Chan, YH

Siu, WC

PolyU Institutional Repository

IEEE Pacific Rim Conference on Communications, Computers and Signal Processing, May 9-10, 1991 New Formulation of Fast Discrete Hartley Transform with the Minimum Number of Multiplications Yuk-Hee CHAN and Wan-Chi SIU Department of Electronic Engineering Hong Kong Polytechnic Hung Hom, Kowloon, Hong Kong Abstract: The discrete Hartley transfom(DHT) is a real-valued tratufom closely related to the discrete Fourier transform (OFT) of a real-valued sequence. It directly maps a rea[-valued sequence to a real-valuedspectrum while preserving some usefulproperties of the Dkcrete Fourier Transfom. In such case, the Discrete Hartley transform can act as an altemative form to the Fourier Tradorm for avoiding complex arithmetic, hence it becomes a valuable tool in digital signalprocessing. In this paper, a simple algorithm is proposed to realize one-dimensional DHT with sequence lengths equal to 2“‘. This algorithm achieves the same multiplicative complaity as Malvar’s algorithm which requires the minimum number of multiplications reported in the literature. However, the present approachgives the advantage of requiring a smallernumber of additions compared with the number that required in Malvar’s algorithm. INTRODUCTION The disc : Hartley transform (DHT) has received growing interest since it was introduced by Bracewell[l] in 1983. One of the main attractions of DHT is that it only involves real computations in contrast to complex computations in the discrete Fourier transform (DFT). Since then, most ofworks that had been done on the problem of computing the discrete Hartley transform[2-9] were based upon the idea of existing fast DlT algorithms. For instance, the so-called “split-radix“ method was originally applied for computing 2m-point DFT[10] and was subsequently applied to the computation of DHT[4,8]. Typically, among these proposed algorithms, Malvar[S,6]’s algorithm can achieve the minimum number of multiplications during the realization of DHT whilst the split-radix approach[4] can achieve the minimum number of additions. In this paper, we propose a new algorithm to realize the DHT with the minimum number of multiplications. The algorithm also achieves the minimum number of additions compared with those algorithms[5,6,9] achieving the same minimum number of multiplications reported in literatures. ALGORITHM DERIVATION Recall that an N-length Discrete Hartley Transform[ 11 for a sequence {x(i):i =0,1 ... N-I} is defined as &ik N-1 i=O X(k) = 2 x(i) cas(,) for k=O,1 ... N-1 (1) If N is even, we have Anik M-1 i=O X(2k) = {x(i)+x(%+i)} cas (F) for k = 0,l ... N/2-1(2) and M-1 i=O X(2k+l) = {x(i)-x(%+i)} cas for k=O,1 ... N/2-1(3) By defining the two sequences Y(k) and Z(k): M-1 1=0 Y(k) = y(i) cos(-) for k=0,1 ... N/4-l(4) M-1 1=0 Z(k) = .E y(i) sin (F) for k = 0,l ... N/4-1 (5)  for i = 0,l ... N/2-l(6) where y(i) =x(i) - x(W2+i) We have X(2k+1) = Y(k) + Z(k) and X(N-2k-1) = Y(k)-Z(k) for k = 0,l ... N/4-l(7) CH2954-6/91-0000-323 $1.00 0 1991 IEEE Authorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on September 30, 2009 at 01:01 from IEEE Xplore.  Restrictions apply. To realize {Y(k):k=O,l..N/4-1}, we can define the following Y(k) + Y(k-1) = G(k) = .z g(i) cos - I = o  for k=0,1 ... N/4-1 (8) where g(0) = 2y(0) g(i) = 2 { y(i) - y(%-i) } COS(W~N) and Y(-1) is defined as Y(0). for i = 1,2 ... N/4-1 (9) Similarly, by defining Z(-1) = -Z(O), we can define the following to take care of the sequence {Z(k):k= 0,l ... N/4-1}, Z(k) - Z(k-1) = for k = 0,l ... N/4-1 (10) where f(0) = 0 f(i) = 2 { y(i) + y(%-i)} sin(&i/N) fori = 1,2 ... N/4-1 (11) Note that G(k) and F(k), defined by eqns. 8 and 10, are respectively the symmetric cosine structure(SCS) and the alternative form of the symmetric cosine structure(SCS6) defined by Chan and Siu[9]. Hence, an N-length DHT can be decomposed into an N/2-length DHT, an N/4-length SCS and an N/4-length SCS6 with a cost of N/2-2 multiplications and 5N/2-4 additions. In particular (See Appendix), we have On the other hand, it is well-known that a 4-length DHT can be realized with 8 additions only and an 8-length DHT can be realized with 2 multiplications and 22 additions[4]. Therefore, the mathematical complexity of the new algorithm is given by the following equations: M(4-DHT) = 0 A(4-DHT) = 8 M(8-DHT) = 2 A(8-DHT) = 22 M(N-DHT) = M(%-DHT) + M(%-SCS) + M(%-SCS6) + N/2-2 A(N-DHT) = A(Wz-DHT) + A(W4-SCS) + A(% -SCS$) + 5N/2 - 4 where N = 2m, m > 3 (14) Table 1 shows the computational complexity of the realization of DHT of different lengths using the present approach and other  efficient techniques[4-6,9]. The computational complexity of other techniques is quoted directly from table 1 of reference(41 and table 1 of reference[6] respectively. The split-radix algorithm[4] always gives the minimum number of additions, whereas Malvar[5,6]'s and Chan and Siu[9]'s algorithms always give the minimum number of multiplications for all lengths shown. The present approach requires the same minimum number of multiplications for all lengths, however, it requires a smaller number of additions than that required by those two approaches. Figures 1 and 2 also illustrate clearly the comparison of the present approach with other major approaches for the realization of the Discrete Hartley Transform. M(N-SCS) = M(N-SCS$) = A(N-SCS) = A(N-SCS;) = M(% -SCS) + M(% -SCS') + N/2 - 1 A(W2 -SCS) + A("&SCS* ) + 3N/2 - 3 f o r N > 4  M(4-SCS) = M(4-SCS;) = I A(4-SCS) = A(4-SCS8) = 7 (12) where M(N-Y) and A(N-Y) are the numbers of multiplications and additions respectively for a length-N structure denoted by Y and SCS' is another alternative form of the symmetric cosine structure, which is defined the same as SCS8 in eqn 10 except that the input data f(0) is not necessary to be zero, whose mathematical complexity is given by M(N-SCS') = M(% -SCS) + M(% -SCS*) + N/2 - 1 A(N-SCS*) = A(% - s c s )  + A(% - scs*)  + 3 ~ / 2  - 1 f o r N > 4  M(4-SCS') = 1 A(CSCS*) = 9 (13) Table 1. Coiiiputational complexity of the realizntioll of UIIY for different leiigtlis using the present approach and other eflicierit tecliiiiques [4,6,9]. N bl  A hl A hl A hl A hl A hl A 8 2  16 I0 32 34 64 98 128 258 256 642 512 153R 11124 3586 21148 8194 22 72 198 12m 2800 14316 31722 sim 6x32 2 Il l  34 98 258 642 1538 3586 8194 29 83 ti7 I35 1269 2931 f a  I 14831 32749 4 20 68 196 516 I284 31176 7172 163M 26 74 I94 482 11.54 2690 6146 13826 30722 2 14 10 I2 42 142 450 124 3311 942 2498 828 1994 ~294 12~02 4668 10698 22 64 I 6 6  416 998 2336 5350 12064 26854 2 I0 34 98 258 642 1538 3586 8194 26 74 206 522 1278 3018 6974 35390 i s m  The basic idea of the proposed algorithm implies a recursive decomposition technique. By using the recursion, a 2m-length DHT (m > 3) is eventually decomposed into an 8-length DHT and a number of 4-length SCS, SCS' and SCS& Hence, the realization modules of 4-length SCS etc. are actually basic elements of the implementation of the algorithm. Consider that the 4-length SCS and the 4-length SCS6 are only variants of the 324 Authorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on September 30, 2009 at 01:01 from IEEE Xplore.  Restrictions apply. 4-length SCS', all of them can be realized with a typical 4-length SCS' module without extra cost. This property is practical and desirable especially when the algorithm is used for the hardware realization of the DHT. In such case, only simple 4-length SCS* hardware modules are required, which makes the proposed algorithm very suitable for VLSI implementation. Some multiplications are required for the generation of sine or cosine coefficients. In general, they can be precomputed and stored in a memory table as they are known values. The table lookup technique can then be applied to save computation effort during the realization. Note that this algorithm requires only a small table for storing values of {sin(&i/N):i= 1,2 ... N/4-1} and {cos(;?ni/N):i = 1,2 ... N/4-1}, the size of which is practical for most applications. our approach: Firstly, we compute the even sequence of the output, {X(2k):k=0,1..7}, via an 8-length D H T  on the sequence {x(i) +x(8 + i):i = 0,1..7} as shown in eqn 2. T o  compute the odd sequence, we have to  compute sequences {Y(k):k=0,1..3} and {Z(k}:k=0,1..3}. By using eqn9, we have where y(i) = x(i) - x(8 + i) for i = 0,1..7 260 400 600 800 1000 1 Length of DHT Fig.1 Comparison of additions per point among different approaches 7 radix-2 6 / I / radix4 I 2 i  I d  I Then, after we compute the 4-length SCS on {g(i):i=O,l..3} with eqn 8 to obtain {G(k):k =0,1..3}, we have (33) -Y(2) and, after computing the 4-length SCS6 on {€(i):i=0,1,.3} with eqn 10 to obtain {F(k):k =0,1..3}, we have Then, we can determine the odd sequence of the final output by using eqn 7: X(1) = Y(O)+z(O) X(5) = Y(2)SZ(2) X(7) = Y(3)+Z(3) X(15) = Y(0)-z(0) X(11) = Y(2)-Z(2) X(9) = Y(3)-z(3) x(3) = Y(l )+z ( l )  and X(13) = Y(l)-z( l )  I 1 Totally, 10 multiplications and 72 additions are required. 0 260 400 600 860 1000 1200 Length of DHT CONCLUSIONS Fig3 Comparison of multiplications per point among different approaches EXAMPLE Let us use a 16-length DHT on {x(i):i=O,l ... lS} to clarify A new algorithm is proposed in this paper to realize a 2m-length Discrete Hartley Transform. This algorithm requires the same minimum number of multiplications as other algorithms reported in the literature[2-9] and requires a smaller number of additions compared with other algorithms[5,6,9]. It is significant to point out that the present algorithm is most suitable for VLSI realization. The resultant structure of the approach is stable and regular. 325 Authorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on September 30, 2009 at 01:01 from IEEE Xplore.  Restrictions apply. whereg'(0) = ~ ' ( 0 )  + xo REFERENCES [ 11 R.N.Bracewell, 'The Discrete Hartley Transform," J. Opt. Soc. Amer., Vo1.73, pp.1832-1835, 1983 [2] R.N. Bracewell, 'The fast Hartley transform," IEEE Proc., V01.72, pp.1010-1018, 1984 (31 R.N.Bracewell, The Hartley Transforms, England: Oxford University Press, 1985. [4] H.V.Sorensen, D.L.Jones, CSBurrus, and M.T.Heideman, "On computing the Discrete Hartley Transform", IEEE Trans. on ASSP., VOLASSP-33, NO.4, pp.1231-1238, OCt 1985. [5] Malvar, "Fast Computation of the Discrete Cosine Transform and the Discrete Hartley Transform", IEEE Trans. on ASSP., Vol.ASSP-35, N0.10, PP.1484-1485, OCt, 1987. [6] Malvar, "Corrections to 'Fast Computation of the Discrete Cosine Transform and the Discrete Hartley Transform"', IEEE Trans.on ASSP., Vo1.36, No.4,pp.610, Apr 1988. [7] H.S.Hou, 'The fast Hartley transform algorithm," IEEE Trans. Comput., V0l.C-36, pp.147-156, 1987. [8] S.C.Pei and J.L.Wu, "Split -radix fast Hartley transform," Electron. Lett., Vo1.22, pp.26-27, 1986. [9]Y.H.Chan, and W.C.Siu, "Modular Structures for Fast Discrete Hartley Transform and Real-valued Discrete Fourier Transform," Proceedings, International Symposium on Signal Processing and Its Applications (ISSPA'90), 27-31 August, 1990, Gold Coast, Australia, pp.221-224. [lO]P.Duhamel and H.Hollmann, "'Split radix' FFT algorithm," Electronics Letters, V01.20, No.1, pp.14-16,5th Jan 1984. APPENDIX An N-length SCS* on sequence {x'(i):i=O,l ... N-1} is defined as follows: N-1 1=0 T(k) = ,E x'(i) cos($) + ( - l ) k h  k=O,l..N-l (Al)  where xO is a real number. This structure forms the basic module of the proposed algorithm. Actually, the N-length SCS and SCS6 are only special cases of this basic structure. In particular, if xO is zero, eqn A1 becomes the definition of an N-length SCS and, when x'(0) is zero, it becomes the definition of an N-length SCS6. If N is even, we can split eqn A1 into two sequences. For the even sequence, we have g'(i) = x'(i) + x'(N-i) for i = 1,2..N/2-1 (A3) and, if we define T(-1) as T(l), we have %-I i = O  T(2k+l)+T(2k-l) = F(k) = e'(i) cos(?) for k = 0,l ... N/2-1 (A4) where e'(0) = 2 { x'(0) - xo } e'(i) = 2 { x'(i) - x'(N-i) } COS(.~/N) for i = 1,2..N/2-1 (A5) Hence, an N-length SCS* can be realized through an N/2-length SCS and an N/2-length SCS' with a cost of N/2-1 multiplications and 3N/2-1 additions. Eqn. 13 gives the mathematical complexity of this basic structure. In particular, if N = 4, it can be realized with 9 additions and 1 multiplication as follows: temp0 = { x'(l)-x'(3) } COS(%) templ = x'(0) + xO temp2 = x'(0) - xO T(0) = templ + $(I) + x'(2) + x'(3) T(3) = temp2 - temp0 T(1) = temp2 + (empO T(2) = templ - x'(2) The substitution of x'(0) =0 and xo = O  into eqns A1 to A5 gives the mathematical complexity of an N-length SCS6 and SCS respectively. Eqn. 12 describes the mathematical complexity of both structures precisely. Acknowledgment The authors wish to acknowledge the support of the Research Sub-committee of the Hong Kong Polytechnic and the Video Technology Computers Ltd. Nh-1 i = O  T(2k) = 2 g'(i) cos €or k=0,1 ... N/2-1 (A2) 326 Authorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on September 30, 2009 at 01:01 from IEEE Xplore.  Restrictions apply. 

New formulation of fast Discrete Hartley transform with the minimum number of multiplications

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