Digital processing of system responses

• Main Author: Rees, David
• Published: University of South Wales 1976
• Online Access: https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.749608

This thesis describes an investigation into the development of techniques for the measurement of system dynamic characteristics based on digital processing methods. The techniques are developed to meet the requirements of rapid measurement time, noise and harmonic rejection capability and ease of interpretation of results. A computational procedure using spectral methods and based on the fast Fourier transform is proposed, which considers a pseudo­ random binary sequence as a series of sine waves of 'discrete' frequencies of well defined amplitudes and phase relationships. Three computational algorithms have been considered, (a) the discrete Fourier transform, {b) the radix-2 fast Fourier transform, and (c) the mixed radix fast Fourier transform. It has been shown that if the radix-2 fast Fourier transform is used unacceptable results are obtained. However, because the sequence length of pseudo-random binary sequences can be expressed as multiples of prime numbers the mixed-radix fast Fourier transform is suitable and has been mechanised successfully. Errors when using the procedure are presented taking quantisation levels, sampling rates, sequence lengths and smoothing techniques into consideration, both with and without the presence of noise in the response. A detailed comparison is made between the crosscorrelation function and fast Fourier transform methods of mechanisation, by comparing modelling estimates given by both procedures for different system conditions. The application of the technique to nonlinear systems has shown. however, the procedure can produce results which are unacceptable. The analysis of a polynomial-type nonlinearity, when subjected to a composite sinusoid signal has led to the derivation of a new test signal. This signal consists of an assemblage of discrete sinusoids of frequencies which are odd and prime number multiples of some fundamental which is itself excluded from the signal. The properties of the new signal are derived. It is shown that an - optimum 'set' of prime sinusoids can be selected to minimise the harmonics generated by a cubic nonlinearity, and that the amplitude probability distribution of the new signal can be modelled to a pre-defined shape. The new signal is successfully applied to a range of highly nonlinear systems, both real and simulated, with and without memory and is shown to be superior to a pseudo random binary sequence in terms of accuracy and noise immunity. Theoretical modelling predictions are presented, based on the single input describing function for the frequency response analyser results and upon the Gaussian input describing function for the results derived from the new test signal. Both of these estimates have been compared with the models obtained experimentally using Powell's optimisation procedure.