Translations:Dictionary:Phase characteristics/1/en

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FIG. P-2. (a) Phase characterization of wavelets having the same amplitude spectrum. (b) Minimum-phase wavelet and its phase spectrum: (1–0.8z)2(1+0.5z)2=1–0.6z–0.71z2+0.24z3+0.16z4. (c) Linear phase: (1–0.8z)(0.8–z)((1+0.5z)(0.5+z)=0.4+0.18z–1.25z2+0.18z3+0.4z4. (d) Maximum phase: (0.8–z)2(0.5+z)2=0.16+0.24z–0.71z2–0.6z3+z4. (e) Zero phase: 0.4z–2+0.18z–1–1.25+0.18z+0.4z2. The zero-phase wavelet is anticipatory, that is, it begins before time zero. Phase curves depend on the time reference. Other mixed-phase wavelets can also be made from these component doublets. (f) Z-plane plot of the roots of the autocorrelation function for the foregoing, all of which have the same autocorrelation: xy(z)=(1–0.8z)2(0.8–z)2(1+0.5z)2(0.5+z)2. In a more general case roots may be complex.

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|.