IEEE TRANSACTIONS ON ACOUSI'ICS. SPEECH, AND SIGNAL PROCESSING. VOL. 37. NO. 2, FEBRUARY 1989 231 Input and Output Index Mappings for a Prime-Factor- Decomposed Computation of Discrete Cosine Transform Abstract-This paper provides a direct derivation of the prime-fac- work and time, still providing a simple and nice structure. tor-decomposed computation algorithm of an N-point discrete cosine However, this technique has not yet been widely utilized transform for the number N decomposable into two relative prime numbers. It also presents input and output index mappings in the form mainly because its input and output index mappings are of tables-namely, ri-, it-, nc-, nK-, and k-tables. The index mapping seemingly too involved. In fact, the mappings are the only tables are useful for practical use of the prime-factor-decomposed barrier to overcome in applying the prime-factor algo- computation of arbitrarily sized discrete cosine transforms. rithm. This paper is therefore intended to provide a simple and organized method to perform the index mappings. In this paper, a formal direct derivation of the prime- I. INTRODUCTION factor-decomposed computation algorithm will be pre- INCE its first introduction in 1974 [l], the discrete sented first. The derivation is a direct one in the sense that Scosine transform (DCT) has found applications in it is based on the real cosine function without resorting to speech and image signal processing [1]-[8] as well as in the DFT expressions or the complex functions. Then, telecommunication signal processing [9], [lo]. The DCT based on the equations obtained during the derivation, in- has been applied for speech and image compression be- put and output index mappings will be introduced in the cause its performance was nearly optimal, yet not being form of tables. This tabulation will enable us to imple- signal dependant. On the other hand, the DCT has been ment any prime-factor-decomposable DCT in a straight- utilized for realizing filter banks in FDM-TDM transmul- forward manner. Finally, the index mapping tables will tiplexers because its real computation was simpler and be demonstrated through the 12-point DCT. faster than the complex computation of the discrete Fou- 11. DIRECTDERIVATION OF PRIMEFACTOR rier transform (DFT). DECOMPOSITION Along with the expanded applications of the DCT, a number of fast computation techniques have been also in- Let x(k), k = 0, 1, - , N - I, be a time-domain troduced [ 111-[ 191. Depending on the number of points sequence and X(n),n = 0, 1, * * - , N - 1, be its trans- N, the computation techniques can be divided into two form-domain data sequence. Then, by definition, the DCT categories: one on the general composite number cases, and the inverse DCT (IDCT), respectively, have the and the other one on the prime-factor cases. For the for- expressions mer case, N of special interest is of 2" type; for the latter 2 N-l case, N is factorizable into two mutually relative prime X(n) = - e(n) c x(k) cos [x(2k + l)n/2N], numbers NI and N2. A recent work reports that the number N k=O of real multiplications for the power-of-two case can re- n = 0,1, **. 9N-1, (1) duce to (N/2)log N, and its structure resembles that of N- 1 the fast Fourier transform (FFT) [ 141. For the prime-fac- x(k) = c e(.) X(n)cos [a(2k + 1) n/2N], tor case, the number of multiplications reduces to N(NI n=O + N2) in its most primitive form, and its structure is sim- k = 0, 1, -*a ,N- 1, ilar to that of the prime-factor algorithm of the DFT [ 191. (2) The prime-factor-decomposed computation of the DCT where was proven to be powerful in reducing the computational [:fa, if n = 0, e(.) = (3) otherwise. Manuscript received September 12. 1987: revised March 5, 1988. The author is with the Department of Electronics Engineering. Seoul Since (1) can be realized simply by transposing the National University. Seoul, 151-742. Korea. IEEE Log Number 8825 133. flowgraph for (2), and since the term e(n)means nothing 'On the prime-factor algorithm of DFT, refer to [20]-[22]. but a slight modification of the data X(n),it is sufficient 0096-35 18/89/0200-0237$01 .OO O 1989 IEEE Authorized licensed use limited to: Seoul National University. Downloaded on February 17,2010 at 03:36:48 EST from IEEE Xplore. Restrictions apply. 238 IEEE TRANSACTIONS ON ACOUSTICS. SPEECH, AND SIGNAL PROCESSING, VOL 37. NO 2. FEBRUARY 1989 for our discussion to consider the IDCT-like equation’ for all nl in XI and all n2 in X2. We denote by & and N- 1 $7, sets of N integers such that x(k) = c X(n) cos [7~(2k+ 1) n/2N], I1 = 0 * = {nln =.f(nl, n2), nI E x1, n2 E ~2). k = 0, 1, . , N - 1. (4) (9a) Throughout this paper we will assume that $7, = {.In =f(nl, n2), nl E rtl, n2 E rt2). N = NlN2, (5) (9b) where NI and N2 are mutually prime integers. Suppose we Then it can be shown that the 2 N integers in the collection could decompose the N-point IDCT in (4)into the cascade of fi and $7, are identical to the 2 N integers in the collec- of N2 NI-point IDCT’s and NI N2-point IDCT’S.~Then, tion of 32 and 32.5This implies that a summation over N the expression for the resulting decomposed transform indexes in X can %plitinto two terms-a summation over would be of the form the N indexes in 3t and a summation over N indexes in $7,. Therefore, we can rewrite (4) as follows: x(k) = 1/2 X(n)cos [7r(2k + 1) n/2N] ncX * COS [7r(2kl + 1) nl/2NI] + 1/2 X(n) COS [a(2k + 1) n/2N]. nE31. COS [n(2kz + 1) n2/2N,] , (6) 1. fork, = 0, 1, . * * , NI - 1, and k2 = 0, 1, . , Nz - We denote, for all (nl, n2)in 32, X 322, 1. The main goal of this section is in deriving (6) from (4) n2) = 44 x(4 I n=f(n,,n*)’ by finding two appropriate mappings: the input mapping connecting X(n), n = 0, 1, * * * , N - 1, to X(ni, n2), * IZI = 0, 1, . * * ,NI - 1, n2 = 0, 1, , N2 - 1, and where the output mapping connecting x(k),k = 0, 1, * . - , N - 1, tox(kl, kz), kl = 0, 1, . ,NI - 1, k2 = 0, 1, 1, if nlNz + n2N, < N, ... , N2 - l.4 The input and output mappings are di- s(n) = - 1, otherwise. rectly tied with the input and output index mappings among the corresponding indexes. Then (10) can be rewritten as We first consider the input mapping which connects N2-1 NI-1 X(n) to X(n1, nz). x(k) = 1/2 c {2(n1,n2) Let 32 denote the set of N integers 0 through N - 1. nz=O n1=0c Similarly, let XIand X2,respectively, denote the sets of * COS [~(2k+ l)(nlN2 + nzNI)/2N] NI integers 0 through NI - 1 and the set of N2 integers 0 through N2 - 1. We define .f and f to be mappings from + qn,, n2) cos [a(2k + 1) Ttl x rt2to 32 such that * (W2 - n,”)/2Nl]. (13) nlN2 + nzNl, ifnlN2 + n2NI < N, hl?n2) = The term s(n)reflects the negative sign appearing in the - otherwise, 2N (nlNz + nzNl), relation COS [a(2k + 1)(2N - n)/2N] = -COS [7~(2k+ 1) n/2N]. (14) We do not need such a term for the second part of (lo), since 1 n1N2 - n2Nl1 < N for all (nl, nz)in 321 X ‘32. We now define X(nl,nz) such that qn,,n2) = qn1, 41, ‘It should be noted that the transposed flowgraph of DCT performs the IDCT function (with e(n) related coefficient modification), while the X(n,, n2) = if n1 = 0 orn2 = 0, (15) transposed flowgraph of DFT still performs the same DFT function. Refer to the arrow marks in Fig. 1 to appear. X(nl, n2) + X(n,, n2), otherwise. ‘Thc term IDCT we encounter hereafter will denote the IDCT in the sense of (4). ‘We name the former input mapping and the latter output mapping, even ‘A proof of this is given in Appendix A. The term collection here in- though the meaning of input and output could be reversed for the forward dicates the set obtained by listing all the elements in two sets. See footnote DCT. 11. Authorized licensed use limited to: Seoul National University. Downloaded on February 17,2010 at 03:36:48 EST from IEEE Xplore. Restrictions apply. LEE: INPUT AND OUTPUT INDEX MAPPINGS 239 Then, since cos [7r(2k + 1) n,/2N2] = cos [7r(2k, + 1) n,/2Nz]. Nz-1 NI-1 ( 22b 1 c c g(nl, n2)cos [a(2k + 1) 112’ I ni = 1 Therefore, we have shown that (4)and (6) are identical if X(n) and X(nl,n2) are connected through (7), (II), - (nlN2 + n2Nl)/2N] = 0, (16a) and (15), and if x (k) and x( kl, k2) are connected through N2-1 NI-1 (20) and (21). The former three equations form the input c c X(nl, n2)cos [7r(2k + 1) nz=l ni=l mapping; and the latter two equations form the output mapping. Thus, we can now perform an N-point IDCT by cascading N2 N,-point IDCT’s and NI N2-point IDCT’s, equation (13) can be written in the form as is demonstrated in Fig.