New high-speed prime-factor algorithm for discrete Hartley transform

Abstract

Fast algorithms for computing the DHT of short transform lengths (N = 2, 3, 4, 5, 7, 8, 9 and 16) are derived. A new prime-factor algorithm is also proposed to compute the long-length DHTs from the short-length DHT algorithms. The short-length algorithms (except for N = 8 and N = 16) are such that the even and the odd parts of the DHT components are obtained directly, without any additional computation. This feature of the short-length algorithms makes the proposed prime-factor DHT algorithm more attractive and efficient. It is found that the proposed algorithm is more efficient compared to the radix-2 FHT in terms of the computational requirements, as well as the execution time for transform lengths higher than 30. It is also observed that the number of operations required for the computation of DHT by the prime-factor FFT algorithm for real-valued data is the same as those of the proposed algorithm for certain transform lengths, e.g. N = 30, 60, 252 etc., which do not contain 8 or 16 as a cofactor. However, for all other transform lengths the proposed algorithm has a lower computational complexity. It is further observed that the proposed algorithm is faster than the prime-factor FFT algorithm for real-valued series.