# Dictionary:Phase characteristics

**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*–*z*^{2}), 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 *n*π (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.