Implicitly Causal Expansions in the Frequency Domain.

Abstract

While the causality statement in the time domain is quite simple, the corresponding statement in the frequency domain involves Hilbert transform relations between the real and imaginary parts of the relevant transfer function. If the transfer function is minimum phase, one can go further and develop Hilbert transform relations between the amplitude and phase. Such relations are important and useful in at least two classes of problems: analysis of frequency‐sweep data and active control. In both problems, one usually wishes to simultaneously enforce the causality statement while achieving some other goals. This lecture will explore various ways of accomplishing this and will focus on implicitly causal expansions of transfer functions in the frequency domain. The expansion functions are built directly from Hilbert transform pairs and therefore the series is guaranteed to be causal. One example is a complex Fourier series in frequency in which the coefficients are required to be real valued. Examples will be...