Abstract: By modifying the conventional decomposition formula for the decimation-in-frequency (DTF) case, a more efficient radix-3 fast Hartley transform(FHT) ...
AN EFFICIENT RADIX-3 FHT ALGORITHM K.M.M. Prabhu Department of Electrical Engineering, Indian Institute of Technology, Madras-600 036, India
Abstract: By modifying the conventional decomposition formula for the decimation-in-frequency (DTF) case, a more efficient radix-3 fast Hartley transform(FHT) algorithm has been introduced resulting in a fewer number multiplications. The reduction in the number of multiplications is of the order of 25-30 percent. It is useful in cases where the sequence length is closer to a power of 3 rather than a power of 2.
1.
INTRODUCTION
The discrete Hartley transform (DHT) [1] can be performed on an N-point real sequence, (x(n), 0 5 n I N-1 }, to obtain another N-point real sequence, { H(k), 0 I k CN-11, as:
Ex(.) N-I
~ ( k=)
('7').
The basic 3-point DHT for a real-valued 3-point sequence, {x(n), 0 In I2 ) , can be obtained from Equation (1) as
cas - 0 I k 2 N-1, .......(1 ) which can be expanded and modified towards the basic 3-point FHT and can be written as
n=o
where, cas(@)= cos(@)+ sin(@)
H(l)= x(O)+x(l)+x(2) ....(3(a))
Therefore, the discrete Hartley transform has the advantage that it is real-valued for a real-valued signal and many of its properties and applications strongly resemble those of the discrete Fourier transform, since the Hartley transform can be considered as a variation on the Fourier Transform. To compute the DHT, researchers have developed fast Hartley transform (FHT) [2] algorithms which are similar to the more popular fast Fourier transform (FFT) algorithms. More efficient radix-2, radix-4, split-radix, Winograd and prime factor FHTs have recently been developed [3]. By modifying the decomposition formula for the decimation-in-time (DIT) case [4], we have introduced a new radix-3 FHT algorithm. In this paper, the decomposition formula for the decimation-in-frequency (DIF) case of radix-3 FHT algorithm is modified, resulting in fewer number of multiplications than in 141 at the cost of additions. Here, instead of the traditional decomposition, the author decomposes the sequence in such a way that some of the twiddle factors of the subsequences can be combined, resulting in fewer number of multiplications. The reduction in the number of multiplications is of the order of 25-30 percent, when compared to the earlier DIT algorithm [4]. When the sequence length is closer to a power of 3, this algorithm is competitive in speed with the radix-2 and radix-4 FHTs and is efficient than the existing radix-3 FHT. 2. BASIC 3-POINT FHT
0-7803-41 37-6/97/$10.0001977 IEEE
....(3(bj)
....(3(cj)
The implementation of Equations 3(a), 3(b) and 3(cj is shown in Figure 1. The basic 3-point FHT in [4] is no1 in-place and requires seven additions and one multiplication, while the 3-point FHT shown in Figurc 1 is in-place and requires only six additions and one multiplication. Multiplication by (1/2) is simply a rightshift and therefore has not been added in the computation of operation count.
3.
DERIVATION OF THE NEW RADIX-3 DIF FHT ALGORITHM
The conventional radix-3 DIF FHT algorithm obtains k 5 the required N-point transform sequence, H(k), 0 I N-I, from the three (N/3)-point transforms H(3k), H(3k+l) and H(3k+2), 0 Ik I (N/3)-1. Obtaining
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H(3k+l) involves multiplication by cos(2nnlN) and sin(2nnlN) while obtaining H(3k+2) involves multiplication by cos(4xn/N) and sin(4nnlN). However, by pairing the rotating factors, evaluation of H(3k-l), instead of H(3k+2), will also involve multiplications by cos(2nn/N)and sin(2nn/N), resulting in an algorithm with fewer number of multiplications. Due to the periodicity of the sequence H(3k-I), with a period of N samples, H(-1) can be interpreted as H(N-1). Expressions €or H(3k), H(3k+l) and H(3k-I) can be derived from Equation ( I ) and can be stated as follows:
. . ..(4(a))
The implementation of the above equations involves the following steps: (i) Compute al(n), a2(n), a3(n) and G(n), 0 5 11 < (N/3)- 1. This involves (4N/3)-2 additions and no multiplication because q(0) = a3(0) and az(0) = a4(0). (ii) Compute b,(n), b,(n), b3(n) and b,(n), 0 S n < (N/3)- 1. For n=O, computing these sequences involves two additions and one multiplication because of the relation b4(0)=( h /2)a4(0). For the other values of n, because the multiplication by (1/2) is simply a right-shift, one can evaluate these sequences with (2N/3)-2 additions and (4N/3)-4 multiplications.
(N/3)-1
~ ( 3 +k 1) =
x3k+l(72) cas IF0
. ...(4(b))
where the sequence xlk(n), xlk+,(n) and ~3k.i(n) can be obtained with the help of the following intermediate sequences:
(iii) Compute cl(n) and c2(n) which require (2N/3)-1 additions and no multiplication. Computing x3k(n), x3k+l(n) and xjk.,(n) require N additions and no multiplication. Hence, by summing up the computations in the above three steps, one can see that the arithmetic required to combine three (N/3)-point transforms into one N-point transform requires (1 1N/3)-3 additions and (4N/3)-3 multiplications. If the number of additions and multiplications required for an N-point FHT are A(N) and M(N), respectively, then A(N) =
(y]+
- 3 3A ):( ....(q a ) )
and
.. . .(5(b)) Using the initial conditions A(3)=6 and M(3)=1, we obtain the expressions for A(N) and M(N) in terms of N
as
(
):(
N-
A(N) = ?)logi
+
....(6(a))
M(N)=
(43 i:) -
log3 N -
.
-
(N-1).
. . . . (6(b))
Equations (6(a)) and (6(b)) are tabulated for various values of N in Table 1. Computations required for [lie radix-3 FHT in [4] are also tabulated in Table 1 for the purpose of comparison. From this, one can see clearly that the radix-3 FHT presented in this paper is more efficient than the radix-3 FHT in [4], since the number of
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multipliers play an important role than adders in the implementation of an algorithm.
4. CONCLUSION A more efficient radix-3 FHT algorithm has been presented. This could be used effectively when the number of samples is closer to a power of 3, rather than a power of 2 or 4.
REFERENCES
[ 11 Bracewell, “Discrete Hartley Transform”, ./. Opt.
Soc. Amer., Vol. 73, No.12 , Dec. 1983, pp.18321835. [2] Bracewell, “The Fast Hartley Transform”, Proc. IEEE, Vol. 72, No.8, Aug. 1984, pp.1010-1018. [3] Sorenson, D.L. Jones, C.S. Burrus and M.T. Heidman, “On Computing the Discrete Hartley Transform”, IEEE Trans. on Acoustic Speech and S i g m l Processing, Vol. ASSP-33, No.4, Oct. 1985, pp.1231-1238. [4] Anupindi, S.B. Narayanan and K.M.M.Prabhu, “New Radix-3 FHT Algorithm”, Electron. Left., Vol. 26, No.18, Aug. 1990, pp. 1537- 1538.
0 WO)
H(1
1
H(2)
-
-
Figure. 1 Flowgraph of a basic 3-point FHT
Table I: Operations count in radix-3 DIF and DIT cases
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