Abstract
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.
Highlights
Hartley presented a new method, the continuous Hartley transform, for the analysis of transmission problems in 1942 [1]
Many researchers have devised methods to improve the computation of the fast Hartley transform (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 discrete Hartley transform (DHT) is commonly used in signal processing, signal compression, image classification, image encryption and communication systems [6, 7, 9]
Summary
Hartley presented a new method, the continuous Hartley transform, for the analysis of transmission problems in 1942 [1]. 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. 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 xL(n) and high-band values xH(n). As in all fast DHT algorithms, only two additions or subtractions are needed for the 2-point FHT, i.e. M2 = 0 and A2 = 2.
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