Bucknell Digital Commons Bucknell Digital Commons Two-Band Fast Hartley Transform Two-Band Fast Hartley Transform

Ef ﬁ cient algorithms have been developed over the past 30 years for computing the forward and inverse discrete Hartley transforms (DHTs). These are similar to the fast Fourier transform (FFT) algor- ithms for computing the discrete Fourier transform (DFT). Most of these methods seek to minimise the complexity of computations and/ or the number of operations. A new approach for the computation of the radix-2 fast Hartley transform (FHT) is presented. The proposed algorithm, based on a two-band decomposition of the input data, pos-sesses a very regular structure, avoids the input or out data shuf ﬂ ing, requires slightly less multiplications than the existing approaches, but increases the number of additions.


Two-band fast Hartley transform
A.N. Skodras, M.F. Aburdene and A.K. Nandi Efficient algorithms have been developed over the past 30 years for computing the forward and inverse discrete Hartley transforms (DHTs). These are similar to the fast Fourier transform (FFT) algorithms for computing the discrete Fourier transform (DFT). Most of these methods seek to minimise the complexity of computations and/ or the number of operations. A new approach for the computation of the radix-2 fast Hartley transform (FHT) is presented. The proposed algorithm, based on a two-band decomposition of the input data, possesses a very regular structure, avoids the input or out data shuffling, requires slightly less multiplications than the existing approaches, but increases the number of additions.
Introduction: Hartley presented a new method, the continuous Hartley transform, for the analysis of transmission problems in 1942 [1]. Subsequently, Bracewell introduced the discrete Hartley transform (DHT) in 1983 [2] and the fast Hartley transform (FHT) in 1984 [3]. In the intervening years, many researchers have devised methods to improve the computation of the FHT and the highly similar inverse FHT [4][5][6][7][8], whereas others have tried to develop recursive [9] and/or parallel methods for computing the FHT [10]. The DHT is commonly used in signal processing, signal compression, image classification, image encryption and communication systems [6,7,9].
This Letter proposes a two-band method, an entirely new approach for computing the FHT, resulting in a highly regular structure with butterflies of constant geometry and a reduced multiplication operations count compared with existing algorithms, while increasing the additions operations count.

Discrete Hartley transform:
The type II DHT of the N-point real-valued data x n , n = 0, 1, 2, …, N − 1 is defined as where cas(·) = cos(·) + sin(·). The transform is linear and its coefficients . Other approaches that appeared have achieved an improvement on the above complexity figures at the expense of a more complicated computational structure [5,8,9]. Fast algorithms are usually in-place, resulting in a shuffling of the input data or the output coefficients.

Discrete Hartley transform:
The proposed method is based on the decomposition of each pair of input data x(2n), x(2n + 1) into low-band values x L (n) and high-band values x H (n). Specifically Starting from the definition of DHT (1), decomposing into even-indexed and odd-indexed data and using (3a) and (3b), leads to and H L N /2 (k) and H H N/2 (k) are the N/2-point DHTs of x L (n) and x H (n), respectively. On the basis of the properties of the cos(·), sin(·) and cas(·) functions, it is easily derived that the DHT coefficients of (6) that are N/2 positions apart from k become Equations (6) and (13) constitute the DHT pair for its fast computation. By noting that the DHT pair of (6) and (13) eventually becomes From (15) and (16), we can realise that the computation of an N-point DHT has been decomposed into two DHTs of length N/2 each, combined with (N − 1) multiplications by twiddle factors, as depicted in the flow graph of Fig. 1. By eliminating the trivial multiplications occurring for k equal to 0, N/4 and (N/2 − k) the flow graph is further simplified.
x 0   The structure is simple, regular, modular and scalable, which facilitates software or hardware implementations. The structure resembles 'constant geometry' of fast transforms. Constant geometry algorithms avoid the area and delay overhead of multiplexing different registers, something that is desirable in high-throughput designs. Computational complexity: On the basis of (15) and (16) and the corresponding flow graphs, we can easily derive the number of operations needed for the computation of the two-band N-point fast DHT. This is calculated by means of the formulae or where M N is the number of multiplications and A N is the number of the total additions and subtractions. Another two multiplications could be saved for k = N/8 and k = (N/2 − N/8). In the first case (k = N/8), we have θ = π/4 and thus cos(θ) = sin(θ), i.e. only one multiplication is needed; in the second case [k = (N/2 − N/8)], θ = 3π/4 and thus cos(θ) = −sin(θ), i.e. only one multiplication is performed. Taking into account this additional saving, the multiplications count reduces to The counts of multiplications and additions/subtractions for different N, where N is a power of 2, are summarised in Table 1. The multiplications counts are less than those of the corresponding well-known radix-2 algorithm of Bracewell [3]. The additions are approximately twice as large. It should be noted that a final multiplication of each coefficient by 1/2 is needed in order for the result to be correct, as dictated by (2).
Conclusions: A new two-band radix-2 algorithm has been proposed for the computation of the fast DHT. The algorithm proceeds by applying the DHT core on the summations and the differences of adjacent samples, i.e. on the low-band and high-band values of adjacent samples. This is equivalent to applying a two-band filter bank followed by a down-sampling by 2. The computational structure is simplified and symmetric, at the expense of increased numbers of additions and subtractions. Multiplications are restricted only to the highs and their count is slightly decreased. The derived structure of the algorithm facilitates its fast implementation in high-level or low-level applications. The computation is in-place and compares favourably with the well-known fast DHT algorithms.