A fast algorithm for computing the Hartley/ Fourier spectrum

MULTIDISCIPLINARY THEMES

Organizer:

ALCIDES N. SIAL

A FAST ALGORITHM FOR COMPUTING THE HARTLEY/ FOURIER SPECTRUM

HELIO M. DE OLIVEIRA AND RICARDO M. C. DE SOUZA

ODEC Grupo de Pesquisas em Comunicações, Departamento de Eletrônica e Sistemas CTG-UFPE, C.P. 7800, 50711-970 Recife-PE, Brazil. Fax: [phone]

Presented by ARON SIMIS

Discrete transforms have been playing a relevant role in several areas, especially in Engineering. An interesting example is the Discrete Fourier Transform (DFT). Another very rich transform related to the DFT is the Discrete Hartley Transform (DHT), the discrete version of the symmetrical, Fourier-like, integral transform introduced by Ralph V.L. Hartley. Besides its numerical side appropriateness, the DHT has proven over the years to be a powerful tool. A decisive factor for applications of the DFT has been the existence of the so-called fast transforms (FT) for computing it. Fast Hartley transforms also exist and are deeply connected to the DHT applications. Recent promising applications of discrete transforms concern the use of finite field Hartley transforms to design digital multiplex systems, efficient multiple access systems and multilevel spread spectrum sequences. Besides being a real transform, the DHT is also involutionary, i.e.; the kernel of the inverse transform is exactly the same as the one of the direct transform (self-inverse transform). Since the DHT is a more symmetrical version of a discrete transform, this symmetry is exploited so as to derive a new FT that requires the minimal number of real floating point multiplications. A FT algorithm for the DHT is also a FT for the DFT and vice versa.

Discrete transforms presenting a low multiplicative complexity have been an object of interest for a long time. The minimal multiplicative complexity, m, of the one-dimensional DFT for all possible sequence lengths, N, can be computed by converting the DFT into a set of multi-dimensional cyclic convolutions. In this work a fast algorithm is derived, which meet the lower bound on the multiplicative complexity of a DFT/DHT for short blocklengths. It is based on a multilayer decomposition of the DHT using Hadamard-Walsh transforms. These new schemes are attractive and easy to implement using a Digital Signal Processor (DSP) and the regularity of the structure allows the design of low-cost high-speed dedicated Integrated Circuits. — ( May 18, 2001 )

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Publication Dates

• Publication in this collection
  08 Oct 2001
• Date of issue
  Sept 2001