Piecewise-quadratic Harmut basis functions and their application to problems in digital signal...

INTERNATIONAL JOURNAL OF COMMUNICATION SYSTEMSInt. J. Commun. Syst. 2010; 23:751–762Published online 15 January 2010 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/dac.1093

Piecewise-quadratic Harmut basis functions and their applicationto problems in digital signal processing

Dhananjay Singh1, Hakimjon Zaynidinov1 and Hoon-Jae Lee2,∗,†

1Department of Ubiquitous IT, Graduate School of Design and IT, Dongseo University, Busan,South Korea 617-716, South Korea

2Division of Information Network Engineering, School of Internet Engineering, Dongseo University, Busan,South Korea 617-716, South Korea

SUMMARY

In this work, the well-known system of orthogonal piecewise-constant Harmut basis functions is investi-gated. As a result of research, their shortcomings are revealed such as weak convergence, discontinuityand others. To eliminate these problems, a new basis of piecewise-quadratic Harmut functions is proposedand a fast spectral transformation algorithm is developed in this basis. For examples of analyticallyset and experimentally verified dependencies, the advantages of the algorithm for spectral transformationsin a basis of piecewise-quadratic Harmut functions are demonstrated. The proposed system and algo-rithm could find wide application in such areas as computer graphics, image processing and restoration,machine vision and multimedia, animation and programming of computer games. Copyright q 2010 JohnWiley & Sons, Ltd.

Received 20 March 2009; Revised 23 October 2009; Accepted 25 October 2009

KEY WORDS: signal; basis; spline; spectrum; fast transformations; piecewise-polynomial

1. INTRODUCTION

In the construction of models for the signals received from real objects, the traditional harmonicfunctions are widely applied. Many signals received from real objects can be easily represented in aFourier analysis as a set of harmonic oscillations [1–5]. This is the result of a transformation fromtime to frequency domain. However, the representation of the time function by harmonic functionsis only one among many possible representations. Any full system of orthogonal functions can beapplied to the decomposition of a series that corresponds to the Fourier series.

∗Correspondence to: Hoon-Jae Lee, Division of Information Network Engineering, School of Internet Engineering,Dongseo University, Busan, South Korea 617-716, South Korea.

†E-mail: [email protected]

Contract/grant sponsor: Dongseo University, Busan, South Korea

Copyright q 2010 John Wiley & Sons, Ltd.

752 D. SINGH, H. ZAYNIDINOV AND H.-J. LEE

As a result of research into the methods for approaching functional dependencies in piecewise-constant bases, their shortcomings are revealed: weak convergence, discontinuity, rather low accu-racy of approximation, and the large volume of memory required to store factors. To eliminatethese shortcomings, we will show that it is necessary to move to piecewise-quadratic bases.The advantages of piecewise-quadratic bases are greater accuracy and smoother approximation,in comparison with piecewise-constant and piecewise-linear bases. These advantages allow highefficiency data structures to be implemented in specialized processors to give a highly accurateapproximation of the piecewise-quadratic bases. The disadvantage of piecewise-quadratic bases isthe absence of fast algorithms for the calculation of factors [6–9].

The current orthogonal piecewise-Harmut’s basic functions have been analyzed in this paper.It is found that the current approach is not suitable due to weak convergence of approximation,discontinuity, and others [5, 10, 11]. To eliminate the previous problems, a new algorithm has beenproposed in this paper to make suitable signal-based applications. The mathematical and simulationresults get to succeed to design fast spectral transformation algorithm.

The paper has designed the method of calculating factor in the piecewise-quadratic basics usingthe previous method and good differential properties of parabolic basic spline. In other words,the requirements of algorithm of fast spectral transformation include minimization of operatingquantity, simplicity of each operation, and minimization of demanded volume of operative memory.The developed method allows the reduction of quantity of arithmetic operations and volume ofnecessary memory. It increases speed because of using orthogonal bases for problems involvingin digital signal processing.

2. THE HARMUT BASIS FUNCTIONS AND PARABOLIC BASIC SPLINES

Technical appendices have been widely distributed listing orthogonal systems of diverging basicfunctions on the valid axis for which fast transformation algorithms exist. They can be broken intotwo classes:

(1) Global basis functions—for which the value is not equal to zero on one sub-interval. TheWalsh functions, the mathematical functions, and the ramp function belong to this class[9, 10, 12].

(2) Local basis functions—for which nonzero values are chosen on the enclosed intervals.Examples are Haar’s functions [2] and Harmut’s functions [1, 4].

Harmut considers systems of piecewise-constant orthogonal basis functions that adopt a positionintermediate between the systems of Walsh and Haar. The most simple are functions of system{hrmk(x)}, obtained from the Haar functions by a double connection of the same underlying formto the next binary piece on the right, with one connection joining with a ‘+’ sign and the otherwith a ‘−‘sign. The system is even or odd depending on the middle binary function pieces and theintegration leads to the creation to a system of piecewise-linear functions, which are either evenor odd [11, 13–15].

hink(x)=2p∫ x

0hrm(r)dr (1)

Graphs of functions of system {hrmk(x)} are shown in Figures 1(a) and (b).

Copyright q 2010 John Wiley & Sons, Ltd. Int. J. Commun. Syst. 2010; 23:751–762DOI: 10.1002/dac

PIECEWISE-QUADRATIC HARMUT BASIS FUNCTIONS 753

0 0.5 1-1

0 1-1

0 1-1

0 1-1

0 1-1

0 1-1

0 1-1

1

1

1

1

1

1

0 0.5 1-1

0-1

0 1-1

0 1-1

0 1-1

0-1

0-1

1hrm1

hrm21

1hrm3

1hrm4

1hrm5

1hrm6

1hrm7

1

1

hin1

hin2

1

hin31

hin41

hin51

hin61

1

hin71

1

0-1

0-1

0-1

0-1

0-1

0-1

0-1

1

1

1

1

1

1

1

hid1

1 x

x

x

x

x

x

hid2

hid3

x

hid4

hid5

hid6

hid7

(a) (b) (c)

Figure 1. Harmut’s basic functions: (a) piecewise-constant;(b) piecewise-linear; and (c) piecewise-quadratic.

In Equation (1), hink(x) represents the integrated Harmut’s functions of x . Here k=number ofHarmut’s functions. hrm(r) is Harmut’s function of r , both of these functions have been establishedby the Harmut matrix [10]. x is the argument value and p is the order of Haar’s functions such as(p=1,2, . . .).

In many practical applications connected with the restoration of functions after readout, contin-uous piecewise-linear bases do not provide enough opportunities. There are basically two reasons:

(1) Low convergence speed, caused by an estimation of the piecewise-linear interpolationerror [3–5], and this often leads to significant computational expenses during the calculationof factors;

ε� 18 max | f ′′(x)|h2 (2)

In Equation (2), ε represents the interpolation error and f ′′(x) is the derivative of f (x)in second order and h is the length of a piece or step of interpolation.

(2) Lack of smoothness (the first derivative of the basis function is discontinuous)—the conceptof curvature is absent causing fundamental restrictions. For example, it is possible to missthe maxima and minima of functions.



A piecewise-quadratic basis can be constructed by means of a double integration of the piecewise-constant orthogonal Harmut functions:

hidk(x)=2p∫ x

0

∫ u

0hrmk(r)dr du=2p

∫ x

0hin(u)du (3)

In Equation (3), hidk(x) is the Harmut’s functions that is integrated twice for piecewise-constant,hrmk(r) is the Harmut’s function of r , hin(u) is the integrated Harmut’s function of u, where r andu are the intermediate variables. x , u are the argument value and k, p have been already describedin Equation (1).

With the purpose of including this basis in the space, we shall exclude linear components offunctions where they are available and we shall designate the resultant system as {hidk(x)}. Itconsists of even and odd functions divided in dynamically rational pieces and these functions go to‘zero’ at the ends of pieces. Piecewise-quadratic Harmut functions {hidk(x)} are shown graphicallyin Figure 1(c).

The wide popularity of spline-approximating methods is explained by the fact that they providea universal instrument for the simulation of functions, and compared with other mathematicalmethods, the information contained in them and the computational expenses ensure that the calcu-lations can be done with great accuracy [16–18].

On an interval [a,b], we shall consider a grid �:

� :a= x0<x1< · · ·<xn =b

The polynomial spline of arbitrary degree m of an imperfection d (d is an integer, 1�d�m) alongthe node0s on a grid � is defined as a function Sm,d(x):

(1) Sm,d(x) =∑ms=0 di,s(x−xi )s

x ∈[xi , xi+1], i=0,1, . . . ,n−1

(2) Sm,d(x)∈Cm−d [a,b]The derivative of a spline about (m−d+1) can diverge at some point on [a,b]. Where m is an

arbitrary degree, an imperfection d (d-integer, 1�d�m) from nodes on a grid � has been definedas the function Sm,d (x); C is the accessory polynomial spline space property that is providedby mathematicians’ results (Figure 2).

Any spline of sufficient smoothness can be represented in terms of basic splines. In particular,for decomposition at d=1, the so-called ‘normalized’ basic splines of degree m (B-splines) areused. They are local (finite), piecewise-polynomial functions.

Their Fourier transform as a finite function is determined by the formula:

Fm(�)=(Sin(�/2)

�/2

)m+1

Combining the theory of basic splines with Harmut’s spectral methods leads to a strategyfor developing new algorithms for the calculation of the factors in piecewise-quadratic Harmutbases [16–18].



2.5-1.5-2-2.5 -1 -0.5 0 0.5 1 1.5 2 3 3.5

0.75

1.0

Bi-1(x) Bi(x) Bi+1(x) Bi+2(x)

x

-1.5-2.5 -0.5

x

i

0.75

1.0

0.5 1.5 2.5 3.5

Bi+1(x) Bi+2(x)Bi (x)Bi-1(x)

-2 -1 1 2 3

x

0

Bi-1(x) Bi(x) Bi+1(x) Bi+2(x)1.0

-2.0

Figure 2. Parabolic B-splines and their derivative of the first and second order.



3. ALGORITHM FOR THE CALCULATION OF FACTORSIN PIECEWISE-QUADRATIC HARMUT BASES

One of the basic features of orthogonal bases is the existence of fast algorithms for the definition ofspectral factors. Fast algorithms allow the number of arithmetic operations as well as the volume ofrequired memory to be reduced. As a result, the speed is increased to the extent that the orthogonalbases can be used for processing digital signals.

We write down the direct formula and return later to the problem of fast spectral transformationfor the readout sequence of a signal {xi} for any valid orthogonal piecewise-constant basis:

Ck = 1

2pn−1∑i=0

x(i) ·�(k, i) (4)

Xi =n−1∑k=0

Ck ·�(k, i) (5)

In Equations (4) and (5),Ck is the kth number of spectral coefficients, p is described in Equation (1),which is the order of Haar’s function, n is the number of samples, x(i) is the entrance signal,�(k, i)is any orthogonal piecewise-constant basis based on Haar’s or Harmut’s basis function. Here k isthe number of coefficients and i is the index number corresponding to a sequence element in avalid readout and Xi is the sample of signal x(i).

The analysis of the computational methods used to calculate the factors in various basis setshas shown that fast algorithms exist only for piecewise-constant and piecewise-linear bases. Factorcalculation algorithms in piecewise-quadratic bases have not been developed.

We investigate how the fast transformation algorithms in bases of orthogonal piecewise-constantfunctions can be adapted to the calculation of the factors in piecewise-linear bases. Formulae areknown for Fourier–Haar.

Fourier–Harmut bases uses integrals of the following kind:

C0 =∫ 1

0x(r)dr ∼=

n−1∑i=0

∫h pj

x(r)dr

Ck =∫ 1

0x(r) ·hark(r)dr

=n−1∑i=0

har(i)∫h pj

x(r)dr , i=1,2, . . . ,n, j =0,1, . . . ,2p−1

(6)

In Equation (6), C0 represents the spectral coefficients of first order, x(r) is the transformablesignal that belongs to a metric space L2, h pj is any binary interval defined by the followingequation:

|hpj|=|h+pj|+|h−

pj|=21−p

p, n, i , and Ck have been already described in Equations (1), (4), and (5). hark (r) is Haar’sfunction concerning argument r and har (i) is Haar’s function concerning argument of i .

The algorithm for calculating factors (6) does not possess the property of a fast transformationand, besides, if it is necessary to take in the values of factors from localization bases, it is morestraightforward to apply the final difference operations directly.



For example, factors in the Shauder basis are calculated in the transformation basis

� fi =n−1∑k=0

Ck ·Shdk(xi+1)−n−1∑k=0

Ck ·Shdk(x, i)

=n−1∑k=0

Ck

(∫hkhark(r)dr−

∫hkhark(r)dr

)= 1

2pn−1∑k=0

Ck hark(xi ) (7)

An analysis of Harmut’s matrix,

[Hrm]=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 1 1 1 1 1 1 1

1 1 1 1 −1 −1 −1 −1

1 1 −1 −1 −1 −1 1 1

1 1 −1 −1 1 1 −1 −1

1 −1 −1 1 0 0 0 0

1 −1 1 −1 0 0 0 0

0 0 0 0 1 −1 −1 1

0 0 0 0 1 −1 1 −1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

leads to a new algorithm for fast transformations.In Equation (7), � fi is a finite difference where i represents the index number corresponding

to a sequence element in a valid readout such as � f = fi+1− fi , Ck , n and k have been alreadydescribed in Equations (3) and (4). Shdk(xi+1) and Shdk(x, i) are Shauder’s functions where k isthe number of Shauder’s functions, x is the argument value, and i is the number of arguments.hark(r) and hark(xi ) are Haar’s functions of arguments values r and xi . i represents the indexnumber of arguments.

Factors C0 and C1 are decomposed with the Harmut basis functions. The zeroth-order andfirst-order terms are defined, respectively, by the following formulae:

C0 =n−1∑i=0

� fi

C1 =n/2−1∑i=0

� fi − ∑i=n/2

� fi

For the second-order basis function factors of the fast Harmut transformation, C2 and C3 arecalculated by grouping the sums of differences

C2 =(n/4−1∑j=0

� fi −n/2−1∑j=n/4

� f j

)−(3n/4−1∑j=n/2

� fi −n−1∑

j=3n/4� f j

)

C3 =(n/4−1∑j=0

� f j −n/2−1∑j=n/4

� f j

)−(3n/4−1∑j=n/2

� f j −n−1∑

j=3n/4� f j

)



Other factors for P�2, k�4 are calculated as the sum of the difference of the following iteration:

Ck =(∑

{ j}� f j− ∑

{h/2p+ j}� f j

)+( ∑

{n/2p−1+ j}� f j− ∑

{3n/2p+ j}� f j

)

Ck =(∑

{ j}� f j− ∑

{n/2p+ j}� f j

)+( ∑

{n/2p−1+ j}� f j− ∑

{3n/2p+ j}� f j

)

Finally, the last factors are calculated with equal number factors:

Cn−2 = (� fn−1+� fn−3)−(� fn−2−� fn−1)

Cn−1 = (� fn−4−� fn−3)−(� fn−2−� fn−1)

Let us consider approximating the number of Harmut piecewise-quadratic functions:

f (x)∼=n−1∑k=0

Ckhidk(x) (8)

In Equation (8), f (x) is the entrance function, hidk(x) is the double integration of Harmut’sfunction where k is the number of integration, Ck, n, and k have been already described inEquations (3)–(5).

The limitation of this series is shown by the absence of a fast algorithm for the calculation ofthe factors. This limitation can be overcome by the application of a parabolic spline. If we takethe second derivative of the parabolic spline interpolation on the interval [0,1] of the functionf (x), it will represent a piecewise-constant function with step changes in units of a spline, leadingto piecewise-constant orthogonal basis functions. For example, we write down the decompositionderivative of a spline in Harmut’s series:

S2′′(x)∼=

n−1∑k=0

Ck hrmk(x)

where hrmk(x) is Harmut’s function.According to theorems of limited convergence and integration of closed systems, by integrating

both parts, we get

S2′(x) = 2p

∫ x

0S′′2 (u)du=

n−1∑k=0

Ck hink(x)+S2′(0)

S2(x) =∫ x

0S2

′(u)du+S2(0)=n−1∑k=0

Ck hidk(x)+S2′(0)+S2(0)

Thus, it follows that the decomposition factors in a series of orthogonal Harmut functionstake the second derivative of the parabolic spline interpolating function in binary-rational unitsand are decomposition factors for the first derivative of a spline of hin-functions and a spline ofhid-functions. The factor at a linear part of the decomposition is defined as the value of the firstderivative of the spline S2(x) at the point x=0, and the constant component is defined as the valueof the spline at that point.



4. EXPERIMENTAL RESULTS

A series of numerical experiments were carried out to investigate piecewise-quadratic bases(Figure 3). The proposed algorithm was used to calculate the coefficients in Haar and Harmutpiecewise-quadratic bases as reported in Table I. The factor of compression Kc is defined by theformula Kc=N/(N−N1), where N is the number of readouts of the function, N1 is the numberof zero factors calculated with the algorithm.

In Table II, results of modeling the function y=arcth(x) over an interval [0,1] are shown forN=128 readouts in a Harmut piecewise-quadratic basis. The numerical experiments allow us toconclude that the number of zero coefficients calculated in bench tests on the Haar and Harmutpiecewise-quadratic bases range: from 5 to 17% in digital signal processing applications, from 5 to

X1

X2

X3

X4

X5

X6

X7

X8

X9

X10

X11

X12

X13

X14

X15

X0 C0

C1

C8

C9

C10

C11

C12

C13

C14

C15

C4

C5

C6

C7

C2

C3

Figure 3. The graph of direct fast transformation of Harmut at N=16.

Table I. Experimental result of fast transformations Haar and Harmut in piecewise-quadratic bases.

Haars fast transformation Harmuts fast transformation

No. Function Kc Zero coefficients (%) Kc Zero coefficients (%)

1 y=ex 2.06 51.5 3.76 73.42 Y =e−2x cos(4�x) 1.09 10.9 2.28 56.23 Y =arcth(x) 2.5 60.1 3.46 71.04 ABT 1.05 5.5 1.21 17.25 AME 1.07 6.3 1.29 24.2

ABT—array received as a result of bench tests.AME—array received on magnetic exploration.



Table II. Executed FT Harmut with function y=arcth(x) and N =128, arrayfactors in Harmut’s piecewise-quadratic basis.

i Ci i Ci i Ci i Ci

1 0.000032 33 0.000000 65 −0.005860 97 0.0000022 0.000000 34 0.000008 66 −0.005860 98 −0.0000003 −0.000012 35 −0.000001 67 −0.000000 99 0.0000014 0.000052 36 0.000011 68 −0.000000 100 −0.0000015 −0.000004 37 −0.000002 69 0.000000 101 0.0000006 0.000000 38 0.000014 70 0.000000 102 0.0000007 −0.000000 39 −0.000003 71 0.000000 103 0.0000008 −0.000000 40 0.000025 72 −0.000000 104 −0.0000029 0.000642 41 −0.000011 73 0.000000 105 0.00000210 0.000508 42 0.000053 74 −0.000000 106 −0.00000211 −0.000278 43 −0.000048 75 0.000001 107 −0.00000112 −0.000200 44 0.000158 76 −0.000001 108 −0.00000313 −0.000550 45 −0.000472 77 0.000000 109 −0.00000114 −0.000424 46 0.000934 78 0.000000 110 −0.00000315 0.000258 47 0.034729 79 0.000002 111 0.00000116 0.000188 48 −0.026103 80 −0.000000 112 −0.00000317 −0.000013 49 0.000000 81 0.000000 113 0.00000018 0.000029 50 0.000000 82 −0.000000 114 −0.00000819 −0.024045 51 −0.000001 83 0.000000 115 0.00000320 0.024275 52 −0.000001 84 −0.000002 116 −0.00001521 0.000001 53 −0.000000 85 0.000001 117 0.00000322 −0.000001 54 0.000000 86 0.000001 118 −0.00002923 0.109830 55 −0.000001 87 0.000000 119 0.00001324 −0.010997 56 −0.000001 88 −0.000000 120 −0.00005925 0.000003 57 0.000001 89 −0.000001 121 0.00004426 −0.000006 58 −0.000003 90 0.000001 122 −0.00016427 0.021101 59 0.000006 91 0.000001 123 0.00025728 −0.021163 60 −0.000012 92 0.000001 124 −0.00066729 0.000001 61 0.000090 93 0.000000 125 0.00349030 −0.000001 62 −0.000144 94 0.000000 126 −0.00601231 −0.010343 63 −0.778097 95 −0.000002 127 −0.09596832 −0.010349 64 0.775555 96 −0.000000 128 0.819582

25% when processing geophysical signals obtained in an exploration of magnetic field, and from 10to 70% while processing elementary functions (or functions consisting of their combinations). Thecoefficients are calculated with an accuracy of 10−4−10−6. It is established that the decomposition(8) enables reception at high speeds in the Haar basis and leads to a large factor of compression inthe Harmut basis. Also, our study shows that when the number N of readouts increases, the valueof the factors decreases according to an exponential law.

5. CONCLUSIONS

Our research into methods for approximating functional reveals some general limitations: weakconvergence, discontinuity, a rather low accuracy of approximation, the large volume of memory



required to store factors. In order to overcome these limitations, the need to use piecewise-quadratic bases was shown. Advantages of piecewise-quadratic bases are greater accuracy andsmoother approximation compared with piecewise-constant and piecewise-linear bases. The limi-tation of piecewise-quadratic basis functions lies in the absence of fast algorithms for calculatingtheir coefficients. In order to overcome this limitation, we have proposed a method for calcu-lating the coefficients in the Harmut piecewise-quadratic bases. The method takes advantageof the well-behaved differential properties of basic splines. It is hardware-focused and allowsthe existing algorithms for fast transformations in bases of orthogonal piecewise-constant func-tions to be used in the calculation of factors for both piecewise-linear and piecewise-quadraticbases.

ACKNOWLEDGEMENTS

This research was supported by Dongseo Frontier project, from the Dongseo University, Busan, SouthKorea.

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AUTHORS’ BIOGRAPHIES

Dhananjay Singh is a PhD student in the Department of Ubiquitous Information Tech-nology, Graduate School of Design & IT at Dongseo University, Busan, South Korea.He received the BTech degree in Computer Science and Engineering from V.B.S.Purvanchal University, Jaunpur, in 2003 and the MTech degree in IT with specializationin Wireless Communication and Computing from the Indian Institute of InformationTechnology, Allahabad, in 2006, India. His current research interests include IP-Basedwireless sensor networks, digital signal processing and global healthcare monitoringapplications. He has published more than 25 papers and 5 patents. He has served as areviewer for five international conferences and journals.

Hakimjon Zaynidinov was a professor of Division of Design and IT at DongseoUniversity, Busan, South Korea from September 2007 to March 2009. Before joiningDSU, he was a professor at the Tashkent University of Information Technology, and inthe Department of Computer Science, computers system of signal processing at TashkentState Technical University, Tashkent, Uzbekistan. He received the Bachelor’s degree inComputer Science from Tashkent Polytechnic University, Uzbekistan, in 1984 and thePhD degree from St. Petersburg State Electro Technical University, Russia, in 1993.His current research interests include digital signal processing, wireless sensor networksand computer graphics. He has published more than 100 papers, 5 text books and 10patents. He has served as a chair person and member for many scientific and academicorganizations. He is a senior member of the IEEE Computer Society, Russia.

Hoon-Jae Lee is an associate professor of the School of Computer and InformationEngineering at Dongseo University, Busan, South Korea. Before joining DSU, he wasa Research Associate at the Agency for Defense Development (ADD) in Korea. Hereceived the BS, MS, and PhD degrees in Electronic Engineering from KyungpookNational University, Daegu, South Korea, in 1985, 1987, and 1998, respectively. Hiscurrent research interests include developing secure communication system, side-channelattack, and USN/RFID security. He has published more than 200 papers and 50 patents.He has served as a reviewer for many international conferences and journals.


Piecewise-quadratic Harmut basis functions and their application to problems in digital signal...

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