Sequency-Ordered Complex Hadamard Transform: Properties, Computational Complexity and Applications

3562 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 8, AUGUST 2008

Sequency-Ordered Complex Hadamard Transform:Properties, Computational Complexity and

ApplicationsAye Aung, Boon Poh Ng, and Susanto Rahardja, Senior Member, IEEE

Abstract—In this paper, the generation of sequency-orderedcomplex Hadamard transform (SCHT) based on the complexRademacher matrices is presented. The exponential form ofSCHT is also derived, and the proof for the unitary property ofSCHT is given. Using the sparse matrix factorization, the fast andefficient algorithm to compute the SCHT transform is developed,and its computation load is described. Certain properties of theSCHT matrices are derived and analyzed with the discussion ofSCHT applications in spectrum analysis and image watermarking.Relations of SCHT with fast Fourier transform (FFT) and unifiedcomplex Hadamard transform (UCHT) are discussed.

Index Terms—Complex Hadamard transforms, discrete or-thogonal transforms, fast algorithms, sequency-ordered complexHadamard transform (SCHT).

I. INTRODUCTION

W ITH the rapid development of digital device tech-nology, discrete orthogonal transforms have been used

extensively in the fields of digital signal processing, communi-cations and logic design. The existence of fast algorithms is animportant issue due to high demand for the efficient computa-tion in the real-time digital signal processing system. One ofthe important orthogonal transforms named Walsh–Hadamardtransform (WHT) had been found in many areas of digitalsignal processing applications such as digital filtering, facerecognition, image watermarking [1]–[3], and communications,for example, multicarrier CDMA (MC-CDMA) systems [4]and multiband OFDM (MB-OFDM) ultrawideband system[5]. Other commonly used orthogonal transforms found in theliterature are the Haar transform (HT), the Karhunen–Loevetransform (KLT), and the discrete cosine transform (DCT) [6],[7]. HT is useful in signal and image processing applications inwhich the real-time implementation is considered as a crucialfactor [8], [9]. It has the lowest computational cost among theexisting discrete orthogonal transforms. The computationalcomplexity can be reduced to to compute thetransform of -point data samples [10]. In [11], the complex

Manuscript received September 26, 2007; revised March 2, 2008. The as-sociate editor coordinating the review of this manuscript and approving it forpublication was Prof. William A. Sethares.

A. Aung and B. P. Ng are with the School of Electrical and ElectronicEngineering, Nanyang Technological University, Singapore 639798 (e-mail:[email protected]; [email protected]).

S. Rahardja is with the Agency for Science, Technology, and Research(A*STAR), Institute of Infocomm Research, Singapore 119613 (e-mail:[email protected]; [email protected]).

Digital Object Identifier 10.1109/TSP.2008.923195

version of Haar transform called complex Haar transform(CHT) is developed, and several of its properties are defined.It has been found that CHT is more suitable for the problemswith complex-valued functions. Even though CHT involves thecomplex arithmetic compared to the real arithmetic for HT,two sequences can be processed simultaneously using CHT.The KLT is known to be a statistically optimal transform atthe price of costly computational requirement. Related appli-cations of such transform include statistical pattern matching,source coding, and certain lossy data compression [12], [13].The DCT is often used in data and image compression due toits superior energy compaction property [6], [13]–[15]. Thecurrent standards for compression of still images, JPEG, andmoving images, MPEG, use the DCT to compress images.

Besides, there exists another orthogonal transform called“complex BIFORE transform” [16] in the literature. The realBIFORE or Hadamard transform is introduced in [17] andseveral of its properties had been developed. It is based onthe Walsh functions, which form an ordered set of rectangularwaveforms taking only two amplitude values . It has beenfound in many useful applications such as signal representationand classification, spectral analysis, and synthesis of voicesignals, etc. [17]. But due to its limitation to the real-valuedinput data sequences, complex BIFORE transform, which canprocess the complex-valued input functions, has been devel-oped in [16]. It is based on four complex values .Basically, the complex BIFORE transform is defined basedon the recursive formula which assigns one class of complexHadamard matrices involving the diagonalization of higherorder matrices and multiple Kronecker products [18].

In [7] and [18], Rahardja and Falkowski proposed a setof complex Hadamard transforms named unified complexHadamard transforms (UCHTs), which are constructed by thedirect matrix operator and recursively generated to the higherdimensional matrices by a single Kronecker product. Amongthe family of UCHT matrices, half of them possess a uniqueproperty called half spectrum property. Therefore, they are ef-ficient to implement the problems with complex integer-valuedfunctions as they require half of the spectral coefficients forthe data analysis. Besides, it has in-place architecture for thefast algorithms, which is greatly helpful to reduce the memoryrequirements as well as the hardware size. UCHTs have beenemployed in many applications of digital signal processing,multiple-valued logic design, and communications. For ex-ample, UCHTs are used as the spectral techniques in logicdesign [19], [20]. In addition, UCHTs are also utilized in image

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AUNG et al.: SEQUENCY-ORDERED COMPLEX HADAMARD TRANSFORM 3563

watermarking [20], [21] and as complex spreading sequences indirect-sequence (DS) CDMA communication systems [22] anddownlink multicarrier DS-CDMA systems [23]. However, nowork has been presented so far mainly focusing on the orderingof complex Hadamard matrices and no applications are directlyrelated to the order of complex Hadamard matrices. Hence, it isan interest to investigate Complex Hadamard Transform withcertain order for some particular applications in the research.

This paper aims to introduce sequency-ordered complexHadamard transform (SCHT) and some of its properties andapplications. Generally, the SCHT matrices are discrete trans-forms which are orthogonal in the complex domain and strictlyconfined to four complex values in the unit circle of acomplex plane. Sequency is analogous to frequency in discreteFourier transform (DFT). In contrast to UCHT, the row vectorsof an SCHT matrix are arranged in ascending order of sequen-cies (zero crossings per unit time interval in the complex plane).Especially, sequency order is favored for communications, andsignal processing such as the spectral analysis and filtering [6].Recently, SCHT has been employed in a DS-CDMA system[24] as the complex spreading sequences which are derivedfrom the row vectors of an SCHT matrix. Due to their lowcross-correlation values, it has been found that the bit-error rate(BER) performance of the SCHT sequence is better than that ofthe existing well-known sequences such as UCHT, WHT, andGold sequences in the asynchronous CDMA system over themultipath fading channel. In this paper, more general propertiesof SCHT are investigated, which are important for its applica-tions in communications as well as digital signal processingsystem. Certain potential applications of SCHT especiallyin spectrum analysis and image watermarking are presentedtogether with some simulation results. The sparse matrix fac-torization of an SCHT matrix leads to the fast algorithm whichis very efficient in terms of computational complexity for theapplications of real-time signal processing system. For an

-point SCHT, the fast algorithm reduces the computationalcomplexity to instead of in directcomputation.

The organization of the paper is as follows. Section II de-scribes the Rademacher function and its extended form, whichis known as the complex Rademacher function. The generationof the SCHT matrices based on the complex Rademacher ma-trices is mentioned, and some of their properties are described inthat section. The exponential form of SCHT is also presented inSection II. Due to the sparseness of the SCHT transform matrix,the fast SCHT algorithm for the efficient computation is derived,and its computational complexity is examined in Section III.Some useful applications of SCHT such as spectrum estimationand image watermarking are suggested with the correspondingsimulation results in Section IV. The relationships of SCHTwith FFT and UCHT are introduced in Section V. Finally, thispaper is concluded in Section VI.

II. BASIC DEFINITIONS AND PROPERTIES OF SCHT

In this section, we will define sequency-ordered complexHadamard transform and some of its properties. Before goingdirectly to the generation of SCHT, we first introduce the

Rademacher functions as well as their extended complex-valued functions named complex Rademacher functions andtheir matrices.

A. Rademacher Functions

Rademacher functions are an important, incomplete butorthogonal function set which represents a series of rectangularpulses or square waves [6]. Although they form an incompleteseries having odd symmetry, it is possible to generate the otherfunction series which exhibit either odd or even symmetry.Therefore, a complete series can be derived from an incompleteset of functions. Rademacher functions have two arguments

and such that RAD has periods of rectangularwaves over a normalized time base . The amplitudesof the functions are and based on the values of . Theycan be derived from the sinusoidal functions which have theidentical zero crossings followed by hard limiting using thesignum function as follows: [6]

RAD (1)

where and . They are important prin-cipally because the other complete series, such as Walsh series,can be derived from them.

B. Complex Rademacher Functions and Matrices

In this subsection, we shall introduce a type of functionsknown as complex Rademacher functions which name followsthe conventional Rademacher functions. We extend the realRademacher functions and define the complex Rademacherfunctions (CRAD) over a normalized time base as

CRAD (2)

and

CRAD CRAD (3)

Half of the values of real Rademacher functions are shifted bythe phase of 90 in CRAD. This holds the orthogonality prop-erty of the Rademacher functions.

For non-negative integer , the complex Rademacher func-tions can be defined based on (1) as follows:

CRAD CRAD (4)

where . This means that CRAD is obtainedby compressing CRAD in the horizontal direction by afactor of .

The complex Rademacher matrices are the discrete versionsof complex Rademacher functions CRAD , and they can begenerated by sampling the complex Rademacher functions. Theth row of a complex Rademacher matrix of the size can

be defined as

CRAD (5)


where and . The rowvectors of the complex Rademacher matrices are orthogonal toeach other in the complex domain as

forfor

(6)

where is the complex conjugate ofand .

Like the real Rademacher functions, the complexRademacher functions are also an incomplete orthogonalset of functions in the complex domain. But they are useful toconstruct the complete set of functions, which exhibits botheven and odd symmetries.

C. SCHT Matrices

With the complex Rademacher matrices defined, the se-quency-ordered complex Hadamard matrices are gener-ated based on the products of the row vectors of complexRademacher matrices as follows:

(7)

where is the ( th, th) element of the complexRademacher matrix

(8)

and or . Let , for , be theth row vector of the complex Rademacher matrix, and also, let

be the operator for element by element vector multiplication.For example, can be expressed as

(9)

Since binary number of is and the index values 2, 1and 0 refer to ones found in the binary bit positions.

D. Properties of SCHT

In this subsection, we shall present the properties of SCHT.Property 1: Exponential Property of SCHT. Let be an

SCHT matrix of the size where . If is anelement of at row and column , where ,then each element of an SCHT matrix can be defined as

(10)

where and are the binary representations ofand respectively, i.e., and

. is a binary representationof the Gray code [6] of the bit reversal of represents thebinary bits of the bit reversal of , and .The example for is given in the Appendix.

Property 2: Unitary. Sequency-ordered complex Hadamardmatrix of order is a square matrix with the ele-

ments , and is said to be orthogonal in the com-plex domain as

forfor

(11)

where denotes the conjugate transpose of the matrix ,and and are the row and column indexes, respectively.

Proof: From Property 1, the elements of an SCHTmatrix can be expressed in exponential form as follows:

(12)

where and are rowand column indexes of expressed in binary notations. Thesum of the products of any two row vectors of can be givenas the binary summation as follows:

(13)

Substituting (12) into (13), we have

(14)

The sum of products can be written as the product of sums.Therefore

(15)

But if , then . Hence,

(16)

If we consider the case that is , then . But thevalues of and only can be zero or one. Then, it becomes

(17)


Fig. 1. Zero crossings of sequency in the unit circle of a complex plane.

If , whenever , there is at least one term in (16)giving the zero sum (i.e., ). It will lead to a zero product.Therefore, (11) is proved. The proof of the unitary property iscompleted here.

Definition 1: Sequency-ordered complex Hadamard trans-form of a signal vector isdefined as

(18)

where is the transformedcolumn vector and is the complex conjugate transpose of

. The values of are the complex numbers. The data se-quence can be recovered uniquely from the inverse transform,that is

(19)

Property 3: Sequency Property. The SCHT matrices showthe form of order that is sequency ordering. The row vectors ofan SCHT matrix are arranged in an ascending order of zerocrossings in the unit circle of a complex plane. Zero crossingsare shown in Fig. 1. Sequency describes the number of timesthat the row vectors of an SCHT matrix crosses the imaginaryaxis in the unit circle over a normalized time base .Sequency is analogous to frequency of DFT in which the Fouriercomponents are arranged in an increasing harmonic number interms of cosine and sine waves. Although the phase shifts arenot the same each time a row vector of an SCHT matrix rotatesin the unit circle of a complex plane, the total phase shifts of therow vectors of the SCHT matrix are the same as that of the DFTmatrix having the same size. This sequency property is usefulto apply SCHT in the implementation of fast DFT using SCHT[25]. In fact, with such property, SCHT can be used as a tool forsignal analysis and synthesis.

Property 4: Symmetry. Let be an SCHT matrix of orderwhere . is symmetrical, that is

(20)

where is the transpose matrix of . The inverse of theSCHT matrix, , is related as its complex conjugate trans-pose matrix, , as follows:

(21)

This property enables us to implement the complex Hadamardtransform more easily as both the forward transform and theinverse transform can share the same hardware architecture.

Property 5: Linearity. Let and be the data vec-tors of the length of and . Theyare assumed to have the same length. Then, SCHT is a lineartransformation that can be expressed as

SCHT SCHT SCHT

(22)

where and are constants. Instead of transforming the twosequences individually, the two sequences can be combined to-gether before taking the transform. It will help to reduce thecomputational time as this technique requires one SCHT trans-formation in contrast to the use of two -point SCHTs to com-pute and independently and combine them.

III. FAST ALGORITHM OF SCHT

Using the sparse matrix factorization, the algorithms for fastand efficient computation of SCHT can be developed. For thereal signals, real addition/subtractions and realmultiplications are required to perform the SCHT transformwithout using the fast SCHT algorithm. The inversed SCHTtransform matrix in (19) is the same as the SCHT matrix itselfso we start with the inversed complex Hadamard transform toderive the fast algorithm.

From (19), the inverse of complex Hadamard transform of an-point data sequence is .Let the vector be divided into two vectors, the one

with entries having even indices, the vector, and the otherwith odd indices, the vector. That is,

and.

Then the transform can be written using the matrix factoriza-tion as follows:

(23)

where .

From (23), we can further factorize as

(24)

Then

(25)

where is the direct sum operator and

Now we define (26) as in [26]

(26)


and

...

(27)

From (24), we can take the transform of the data vectors,and , separately. That is

(28)

and

(29)

The above two (28) and (29) can be written in the compactform as follows:

(30)

Substituting (30) into (25), we have

(31)

Consequently, can be further reduced to the dimension of2 2 which involves no more multiplications. Finally, we canhave

(32)

where is a bit reversal matrix which shuffles the vector, ,into the bit-reversed sequence and

...

(33)

As an example, we consider . Then (31) can be expressedin the compact form as follows:

(34)

Fig. 2. Flow graph of the final decomposition of fast inversed SCHT algorithm�� .

The decomposition of an 8-point SCHT for the efficient com-putations is illustrated by the signal flow graph in Fig. 2.

Fig. 2 shows the flow graph of the final decomposition offast inversed SCHT algorithm. As shown in Fig. 2, an 8-pointSCHT can be decomposed into three stages, and each stageneeds eight addition/subtractions. In each stage, two complexmultiplications by are required except for the first stageand four complex multiplications are required to perform the8-point SCHT operation. In general, the total computationalrequirements are complex addition/subtractions and

complex multiplications for the trivialtwiddle factors where and is the number of stages.Actually, the trivial complex multiplications can be performedby using the swapping and negating operations in the hardwareimplementation. The forward SCHT has the same compu-tational operation as that of the inversed SCHT except theconjugate of the transform matrix and the scaling factor of

which can be easily performed by the binary shiftingoperation without requiring any arithmetic operation.

The WHT matrices following the frequencies of the sinu-soidal waves in DFT are the sequency-ordered Walsh transform(SOWT) matrices. They can be obtained by rearranging the rowvectors of a WHT matrix through a Gray code and bit reversaloperation [6]. While the Fourier bases are sinusoids with har-monic frequencies, the SOWT bases are the Walsh functions.Actually, SOWT is an approximation of sine and cosine wavesof DFT by using the rectangular waves. It can be realized thatSCHT is an extended complex version of SOWT and it is an-other close approximation to the sinusoidal waves in Fourieranalysis. Since SOWT is a sequency-ordered version of WHT,several well-known techniques in developing the fast algorithmsfor this type of transform are applicable [6]. In general, for an

-point SOWT, the fast algorithms reduce the computationalcomplexity to . This is similar to the computa-tional load of an -point fast SCHT except that the latter needsthe trivial complex multiplications by factor , which do notrequire any arithmetic operation as mentioned earlier. Whileboth the signal and the transform are restricted to real numbersin SOWT, SCHT is suitable to analyze and process the com-plex-valued signals.

The fast SCHT algorithm is very much similar to the radix-2FFT algorithm because they have almost the same signal flowgraphs. Unlike the radix-2 FFT, however, the former does not


Fig. 3. Magnitude spectra of DFT, SCHT, and SOWT for the sinusoidal waveof 20 Hz.

need any nontrivial twiddle factor multiplications, which re-sults in significant savings in computational time. Similar tothe radix-2 FFT, this fast SCHT algorithm is suitable to adopta pipelined hardware structure to efficiently map the arithmeticoperations to the hardware. At the permissible expense of someloss in parameter estimation accuracy, one can replace FFT withSCHT with simpler implementation and consequent savings inthe computational cost in some applications. The following sec-tion will suggest some useful applications of SCHT.

IV. APPLICATIONS OF SCHT

In this section, we will show some potential applications forSCHT with supporting simulation results.

A. Signal Analysis and Synthesis

SCHT is applied in spectrum estimation of a sinusoidal signalas an example. Fig. 3 presents the normalized magnitude spectraof 64-point DFT, SCHT, and SOWT of a sine wave of 20 Hzwith 64 samples per second. It can be seen from the figure thatthe SCHT magnitude spectrum is closely matched with that ofDFT with a few side lobes as compared to that of SOWT. Thedesired peaks occur in the same locations as that of DFT eventhough there are some spurious spikes in the SCHT spectrum.Therefore, it is worth considering SCHT in spectral analysiswhen it is necessary to achieve significant hardware savings andreduced computational time.

It is also found that the SCHT magnitude spectrum is rela-tively more invariant to the circular phase shift of signals in timedomain than that of SOWT. Fig. 4 shows the SCHT and SOWTnormalized magnitude spectra due to 90 circular phase shiftof the signal in time domain. While the circular phase shift intime domain does not change the SCHT magnitude spectrum,the change is apparent in the SOWT spectrum as shown in thefigure. The SCHT of a real signal generates a series of complexnumbers in the transformed domain. It can be concluded that theeffect of the circular phase shift changes mostly the phase an-gles of these complex numbers in the sequency domain rather

Fig. 4. Magnitude spectra of SCHT and SOWT with 90 circular phase shiftin time domain.

than their magnitudes whereas the circular phase shift affectsthe magnitudes of the SOWT spectrum.

B. Image Watermarking

SCHT is suitable for use in the transform domain image wa-termarking. Similar to the DFT-based watermarking [27], thephase components of the selected SCHT coefficients are al-tered to convey the watermark information using the phase-shift-keying (PSK) modulation. The reasons for selecting thephase components rather than the amplitudes are that a water-mark embedded in the phase domain of the SCHT coefficientsis more robust to tampering, and besides, it is well known fromthe communication theory that phase modulation can achievebetter performance and superior noise immunity than amplitudemodulation.

In this paper, the SCHT watermarking scheme is evaluatedusing the well-known grayscale Lena image of 256 256 pixelsas compared with the other transforms. First, the image is seg-mented into the 8 8 blocks, and then two-dimensional (2-D)SCHT is applied on these image blocks as follows:

(35)

where represents the transpose ofis an SCHT matrix of size 8 8, is

the 8 8 real image block, and is the 8 8 2-D SCHTcomplex-valued coefficients in the sequency domain.

After transforming the whole image, the suitable SCHT co-efficients are selected to insert the watermark. By adopting themethod used in [27], we embed the secret bits in the lower fre-quency (sequency) components, which are the coeffi-cient of each 8 8 transformed image block in the sequency do-main to enhance the robustness. One bit of watermark is insertedin every 8 8 block, hence, a total of

secret bits are embedded in the Lena watermarked image.As described earlier, the phases of the selected coefficients aremodified to express the watermark information as mentioned in


Fig. 5. (a) Original image of Lena; the watermarked images of Lena using (b)SCHT, (c) DFT, and (d) UCHT.

Fig. 6. (a) Original watermark of 32� 32 pixels; the extracted watermarksfrom the attacked watermarked images of (b) SCHT, (c) DFT, and (d) UCHT.

[27]. Then, the inverse 2-D SCHT is performed to obtain thewatermarked copy of the original image.

Fig. 5 shows the original Lena image and the watermarkedimages of SCHT, DFT, and UCHT. The peak signal-to-noiseratios (PSNRs) of the SCHT, DFT, and UCHT watermarkedimages with respect to the original image are 34.1515, 33.7410,and 23.0243 dB, respectively. We can see from the figure thatdespite the presence of the watermark, the SCHT watermarkedimage does not contain any visible artifacts as the DFT wa-termarked image. One could hardly perceive the differencebetween the original image and the watermarked images ofSCHT and DFT whereas the visible degradation occurs in theUCHT watermarked image. Therefore, UCHT is not suitablefor inserting watermark in the direct transformed coefficients,whereas the multiresolution integer-valued Walsh transformis first used to decompose the image before taking the UCHTtransform [21].

In the PSK demodulation, the watermark information isextracted from the phases of the selected SCHT coefficients foreach block. The original host image is not needed during thewatermark extraction process in this watermarking scheme. TheSCHT watermarked image is robust to a series of attacks likethe DFT watermarked image. As an example, the additive whiteGaussian noise (AWGN) is introduced to the watermarkedimage. The watermark can still be extracted from the alteredimage as shown in Fig. 6.

V. RELATIONSHIPS OF SCHT WITH FFT AND UCHT

In [28], it has been shown that the radix-2 FFT can be ob-tained recursively from WHT and some matrices which includethe twiddle factors. In this section, we discuss the relations ofSCHT with the radix-2 FFT and UCHT. First we define dec-imation-in-time radix-2 FFT. The DFT of the data sequence

of length can be defined as [29]

(36)

where .Let the data vector be divided into two vector sequences,

the one with entries having odd indexes and the other witheven indices. Then from (36), we have the following FFT form[28], [29]:

(37)

(38)

(39)

In order to express the relationship with SCHT, (37)–(39) canbe rewritten in the matrix form as follows [26]:

(40)

where

and is the DFT matrix of dimension . Ingeneral, the radix-2 FFT can be written as

(41)

where with

...

(42)


and is a permutation matrix of the radix-2 FFT,which scrambles the data vector into the bit-reversal order.

Since, in general, , we can compute the

radix-2 FFT using the SCHT matrix as (43), shown at the bottomof the page, where and is the SCHTmatrix of dimension 4 4 with

and

is a permutation matrix to permute the columns of .Hence, the radix-2 FFT can be computed using the SCHT

matrix with the same computational load by the sparse matrixfactorization.

In addition, SCHT can also be obtained using WHT andUCHT. From Definition 1, the SCHT of an -point datavector is and the inverse transform is

. By using the matrix factorization technique,the SCHT matrix can be expressed as

(44)

where

and is the bit-reversal permutation matrix. Therefore, theSCHT matrix can be written using the WHT and UCHTmatrices as

(45)

where is a real Walsh–Hadamard matrix, and

is a UCHT matrix. Hence, SCHT can be com-

puted recursively from WHT and UCHT.The SCHT matrices are constructed based on the complex

Rademacher matrices as mentioned in Section II while the gen-eration of UCHT is based on WHT and successive Kroneckerproducts [7]. Even though they are generated differently, SCHTcan be obtained recursively from the WHT and UCHT matricesof dimension 2 2, and some sparse matrices which consistsof the identity matrices. Both transforms are suitable for prob-lems with complex-valued functions since they are complex in-teger-valued transforms. The UCHTs with half spectrum prop-erty are suitable and advantageous for use in the classificationand synthesis/analysis of logical circuits [20]. SCHT is more fa-vored for use in spectrum analysis, filtering, image signal pro-cessing and communications due to its sequency-ordering.

VI. CONCLUSION

The sequency-ordered complex Hadamard transform is dis-cussed in this paper and some of its important properties are de-scribed and analyzed with the discussion of some potential ap-plications. It has been shown that the SCHT matrices can be con-structed based on the complex Rademacher matrices. The realRademacher functions are extended to the complex Rademacherfunctions which form an orthogonal set of rectangular wave-forms taking the four complex values . The exponen-tial form of SCHT is presented and the proof of the unitaryproperty of SCHT is also given. The existence of the fast algo-rithm of SCHT based on the sparse matrix factorization (whichis similar to the radix-2 FFT) is derived and its computationalcomplexity is presented. SCHT is shown to have certain signif-icance in signal analysis and synthesis with the availability ofits fast algorithm. SCHT is also applied in image watermarking,and according to the simulation results, it has been observedthat SCHT can be considered as a good candidate for such ap-plication. The radix-2 FFT is mentioned in this paper and it hasbeen found that the radix-2 FFT can be obtained from the SCHTmatrix of size 4 4. The relationship of SCHT with UCHT isalso described. We have shown that SCHT can be computedrecursively from the UCHT and WHT matrices of dimension2 2 and some of the sparse matrices. SCHT shows sequencyordering which is similar to frequency where we find in DFT.Therefore, SCHT has the same potential in different applica-tions as DFT, and it may be of interest to do the further researchin the areas where DFT has been applied. Moreover, as the se-quency ordering is especially favored for communication, dig-ital filtering and spectral analysis [6], and due to the existenceof fast and efficient SCHT algorithm, the research should alsobe further explored in such areas of applications.

ifif

(43)


TABLE IBINARY REPRESENTATIONS OF ��

TABLE IIBINARY NOTATIONS FOR ��

APPENDIX

EXPONENTIAL FORM OF SCHT

From Property 1, the ( th, th) element, , of an SCHTmatrix is expressed as follows:

(46)

For example, we consider , hence, . Thebinary representations of and can be found in Tables I and II.

From (46), sequency-ordered complex Hadamard matricesare

and

ACKNOWLEDGMENT

The authors would like to thank all the anonymous refereesfor their valuable and helpful comments.

REFERENCES

[1] G. Peceli and B. Feher, “Digital filters based on recursive Walsh-Hadamard transformation,” IEEE Trans. Circuit Syst., vol. 37, no. 1,pp. 150–152, Jan. 1990.

[2] F. Fonda and S. Pastore, “Innovative image watermarking technique forimage authentication in surveillance applications,” in Proc. IEEE Int.Workshop Imaging Systems Techniques, Niagara Falls, ON, Canada,2005, pp. 32–5.

[3] M. Faundez-Zanuy, J. Roure, V. Espinosa-Duro, and J. A. Ortega, “Anefficient face verification method in a transformed domain,” PatternRecognit. Lett., vol. 28, no. 7, pp. 854–858, 2007.

[4] S. H. Tsai, Y. P. Lin, and C. C. J. Kuo, “MAI-free MC-CDMA systemsbased on Hadamard-Walsh codes,” IEEE Trans. Signal Process., vol.54, no. 8, pp. 3166–79, Aug. 2006.

[5] B. Gaffhey and A. D. Pagan, “Walsh Hadamard transform precededMB-OFDM: An improved high data rate ultrawideband system,” inProc. IEEE 17th Int. Symp. Personal, Indoor, Mobile Radio Commu-nications, Helsinki, Finland, 2006, p. 5.

[6] K. G. Beauchamp, Applications of Walsh and Related Functions, withan Introduction to Sequency Theory. New York: Academic, 1984.

[7] B. J. Falkowski and S. Rahardja, “Complex Hadamard transforms:Properties, relations and architecture,” IEICE Trans. Fundam. Elec-tron. Commun. Comput. Sci., vol. E87A, no. 8, pp. 2077–2083, 2004.

[8] E. Pogossova, K. Egiazarian, and J. Astola, “Signal denoising in tree-structured Haar basis,” in Proc. 3rd Int. Symp. Image Signal ProcessingAnalysis, Rome, Italy, 2003, vol. 2, pp. 736–9.

[9] A. Dziech, P. Slusarczyk, and B. Tibken, “Methods of image compres-sion by PHL, transform,” J. Intell. Robot. Syst., Theory Appl., vol. 39,no. 4, pp. 447–458, 2004.

[10] J. A. R. Macias and A. G. Exposito, “Efficient computation of the run-ning discrete Haar transform,” IEEE Trans. Power Del., vol. 21, no. 1,pp. 504–5, 2006.

[11] K. R. Rao, K. Revuluri, M. A. Narasimhan, and N. Ahmed, “ComplexHaar transform,” IEEE Trans. Acoust., Speech, Signal Process., vol. 24,pp. 102–104, Feb. 1976.

[12] M. Effros, F. Hanying, and K. Zeger, “Suboptimality of theKarhunen–Loeve transform for transform coding,” IEEE Trans.Inf. Theory, vol. 50, no. 8, pp. 1605–19, 2004.

[13] B. Penna, T. Tillo, E. Magli, and G. Olmo, “Transform coding tech-niques for lossy hyperspectral data compression,” IEEE Trans. Geosci.Remote Sens., vol. 45, no. 5, pp. 1408–1421, 2007.

[14] N. N. Ponomarenko, K. O. Egiazarian, V. V. Lukin, and J. T. As-tola, “High-quality DCT-based image compression using partitionschemes,” IEEE Signal Process. Lett., vol. 14, no. 2, pp. 105–108,2007.

[15] S. Singh, V. Kumar, and H. K. Verrna, “DWT-DCT hybrid scheme formedical image compression,” J. Med. Eng. Technol., vol. 31, no. 2, pp.109–22, 2007.

[16] K. R. Rao and N. Ahmed, “Complex BEFORE transform,” Int. J. Syst.Sci., vol. 2, no. 2, pp. 149–162, 1971.

[17] N. Ahmed, K. R. Rao, and Abdussattar, “BIFORE or Hadamardtransform,” IEEE Trans. Audio Electroacoust., vol. AU-19, no. 3, pp.225–234, Sep. 1971.

[18] S. Rahardja and B. J. Falkowski, “Family of unified complexHadamard transforms,” IEEE Trans. Circuits Syst. II, Analog Digit.Signal Process., vol. 46, no. 8, pp. 1094–1100, 1999.

[19] B. J. Falkowski and S. Rahardja, “Complex decision diagrams,” Elec-tron. Lett., vol. 32, no. 4, pp. 290–291, 1996.

[20] S. Rahardja and B. J. Falkowski, “Complex composite spectra of uni-fied complex Hadamard transform for logic functions,” IEEE Trans.Circuits Syst. II, Analog Digit. Signal Process., vol. 47, no. 11, pp.1291–1297, 2000.

[21] B. J. Falkowski and L. S. Lim, “Image watermarking using Hadamardtransforms,” Electron. Lett., vol. 36, no. 3, pp. 211–213, 2000.

[22] S. Rahardja, W. Ser, and Z. N. Lin, “UCHT-based complex sequencesfor asynchronous CDMA system,” IEEE Trans. Commun., vol. 51, no.4, pp. 618–626, 2003.

[23] Z. Gu, S. Xie, and S. Rahardja, “UCHT complex sequences for down-link multi-carrier DS-CDMA systems,” in Proc. IEEE 61st VehicularTechnology Conf., Stockholm, Sweden, 2005, vol. 61, pp. 392–395.

[24] A. Aung, B. P. Ng, and S. Rahardja, “Performance of SCHT sequencesin asynchronous CDMA system,” IEEE Commun. Lett., vol. 11, no. 8,pp. 641–643, Aug. 2007.

[25] A. Aung and B. P. Ng, “Fast DFT implementation using SCHT,”Nanyang Technological Univ., Singapore, Sep. 2007, Tech. Rep..

[26] R. Yarlagadda and J. Hershey, Hadamard Matrix Analysis and Syn-thesis with Applications to Communications and Signal/Image Pro-cessing. Norwell, MA: Kluwer, 1997.


[27] W. Y. Chen and C. H. Chen, “A robust watermarking scheme usingphase shift keying with the combination of amplitude boost and lowamplitude block selection,” Pattern Recognit., vol. 38, no. 4, pp.587–598, 2005.

[28] S. Xie, S. Rahardja, and Z. Gu, “Relationship between UCHT andFFT,” in Proc. 33rd Int. Symp. Multiple-Valued Logic, May 2003, pp.154–159.

[29] S. M. Kuo and W. S. Gan, Digital Signal Processors: Architecture,Implementations, and Applications, Int. ed. Englewood Cliffs, NJ:Pearson Prentice-Hall, 2005.

Aye Aung received the B.Eng. degree in electricaland electronic engineering from the Nanyang Tech-nological University (NTU), Singapore, in 2005.Since 2006, he has been working towards the Ph.D.degree at NTU, Singapore.

His research interests include digital signalprocessing for communications, wireless communi-cations, signal analysis and synthesis, fast algorithmsfor various transforms, image signal processing ingeneral, and multimedia digital watermarking.

Boon Poh Ng received the B.Eng. degree in elec-trical engineering from the Nanyang TechnologicalInstitute, Singapore, in 1987, the D.I.C. and M.Sc.degrees in communications and signal processingfrom the Imperial College, University of London,London, U.K., in 1991, and the Ph.D. degree fromthe Nanyang Technological University (NTU),Singapore, in 1995.

He was a Lecturer with the Department of Elec-tronics and Communication Engineering, SingaporePolytechnic, from 1987 to 1996. From 1996 to 1999,

he was a Senior Research Fellow at the Centre for Signal Processing, NTU. Heis currently an Associate Professor at the School of Electrical and Electronic En-gineering, NTU. His research interests include array synthesis, adaptive arrayprocessing, spectral estimation, and digital signal processing in general.

Susanto Rahardja (SM’07) received the B.Eng.degree from the National University of Singapore,Singapore, and the M.Eng. and Ph.D. degrees fromthe Nanyang Technological University (NTU), Sin-gapore, all in electrical and electronic engineering.

He is the Director of the Personal 3-D Entertain-ment System program and head of the Signal Pro-cessing Department at the Institute for Infocomm Re-search (I R), Singapore. He is also an Associate Pro-fessor at the School of Electrical and Electronic En-gineering in NTU. His research interests are in audio/

video signal processing, spread spectrum and multiuser detection techniques forCDMA applications, digital signal processing algorithms and implementations,and logic synthesis, of which he has more than 200 publications in internation-ally refereed journals and conferences.

Dr. Rahardja was the recipient of the IEE Hartree Premium Award in 2002.In 2003, he received the prestigious Tan Kah Kee Young Inventors’ Gold Awardin the Open Category, for his contributions on scalable to lossless audio com-pression technology. Since 2002, he actively participated in the internationalISO/IEC JTC1/SC29/WG11 (Moving Picture Expert Group, or MPEG) wherehe contributed to MPEG-4 Scalable to Lossless System (SLS), technology incor-porated in ISO/IEC 14496-3:2005/Amd.3:2006. He also contributed technologyto the MPEG-4 Audio Lossless System (ALS), where it is now incorporated inISO/IEC 14496-3:2005/Amd.2:2006. In recognition for his contributions to thenational standardization program, he was awarded the Standards Council MeritAward by SPRING Singapore in 2006. For his leadership and technical con-tribution to advancement of digital audio signal processing and its adoption tothe MPEG, he was awarded the National Technology Award in 2007. He hasserved on several boards, advisory, and technical committees in various IEEE-and SPIE-related professional activities in the areas of multimedia. He is anelected member of the Technical Committee of the Visual Signal Processingand Communications, Circuits and Systems for Communications and Multi-media Systems and Applications of the IEEE Circuits and Systems Society.He is currently serving as an Associate Editor for the IEEE TRANSACTIONS ON

AUDIO, SPEECH AND LANGUAGE PROCESSING and the IEEE TRANSACTIONS ON

MULTIMEDIA as well as the Journal of Visual Communication and Image Repre-sentation. He received the prestigious A STAR Most Inspiring Mentor Awardin March 2008.

Documents

Sequency-Ordered Complex Hadamard Transform: Properties, Computational Complexity and Applications