Dictionary:Fig P-2

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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.