Z. Wang. Fast Algorithms for the Discrete W Transform and for the Discrete Fourier Transform. IEEE Trans. on Acoustics, Speech, and Signal Processing, ASSP-32(4):803--816, 1984.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the Department of Electrical and Computer Engineering, The Johns Hopkins University, Baltimore, MD 21218 USA (e mail: jieliang jhu.edu; trac jhu.edu) Publisher Item Identifier S 1053 587X(01)10495 2. the sparse factorizations of the DCT matrix [12] 17] and many of them are recursive [12] [14], 16] 17] Besides one dimensional (1 D) algorithms, two dimensional (2 D) DCT algorithms have also been investigated extensively [6] 18] 21] generally leading to less computational complexity than the row column application of the 1 D methods. However, the implementation of the direct 2 D ....

.... into plane rotations and butterflies [12] 13] The factorization has a very regular structure and is six times as fast as the DFT based fast DCT algorithm [1] The method was generalized by Wang to all versions of DCT, DST, the discrete transform, as well as the DFT with the size of power of 2 [14]. Similar results were also reported in [41] In this paper, we will concentrate on the four point, eightpoint and 16 point transforms since they are the most useful ones in practical applications. Block transforms of other sizes can be designed in a similar fashion. The factorization of the ....

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Z. Wang, "Fast algorithm for the discrete W transform and for the discrete Fourier transform," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-32, pp. 803--816, Aug. 1984.

....of the reasons to look for preconditioners which can be diagonalized by trigonometric matrices corresponding to fast trigonometric transforms instead of the Fourier matrix F N . In practice, four discrete sine transforms (DST I IV) and four discrete cosine transforms (DCT I IV) were used (see [25]) Any of these eight trigonometric transforms can be realized with log N) arithmetical operations. Likewise, we can define preconditioners with respect to any of these transforms. In this paper, we restrict our attention to the so called discrete cosine transform of type II (DCT II) and ....

Z. Wang, Fast algorithms for the discrete W transform and for the discrete Fourier transform, IEEE Trans. Acoust. Speech Signal Process, 32 (1984), pp. 803--816.

....of the transform matrix C n into a product of real, sparse matrices. The trigonometric approximation algorithm of Runge [18] can be considered as a rst example of this approach. Results on direct matrix factorization of C n into orthogonal, sparse matrices are due to Chen et al. 4] and Wang [26]. Note that various factorizations of C n use non orthogonal matrix factors (see [17] pp. 53 62, 3, 12, 13] Many results were published without proofs. A direct orthogonal factorization of the cosine matrix of type II and of order 8 was given by Loe er et al. 14] and is used even today ....

....factors (see [17] pp. 53 62, 3, 12, 13] Many results were published without proofs. A direct orthogonal factorization of the cosine matrix of type II and of order 8 was given by Loe er et al. 14] and is used even today in JPEG standard (cf. Example 2. 8) Improving the earlier results in [4, 26], Schreiber has given a constructive proof of a factorization of some cosine matrices of size 2 into a product of sparse, orthogonal matrices in [19] Unfortunately, the construction of this factorization is not simple. However, another important result in [19] see also [23] says that a fast ....

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Z. Wang, Fast algorithms for the discrete W transform and the discrete Fourier transform, IEEE Trans. Acoust. Speech Signal Process. 32 (1984), 803 - 816.

.... introduced in this work represent a new class of QN algorithms with memory involving suitable approximations of the whole Hessian matrix, determined in an algebra of matrices diagonalized by a fast unitary transform (FFT, Hartley type [6] 7] 9] 17] 21] or trigonometric [8] 26] 32] [40]) The most interesting property of QN methods depends upon the fact that they can be implemented in flops per step (i.e. the same computational cost of the fast transforms involved) and memory allocations. The strong reduction of space complexity is obtained since all iterative formulas ....

....We shall see that if the unitary matrix associated with defines a fast transform, then each step of these methods can be performed in flops. IV. COMPUTATIONAL COMPLEXITY In Theorem 4. 1, it is stated that, if defines a fast discrete transform, as for instance Fourier, Jacobi type [8] 26] 32] [40] Hartley type [6] 7] 9] 17] then flops are sufficient to perform one step of both and QN methods. Moreover, the number of memory allocations required to implement the same methods is . An obvious example of such special matrices is given by the Fourier matrix , Since the BFGS, the BFGS ....

Z. Wang, "Fast algorithms for the discrete W transform and for the discrete Fourier transform," IEEE Trans. Acoust, Speech, Signal Processing, vol. ASSP-32, pp. 803--816, Aug. 1984.

....level. An example of a fast exact algorithm is shown in Figure 1 [2] for a size 8 1 D DCT. This algorithm requires only 13 multipli cations and 29 additions. The theoretical bound on the number of nonrational multiplication for size 8 1 D DCT is 11 [3] which is achieved as in [4] and [5] via integer arithmetic and is adopted in many well known implementation of DCT based coding standards ( 6] 7] For situations where even a fast fixed complexity DCT algorithm is too complex (e.g. computation of large DCTs or complexity constrained encoding situ ations) it is possible to ....

Z. Wang, "Fast algorithms for the discrete W- transform and for the Discrete Fourier Transform," IEEE Trans. on Acoustics, Speech and Signal Proc., vol. ASSP-32, Aug 1984.

....but simply indicate their type using the notation introduced in Section 2. The symbols P; D;R;S refer to permutation, diagonal, rotation, and other sparse matrices, respectively. The same symbols can have different meanings in different rule The exact form of the occurring matrices can be found in [8, 9, 10]. We note the following important facts. Base case rules expand transforms of trivial size (e.g. Rules (11) and (13) Recursive rules expand a transform in terms of similar (e.g. Rules (4) and (18) or different (e.g. Rules (12) and (8) transforms of smaller size. Transformation ....

Z. Wang, " Fast Algorithms for the Discrete W Transform and for the Discrete Fourier Transform ," IEEE Trans. on Acoustics, Speech, and Signal Processing, vol. ASSP-32, no. 4, pp. 803--816, 1984.

....on the parameters (e.g. size) of the transform. Examples of breakdown rules for DCT and DCT (Iv) are (II) IV) t tt DCT = P, DCT, 2 DCT, 2 ) P. I, 2 DFT2) P, and DCT(n IV) S. DCT(n II) D. where P, Pn , Pn are permutation matrices, is bidiagonal, and D is a diagonal matrix (see [13] for details) A transform usually has several different rules. Rules for the DFT that we can capture from fast algorithms as they are given in literature, include the Cooley Tukey rule (n = r.s composite) Rader s rule (n prime) Good Thomas rule (n = r .s, gcd(r, s) 1) and several others ....

Z. Wang, "Fast Algorithms for the Discrete W Transform and for the Discrete Fourier Transform ," IEEE Trans. on Acoustics, Speech, and Signal Processing, vol. ASSP-32, no. 4, pp. 803 816, 1984.

....algebras rather than matrix entries. This makes the derivation more transparent and explains their structure. Our results show that the relationship between signal processing and algebra is stronger than previously understood. 1. INTRODUCTION There is a large number (several hundred, e.g. [1, 2]) of publications on fast algorithms for the family of 16 discrete trigonometric transforms (DTTs) comprising 8 cosine and 8 sine transforms (DCTs and DSTs) With very few exceptions (including [3, 4, 5] each of these algorithms has been found by insightful manipulation of the transform matrix ....

....formulas, we need the following building blocks. The base change matrices B 2m in (31) and B 2m 1 = 0 1 0 5 ; and the permutation matrices P 2m : i 7 mi mod (2m 1) 0 i 2m 1; 2m 1 7 2m 1; P 2m 1 : i 7 i(m 1) mod (2m 1) 0 i 2m: Based on U 2m 1 = 2Um 1 Tm we get (e.g. [2, 11]) DST 1 2m 1 = P 2m 1 (DST 3m DST 1m 1 )B 2m 1 : DCT 2 2m = P 2m(DCT 2m DCT 4m )B 2m : Based on U 2m = VmWm we get DCT 1 2m = P 2m(DCT 5m DCT 7m )B 2m : DST 2 2m 1 = P 2m 1(DST 8m 1 DST 6m )B 2m 1 : We did not find these in the literature. Decomposition by Polynomial ....

Z. Wang, "Fast Algorithms for the Discrete W Transform and for the Discrete Fourier Transform," IEEE Trans. on Acoustics, Speech, and Signal Processing, vol. ASSP-32, no. 4, pp. 803--816, 1984.

....cosine transform (DCT) and four types of the discrete sine transform (DST) Table I gives a few example break down rules for some of these. The operator denotes matrix direct sum and exponentiation denotes matrix conjugation. The discrete trigonometric transforms (DTTs) DCT and DST, 6] 7] [8], 9] are considerably different from the 6 TABLE I A FEW EXAMPLE BREAK DOWN RULES. DIAGONAL MATRICES ARE REPRESENTED BY D, PERMUTATIONS BY P AND L, ROTATION MATRICES BY R s AND OTHER SPARSE MATRICES BY M AND T. WHT. Specifically, the following differences are of importance: While we have ....

Z. Wang, "Fast algorithms for the discrete w transform and for the discrete fourier transform," IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. ASSP-32, no. 4, pp. 803 816, 1984.

.... are DCT (II) n = Pn Delta (DCT (II) n=2 Phi DCT (IV) n=2 ) Delta P 0 n Delta (I n=2 Omega DFT 2 ) Delta P 00 n ; and DCT (IV) n = Sn Delta DCT (II) n DeltaD n ; where Pn ; P 0 n ; P 00 n are permutation matrices, Sn is bidiagonal, and Dn is a diagonal matrix (see [13] for details) A transform usually has several different rules. Rules for the DFT that we can capture from fast algorithms as they are given in literature, include the Cooley Tukey rule (n = r Delta s composite) Rader s rule (n prime) Good Thomas rule (n = r Delta s, gcd(r; s) 1) and ....

Z. Wang, " Fast Algorithms for the Discrete W Transform and for the Discrete Fourier Transform ," IEEE Trans. on Acoustics, Speech, and Signal Processing, vol. ASSP-32, no. 4, pp. 803--816, 1984.

.... by Cooley and Tukey (first discovered by Gauss [2] 3] Rader s algorithm for prime size [4] Winograd s algorithms [5] as well as [6] 8] An overview on FFT algorithms can be found in [9] or [10] Important algorithms for the trigonometric transforms were found by Chen et al. 11] Wang [12], Yip and Rao [13] 14] Vetterli and Nussbaumer Manuscript received June 1, 2000; revised May 30, 2001. M. Pschel was supported by the National Science Foundation through Award 9988296 and by the Defense Advanced Research Projects Agency through research Grant DABT63 98 1 0004, administered by ....

....DCT , the first four lines give the matrix from Algorithm 1, the last line contains the permutation matrix (which makes the block structure of and explicit) and the fifth line gives the matrix . The algorithms for DCT and DCT have the same arithmetic cost as the best known algorithms [8] 11] [12], 15] 17] 19] 39] Note that those who use only 12 multiplications do not normalize the first row of the DCT , which saves one multiplication. The only algorithm that claims 11 multiplications [40] considers a scaled version of the DCT matrix DCT DCT Multiplying by scalars conserves the ....

[Article contains additional citation context not shown here]

Z. Wang, "Fast algorithms for the discrete W transform and for the discrete Fourier transform," IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-32, pp. 803--816, Apr. 1984.

.... of the relationships between the DCT and various existing fast transforms, including the FFT [1] 6] 7] 8] the Walsh Hadamard transform (WHT) 9] 10] and the discrete Hartley transform (DHT) 11] Some algorithms are based on the sparse factorizations of the DCT matrix [12] 13] [14], 15] 16] 17] and many of them are recursive [12] 14] 16] 17] Besides one dimensional algorithms, two dimensional DCT algorithms have also been investigated extensively [6] 18] 19] 20] 21] generally leading to less computational complexity than the row column application of ....

.... fast transforms, including the FFT [1] 6] 7] 8] the Walsh Hadamard transform (WHT) 9] 10] and the discrete Hartley transform (DHT) 11] Some algorithms are based on the sparse factorizations of the DCT matrix [12] 13] 14] 15] 16] 17] and many of them are recursive [12] [14], 16] 17] Besides one dimensional algorithms, two dimensional DCT algorithms have also been investigated extensively [6] 18] 19] 20] 21] generally leading to less computational complexity than the row column application of the 1 D methods. However, the implementation of the direct 2 D ....

[Article contains additional citation context not shown here]

Z. Wang, \Fast algorithm for the discrete w transform and for the discrete fourier transform," IEEE Trans. ASSP, vol. ASSP-32 (4), pp. 803-816, Aug. 1984.

....the reasons to look for preconditioners which can be diagonalized by trigonometric matrices corresponding to fast trigonometric transforms instead of the Fourier matrix F N . In practice, four discrete sine transforms (DST I IV) and four discrete cosine transforms (DCT I IV) were used (see [21]) Any of these eight trigonometric transforms can be realized with O(N log N) arithmetical operations. Likewise, we can define preconditioners with respect to any of these transforms. In this paper, we restrict our attention to the so called discrete cosine transform of type II (DCT II) and ....

Z. Wang. Fast algorithms for the discrete W transform and for the discrete Fourier transform. IEEE Trans. Acoust. Speech Signal Process, 32:803 -- 816, 1984.

....more zeros result in potentially faster IDCT implementations as operations involving zeros can be skipped. For example an all zero input (i.e. Class 0) would require no operations while a Class 255 input would require the maximum number of operations. We first select the Chen Wang (CW) algorithm [14] 3 as our baseline algorithm. For each input class one can find an exact IDCT implementation with minimum number of operations which will be referred to as the reduced IDCT version for a given class. The baseline IDCT (see Fig. 1) structure involves several stages at which groups of 2, 4 and 8 ....

....involved (additions, multiplications, logical operations, etc) in our optimization. Fig. 3 indicates that the experimental results in (b) are sufficiently close to those predicted by the model (a) 4 . In Fig. 3 all the values are normalized by the complexity of the baseline CW algorithm [14] without any zero tests. Note that significant reductions in complexity are possible. Other than our optimized algorithm and FW [12] we give results for the CW algorithm with all zero test, and CW with all zero test for the first 1 D IDCT and aczero test for the second 1 D IDCT (since after the ....

Z. Wang, "Fast algorithms for the discrete w transform and for the discrete fourier transform," IEEE Trans. on Signal Proc., vol. ASSP-32, pp. 803--816, Aug. 1984.

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Z. Wang. Fast Algorithms for the Discrete W Transform and for the Discrete Fourier Transform. IEEE Trans. on Acoustics, Speech, and Signal Processing, ASSP-32(4):803--816, 1984.

No context found.

Z. Wang. Fast Algorithms for the Discrete W Transform and for the Discrete Fourier Transform. IEEE Trans. on Acoustics, Speech, and Signal Processing, ASSP-32(4):803--816, 1984.

No context found.

Z. Wang. Fast algorithms for the discrete W transform and for the discrete Fourier transform. IEEE Trans. Acoust., Speech and Signal Process., vol. ASSP-32, pp. 803-816, Aug. 1984.

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Z. Wang, "Fast algorithms for discrete W transform and for the discrete Fourier transform," IEEE Trans. on Acoust., Speech, Signal Processing, vol. 32, pp. 803--816, Nov. 1984.

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Zhongde Wang, Fast Algorithms for the Discrete W Transform and the Discrete Fourier Transform, IEEE Transactions on Acoustic, Speech, and Signal Processing, Vol. ASSP-32, No. 4, 1994

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Wang, Z. "Fast algorithms for the discrete W transform and for the discrete Fourier transform," IEEE Trans. Acoust., Speech, Signal Process. vol. ASSP-32, pp.803-816, 1984.

No context found.

Z. Wang. Fast Algorithms for the Discrete W Transform and for the Discrete Fourier Transform. IEEE Trans. on Acoustics, Speech, and Signal Processing, ASSP-32(4):803--816, 1984.

No context found.

Z. Wang. Fast algorithms for the discrete W transform and for the discrete Fourier transform. IEEE Trans. Acoust. Speech Signal Process, 32:803 -- 816, 1984.

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Z. Wang, "Fast algorithm for the discrete W transform and for the discrete Fourier transform," IEEE Trans. Accoust., Speech, Signal Processing, vol. ASSP-32, pp. 803--816, Aug. 1984.

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Z. Wang: "Fast Algorithms for discrete W transform and for the discrete Fourier Transform", IEEE Transactions on Acoustic, Speech, and Signal Processing, Vol. 32, No. 8, pp.803-816, 1994.

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Z. Wang,"Fast Algorithms for the Discrete W Transform and the Discrete Fourier Transform", IEEE Trans. on ASSP, Vol.ASSP-32, pp.803-816, Aug. 1984.

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