Matrix decomposition and butterfly diagrams for mutual relations between Hadamard-Haar and arithmetic spectra

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 53, NO. 5, MAY 2006 1119

Matrix Decomposition and Butterfly Diagrams forMutual Relations Between Hadamard–Haar and

Arithmetic SpectraBogdan J. Falkowski, Senior Member, IEEE, and Shixing Yan, Student Member, IEEE

Abstract—The mutual relationships between Hadamard–Haarand Arithmetic transforms and their corresponding spectra in theform of matrix decomposition as layered vertical and horizontalKronecker matrices are discussed here together with their proofs,fast algorithms, and computational costs. The new relationsapply to an arbitrary dimension of the transform matrices andallow performing direct conversions between Arithmetic andHadamard–Haar functions and their corresponding spectra. Inaddition, analysis of butterfly diagrams for these new relations isalso introduced and it is shown that they are more efficient thanthe matrix decomposition method.

Index Terms—Arithmetic transform, discrete transforms, fasttransforms, Hadamard–Haar transform, spectral techniques.

I. INTRODUCTION

DURING THE LAST decade there has been increasinginterest in applications of different transforms used in dig-

ital signal processing for spectral analysis, synthesis and testingof discrete functions [1]–[16]. The recent interest in applicationof various transforms in electronic design automation (EDA) isalso caused by development of efficient methods of their calcu-lation [1]–[19]. Discrete orthogonal Walsh-like transforms suchas Walsh transform, Haar transform, Hadamard–Haar trans-form, and others like the discrete transform, slant transformand Fourier transform, have been used in data transmission,multiplexing, filtering, speech processing, image enhancement,pattern recognition, control theory, communication systems,statistical analysis, solving differential equations, and theanalysis, synthesis, classification and testing of logical circuits[1]–[16], [20]–[23].

Discrete Haar transform is the simplest example of waveletexpansion and attracts much attention in engineering practicefor its peculiar properties [4], [10]–[12], [15]–[19]. Due toits low computing requirements, the Haar transform has beenmainly used for pattern recognition and image processing [16].For real time applications, hardware-based fast Haar chipshave been developed [16]. A brief discussion of various otherapplications, where the use of Haar and Walsh functions offerssome advantages compared to the Fourier transform, is givenin [16]. The authors of [22], presented a set of EDA tools toperform a switch-level fault detection and diagnosis of physical

Manuscript received December 6, 2004; revised May 6, 2005, and August 14,2005. This paper was recommended by Associate Editor M. Chakraborty.

The authors are with the School of Electrical and Electronic Engi-neering, Nanyang Technological University, Singapore 639798 (e-mail:[email protected]; [email protected]).

Digital Object Identifier 10.1109/TCSI.2006.869899

faults for practical MOS digital circuits using a reduced Haarspectrum analysis. The advantage of using Haar functionsinstead of Walsh functions in EDA system based on spectralmethods for some classes of Boolean functions was shown in[4], [16]. In [13], a method for probabilistically determining theequivalence of two switching functions through Haar spectralcoefficients has been developed. The method is reported asan alternative for equivalence checking of function that aredifficult to represent completely and is based on binary decisiondiagrams and Haar spectral diagrams [19].

The Arithmetic transform has the sets of basis functions thatcan be obtained from its inverse transform while the details of ap-plication ofArithmetic transformwere shown in [1], [6]. Detailedrelations between Arithmetic and Haar functions were given in[24]. Heidtman proposed an approach to derive the signatures forstuck-at faults in irredundant combinational networks by usingArithmetic coefficients of the Arithmetic polynomial expansion[2]. Rahardja and Falkowski [25] have shown that Arithmeticpolynomial logic is advantageous over the standard zero polarityArithmetic transformusedbyHeidtman in testing. Inparticular, itwas shown in [25] that the number of spectral tests is significantlyreduced for both stuck-at and bridging faults for the majority ofthe classes of digital circuits.

The Hadamard–Haar transform is based on a hybrid versionof the Haar and Walsh transforms [20]. This transform is de-rived from different linear combinations of the basis Haar func-tions with an appropriate scaling factor. Such a combination ofbasis functions have been found advantageous for feature selec-tion and pattern recognition. Another version of this transformnamed ‘Rationalized Hadamard–Haar transform’ has also beenintroduced [21]. Walsh transforms in Gray code ordering was in-troduced in [26] and different and efficient ways of generatingsuch Walsh transforms have been presented.

While majority of research has been done using Walshtransform [3], [4], [11], [15], [16], dependent on a case sometransforms are better suited than others. Many authors haveconsidered the mutual relationships between Haar and Walshfunctions. The relations between normalized and nonnor-malized Haar and Arithmetic transforms were given withoutproofs in [24]. The equations for Haar transform and cor-responding butterfly diagrams were given in [19] while therelations between Hadamard and natural order Haar transformwere presented in [14], [15]. Frequently it is advantageousto apply more than one transform in a given task based onthe local properties of a data function [3], [4], [8]–[13], [15],[16], [19]. For example, if the logic function has many zeros

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1120 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 53, NO. 5, MAY 2006

in its truth vector, it is better to apply local transform that hasnonzero entries in its own transformation matrix that almostoverlap with nonzero elements in the truth vector. In the lattercase, it is also of interest to investigate mutual relations be-tween various local discrete transforms such as, for example,Hadamard–Haar and Arithmetic transform. Therefore, it is notonly interesting theoretically, but also practically to state theirmutual relations. In this work, we extended our previous resultsfrom [27] by showing mutual relations between Arithmeticand Hadamard–Haar transforms, their proofs, fast algorithmsand computational costs for arbitrary transform size in theform of matrix decomposition and as layered vertical andhorizontal Kronecker product based matrices for an arbitrarytransform matrix order size. From the detailed references forall related transforms as given here, it is very clear that onlyfor the first time such mutual relations between Arithmetic andHadamard–Haar transforms are given in our article. In addition,due to the fact that the Hadamard–Haar transform covers bothWalsh and Haar transforms, our results and computational costscan be also used as proofs for relations between Walsh, Haarand Arithmetic transforms.

II. BASIC DEFINITIONS

In all equations inside this paper, the symbols and rep-

resent the Kronecker product [3], [4], [10], [11], [16] of andtwo matrices, respectively.

Definition 1: The Walsh transform matrix of order is de-fined as

(1)

Also (2)

Definition 2: The normalized Haar transform matrix of orderis defined as

(3)

where is identify matrix.The nonnormalized Haar transform is obtained from re-

placing the nonzero entries of normalized Haar matrix withtheir arithmetic signs. The nonnormalized Haar transformpreserves all the properties of the normalized Haar transform.Its matrix of order is defined as

(4)

Definition 3: The matrix of order for the Arithmetic trans-form is defined as

(5)

Also (6)

Definition 4: The th-order normalized Hadamard–Haartransform and its inverse are defined as [20]

and (7)

where is the Walsh transform matrix,is the normalized Haar transform matrix. The hy-brid transform of the Haar and Walsh transform is also orthog-onal.

Similar to Haar transform, the nonnormalizedHadamard–Haar transform can be obtained from re-placing the nonzero entries of normalized Hadamard–Haarmatrix with their arithmetic signs. Its matrix is defined as [21]

and (8)

III. RELATIONSHIP BETWEEN HADAMARD-HAAR AND

ARITHMETIC TRANSFORM

The following relations will be given only forHadamard–Haar and Arithmetic transforms. SinceHadamard–Haar transform is the hybrid of Walsh andHaar transform, the relations between Hadamard–Haar andArithmetic transforms can be derived from their definitions.

Property 1: Let and represent the Arith-metic and normalized Hadamard–Haar spectra. The mutual rela-tions between th-order normalized Hadamard–Haar and Arith-metic transforms for the general case of arbitrary are as fol-lows:

(9)

(10)

where (11) and (12), shown at the bottom of the next page aretrue, where , and .

In the above equations, and are matrices,the vertical dotted lines denote the layered vertical Kroneckermatrices, and the horizontal dashed lines denote the layered hor-izontal Kronecker matrices, respectively. A layered horizontalKronecker matrix is defined as the horizontal sum of Kroneckermatrices while a layered Kronecker vertical matrix is definedas the vertical sum of Kronecker matrices [24]. When the Kro-necker direct product of matrices is carried out for the above

equations for then the term disappears from the

above equations. The meaning of the symbols and the restric-

tion on the term is the same as above also for the following

equations.Proof: Let the vector in function domain be represented

by , we have: and. Hence

From (7), we can get the Arithmetic spectrum from normalizedHadamard–Haar spectrum as follows:

FALKOWSKI AND YAN: MATRIX DECOMPOSITION AND BUTTERFLY DIAGRAMS 1121

The inverse Walsh transform is known as:, and the inverse normalized Haar transform is

known as: where the superscriptis the matrix transpose operator. Using (2), (6), and the

inverse transforms, we have

According to the properties of Kronecker product, we have

(13)

...

...

...

...

...

...

...

(11)

(12)


where and

. Comparing (9), (11), and (13), we need to prove

for the proof of (9) whereand

............

............

The argument will be proved byinduction.

Basis: From (3), we have...

Hence, , then

Induction Step: Assume . From (6),we have

...

...

...

...

...

...

...

...

...

...

...

...

...

...

As is proved by induction, theproof of (9) is completed.

Similarly, we can prove (10). From the inverse Arithmetic

transform known as: where

, we can get the normalized Hadamard–Haar spectrum

from Arithmetic spectrum as follows:

(14)

where and

. Comparing (10), (12), and (14), we need to prove

for the proof of (10) whereand

The argument will be proved byinduction.

Basis: From (3), we have: , thus


Induction Step: Assume . We have

This completes the proof of (10).Example 1: For and , the relationships between

the normalized Hadamard–Haar and Arithmetic transforms canbe described by using Property 1. The corresponding and

are

...

and

Hence, the transformations between the normalizedHadamard–Haar spectrum where andArithmetic spectrum can be described by usingand as follows:

and

The relationships between nonnormalized Hadamard–Haarand Arithmetic transforms are shown in Property 2. The rela-tionships between nonnormalized Hadamard–Haar and Arith-metic transforms can also be derived similarly as the nonnormal-ized Hadamard–Haar transform is obtained from the normalizedversion by using signs only.

Property 2: Let represent the nonnormalizedHadamard–Haar spectra. The mutual relations betweenth-order nonnormalized Hadamard–Haar and Arithmetic

transforms for the general case of arbitrary are as follows:

(15)

and (16)

where (17) and (18) at the bottom of the next page, hold, where, and .

Proof: Similar to the proof of Property 1.

Since Walsh and Haar transforms can be seen as the spe-cial cases (when and ) of Hadamard–Haartransform, the Walsh and Haar domains are subdomainsof Hadamard–Haar domain. Hence, the relationshipsshown in Property 1 and Property 2 can be applied forWalsh, Haar and Arithmetic transforms as well. Forexample, the relationships between Walsh and Arith-metic transforms can be derived from (9) as follows:

andwhere is

the corresponding Walsh spectrum. The relationships betweennormalized Haar and Arithmetic transforms can be also derivedfrom (9) as follows:

and .Example 2: For and , the relationships between

the nonnormalized Hadamard–Haar and Arithmetic transformscan be described by using Property 2 as follows:

and

where the corresponding and are

...

and

IV. FAST ALGORITHMS

The fast algorithms for different discrete transforms havebeen developed to reduce the computational costs, such as fastFourier transform, fast Walsh transform, fast Haar transform,fast Arithmetic transform, fast Hadamard–Haar transform [3],[4], [10]–[12], [14]–[16], [20], [21]. The matrix transformationmethods between Hadamard–Haar and Arithmetic domainshave been shown in Section III. The fast algorithms for thetransformations between Hadamard–Haar and Arithmetic do-mains will be shown in this section.

Property 3: For the relations between th-order normalizedHadamard–Haar and Arithmetic transforms, andin Property 1 can be expressed by the products of factorizedmatrices as follows:

(19)

and (20)

where and are defined in (21) and (22), respec-tively, ,

, and are the same as in the proof


of Property 1, [see (21) and (22), shown at the bottom of thepage].

Proof: For , we have

Since where

...

we need to prove by inductive method for

...

...

...

...

...

...

...

(17)

(18)

for

... for

(21)for

for(22)


the proof of (19) where .

Basis: .

Induction Step: Assume . From the struc-

ture of , it can be noticed that

Therefore

...

...

This completes the proof of (19).Similarly, for , we have

Then we need to prove by induction for

the proof of (20) where

Basis: .

Induction Step: Assume .

Since , we have

This completes the proof of (20).Property 4: For the relations between th-order nonnormal-

ized Hadamard–Haar and Arithmetic transforms, the andin Property 2 can be expressed by the products of factor-

ized matrices as follows:

(23)

and (24)

where and are defined in (25) and (26), respec-tively, shown at the bottom of the page.

Proof: Similar to the proof of Property 3.The fast algorithms shown in Property 3 and Property 4 can

also be used for Walsh or Haar transform when is set to forthe first transform or 0 for the second transform.

Example 3: We can use the fast algorithms to factorizein Example 2 as follows:

for

... for

(25)for

for(26)


Fig. 1. The fast flow diagrams for the transformation from nonnormalizedHadamard–Haar spectrum to Arithmetic spectrum using proposed methodwhere r = 1 and n = 3.

Since

, the butterfly diagram for the transformation betweenthe nonnormalized Hadamard–Haar and Arithmetic spectra canbe constructed based on the factorized matrices as shown inFig. 1, where corresponds to the rightmost stage,corresponds to the middle stage, and corresponds to theleftmost stage of the butterfly diagram.

The same transformation using conventional method is alsogiven here for the comparison. Conventionally, the transfor-mation from the nonnormalized Hadamard–Haar spectrumto Arithmetic spectrum is calculated by using fast inverseHadamard–Haar transform and fast Arithmetic transform. Weneed to transform the nonnormalized Hadamard–Haar spectrumto the functional domain using fast inverse Hadamard–Haartransform, then the corresponding Arithmetic spectrum can becalculated from the functional domain using fast Arithmetictransform. The corresponding fast butterfly diagram usingconventional methods is shown in Fig. 2, where the left threestages in the butterfly diagram correspond to the fast inverseHadamard–Haar transform which transforms the nonnormal-ized Hadamard–Haar spectrum to the functionaldomain, and the right three stages correspond to the fast Arith-metic spectrum which transforms the data in functional domainto the corresponding Arithmetic spectrum .

As shown in the butterfly diagrams in Figs. 1 and 2, the trans-formation is processed from the left side to the right side whenthe calculations in the same stage are parallel processed. Sincethe calculations in every stage require the output results of its leftstage as the inputs, it can be noted the number of the stages inthe butterfly diagram determines the processing time of the but-terfly diagram. We can notice that there are three stages of cal-culations in the butterfly diagrams shown in Fig. 1, while thereare six stages of calculations in Fig. 2. Therefore, the transfor-mation using proposed method is much efficient than the onethat is using conventional method. The more detailed analysisfor the comparison of the proposed and conventional methodsis shown in Section V.

V. ANALYSIS OF RUNNING TIME OF THE ALGORITHMS

The running time of the algorithms depends on the com-putational costs required by the calculations which includesadditions, subtractions, and multiplications. The running timeof the conventional methods and proposed methods for thetransformations between Hadamard–Haar and Arithmetic trans-forms when using fast algorithms will be compared in thissection.

It has been known that there are arithmetic operations(additions and subtractions) required for the computation of

-point Walsh transform. The computation of -point nor-malized Haar transform requires arithmetic operationsand multiplications, and the computation of -pointnonnormalized Haar transform does not require multiplica-tions. Since Hadamard–Haar transform includes Walsh andHaar parts, we can develop the computational costs for thecomputations of two versions of Hadamard–Haar transforms.

The number of arithmetic operations required by the calcu-lation of forward or inverse Hadamard–Haar transform can becalculated as follows: .The numbers of multiplications required by calculating for-ward and inverse normalized Hadamard–Haar transform are:

and ,respectively. There are multiplications required by the cal-culation of inverse nonnormalized Hadamard–Haar transform.

It is also known that there are arithmetic operationsrequired by the calculation of forward or inverse Arithmetictransform. So we can derive the computational costs for thetransformations between Hadamard–Haar and Arithmeticspectra using conventional methods based on the computa-tional costs of forward (inverse) Hadamard–Haar and inverse(forward) Arithmetic transforms. The computational costs ofnew proposed methods can be obtained from (19)–(26) and areshown in Table I for different cases. The numbers of arithmeticoperations when using proposed methods are always less thanthe numbers when using conventional methods. It should benoticed that there are additional multiplications while usingproposed methods than conventional methods in two cases forthe normalized version, but the numerical value of the multi-pliers in the additional multiplications are all powers-of-two,i.e., the multiplier is . The multiplications in the compu-tations when using conventional and proposed methods forthe nonnormalized version are also . Such multiplicationscan be easily implemented in hardware by bit-shift operations(plus invertion of sign digit), so this kind of multiplicationsare even more efficient than the arithmetic operations. Forthe convenience of comparing the computational costs whenusing conventional and proposed methods, we consider onlythe cases for the nonnormalized versions and combine thearithmetic operations and multiplications, i.e., the sum ofthe arithmetic operations and multiplications. The proposedmethods show computational advantages over the traditionalmethod due to less computational costs. For example, thecomputational costs for the transformation from nonnormalizedHadamard–Haar spectrum to Arithmetic spectrum using con-ventional and proposed methods are:and , respectively. Therefore, the


Fig. 2. The fast flow diagrams for the transformation from nonnormalized Hadamard–Haar spectrum to Arithmetic spectrum using conventional method wherer = 1 and n = 3.

TABLE ICOMPARISON OF COMPUTATIONAL COSTS USING CONVENTIONAL AND PROPOSED METHODS

Fig. 3. The reduction in running time when using proposed method for thetransformation from nonnormalized Hadamard–Haar spectrum to Arithmeticspectrum for some cases.

reduction in computational costs when using proposed methodis

From the reductions for several casesshown in Fig. 3, we can notice that most reductions in Fig. 3are larger than 20%. The reductions in computational costs,which can also be seen as the reductions in running time, when

Fig. 4. The reduction in running time when using proposed method for thetransformation from nonnormalized Hadamard–Haar spectrum to Arithmeticspectrum.

using proposed method for the transformations between non-normalized Hadamard–Haar spectrum and Arithmetic spectrumfor general cases (all ) where are shown in Fig. 4 andFig. 5, respectively.

We will now show the significant savings in the time com-plexity when using proposed methods. Generally, there are(or ) stages required in the fast diagrams for the compu-tations when using conventional methods, while there are only

(or ) stages required when using proposed methods.Hence, the proposed methods carry about 50% saving in timecomplexity compared with the conventional methods.


Fig. 5. The reduction in running time when using proposed method for thetransformation from Arithmetic spectrum to nonnormalized Hadamard–Haarspectrum.

The efficiency of proposed methods in the running time of thealgorithms has been shown, which also means the simplicity ofthe design and the reduction of hardware costs, especially whenall the operations are performed in sign complement arithmetic.

VI. CONCLUSION

In this article, the mutual conversions betweenHadamard–Haar and Arithmetic transforms for an arbitrarytransform size using matrix decomposition methods andbutterfly diagrams as well as their computational costs havebeen developed. The relations between both transforms areintroduced through recursive equations in the form of layeredvertical and horizontal Kronecker products. The presentedrelations allow transferring known results of spectral logicdesign in Arithmetic domain to Hadamard–Haar domainand vice verse and compare efficiency of both approachesin different applications for large discrete functions. Finallyit should be also noticed that presented derivations based onlayered matrices and corresponding butterfly diagrams can beefficiently implemented in the form of operations on spectraldecision diagrams [11], [13], [15], [17]–[19], using softwareor as hardware operations using look-up table cascades ina manner similar to Walsh transform [28]. Hence, all thepresented results are not only very interesting theoretically butalso are very important for the practical applications of Haar,Walsh, Hadamard–Haar and Arithmetic transforms in manyresearch areas.

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Bogdan J. Falkowski (S’88–M’90–SM’94) receivedthe M.S.E.E. degree and the Ph.D. degree in electricaland computer engineering from the Warsaw Univer-sity of Technology, Warsaw, Poland, and the PortlandState University, Portland, OR, in 1978 and 1991, re-spectively.

His industrial experience includes research and de-velopment positions at several companies. He thenjoined the Electrical and Computer Engineering De-partment, Portland State University. Since 1992, hehas been with the School of Electrical and Electronic

Engineering, Nanyang Technological University, Singapore, where he is cur-rently an Associate Professor. His research interests include very large-scale in-tegrated systems and design, switching circuits, testing, digital signal and imageprocessing, and design of algorithms. He specializes in the design of digital cir-cuits with the use of spectral methods and has published three book chaptersand over 270 refereed journal and conference papers in this area.

Dr. Falkowski received the Hartree Premium Award that recognizes the out-standing paper published in 2001 in IEE Proceedings on Computers and Dig-ital Techniques in 2002. He was a Guest Editor of the VLSI Design Journal, andTechnical Chair for IEEE International Conference on Information, Communi-cation and Signal Processing, Singapore, in 1999. He is listed in Marquis Whois Who in Science and Engineering as well as in the International BiographicalCentre, Cambridge, UK. He was elected an Executive Member of IEEE Com-puter Society Technical Committee in Multiple-Valued Logic for 2004–2006.

Shixing Yan (S’03) received the B.Eng. degree in in-formation engineering from Xi’an Jiaotong Univer-sity, Xi’an, China, in 2001. He is currently workingtoward the Ph.D. degree at Nanyang TechnologicalUniversity, Singapore.

From 2001 to 2002, he was an R&D Engineerin the Department of Information Technology, In-dustrial and Commercial Bank of China, Shenzhen,China. His research interests include logic synthesis,digital signal and image processing, and spectralmethods in VLSI design.

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Matrix decomposition and butterfly diagrams for mutual relations between Hadamard-Haar and arithmetic spectra