Introduction: Pulse Train, Resonance, and Zeros

Poles and zeros might be the key words to transfer functions of acoustic systems from the perspective of periodic responses. Chapter 1 reviews the transfer functions so that magnitude and phase responses might be well defined by the poles and zeros. In particular, the phase responses would be well organized by the minimum-phase or non-minimum phase zeros including poles through group delay functions. The group delay, which gives the delay of modulation envelopes, may be also the key to perception of the phase for reverberation in rooms. Reverberation may be, more or less, characterized by the all-pass properties in which local phase variabilities such as the group delay are informative rather than magnitude properties. The group delay functions for the poles and zeros may graphically develop a decomposition of the transfer function into a product of the minimum-phase and all-pass components, where the magnitude and phase responses could not be independent to each other for the minimum-phase transfer function. This chapter would present a nice theoretical background to the readers for this book, even who may be well familiar with discrete signal processing, because the phase for acoustic systems might be paid less attention.