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1004 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-25, NO. 9, SEPTEMBER 1977

After his military servicehe joined N. V. Peter W. Millenaar was born in Jutphaas, The Philips’ Gloeilampenfabrieken, Eindhoven, Netherlands, on September 6, 1946. He re- where he worked in the fields of electronic de- ceived the degreein electrical engineering at sign of broadcast radio sets, integrated circuits the School of Technology, Rotterdam, in 1967. and optical character recognition. Since 1970 In 1967 he joined the Philips Research Lab- he hasbeen atthe PhilipsResearch Labora- oratories, Eindhoven, The Netherlands, where tories, where he is now working on digital com- he worked in the field of data transmission and munication systems. on the synchronizationaspects of a videophone Mr. Roza is a member of the Netherlands system. Since 1972 he has been engaged in sys- Electronics and Radio Society. tem design and electronics of high speed digital transmission.

Concise Papers

A Fast Computational forthe Discrete Cosine elements to a form which preserves a recognizable bit- Transform reversedpattern at every other node. The generalization is not unique-several alternate methods have been discovered- but the method described herein appears bethe simplest WEN-HSIUNG CHEN, C. HARRISON SMITH, AND S. C. FRALICK to to interpret.It is not necessarily themost efficient FDCT which could be constructed but represents one technique for Abstruct-A Fast DiscreteCosine Transform algorithm hasbeen methodical extension. The method takes (3N/2)(log2 N- 1) + developed which provides a factor of six improvement in computational 2 real additions and N logz N - 3N/2 -t 4 real multiplications: complexity when compared to conventional Discrete Cosine Transform this is approximatelysix times as fast as the conventional using the Fast . The algorithm is derived approach using a double size FFT. in the form of matrices and illustrated by a signal-flow graph, which may be readily translated to hardware or software implementations. DISCRETE COSINE TRANSFORM INTRODUCTION The discrete cosine transform ofa discrete functionfc), j = The Discrete Cosine Transform (DCT) has been successfully 0, 1, *-, N - 1 is defi.ned as [ 1 ] applied to the coding of high resolution imagery [ 1-51. The conventionalmethod of implementingthe DCT utilized a double size Fast Fourier Transform (FFT) algorithm employing complex arithmetic throughout the computation [ 11. Use of theDCT in a widevariety of applications has not been as k = 0, 1, -., N extensive as its properties would imply due to the lack of an efficientalgorithm. This report describes a moreefficient and the inverse transformis algorithminvolving only real operations for computing the Fast Discrete Cosine Transform (FDCT) of a set of N points. N-1 (2j + 1)kn The algorithm can be extended to any desired value of N = 2”’, m 2 2. The generalization consists of alternating cosine/ k=O [ 2N 1’ sinebutterfly matrices with binary matrices to reorder the j = 0, 1, .-, N Paper approved by the Editor for Communication Theory of the where IEEE Communications Society for publication without oral presenta- tion. Manuscript received January 24, 1977. 1 W.-H.Chen was with the Western Development Laboratories Divi- c(k) = 3 for k = 0 sion, Ford Aerospace and Communications Corporation, Palo Alto, CA 94303. He is now with Compression Laboratories, Inc., Campbell, CA 95008. =1 fork = 1, 2, -., N - 1. C. H. Smith is with the Western Development Laboratories Division, Ford Aerospace and Communications Corporation, Palo Alto, CA Thetransform possesses a highenergy compaction property 94303. which is superiorto any known transform with a fast S. C. Fralick was with the Western Development Laboratories Divi- computational algorithm. [ 1-51 The transform also possesses a sion, Ford Aerospace and Communications Corporation, Palo Alto, CA 94303. He is now with Comtech Advanced Systems, Inc., Sunnyvale, circularconvolution-multiplication relationship which can CA 94086. readily be used in linear system theory.[6] CONCISE PAPERS 1005

A FAST COMPUTATIONAL ALGORITHM where The discrete cosine transform of an N x 1 data vector [f] N can be expressed in a matrix form as [IN/21 is an identity matrix of order - 2

[jN/21 is the opposite diagonal identity matrix

kn where [AN] = [c(k)cos (2j + l)kn/2N]; j, k = 0, 1, -*, (N - [Sik]= sin - [I,,,,,~] 1) as defined in equation (1) and [ F] is the N X 1 transformed 2 vector.The fast computational algorithm to be presented hereis based upon the of the [AN] matrix. As shown below, this matrix can first be written into the following recursive form:

where [BN 1 is defined in equation (7), (8)

In equations (8) above, the identity matrices [IN/2i] specify the order of the diagonal sine or cosine matrices [Sik],[gik 1, [Cik],[Fik] with the condition that [IN/zi]E 1 for i > N/2. N The four types of matrices may now be described in detail j, k = 0, 1, -., - - 1 L with reference to the right hand sideof equation (9).

and [PN]is an N x N permutation matrix which permutes the TYPE 1 transformed vector from a bit reversed order to a natural order. As in all unitary transforms the 2 x 2 DCT can be written as The firstmatrix [MI]is formedby concatenating S2Naj matrices(of order 1) along the upper left to middle of the main diagonal and C2Naj matrices along the middle to lower right.The opposite diagonal is formed by C2~’jmatrices along the upper right to middle and 32~’jmatrices along the It can be seen from the recursive nature of equation (4) that middle to lower left. For this type matrix the values of ai are [A2] canbe extended into higher order matrices as longas the binary bit-reversed representation of N/2 + j - 1 for j = there is ageneralized method of decomposingthe [RN/2] 1, 2, *..,NJ2. matrix. The followingdiscussion presents one systematic way of decomposingthe [RN/2] matrix.It is emphasized that this TYPE 2 method of decomposition is not unique and is not optimum. The last matrix [M(2 log2 N - 3)] is formed by concate- Several methods havebeen found which require fewer nating zN/8, - zN/8 matrices along the upper leftto computationalsteps but with no apparent generalization to c41,c41, lowerright of the main diagonal and concatenating 0~18, larger sizes. C4l, F4l, ON/8 matrices along the upper right to lower left The [RN/2] matrix isdecomposed into (2 log2 N - 3) of the opposite diagonal.* matrices in the following manner:

TYPE 3

The matrices areof four distinct types. The remaining odd matrices [Mq] are formed by repeated k. concatenation of thematrix zN/2i, - cikj,- si I Type 1 : [Ml ] ,the first matrix and Z~/ziwhere i = N/(2(q-1)/2) for j = 1, 2, ..*, i/8 along Type 2: [M(2 log2 N - 3)],the last matrix the upper left to middle of the main diagonal and the matrix Type 3:[Mql , theremaining odd numbered matrices sequence zN/2i. cikj,sikj and zN/2i for j = iJ8 + 1, -*, i/4 [M31, [MS],etc. alongthe middle to lowerright. The opposite diagonal is Type 4: [Mpl,the evennumbered matrices [M21, [M4], formedsimilarly, using the matrix sequence ON/2i, Sik’, - etc. Fi’j, ON/2i along the upper right to middle and the matrix Beforedescribing the four types of matricesin detail, the sequence ON/2i, - Sikj, Ciki,ON/2i alongthe middle to following definitions are provided for notational efficiency: lowerleft. Repeated concatenation of amatrix sequence alonga diagonal is clearly illustrated in equation (9), where for clarity the havebeen replaced by and etc., zN/2 zN/2 kj’s bj’s ti's, BN =[- becausethe value of kj depends on the matrix index q. For IN12 --IN12 ] this type matrix, thevalues of the k, are the binary bit-reversed variables (i/4) + j - 1. BN*=[- -IN/, IN12 IN12IN12- ] * Oj is a null matrix of order j. 1006 TRANSACTIONSIEEE ON COMMUNICATIONS, VOL.COM-25, NO. 9, SEPTEMBER 1977 - al -al - S2N a 2n 2 '2N . 0 SaN/4 caN/4 ' 2N 2N 2N

a' - N/2-1 -S CaN12-1 2N 2N B2 -aN/2 CaN/2 -9 2N 2N.

0 bl I2 c -C -c 1 - c1 NI2 Nl4 'n14 1 c1 -S bi -S -C- N/2 n14 n14 1 0 12. 0 .. ..- :

0- -5bN/8 N/2 'b - 118 ,bNi8 Nl2 NI2

B8

B8 * B8 . CONCISE PAPERS 1007

TYPE 4 occupiedhalf are oppositethe diagonalson log2 and N - stages are fully occupied on both diagonals. Therefore The evennumbered matrices [Mp] arebinary matrices formed by alternating Bl and Bl* matrices along the upper left N N KRN i2 = (10g2 N - 2) + -(log2 N - 1) to lower right of the main diagonal. The subscript 1 indicates 7 2 the order of the B or B* matrix, and takes on the value of 2F.12. -_3N (1 24 A specificexample of [RN/2]for N = 16 is shownin Eq. - 4 log2 N- N N>4. (10).

.1 1.1

11

3n 3n 1 -sir 1 32 co"32 ["3 = 1In In 1-.1 -sir1 cor 32 32 7n 7n -1 1 -sin- 32 co"32 151-1 11 -si- 32 -- 0 1 1

s i+ 1 1 n - '05 1 -1

0 1 -1

1 -1 3Tl cos- -1 1 8 3n COT 1 1

1 1 dL

0

1 0 n n -coq coq

nll -cor44 cor nn cos- coy 4 n n coq co"i: 1 I 0 lo 1 The computational steps required for [F]of equation (3) As for the number of multiplications, only the odd matrices can be found from equation (4) with the following recursive consistof multiplicative terms. In these matrices the first relations: matrixconsists of N multipliers,the last matrix consists of (1 la) N/4 multipliers, and the rest of the log; N - 3 matrices each KA~= -k KAN/2 + KR~/2 consists of N/2 multipliers. Thus KA~'=KA~~; 'KR~/2' 1b) (1 NN whereKAi and are the number of additionsand KR~,~'= N + -- -(log2 N - 3) multiplicationsfor [qi],and KR~and KRi'numberthe are 42 of additionsand multiplications for Thenumber of [Ri]. N N additions for [RN!2] can easily be determined from equation =-logzN--; N28. (9) by noting that log2 N - 2 stages of decomposed matrices 2 4 1008 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-25, NO. 9, SEPTEMBER 1917

5000 - - 5000 - -

1000 1000 - - - ConventioMl 500 - 500 - .a - 0 4 u - a - 3 3 a 4 4 - u - ze 3 "4 v

5 100 - 8. 100 - E P - I - 50 50 ------

10 10 - - - - I I I I I I 1 I 4 8 16 32 64 128 4 8 16 32 b& 128 Transform Size Transform Size (a) (b) Figure 1. Comparison of Computational Steps for Conventional FDCT, FFT and FDCT. (a) Additions. (b) Multiplications.

natural order from top to bottom. For every N, the output transform coefficients are in bit-reversed order. Note that as N increasesthe even coefficients of each successive transform are obtained directly from the coefficients of the prior transform by doubling the subscript of the prior coefficients.

It~~ can be seen that extensionof the signal-flow graph to the next power of 2 merely involves adding a set of +1 butterflies to accommodate the new set of input samples and a series of alternatingcosine/sine butterflies and +1 butterflies to yield the new set of odd transform coefficients. In Figure 2, the coefficients have not been normalized. TO obtain.the normalized N-point DCFT coefficients, the appropriateterminal points of theflow graph of Figure2 shouldeach be multiplied by 2/N. This signal-flowgraph representsthe forward transform matrix [AN]. The inverse For purposes of comparison the conventional approach of transform matrix [AN]-1 is simply (N/~)[AN'IT. computing the DCT utilizing an FFT takes 2N log 2N complex 'Thus, except for .normalizatiop factor these FDCT signal- additions and N(1og 2N + 1) complex multiplications (which flow .graphs are bidirectional, i.e., the inverse transform may are equivalent to 6N log 2N -I- 2N additions and 4N(log 2N-k be computed by introducing the vector w]at the output and 1)real multiplications). Figure 1 plotsthe number of recovering the vector [f] at the input. This follows from the computationalsteps versus the transform sizes for'the factthat every butterfly pair in the signal-flow graph is a conventionalalgorithm and the' algorithm presented in this unitary matrix except for a normalization factor.' paper. It can easily be seen that the algorithm presented here takes less than 1/6 as many steps as the conventional algorithm. SUMMARY Also plotted in the figure are the computational steps forFFT. It can be seen that the new algorithm takes only 1/3 as many A FastDiscrete Cosine Transform (FDCT) algorithm has steps as the FFT. been developed which may be extended to any desired value Figure 2 is a signal-flow graph forN = 4, 8, 16, 32 arranged of N = 2m 2 2. The algorithm has been interpreted 'in the in the fashion described. Note that the input samples are in form of matricesand illustrated by a signal-flow graph. The CONCISE PAPERS 1009

= 32

signal-flow graphmay be readily translated to hardware(or Incoherent Adaptive Reception of Signal with Unknown softw are) implementation.software) computational Theof number Envelope steps has been shown to be less than 1/6 of the conventional DCT algorithmemploying a 2-sided FFT. SOLOMON M. FLEISHER,MEMBER,IEEE

ACKNOWLEDGMENT Absfracf-Thispaper deals with the designof adaptive algorithms for reception of a narrowband signal with an unknown envelope in a The authors are deeply indebted to Miss J. Del Mastro for noisy channel (Gaussian noise). We here consider a system (with a “real her patience and skillin typing this paper. teacher”) which learns from the samples classified by this self-learning system(decision directed adaptive receiver). By usingthose samples which are accepted as learning samples, the parameters of the unknown REFERENCES envelope are estimated. The envelope’s parameters appear in the form of coefficients of the generalized expansion of the signal 1. N. Ahmed,T. Natarjan, and K. . Rao,“Discrete Cosine Trans- (withrespect to eigenfunctions of appropriateintegral equation). It form,” IEEE Transacfionson Computer, January 1974, pp. 90-93. is possible to utilize any orthonormal set with respect to the interval 2. A. Habibi, “Hybrid Coding of Pictorial Data,” IEEE Transactions on (0, T) under the usualassumption, that the complexautocovariance Communications, Vol. COM-22, No. 5,May 1974, pp.614-624. 3. J. A.Rose, W. K. Pratt, G. S. Robinson, and A. Habibi, “Inter- function is R(T)= N~(T)(i.e., that the noise bandwidth is much greater frame and Predictive Coding Methods,” Proceed- than both l/T and the signal bandwidth and N is the unilateral spectral ings of KC 75, IEEECatalog No. 75CH0971-2GSCB, pp. 23.17- density of the noise in the neighborhood of the signal spectrum). We 23.21. present expressions that enable the upper bound estimates of the error 4. W. Chen and C. H. Smith, “Adaptive Coding of Color Images Using probability to be found for the derived algorithms. The results obtained CosineTransform,” Proceedings of ICC 76, IEEECatalog No. for the binary detection are readily generalized to the case of an M-ary 76CH1085-0 CSCB, pp. 47-7 to 47-13. signal. 5. J. R. Persons and A. G. Tescher, “An Investigation of MSE Contri- butions in TransformImage Coding Schemes,” Proceedings of SPIE, August 1975, pp. 196-206. Paperapproved by the Editor for Communication Theory of the 6. W. Chenand S. C.Fralick, “Image Enhancement UsingCosine IEEE Communications Society for publication without oral presenta- TransformFiltering,” Proceedings of theSymposium on Current tion. Manuscript received June 1, 1976;revised January 30, 1977. Mathematical Problems in Image Science, Montery, Ca., November The author is with the School of Education, Center for Technologi- 1976, pp. 186-192. cal Education, Tel Aviv University, Holon, Israel.