Discrete Fourier transform computation for digital relaying A Gbmez Expbsito and J A Rosendo Mac|as Departamento de Ingenieria El~ctrica, Universidad de Sevilla. Avda. Reina Mercedes s/n. 41 012 Sevilla, Spain J L Ruis Macias Sevillana de Electricidad, Avda. Borbolla 5, 41004 Sevilla, Spain In this paper several ways of computing the discrete Fourier transform are presented. Starting from the conventional non-recursive version, three recursive algorithms are obtained. Even though these recursive filters are computationally efficient their steady-state response is not always accurate. So, an efficient non-recursive version is proposed which requires computing only the imaginary component of the transform. An example is included in order to compare the operation counts of the different schemes. Keywords: on-line computer systems, data gathering and analysis, software design I. Introduction Currently, many protection devices based on digital technology are commercially available. It is hoped that, in the future, the use of microprocessor-based relays will increase gradually, as they offer several advantages such as reduced cost, better performance, flexibility, adapt- ability, self-checking capability and the possibility of integration in the digital environment of future substations 1'5. Computer relays rely on numerical methods to model the input waveforms and/or the differential equations of the system being protected ~. Some of the numerical techniques proposed for waveform modelling2'3'6'7 are • least-squares fitting, • discrete Fourier transform (DFT), • rectangular Fourier transform, • Walsh transform, • Haar transform, • Kalman filtering. The Kalman filter is a sophisticated tool that should be used in those situations where the covariance of the measurement noise is not constant (Reference 3, p 105). Received 30 December 1992; accepted 11 November 1993 Curve fitting algorithms may be advantageous if certain parameters, such as DC offset time constants, are of interest. Otherwise, when only fundamental and harmonic phasors are required, transform-based methods are usually preferable in terms of computational cost and frequency response. Although rectangular-type transforms are imple- mented using only additions and subtractions, a certain number of multiplications must be carried out to obtain Fourier coefficients from the rectangular coefficients. In fact, based on the comparative analysis of Reference 7, it can be concluded that the DFT is the best choice for computing the fundamental and harmonic components of input waveforms. This paper is exclusively devoted to the DFT. Starting from the conventional, non-recursive version, several recursive algorithms are obtained. It is shown that the real and imaginary components of the recursive DFT lead to a certain type of second-order bandpass filters. These recursive expressions are finally used to obtain an improved non-recursive scheme which resorts to only two consecutive values of the imaginary component of the DFT. An example will be discussed along with each version of the DFT in order to compare the operation counts. II. Standard non-recursive DFT To begin with, the standard non-recursive form of the DFT will be reviewed. This form will be later used for comparison purposes. Let Xo, xl ..... xN_ 1 be the N real samples of the signal whose spectrum is required. The kth harmonic is given by 8 N-I ~k= ~ xie-~Ol (1) i=0 with 2nk 0 - (2) N Volume 1 6 Number 4 1994 0142-0615/94/04/0229-05 © 1994 Butterworth-Heinemann Ltd 229