A New Split-Radix FHT Algorithm for Length-<tex>$q*2^m$</tex>DHTs

Abstract

In this paper, a new split-radix fast Hartley transform (FHT) algorithm is proposed for computing the discrete Hartley transform (DHT) of an arbitrary length = 2 , where is an odd integer. The basic idea behind the proposed FHT algorithm is that a mixture of radix-2 and radix-8 index maps is used in the decomposition of the DHT. This idea and the use of an efficient indexing process lead to a new decomposition different from that of the existing split-radix FHT algorithms, since the existing ones are all based on the use of a mixture of radix-2 and radix-4 index maps. The proposed algorithm reduces substantially the operations such as data transfer, address generation, and twiddle factor evaluation or access to the lookup table, which contribute significantly to the execution time of FHT algorithms. It is shown that the arithmetic complexity (multiplications+additions) of the proposed algorithm is, in almost all cases, the same as that of the existing split-radix FHT algorithm for length-2 DHTs. Since the proposed algorithm is expressed in a simple matrix form, it facilitates an easy implementation of the algorithm, and allows for an extension to the multidimensional case. Index Terms-Discrete Hartley transform (DHT), fast Hartley transform (FHT) algorithms, mixed radix, split radix.

Direct computation of length-N DHT from three adjacent length-N/3 DHT coefficients

“…According to the scheme, three length-N/3 inverse DHTs (IDHTs) and one length-N DHT are required. In this section, we apply two different schemes, based respectively on the algorithms presented in [8] and [11], to implement the length-N DHT according to the different values of N. …”

“…When the sequence length N = q×2 m , where q is an odd integer, we can use Bouguezel's split-radix FHT algorithm [11] to compute the length-N DHT directly. In the following, we assume that a butterfly computation is implemented by using 3 multiplications and 3 additions.…”

“…The computational complexity of Bouguezel's algorithm for length-N = q×2 m is given by N>8q. (9) Detailed computational complexity for the DHT with length-N = q×2 m , q = 1, 3, 9, 15…, can be found in [11]. Note that , , , and ..…”

“…In this case, the split radix FHT algorithm [11] can be used to efficiently compute length-N/3 = 2 m DHTs where the computational complexity is shown in (9). The comparison results of the proposed algorithm and this scheme are shown in table I.…”

“…Bi [10] suggested a radix-3/9 FHT algorithm for further speeding up the computation of DHT. Bouguezel, Ahmad, and Swamy [11] proposed an optimized split-radix FHT algorithm, taking arithmetic complexity, the number of data transfers, address generations, and twiddle factor evaluations into consideration, for computing the DHT of sequences with length N = q×2 m , where q is an odd integer.…”