New formulation of fast Discrete Hartley transform with the minimum number of multiplications

Abstract

The discrete Hartley transfom(DHT) is a real-valued transform closely related to the discrete Fourier transform (DFT) of a real-valued sequence. It directly maps a real-valued sequence to a real-valued spectrum while preserving some useful properties of the Discrete Fourier Transform. In such case, the Discrete Hartley transform can act as an alternative form to the Fourier Transform for avoiding complex arithmetic, hence it becomes a valuable tool in digital signal processing. In this paper, a simple algorithm is proposed to realize one-dimensional DHT with sequence lengths equal to 2[sup m]. This algorithm achieves the same multiplicative complexity as Malvar’s algorithm which requires the minimum number of multiplications reported in the literature. However, the present approach gives the advantage of requiring a smaller number of additions compared with the number that required in Malvar’s algorithm.