Transfer Functions and Sequences in the Time Domain

Abstract

The transfer function of a discrete system is defined by the $z-$transform of the impulse response. This chapter explores the magnitude and phase characteristics by introducing the notion of poles and zeros of a complex function. Phase responses are crucial in establishing the time delay sequences representing the impulse responses. Accumulated phase functions (continuous phase functions of the angular frequency) are defined by integration of the group delay functions. In particular, the linear phase is rendered by a pair of symmetric zeros with respect to the unit circle in the complex frequency plane. Auto-correlation sequences and their Fourier transforms (power spectral functions) are represented by the symmetric pairs of zeros. Time sequences such as impulse responses are identified from the power spectral functions by properly selecting the zeros from the symmetric pairs, subject to the initial value of the time sequence being normalized to unity. Poles, which are identified by the frequencies of the free oscillation or eigenfrequencies of the linear system, determine the transient and decaying time sequences. The cumulative spectral analysis (CSA) is used for estimation of the resonance frequencies from transient responses. CSA is helpful in condition monitoring and/or predicting howling frequencies corresponding to unstable poles. Symmetrically or closely located pairs of poles and zeros are important as these combinations govern the behavior of linear filters such as notch, all-pass, and inverse filters.