1. Of the set of all those wavelets, filters, or systems that have the same amplitude spectrum or autocorrelation, particular members can be characterized by their phase spectra (phase as a function of frequency). (They can also be characterized in other ways, for example by the location of their roots in the z-domain; see Figure P-2.) The principal feature of minimum phase is that the energy arrives as early as possible. The phase of a minimum-phase wavelet is smaller and its energy builds up faster (i.e., it is minimum delay) than for any other causal wavelet with the same amplitude spectrum (or same autocorrelation). A two-term wavelet (or doublet) [a,b] is minimum phase (minimum delay) if |a|>|b|.
Any wavelet may be represented as the convolution of doublets and a wavelet is minimum phase if all of its doublet factors are minimum phase. For example, the z-transform of a wavelet might be (6+z–z2), which can be expressed as (3–z)(2+z), each of which is minimum phase; hence the wavelet is minimum phase. Minimum phase is sometimes expressed as having all roots outside the unit circle in the z-plane, or as having no zeros in the right half of the Laplace transform S-plane. A maximum-phase or maximum-delay doublet [a,b] has |a|<|b|. Maximum-phase wavelets have all their roots inside the unit circle in the z-plane. For a linear-phase wavelet, the phase-frequency plot is linear. If its intercept is nπ (where n is any integer), such a wavelet is symmetrical.
A zero-phase wavelet has phase identically zero; it is symmetrical about zero but is not causal.