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2 Orthogonal Discrete Fourier and osine Matrices for Signal Processing 8 Daechul Park and Moon Ho Lee Hannam University, honbuk ational University Korea. Introduction he A DF (Discrete Fourier ransform) has seen studied and applied to signal processing and communication theory. he relation between the Fourier matrix and the Hadamard transform was developed in [Ahmed & Rao, 975; helchel & Guinn, 968] for signal representation and classification and the Fast Fourier-Hadamand ransform(ffh) was proposed. his idea was further investigated in [Lee & Lee, 998] as an extension of the conventional Hadamard matrix. Lee et al [Lee & Lee, 998] has proposed the Reverse Jacket ransform(rj) based on the decomposition of the Hadamard matrix into the Hadamard matrix(unitary matrix) itself and a sparse matrix. Interestingly, the Reverse Jacket(RJ) matrix has a strong geometric structure that reveals a circulant expansion and contraction properties from a basic x sparse matrix. he discrete Fourier transform (DF) is an orthogonal matrix with highly practical value for representing signals and images [Ahmed & Rao, 975; Lee, 99; Lee, ]. Recently, the Jacket matrices which generalize the weighted Hadamard matrix were introduced in [Lee, ], [Lee & Kim, 984, Lee, 989, Lee & Yi, ; Fan & Yang, 998]. he Jacket matrix is an abbreviated name of a reverse Jacket geometric structure. It includes the conventional Hadamard matrix [Lee, 99; Lee, ; Lee et al., ; Hou et. al., ], but has the weights, ω, that are j or k, where k is an integer, and j, located in the central part of Hadamard matrix. he weighted elements' positions of the forward matrix can be replaced by the non-weighted elements of its inverse matrix and the signs of them do not change between the forward and inverse matrices, and they are only as element inverse and transpose. his reveals an interesting complementary matrix relation. Definition : If a matrix J of size m m has nonzero elements m j, j,. j, m j, j,. j, m J, (6-) m jm, jm,. jm, m

3 4 Fourier ransforms - Approach to Scientific Principles / j, / j,. / j, m / j, / j,. / j, m J m / jm, / jm,. / jm, m (6-) where is the normalizing constant, and is of matrix transposition, then the matrix J m is called a Jacket matrix [helchel & Guinn, 968],[Lee et al., ],[Lee et al, 8; hen et al., 8]. Especially orthogonal matrices, such as Hadamard, DF, D, Haar, and Slant matrices belong to the Jacket matrices family [Lee et al., ]. In addition, the Jacket matrices are associated with many kind of matrices, such as unitary matrices, and Hermitian matrices which are very important in communication (e.g., encoding), mathematics, and physics. In section DF matrix is revisited in the sense of sparse matrix factorization. Section presents recursive factorization algorithms of DF and D matrix for fast computation. Section 4 proposes a hybrid architecture for implentation of algorithms simply by adding a switching device on a single chip module. Lastly, conclusions were drawn in section 4.. Preliminary of DF presentation he discrete Fourier transform (DF) is a Fourier representation of a given sequence x(m), m and is defined as π nm Xn ( ) xm ( ), n, (6-) m where e j. Let s denote -point DF matrix as F π nm, nm,, {,,,, } where e j (see about DF in appendix), and the Sylvester Hadamard matrix as H, respectively. he Sylvester Hadamard matrix is generated recursively by successive Kronecker products, H H H, (6-4) for 4, 8,, and H. For the remainder of this chapter, analysis will be concerned only with k, k,,, as the dimensionality of both the F and H matrices. Definition : A sparse matrix S, which relates F and H, can be computed from the factorization of F based on H. he structure of the S matrix is rather obscure. However, a much less complex and more appealing relationship will be identified for S [Park et al., 999]. o illustrate the DF using direct product we alter the denotation of to lower case w e jπ n, so that w becomes the n-th root of unit for -point. For instance, the DF matrix of dimension is given by:

4 Orthogonal Discrete Fourier and osine Matrices for Signal Processing 4 Let s define w w F H w w. (6-5) and in general, w 4 w 4 j e define and j E F, (6-6) j and ( ) / / / / (,,,, ) diag w w w w E F [ P] [ F ] [ ]. (6-7) where [ P] is a permutation matrix and [ F ] [ P] [ F] is a permuted version of DF matrix [ F ].. A sparse matrix factorization of orthogonal transforms. A sparse matrix analysis of discrete Fourier transform ow we will present the Jacket matrix from a direct product of a sparse matrix computation and representation given by [Lee, 989], [Lee & Finlayson, 7] J H S m m m m, (6-8) k + where m, k {,,,4,} and S Jacket matrix can be simply written as m is sparse matrix of m J. hus the inverse of the ( ) ( ) J S H. (6-9) m m m As mentioned previously, the DF matrix is also a Jacket matrix. By considering the sparse matrix for the 4-piont DF matrix F, 4 we can rewrite π π π j j j e e e j j F π π π j j 4 j 6. e e e 6 9 π π π j j j j 6 j 9 e e e F 4 by using permutations as

5 4 Fourier ransforms - Approach to Scientific Principles j j [ F] [ F] F Pr F j j j j j j E E I I [ F ] I I E, (6-) where E j, its inverse matrix is from element-inverse, such that j In general, we can write that ( E ) / / j j j / / j. (6-) [ F] [ F] I I [ F] / / / / / F Pr F E I / E / / I/ E /, (6-) where F F. And the submatrix E could be written from (6-7) by E [ F] [ ] Pr F, (6-) where, and is the complex unit for point DF matrix. j For example, E can be calculated by using j j E Pr F. (6-4) j j By using the results from (6-) and (6-), we have a new DF matrix decomposition as I I [ F ] / / / F Pr F I/ I/ E / [ F ] I I / / / E I / / I/

6 Orthogonal Discrete Fourier and osine Matrices for Signal Processing 4 [ F ] / I/ I/ F Pr / [ F ] I / / / I/ I/ [ F ] / I/ I/ I/. (6-5) Pr/ [ F ] / / I/ I/ Finally, based on the recursive form we have I/ [ F ] / I/ I/ I/ F ( Pr ) F ( Pr ) Pr/ [ F ] / / I/ I/ I / I ( Pr ) I/4 / Pr / Pr I F. (6-) I I I I / I/ I/ I/4 I/4.. I I / I/ I/ Using (6-) butterfly data flow diagram for DF transform is drawn from left to right to perform X [ F] x. [ F] Pr / Pr / 4 Pr / 4 Pr Pr Pr Pr / [ ] 4 / 4 / 4 / / [ ] / 4 Fig.. Butterfly data flow diagram of proposed DF matrix with order / 4 / [ ] /. A Sparse matrix analysis of discrete cosine transform Similar to the section., we will present the D matrices by using the element inverse or block inverse Jacket like sparse matrix [Lee, ; Park et al., 999] decomposition. In this

7 44 Fourier ransforms - Approach to Scientific Principles section, the simple construction and fast computation for forward and inverse calculations and analysis of the sparse matrices, was very useful for developing the fast algorithms and orthogonal codes design. Discrete osine ransform (D) is widely used in image processing, and orthogonal transform. here are four typical D matrices [Rao & Yip, 99; Rao & Hwang, 996], I D - I: + m n mn, mnπ k k cos, mn,,,, ; (6-7) D - II: D - III: II mn, III mn, mn ( + ) π k cos m, mn,,,, ; (6-8) ( m+ ) nπ k cos n, mn,,,, ; (6-9) D - IV: IV mn, ( m+ )( n+ ) π cos, mn,,,,, (6-) where k j, j,,,., j, o describe the computations of D, in this chapter, we will focus on the D - II algorithm, and introduce the sparse matrix decomposition and fast computations. he -by- D - II matrix can be simply written as 4 4, (6-) where can be seen as a special element inverse matrix of order, its inverse is, and i cos( iπ / l) is the cosine unit for D computations. l Furthermore, 4-by-4 D - II matrix is of the form , (6-)

8 Orthogonal Discrete Fourier and osine Matrices for Signal Processing 45 we can write, (6-) P where P 4 which has the form is permutation matrix. P permutation matrix is a special case where with P I, and P P pr i, j,, 4, pri, j, if i j, j, pri, j, if i ( j + )mod, j, pri, j, others. where i, j {,,, } Since,,, we rewrite (6-) as P , (6-4)

9 46 Fourier ransforms - Approach to Scientific Principles and let us define a column permutation matrix reversible permutation matrix which is defined by Pc, and 4 Pc is a Pc I, and hus we have I /4 I /4 Pc, 4. I /4 I /4 6 6 P 4 4Pc , (6-5) B B where write that, the -by- D - II matrix and 8 8 B. hus we can 8 8 I I P 4 Pc 4 4 B B I I B, (6-6) it is clear that B is a block inverse matrix, which has ( ). (6-7) B ( B )

10 Orthogonal Discrete Fourier and osine Matrices for Signal Processing 47 he (6-7) is Jacket like sparse matrix with block inverse. In general the permuted D - II matrix can be constructed recursively by using I I / / / / / P Pc B/ B / I/ I/ B /. (6-8) where denotes the / D - II matrix, and B can be calculated by using / f( m, n) ( ) B / mn, /, (6-9) where f( m,) m, f( m, n+ ) f( m, n) + f( m,), mn, {,,, / }. (6-) For example, in the 4-by-4 permuted D - II matrix, B could be calculated by using 4 f (,), f (,), f(,) f(,) + f(,), and f(,) f(,) + f(,) 9, f(,) f(,) ( 8 ) ( 8 ) f f ( 8 ) ( 8 ) f( m, n),, B ( 8 ) mn, 4 (,) (,). (6-),, and its inverse is of can be simply computed from the block inverse I/ I/ / I/ I/ B / ( P Pc ) ( ) / / / I I ( B / ) I/ I /. (6-) For example, the 8 8 D - II matrix has ( / ) I/ I/ I/ I / ( B / )

11 48 Fourier ransforms - Approach to Scientific Principles [ ] and it can be represented by , P Pc B4 B Additionally, it is clearly that the function (6-8) also can be recursively constructed by using different permutations matrices P and Pc, as I I / / / / / P Pc B I / B / / I/ B /, (6-) where and / B are the permutated cases of / new permutation matrices have the form and / B /, respectively. he Easily, we can check that [ P] / P I /, and [ Pc], > 4. (6-4) / Pc [ Pc] / P Pc / / / / P Pc I/ B/ B/ Pc/ P/ / Pr/ / Pc/ I B I B Pc / / / / /

12 Orthogonal Discrete Fourier and osine Matrices for Signal Processing 49 P Pc P Pc / / / / / / / / P Pc I/ B/ Pc/ I/ B/ Pc/ B / B /, (6-5) where B B Pc. / / / For example, the 4-by-4 D - II case has. (6-6) B B 4/ 4/Pc 4/ Moreover, the matrix matrix as 8 8 B 8 8 can be decomposed by using the -by- D - II B K D, (6-7) where K is a upper triangular matrix, and we use the cosine related function 8 D 8 is a diagonal matrix, where φ m is m-th angle. In a general case, we have cos(k+ ) φ cos( kφ )cosφ cos(k ) φ. (6-8) m m m m where B K D, (6-9) K, Φ 4 Φ 4 D, (6-4) Φ 4 and Φ i +, i {,,,, }. See appendix for proof of (6-9). i By using the results from (6-8) and (6-9), we have a new form for D - II matrix I/ I/ / / I/ I/ Pr Pc I/ I/ B / B/ I/ I /

13 5 Fourier ransforms - Approach to Scientific Principles I I / / / K/ / D/ I/ I/. (6-4) I/ / I/ I/ I/ K D I I / / / / / Given the recursive form of (6-4), we can write I / I ( Pr ) I/4 ( Pr 4 ) I /4 I/ K / K I I I I I I Pc I/ I/ I/ ( Pc ) D/ I/ I / ( ) /4 /4 /4 4 D I I By taking all permutation matrices outside, we can rewrite (6-4) as ( ) I / I Pr I /4 I/ K / K ( ) I I I I I I I I Pc / / / /4 /4.. D I D I / I/ I /. (6-4). (6-4) Using (6-4) butterfly data flow diagram for D-II transform is drawn as Fig. from left to right to perform X[] x. [ ] K / K / 4 K / 4 K K K K [ D] / / [ D] / [ D] / Fig.. Butterfly data flow diagram of proposed D - II matrix with order

14 Orthogonal Discrete Fourier and osine Matrices for Signal Processing 5. Hybrid D/DF architecture on element inverse matrices It is clear that the form of (6-4) is the same as that of (6-), where we only need change K l to Pr l and D l to l, with l {,4,8,, / }. onsequently, the results show that the D - II and DF can be unified by using same algorithm and architecture within some characters changed. As illustrated in Fig., and Fig., we find that the DF calculations can be obtained from the architecture of D by replacing the D to, and a permutation matrix Pr to K. Hence a unified function block diagram for D/DF hybrid architecture algorithm can be drawn as Fig.. In this figure, we can joint D and DF in one chip or one processing architecture, and use one switching box to control the output data flow. It will be useful to developing the unified chip or generalized form for D and DF together. Fig.. A unified function block diagram for proposed D/DF hybrid architecture algorithm 4. onclusion e propose a new representation of D/DF matrices via the Jacket transform based on the block-inverse processing. Following on the factorization method of the Jacket transform, we show that the inverse cases of D/DF matrices are related to their block inverse sparse matrices and the permutations. Generally, D/DF can be represented by using the same architecture based on element inverse or block inverse decomposition. Linking between two transforms was derived based on matrix recursion formula. Discrete osine ransform (D) has applications in signal classification and representation, image coding, and synthesis of video signals. he D-II is a popular structure and it is usually accepted as the best suboptimal transformation that its

15 5 Fourier ransforms - Approach to Scientific Principles performance is very close to that of the statistically optimal Karhunen-Loeve transform for picture coding. Further, the discrete Fourier transform (DF) is also a popular algorithm for signal processing and communications, such as OFDM transmission and orthogonal code designs. Being combined these two different transforms, a unified fast processing module to implement D/DF hybrid architecture algorithm can be designed by adding switching device to control either D or DF processing depending on mode of operation. Further investigation is needed for unified treatment of recursive decomposition of orthogonal transform matrices exploiting the properties of Jacket-like sparse matrix architecture for fast trigonometric transform computation. 5. Acknowledgement his work was supported by orld lass Univ. R-8--4-, RF, Korea. 6. Appendix 6. Appendix he DF matrix brings higher powers of w, and the problem turns out to be and the inverse form. c y n w w. w c y 4 ( n ) w w. w c y. (A-) n ( n ) ( n ) w w. w c n y n F. w w w. 4. w w w n..... w w. w ( n ) ( n ) ( n ) ( n ) ( n ) hen, we can define the Fourier matrix as follows. Definition A.: An n n matrix F a ij is a Fourier matrix if. (A-) a w ( i )( j ) ij, and the inverse form ( ) ( i )( j ) F aij ( w ) w i / e π n, and i, j {,,, n}. (A-). n n For example, in the cases n and n, and inverse is an element-wise inverse like Jacket matrix, then, we have w w F, w w F w w w w

16 Orthogonal Discrete Fourier and osine Matrices for Signal Processing 5 and e need to confirm that Row j of F times column j of iπ i4π F e e, i4π i8π e e iπ i4π F e e. i4π i8π e e FF equals the identity matrix. On the main diagonal that is clear. F is ( ) off the diagonal, to show that row j of F times column k of / n (+ + + ), which is. he harder part is F gives zero: he key is to notice that those terms are the powers of :, if j k. (A-4) (A-5) n 6. Appendix In a general case, we have where B K D, K, and Φ i +, i {,,,, }. i Proof: In case of D - II matrix, Φ 4 Φ 4 D, Φ 4, it can be represented by using the form as where k i+, i {,,,}. i According to (6-7), a matrix kφ kφ kφ kφ kφ kφ kφ kφ kφ kφ kφ kφ kφ kφ kφ k Φ k Φ k Φ k Φ k Φ B from can be simply presented by, (A-6)

17 54 Fourier ransforms - Approach to Scientific Principles B And based on (6-78), we have the formula Φ Φ Φ Φ (k+ ) Φ (k+ ) Φ (k+ ) Φ (k+ ) Φ (k+ ) Φ (k+ ) Φ (k+ ) Φ (k+ ) Φ (k + ) Φ (k + ) Φ (k + ) Φ (k + ) Φ (A-7) where m {,,,.}. hus we can calculate that +, (A-8) (ki + ) Φm kiφm Φm (ki ) Φm (ki ) Φm kiφm Φm K D kφ kφ kφ k Φ k Φ k Φ k Φ k Φ k Φ k Φ Φ 4 Φ 4 Φ 4 Φ 4 kφ + 4 k Φ 4 + Φ 4 kφ 4 Φ 4 kφ kφ kφ kφ kφ kφ kφ kφ kφ kφ kφ 4 kφ 4 kφ kφ 4 kφ 4 + Φ Φ Φ Φ kφ Φ Φ kφ Φ Φ kφ Φ Φ kφ Φ kφ Φ Φ kφ Φ kφ Φ, (A-9) kφ 4,,

18 Orthogonal Discrete Fourier and osine Matrices for Signal Processing 55 Since k, we get + +, (A-) Φm kφm Φm (k ) Φm kφm Φ m (k+ ) Φm (A-) Φm kφm Φm (k ) Φm kφm Φ m (k+ ) Φm and ( ) 4 In case of k i i +, we have (k ) (( k ) ) ( k ) + Φ + Φ Φ, then we get i m i m i m + + Φm ki Φm Φm kiφm Φ m (ki + ) Φm kiφm Φm (ki ) Φm kiφm Φ m (ki + ) Φm aking the (A-)-(A-) to (A-9), we can rewrite that. (A-) [ K ] [ ] [ D] Φ 4 Φ 4 kφ 4 + Φ 4 kφ 4 Φ 4 + Φ 4 kφ 4 Φ 4 Φ 4 Φ 4 kφ 4 + Φ 4 kφ 4 Φ 4 + Φ 4 kφ 4 Φ 4 Φ 4 + Φ 4 kφ 4 Φ 4 Φ 4 (k + ) Φ 4 (k+ ) Φ 4 Φ 4 (k + ) Φ 4 (k+ ) Φ 4 Φ 4 (k + ) Φ 4 (k+ ) Φ 4 [ B]. (A-) he proof is completed. 7. References Ahmed,. & K.R. Rao(975), Orthogonal ransforms for Digital Signal Processing, , Berlin, Germany: Springer-Verlag hen, Z.; M.H. Lee, & Guihua Zeng(8), Fast ocylic Jacket ransform, IEEE rans. on Signal Proc. Vol. 56, o.5, 8, (4-48),5-587X Fan,.-P. & Jar-Ferr Yang(998), Fast enter eighted Hadamard ranform Algorithm, IEEE rans.on AS-II Vol. 45, o., (998) (49-4), 57-7 Hou, Jia; M.H. Lee, & Ju Yong Park(), ew Polynomial onstruction of Jacket ransform, IEIE rans. Fund. Vol. E86A, o., () (65-66), Lee, M.H.(989), he enter eighted Hadamard ransform, IEEE rans. on ircuits and Syst. 6(9), (989) (47-49), Lee, M.H.(99), High Speed Multidimensional Systolic Arrays for Discrete Fourier ransform, IEEE rans. on.ircuits and Syst.II 9 (), (99) ( ), 57-7 Lee, M.H.(), A ew Reverse Jacket ransform and Its Fast Algorithm, IEEE, rans. on ircuit and System 47(),, (9-47), 57-7 Lee, M.H. & Ken Finlayson(7), A simple element inverse Jacket transform coding, IEEE Signal Processing Lett. 4 (5), 7, (5-8), 7-998

19 56 Fourier ransforms - Approach to Scientific Principles Lee, M. H. & D.Y.Kim(984), eighted Hadamard ransform for S/ Ratio Enhancement in Image ransform, Proc. of IEEE ISAS 84, 984, pp Lee, S.R. & M.H. Lee(998), On the Reverse Jacket Matrix for eighted Hadamard ransform, IEEE rans. On AS-II, Vol. 45, o., (998) (46-44), 57-7 Lee, M.H.;.L.Manev & Xiao-Dong Zhang (8), Jacket transform eigenvalue decomposition, Appl. Math.omput.98, (May 8). ( ) Lee, M.H.; B. S. Rajan, & Ju Yong Park(), A Generalized Reverse Jacket ransform, IEEE rans. on ircuits and Syst. II 48 (7),() (684-69), 57-7 Lee, S. R. & J. H. Yi(), Fast Reverse Jacket ransform as an Altenative Representation of point Fast Fourier ransform, Journal of Mathematical Imaging and Vision Vol., o.,,() (4-4), Park, D.; M.H. Lee & Euna hoi(999), Revisited DF matrix via the reverse jacket transform and its application to communication, he nd symposium on Information theory and its applications (SIA 99), 999, Yuzawa, iigata, Japan Rao, K.R. & P. Yip(99), Discrete osine ransform Algorithms, Advantages, Applications, X, Academic Press, San Diego, A, USA Rao, K.R. & J.J. Hwang(996), echniques & Standards for Image Video & Audio oding, , Prentice Hall, Upper Saddle River, J, USA helchel, J.E. & D.F. Guinn(968), he fast fourier hadamard transform and its use in signal representation and classification, Proc. EASO 68, 968, pp

20 Fourier ransforms - Approach to Scientific Principles Edited by Prof. Goran ikolic ISB Hard cover, 468 pages Publisher Inech Published online, April, Published in print edition April, his book aims to provide information about Fourier transform to those needing to use infrared spectroscopy, by explaining the fundamental aspects of the Fourier transform, and techniques for analyzing infrared data obtained for a wide number of materials. It summarizes the theory, instrumentation, methodology, techniques and application of FIR spectroscopy, and improves the performance and quality of FIR spectrophotometers. How to reference In order to correctly reference this scholarly work, feel free to copy and paste the following: Daechul Park and Moon Ho Lee (). Orthogonal Discrete Fourier and osine Matrices for Signal Processing, Fourier ransforms - Approach to Scientific Principles, Prof. Goran ikolic (Ed.), ISB: , Inech, Available from: Inech Europe University ampus SeP Ri Slavka Krautzeka 8/A 5 Rijeka, roatia Phone: +85 (5) Fax: +85 (5) Inech hina Unit 45, Office Block, Hotel Equatorial Shanghai o.65, Yan An Road (est), Shanghai, 4, hina Phone: Fax: