhein Foundation awardedKotelnikov their Basic Research Award “for the first the-oretically exact formulation of the sampling theorem.”

10.1 Other discoverers

Others who have independently discovered or playedroles in the development of the sampling theorem havebeen discussed in several historical articles, for exampleby Jerri[9] and by Lüke.[10] For example, Lüke points outthat H. Raabe, an assistant to Küpfmüller, proved the the-orem in his 1939 Ph.D. dissertation; the term Raabe con-dition came to be associated with the criterion for unam-biguous representation (sampling rate greater than twicethe bandwidth).Meijering[11] mentions several other discoverers andnames in a paragraph and pair of footnotes:

As pointed out by Higgins [135], the sam-pling theorem should really be considered intwo parts, as done above: the first stating thefact that a bandlimited function is completelydetermined by its samples, the second describ-ing how to reconstruct the function using itssamples. Both parts of the sampling theoremwere given in a somewhat different form byJ. M. Whittaker [350, 351, 353] and beforehim also by Ogura [241, 242]. They wereprobably not aware of the fact that the firstpart of the theorem had been stated as earlyas 1897 by Borel [25].27 As we have seen,Borel also used around that time what becameknown as the cardinal series. However, he ap-pears not to have made the link [135]. In lateryears it became known that the sampling the-orem had been presented before Shannon tothe Russian communication community by Ko-tel'nikov [173]. In more implicit, verbal form,it had also been described in the German litera-ture by Raabe [257]. Several authors [33, 205]have mentioned that Someya [296] introducedthe theorem in the Japanese literature parallelto Shannon. In the English literature, Weston[347] introduced it independently of Shannonaround the same time.28

27 Several authors, following Black [16],have claimed that this first part of the samplingtheorem was stated even earlier by Cauchy, ina paper [41] published in 1841. However, thepaper of Cauchy does not contain such a state-ment, as has been pointed out by Higgins [135].

28 As a consequence of the discoveryof the several independent introductions ofthe sampling theorem, people started to re-fer to the theorem by including the names ofthe aforementioned authors, resulting in suchcatchphrases as “the Whittaker–Kotel’nikov–Shannon (WKS) sampling theorem” [155]or even “the Whittaker–Kotel'nikov–Raabe–Shannon–Someya sampling theorem” [33]. Toavoid confusion, perhaps the best thing to do isto refer to it as the sampling theorem, “ratherthan trying to find a title that does justice to allclaimants” [136].

10.2 Why Nyquist?

Exactly how, when, or why Harry Nyquist had his nameattached to the sampling theorem remains obscure. Theterm Nyquist Sampling Theorem (capitalized thus) ap-peared as early as 1959 in a book from his former em-ployer, Bell Labs,[12] and appeared again in 1963,[13] andnot capitalized in 1965.[14] It had been called the Shan-non Sampling Theorem as early as 1954,[15] but also justthe sampling theorem by several other books in the early1950s.In 1958, Blackman and Tukey cited Nyquist’s 1928 pa-per as a reference for the sampling theorem of informa-tion theory,[16] even though that paper does not treat sam-pling and reconstruction of continuous signals as othersdid. Their glossary of terms includes these entries:

Sampling theorem (of information theory)

Nyquist’s result that equi-spaced data, withtwo or more points per cycle of highest fre-quency, allows reconstruction of band-limitedfunctions. (See Cardinal theorem.)

Cardinal theorem (of interpolation theory)

A precise statement of the conditions underwhich values given at a doubly infinite set ofequally spaced points can be interpolated toyield a continuous band-limited function withthe aid of the function

sin(x− xi)

x− xi.

Exactly what “Nyquist’s result” they are referring to re-mains mysterious.When Shannon stated and proved the sampling theoremin his 1949 paper, according to Meijering[11] “he referredto the critical sampling interval T = 1/(2W) as the Nyquist

8 13 REFERENCES

interval corresponding to the band W, in recognition ofNyquist’s discovery of the fundamental importance ofthis interval in connection with telegraphy.” This explainsNyquist’s name on the critical interval, but not on the the-orem.Similarly, Nyquist’s name was attached to Nyquist rate in1953 by Harold S. Black:[17]

“If the essential frequency range is limited toB cycles per second, 2B was given by Nyquistas the maximum number of code elementsper second that could be unambiguously re-solved, assuming the peak interference is lesshalf a quantum step. This rate is generallyreferred to as signaling at the Nyquist rateand 1/(2B) has been termed a Nyquist interval.”(bold added for emphasis; italics as in the orig-inal)

According to the OED, this may be the origin of the termNyquist rate. In Black’s usage, it is not a sampling rate,but a signaling rate.

11 See also

• Balian–Low theorem, a similar theoretical lower-bound on sampling rates, but which applies to time–frequency transforms.

• The Cheung–Marks theorem specifies conditionswhere restoration of a signal by the sampling the-orem can become ill-posed.

• Hartley’s law

• Nyquist ISI criterion

• Reconstruction from zero crossings

• Zero-order hold

12 Notes[1] Nemirovsky, Jonathan; Shimron, Efrat (2015). “Utilizing

Bochners Theorem for Constrained Evaluation of MissingFourier Data”. arXiv:1506.03300 [physics.med-ph].

[2] , “Communication in the presence of noise”, Proc. Insti-tute of Radio Engineers, vol. 37, no. 1, pp. 10–21, Jan.1949. Reprint as classic paper in: Proc. IEEE, vol. 86,no. 2, (Feb. 1998)

[3] The sinc function follows from rows 202 and 102 of thetransform tables

[4] Nonuniform Sampling, Theory and Practice (ed. F. Mar-vasti), Kluwer Academic/Plenum Publishers, New York,2000

[5] Landau, H. J. (1967). “Necessary density conditions forsampling and interpolation of certain entire functions”.Acta Math. 117 (1): 37–52. doi:10.1007/BF02395039.

[6] see, e.g., Feng, P. (1997). Universal minimum-rate sam-pling and spectrum-blind reconstruction for multiband sig-nals. Ph.D. dissertation, University of Illinois at Urbana-Champaign.

[7] Mishali, Moshe; Eldar, Yonina C. (March 2009). “BlindMultiband Signal Reconstruction: Compressed Sensingfor Analog Signals”. IEEE Trans. Signal Processing 57(3). CiteSeerX: 10 .1 .1 .154 .4255.

[8] Küpfmüller, Karl (1928). "Über die Dynamik der selbst-tätigen Verstärkungsregler”. Elektrische Nachrichtentech-nik (in German) 5 (11): 459–467. (English translation2005).

[9] Abdul Jerri, The Shannon Sampling Theorem—Its Vari-ous Extensions and Applications: A Tutorial Review, Pro-ceedings of the IEEE, 65:1565–1595, Nov. 1977. See alsoCorrection to “The Shannon sampling theorem—Its vari-ous extensions and applications: A tutorial review”, Pro-ceedings of the IEEE, 67:695, April 1979

[10] Hans Dieter Lüke, The Origins of the Sampling Theo-rem, IEEE Communications Magazine, pp.106–108, April1999.

[11] Erik Meijering, A Chronology of Interpolation From An-cient Astronomy to Modern Signal and Image Processing,Proc. IEEE, 90, 2002.

[12] Members of the Technical Staff of Bell Telephone Labab-oratories (1959). Transmission Systems for Communica-tions. AT&T. pp. 26–4 (Vol.2).

[13] Ernst Adolph Guillemin (1963). Theory of Linear Physi-cal Systems. Wiley.

[14] Roberts, Richard A.; Barton, Ben F. (1965). Theory ofSignal Detectability: Composite Deferred Decision Theory.

[15] Gray, Truman S. (1954). Applied Electronics: A FirstCourse in Electronics, Electron Tubes, and Associated Cir-cuits.

[16] Blackman, R. B.; Tukey, J. W. (1958). The Measurementof Power Spectra : From the Point of View of Communi-cations Engineering (PDF). New York: Dover.

[17] Black, Harold S. (1953). Modulation Theory.

13 References• J. R. Higgins: Five short stories about the cardinal

series, Bulletin of the AMS 12(1985)

• V. A. Kotelnikov, “On the carrying capacity ofthe ether and wire in telecommunications”, Mate-rial for the First All-Union Conference on Ques-tions of Communication, Izd. Red. Upr. SvyaziRKKA, Moscow, 1933 (Russian). (english transla-tion, PDF)

9

• Karl Küpfmüller, “Utjämningsförlopp inomTelegraf- och Telefontekniken”, (“Transients intelegraph and telephone engineering”), TekniskTidskrift, no. 9 pp. 153–160 and 10 pp. 178–182,1931.

• R.J. Marks II: Introduction to Shannon Sampling andInterpolation Theory, Springer-Verlag, 1991.

• R.J. Marks II, Editor: Advanced Topics in Shan-non Sampling and Interpolation Theory, Springer-Verlag, 1993.

• R.J. Marks II, Handbook of Fourier Analysis andIts Applications, Oxford University Press, (2009),Chapters 5-8. Google books.

• H. Nyquist, “Certain topics in telegraph transmis-sion theory”, Trans. AIEE, vol. 47, pp. 617–644,Apr. 1928 Reprint as classic paper in: Proc. IEEE,Vol. 90, No. 2, Feb 2002.

• Press, WH; Teukolsky, SA; Vetterling, WT; Flan-nery, BP (2007), “Section 13.11. Numerical Useof the Sampling Theorem”, Numerical Recipes:The Art of Scientific Computing (3rd ed.), NewYork: Cambridge University Press, ISBN 978-0-521-88068-8

• C. E. Shannon, “Communication in the presence ofnoise”, Proc. Institute of Radio Engineers, vol. 37,no.1, pp. 10–21, Jan. 1949. Reprint as classic paperin: Proc. IEEE, Vol. 86, No. 2, (Feb 1998)

• Michael Unser: Sampling-50 Years after Shannon,Proc. IEEE, vol. 88, no. 4, pp. 569–587, April2000

• E. T. Whittaker, “On the Functions Which are Rep-resented by the Expansions of the Interpolation The-ory”, Proc. Royal Soc. Edinburgh, Sec. A, vol.35,pp. 181–194, 1915

• J. M. Whittaker, Interpolatory Function Theory,Cambridge Univ. Press, Cambridge, England,1935.

14 External links

• Learning by Simulations Interactive simulation ofthe effects of inadequate sampling

• Interactive presentation of the sampling and recon-struction in a web-demo Institute of Telecommuni-cations, University of Stuttgart

• Undersampling and an application of it

• Sampling Theory For Digital Audio

• Journal devoted to Sampling Theory

• Sampling Theorem with Constant Amplitude Vari-able Width Pulse

• Hans Dieter Lüke (April 1999). “The Origins of theSampling Theorem”. IEEE Communications Maga-zine. CiteSeerX: 10 .1 .1 .163 .2887.

10 15 TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES

15 Text and image sources, contributors, and licenses

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