Minimum-Phase Systems—Deconvolution

Abstract

In this chapter we introduce the notion of the minimum-phase system. We show with simple examples for two causal FIR systems having the same amplitude of the frequency gain, that a filter whose zeros are located within the unit circle will have a lower variation of phase with frequency. It follows that the impulse response of this filter is earlier. Since a minimum-phase causal filter has its zeros inside the unit circle, its inverse will be causal with its poles inside the unit circle, resulting in its stability. Deconvolving a signal is thus possible, that is to say, finding back the input signal of a filter by filtering the output signal of that filter. Then, we present the general problem of deconvolution with its frequency and time aspects. Deconvolution by the complex cepstrum method is introduced. It is illustrated with an example inspired from seismic measurements.