On Fast Algorithm for Computing Even-Length DCT

1 On F ast Algorithm for Computing Ev en-Length DCT Y uriy A. Reznik Abstract W e study recursive algorithm fo r comp uting DCT of length s N = q 2 m ( m, q ∈ N , q is odd) due to C. W . K ok [16]. W e show that this algorith m has the same mu ltiplicative complexity as theoretically achiev able b y the prime factor d ecompo sition, when m 6 2 . W e also show th at C. W . K ok’ s factorization allows a simple con version to a scaled fo rm. W e analyze complexity of suc h a scaled factorization , and show that fo r som e leng ths it achieves lower multiplicative complexity than one of kn own prime factor-based scaled transfor ms [14]. Index T erms Discrete cosine transform, DCT , scaled tra nsform, factorization, multiplicative complexity . I . I N T RO D U C T I O N The discrete cosine transfor m (DCT) [1]–[3] is a fund amental a nd frequently us ed ope ration in modern digital signal p rocessing . It finds ap plications in da ta compress ion, filter design , image recognition, e tc. Scaled DCT is a mo dified version of this trans form, allowing the output to b e scaled in way that s implifies its comp utation [6]. Scaled DCTs are particularly popular in data c ompression, wh ere sca ling of DCT output ca n usua lly b e don e jointly w ith qua ntization, therefore reducing the complexity of the entire algorithm [11], [12]. Since its disc overy in early 1970s, DCT has been a subjec t of extensiv e research, focus ing, in part, on the design of fast algorithms for its computation [2], [3]. The class o f DCT of type II (DCT -II) with dyad ic lengths N = 2 m , m ∈ N has been s tudied particularly we ll. Both the oretical complexity estimates [4]–[6] and a number of ef ficient algo rithms for their construction hav e been de ri ved [7]–[10], The author i s with Qualcomm Inc., S an Di ego, CA, 92121 USA; e-mail: yreznik@qualcomm.com. December 29, 2009 DRAFT 2 [13]. The construction of scaled DCT -II of dyad ic leng ths ha s a lso been studied [6], [11], [12]. Scaled factorizations of Y . Arai, T .A gui and M. Nakajima [11] and E . Feig an d S. W inograd [6] are among best-known algorithms from this class. The con struction of odd -length transforms has b een s tudied by M. He ideman [17], and S. Cha n and K. Ho [18], uncovering, in p art, a n elegant c onnec tion b etween real valued DFT and DCT -II o f same len gths. T he construction of DCT of composite sizes, such a s N = p q , where p and q are c o-prime, was s tudied by P . Y an g and M. Narasimha [19], B.G. L ee [20], and others, resulting in the development of the prime-factor de compos ition of the D CT -II. E. Feig an d E. Linz er have further exten ded the prime-factor technique for c omputing sc aled transforms [14]. The cons truction of DCT of ev en (but not dyadic) lengths has been addresse d b y a variety of techniques , ranging from prime- factor de compositions [19], [20] to generaliza tions of the radix- 2 DCT algo rithm [16]. Re ference [16 ] contains comparison of several such approa ches. In this c orresponde nce we take a nother look at the recursive algorithm for c omputing of DCT of lengths N = q 2 m , m, q ∈ N , proposed by C. W . Kok [16]. W e off er a n a lternati ve matrix formulation of this algorithm, its detailed complexity a nalysis, a nd a modification allowi ng to compute a sca led DCT . W e s how , that C. W . K ok’ s algo rithm ach iev es the same multiplicati ve comp lexity as one theoretically attainable by the p rime factor dec omposition w hen m 6 2 . W e also show that for some leng ths our proposed scaled version of C. W . Kok’ s algorithm achieves lower multiplicati ve complexity than one of known sca led prime factor a lgorithm-based factorizations [14]. W e accomp any our pres entation with several examples of s caled factorizations co nstructed by using de scribed a lgorithms, an d c omplexity comparison plots that can be of interes t to the enginee ring commu nity . This correspond ence is or ganized as follo ws. In Se ction II , we introduce no tation and survey relev ant results. In Section III, we offer matrix formulation of C. W . Kok’ s algorithm and its complexity an al- ysis. An mod ified (sc aled) version of C. W . Kok’ s factorization is d escribed in Sec tion IV. Section V contains comparison with prime factor- base d implementations . Section VI brings remarks on normalized multiplicati ve complexity of comp osite-length trans forms. Conclus ions are d rawn in S ection VII. December 29, 2009 DRAFT 3 I I . N OTA T I O N A N D S O M E B A S I C F AC T S By C I I N , C I I I N and C I V N we will den ote matrices o f N -point DCT -II, DCT -III, and DCT -IV trans forms correspond ingly 1  C I I N  n,k = cos  π (2 n +1) k 2 N   C I I I N  n,k = cos  π n (2 k +1) 2 N   C I V N  n,k = cos  π (2 n +1)( k +1) 4 N             n, k = 0 , ..., N − 1 . Among these transforms, the DCT of type II (DCT -II) is the on e that we will need to comp ute. It is well known (see , e.g. [2]), that the DCT -III is simply an in verse (or trans pose) of DCT -II C I I I N =  C I I N  T =  C I I N  − 1 , and that DCT -IV is in v olutary (self-in verse, se lf-transpose) C I V N =  C I V N  T =  C I V N  − 1 . It is also kn own (cf. S. C. Chan a nd K.L. Ho [15], C. W . K ok [16]), that DCT -IV and DC T -II a re connec ted as follows C I V N = R N C I I N D N , (1) where R N is a matrix of recursive subtractions R N =            1 2 0 0 . . . 0 − 1 2 1 0 . . . 0 1 2 − 1 1 . . . 0 . . . . . . . . . . . . . . . − 1 2 1 − 1 . . . 1            , and D N is a diago nal matrix D N =         2 cos  π 4 N  0 2 cos  3 π 4 N  . . . 0 2 cos  (2 N − 1) π 4 N          . 1 For simplicity , we omit normalization factors [2]. December 29, 2009 DRAFT 4 Further , it is known (cf. W . H. Chen, et. a l. [7], Z. W ang [8]) tha t if length of DCT is even, then it can be factored into two ha lf-length transforms C I I N = P N   C I I N/ 2 0 0 C I V N/ 2 J N/ 2   B N , (2) where P N is a permutation matrix, produc ing reorde ring ˜ x i = x 2 i , ˜ x N/ 2+ i = x 2 i +1 , i = 0 , . . . , N / 2 − 1 , B N is a butterfly B N =   I N/ 2 J N/ 2 J N/ 2 − I N/ 2   , and where I N/ 2 and J N/ 2 denote N/ 2 × N / 2 ide ntity and order reversal matrices correspond ingly . I I I . R E C U R S I V E D C T - I I C O M P U TA T I O N . A L G O R I T H M O F C . W . K O K [ 1 6 ] W e note that factorization (2) is n ot fully rec ursiv e: it us es DCT -II and DCT -IV of leng th N / 2 as buil ding blocks, but sub seque nt factorization of DCT -IV is not defined. One poss ible way of closing the recursion is to simply replace DCT -IV with D CT -II in ac cordance with (1). This way we arriv e at the follo wing factorization: C I I N = P N   C I I N/ 2 0 0 R N/ 2 C I I N/ 2 D N/ 2 J N/ 2   B N . (3) This factorization can be app lied recursi vely , producing a s imple algorithm for comp uting of DCT -II of ev en lengths, known as C. W . K ok’ s algorithm [16]. W e s how the flowgraph of this a lgorithm in Fig. 1 . As cus tomary , dashed lines in the flowgraph denote s ign inv ersions, circles indicate additions, and c onstants a bove lines indica te multiplications by the correspo nding factors. Based on Fig. 1, it can be ob served that the numbe rs of multiplications µ ( N ) , add itions and subtrac- tions α ( N ) , and shifts (multiplications by dya dic factors) σ ( N ) sa tisfy µ ( N ) = 2 µ ( N/ 2) + 1 2 N , α ( N ) = 2 α ( N/ 2) + 3 2 N − 1 , σ ( N ) = 2 σ ( N / 2) + 1 . By applying this deco mposition rec ursiv ely m -times, we arri ve at the follo wing result (cf. [16]). December 29, 2009 DRAFT 5 3 N X − / 2 N x / 2 1 N x + 2 N x − 1 N x − 1 X 3 X 1 N X − 4 N X − 0 x 1 x / 2 2 N x − / 2 1 N x − 2 N X − 0 X 2 X 1 2 2 2 cos N π ⋮ ⋮ ⋮ ⋮ 3 2 2 cos N π ⋮ ( 1) 2 2 cos N N π − ( 2 ) 2 2 cos N N π − ⋮ ⋮ N/2 - point DCT-II N/2 - point DCT-II Fig. 1. C. W . Kok’ s factorization of ev en-length DCT -II [16]. 3 N X − / 2 N x / 2 1 N x + 2 N x − 1 N x − 1 X 3 X 1 N X − 4 N X − 0 x 1 x / 2 2 N x − / 2 1 N x − 2 N X − 0 X 2 X 1 2 2 2 cos N π ⋮ ⋮ ⋰ ⋮ ⋮ 3 2 2 cos N π ⋮ ( 1) 2 2 cos N N π − ( 2) 2 2 cos N N π − ⋮ ⋮ N/2 - point DCT-II I (or S caled DCT-I I) N/2 - point DCT-I I Fig. 2. Proposed alternati ve factorization of ev en-length DCT - II. Proposition 1 (C. W . Kok, 1 997) . The n umbers of arithmetic operations ( µ, α, σ ) n eeded for computing DCT -II of length N = q 2 m using C. W . K ok’ s algorithm, satisfy: µ ( N ) = 2 m µ ( q ) + m 2 N , (4) α ( N ) = 2 m α ( q ) + 3 m 2 N − 2 m + 1 , σ ( N ) = 2 m σ ( q ) + 2 m − 1 . December 29, 2009 DRAFT 6 I V . P RO P O S E D A L T E R NA T I V E ( S C A L E D ) F AC T O R I Z A T I O N Consider DCT -II f actorization (2) one more time. Since DC T -IV is in volutary , we can compu te it in a transposed fashion, producing (cf. (1)): C I V N =  R N C I I N D N  T = D N C I I I N R T N . (5) By plugging this expression in (2), we arri ve at the follo wing alternative decimation sche me: C I I N = P N   C I I N/ 2 0 0 D N/ 2 C I I I N/ 2 R T N/ 2 J N/ 2   B N . (6) Since only the order of operations has cha nged, the complexity of this d ecimation sche me and one used in C. W . Kok’ s algorithm (3) must be exactly the same. At the same time, as s hown in Fig. 2 , this modified factorizations moves all the factors assoc iated with matrix D N/ 2 to the last stage. This me ans, that if it is s uf ficient to compute a sc aled version of the transform, such multiplications c an be a voided. Proposed factorization, therefore, is well suitable for implementation of sc aled transforms. Hereafter , we will say that DCT factorization is scaled , if it ca n b e presented as C I I N = Π N ∆ N ˜ C I I N , (7) where Π N is a r eordering ma trix , and ∆ N is a diagonal matrix of scale factors , and ˜ C I I N is a matrix of the scaled transform . By using suc h rep resentation, we can re write (6) as Π N ∆ N ˜ C I I N =   Π N/ 2 ∆ N/ 2 ˜ C I I N/ 2 0 0 D N/ 2 C I I I N/ 2 R T N/ 2 J N/ 2   B N , implying, that sca led pa rt of the transform can be computed recursively as follows ˜ C I I N =   ˜ C I I N/ 2 0 0 C I I I N/ 2 R T N/ 2 J N/ 2   B N . (8) The assoc iated reordering a nd s caling matrices can also be compu ted recursiv ely b y using Π N = P N   Π N/ 2 0 0 I N/ 2   , ∆ N =   ∆ N/ 2 0 0 D N/ 2   . (9) In o rder to co mpute the rema ining DCT -III block in (8), we can either pick so me existing (non-sca led) factorization, o r reuse our sca led d esign (8-9) follo wed by co n version to full (non-scaled) transform C I I I N =  Π N ∆ N ˜ C I I N  T = ˜ C I I I N ∆ N Π T N . (10) December 29, 2009 DRAFT 7 A. Complexity Analysis As already noticed, the co mplexity of computing DCT -II by using our factorization (6) is identical to one of C.W .Kok’ s algorithm (4). Howev er , when only a scaled transform (8) n eeds to be compu ted, s ome operations can be saved. Base d on Fig. 2, we can establish the followi ng relations: ˜ µ ( N ) = ˜ µ ( N/ 2) + µ ( N/ 2) , ˜ α ( N ) = ˜ α ( N/ 2) + α ( N / 2) + 3 2 N − 1 , ˜ σ ( N ) = ˜ σ ( N / 2) + σ ( N/ 2) + 1 , where ˜ µ , ˜ α , and ˜ σ denote the number of multiplications, a dditions, and shift ope rations correspond ingly needed for co mputing s caled transforms ˜ C I I , a nd w here µ , α , and σ rep resent numb ers of ope rations needed for computing lo we r non-scaled blocks C I I I . By applying this decomposition r ecursively m -times, we arri ve at the follo wing result. Proposition 2 . Th e numbers of a rithmetic operations ( µ, α, σ ) need ed for c omputing of sca led DCT -II of length N = q 2 m using factorization (8) sa tisfy: ˜ µ ( N ) = ˜ µ ( q ) + (2 m − 1) µ ( q ) +  m 2 − 1 + 2 − m  N , (11) ˜ α ( N ) = ˜ α ( q ) + (2 m − 1) α ( q ) + 3 m 2 N − 2 m + 1 , ˜ σ ( N ) = ˜ σ ( q ) + (2 m − 1) σ ( q ) + 2 m − 1 . By c omparing (11) with the numbe r of multiplications requ ired in C. W . K ok ’ s algorithm (4), we can conclude that the use of our p roposed s caled factorization sav es at least µ ( N ) − ˜ µ ( N ) = µ ( q ) − ˜ µ ( q ) −  1 − 2 − m  N >  1 − 2 − m  N multiplications. When number of iterations m is large, it can be further obs erved that µ ( N ) − ˜ µ ( N ) → N approach ing the well kn own upper bo und for multiplicativ e c omplexity reduction realizable b y sc aled transforms [6]. B. Cons truction E xamples W e note tha t in many prac tical situations, the multiplications b y factors 1 / 2 in our sche me can b e av o ided. Belo w , we provide two examples showing how this ca n be accomplishe d. December 29, 2009 DRAFT 8 0 x 1 x 1 2 2 X 1 0 2 X 2 1 6 2 X 1 4 2 X 8 2 cos π 3 8 2 cos π 2 x 3 x 1 2 2 4 x 5 x 1 5 2 X 1 1 2 X 1 7 2 X 1 3 2 X 6 x 7 x 1 2 16 2 cos π 3 16 2 cos π 5 16 2 cos π 7 16 2 cos π 2 8 2 cos π 3 8 2 cos π 1 2 2 0 1 0 1 2 3 2-point DCT-I I 2-point DCT-II I 2-point DCT-III 2-point DCT-I I 4-point DCT-I I 4-point DCT-I II Fig. 3. DCT -II of lengths N = 2 m ( m = 1 , 2 , 3 ) f actorized by our algorithm. Dashed arro ws show fac tors that can be merg ed. 1 9 2 X 1 5 2 X 6 x 7 x 8 x 9 x 10 x 11 x 1 7 2 X 1 1 2 X 1 3 2 X 1 11 2 X 1 8 2 X 1 4 2 X 0 x 1 x 2 x 3 x 4 x 5 x 1 6 2 X 1 0 2 X 1 2 2 X 1 10 2 X 2 6 2 cos π 2 6 2 cos π 12 2 cos π 3 12 2 cos π 5 12 2 co s π 1 2 24 2 cos π 3 24 2 cos π 5 24 2 cos π 7 24 2 c os π 9 24 2 cos π 11 24 2 cos π 2 0 1 2 1 2 2 3-point DCT-III 3-point DCT-II 0 2 4 1 5 3 6-point DCT-II 2 6 2 cos π 2 6 2 cos π 12 2 cos π 3 12 2 c os π 5 12 2 c os π 2 1 2 2 3-point DCT-II 3-point DCT-III 6-point DCT-III Fig. 4. DCT -II of lengths N = 3 · 2 m ( m = 1 , 2 , 3 ) factorized by our algorithm. Dashed arrows show canceling factors. 1) Sc aled DCT of lengths N = 2 m : W e scale the matrix of 2-point DCT -II as follo ws C I I 2 =   1 1 1 √ 2 − 1 √ 2   = 1 √ 2   √ 2 √ 2 1 − 1   . This moves factors √ 2 in DC paths, allo wing them to be subseq uently merged with factors 1 / 2 in ou r algorithm. W e show the resu lting flowgraphs in Fig. 3. Simple calculations show the nu mber of operations in suc h sc aled factorizations sa tisfy ˜ µ (2 m ) = m 2 m − 1 − 2 m + 1 , ˜ α (2 m ) = 3 m 2 m − 1 − 2 m + 1 , ˜ σ (2 m ) = 0 . For example, when N = 8 (largest size shown in Fig. 3) o ur a lgorithm p roduces factorization with December 29, 2009 DRAFT 9 just 5 multiplications a nd 29 ad ditions. T his matches the pe rformance of the well-known scaled DCT factorization of Y . Arai, T . Agui, and M. Nakajima [11]. 2) Sc aled DCT -II of lengths N = 3 2 m : W e scale the matrix of 3-point DCT -II as follo ws C I I 3 =      1 1 1 cos π 6 0 − cos π 6 1 2 − 1 1 2      = 1 2      2 2 2 2 cos π 6 0 − 2 cos π 6 1 − 2 1      . This brings factor 2 to the DC path, leading to cancelation of factors 1 / 2 in our algorithm. The resulting flowgraphs are shown in Fig. 4. It can be readily verified that the numbe rs of op erations in such scaled factorizations are ˜ µ (3 2 m ) = 3 m 2 m − 1 − 2 m +1 + 2 , ˜ α (3 2 m ) = 9 m 2 m − 1 + 3 2 m + 1 , ˜ σ (3 2 m ) = 2 m . For example, a s caled trans form of leng th N = 6 s hown in Fig. 4 use s only 1 multiplication, 16 additions, 2 shifts. V . C O M P A R I S O N W I T H T H E P R I M E F AC T O R A L G O R I T H M - B A S E D I M P L E M E N T AT I O N S It is known that DCT -II of length N = p q , where p and q are relati vely prime, can be computed as a c ascad e of p transforms of length q followed by q transforms o f length p [2], [19 ], [20]. Such a decompo sition is commonly called a prime factor algorithm (PF A). When one of the p rime factors, for example p , is dyadic, we arri ve at lengths N = q 2 m , implying that PF A is an alternative tec hnique for computing such transforms. Hence, we are interested in co mparison of PF A vs. C. W . Kok’ s algorithm. W e report the followi ng res ult. Theorem 1. Multiplicative c omplexity of DCT -II of length N = q 2 m construc ted by using C. W . K ok ’ s algorithm matches o ne theoretically achievable by using prime-factor DCT -II factorization, if f m 6 2 . Pr o of: B ased on PF A structure, the numb er of multiplications need ed to implement transform o f length N = q 2 m satisfies (cf. [14], [19]) µ ( N ) = 2 m µ ( q ) + q µ (2 m ) . F urthermore, from c omplexity study of dyad ic-length transforms [4]–[6] we kn ow that µ (2 m ) > 2 m +1 − m − 2 . Co mbining these formulae, we obtain µ ( N ) > 2 m µ ( q ) + q  2 m +1 − m − 2  . December 29, 2009 DRAFT 10 T ABLE I C O M P O N E N T S H O RT - L E N G T H D C T - I I [ 6 ] , [ 1 4 ] , [ 1 7 ] , [ 2 1 ] , [ 2 2 ] N DCT Scaled DCT µ α σ ˜ µ ˜ α ˜ σ 3 1 4 1 0 4 1 5 4 13 1 2 13 1 15 14 70 4 10 67 8 2 1 2 0 0 2 0 4 4 9 0 1 9 0 8 11 29 0 5 29 0 16 26 81 0 16 81 0 By compa ring this resu lt with complexity e stimate for C.W .K ok’ s a lgorithm (4): µ ( N ) = 2 m µ ( q ) + q 2 m m 2 , we arri ve at the statement of the theorem. W e now turn o ur attention to complexity comparison for scaled transforms. Proposition 3. Multiplicative co mplexity of PF A-based sc aled DCT -II of length N = q 2 m satisfies: ˜ µ ( N ) 6 2 m ˜ µ ( q ) + 5 2 N − q m ( m +3)+5 2 − 2 m − 1 + 1 2 . (12) Pr o of: W e use scaled PF A construction of Feig and Linzer [14], wh ich yields: ˜ µ ( N ) 6 2 m ˜ µ ( q ) + q ˜ µ (2 m ) + 1 2 ( N − 2 m − q + 1) . W e then apply F eig-W inograd algorithm for co mputing sca led DCT of dyadic lengths [6], for which: ˜ µ (2 m ) = 2 m +1 − m ( m +3) 2 − 2 . W e note, tha t in order to compare the obtained expression (12) with one c orresponding to ou r scaled version of C.W .Kok’ s algorithm (11): ˜ µ ( N ) = (2 m − 1) µ ( q ) + ˜ µ ( q ) +  m 2 − 1 + 2 − m  N . we need to kn ow complexities o f both sca led a nd n on-scaled transforms of length q . For this pu rpose, we will use several short-length DCT -II modules with complexity numbers shown in T able 1. Su ch odd-length transforms c an be found in [17] ( N = 3 ), [21 ] ( N = 5 ), a nd [14], [22] ( N = 15 ). Listed co mplexity numbers for dyadic-leng th transforms a re from [6], [14]. In T able 2 we provide co mparison of the resu lting transforms of c omposite lengths . Bold font is used to highlight b est comp lexity numb ers. It ca n be obs erved, that for q = 3 , 5 our propos ed a lgorithm shows December 29, 2009 DRAFT 11 T ABLE II C O M P L E X I T Y O F S C A L E D D C T- I I F AC T O R I Z A T I O N S O F L E N G T H S N = q 2 m q m N Proposed algorithm Feig and Linzer [14] ˜ µ ˜ α ˜ σ ˜ µ ˜ α ˜ σ 3 1 6 1 16 2 1 16 2 2 12 6 49 4 6 49 4 3 24 22 13 3 8 22 133 8 4 48 66 33 7 16 63 337 16 5 1 10 6 40 3 6 40 2 2 20 19 10 9 7 19 109 4 3 40 55 27 7 15 55 277 8 4 80 147 673 31 142 673 16 15 1 30 24 181 13 27 178 16 2 60 67 454 23 76 445 32 3 120 183 1090 43 204 1069 64 4 240 475 2542 83 505 2497 128 identical co mplexity to Feig-Linzer scaled PF A implementations when m 6 3 . It become s mo re c omplex for highe r m . For q = 15 and m 6 4 it is s hown that our propo sed algorithm is more e f ficient (in multiplicati ve complexity sens e) than scaled PF A implementations. V I . O N N O R M A L I Z E D M U L T I P L I C A T I V E C O M P L E X I T Y O F S C A L E D T R A N S F O R M S W e complement our presentation by providing plots of normalized multiplicati ve complexity ˜ µ ( N ) / N of scaled DCT of lengths N = [2 m , 3 2 m , 5 2 m , 15 2 m ] . W e pr esen t the se plots in Fig. 5. It can be observed, that amon g short-length transforms ( N 6 128 ), sc aled dyadic-length transforms are more c omplex than transforms with nearest composite len gths fr om seque nces N = 3 2 m or N = 15 2 m . W e believe that the use of su ch c omposite-length trans forms can o f fer appreciab le complexity s avings in many prac tical applications. V I I . C O N C L U S I O N S An alternative deriv ation and detailed comp lexity ana lysis of C. W . K ok’ s algorithm for computing DCT of lengths leng ths N = q 2 m ( m, q ∈ N , q is od d) is offered. It is shown that this algo rithm has the same multiplicativ e complexity as theoretically ac hiev able b y the p rime factor deco mposition, when m 6 2 . Ad ditionally , a sc aled DCT factorization b ased on C. W . Kok’ s algorithm is propo sed. It is December 29, 2009 DRAFT 12 N=2^m: Feig-Winograd N=3*2^m: Feig-Linzer/Proposed (m<=3) N=5*2^m: Feig-Linzer/Proposed (m<=3) N=15*2^m: Proposed algor ithm 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 mu(N)/N 50 100 150 200 250 N Fig. 5. Normalized multiplicative comple xity ˜ µ ( N ) / N of scaled DCT factorizations of lengths N = [2 m , 3 2 m , 5 2 m , 15 2 m ] . shown, that for some lengths this sc aled factorization achieves lower multiplicati ve complexity than on e of known prime factor- base d scaled transforms. R E F E R E N C E S [1] N. Ahmed, T . Natarajan, and K . R. R ao, “Discrete Cosine Transform, ” IE EE T rans. Computers , vol. X, pp. 90–93, Jan. 19 74. [2] K.R. Rao, and P . 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