# Difference between revisions of "Dictionary:Phase characteristics"

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[[File:Segp2.jpg|thumb|FIG. P-2. (a) <b>Phase characterization of wavelets</b> having the same amplitude spectrum. (b) <b>Minimum-phase</b> wavelet and its phase spectrum: (1–0.8''z'')<sup>2</sup>(1+0.5''z'')<sup>2</sup>=1–0.6''z''–0.71''z''<sup>2</sup>+0.24''z''<sup>3</sup>+0.16''z''<sup>4</sup>. (c) <b>Linear phase</b>: (1–0.8''z'')(0.8–''z'')((1+0.5''z'')(0.5+''z'')=0.4+0.18''z''–1.25''z''<sup>2</sup>+0.18''z''<sup>3</sup>+0.4''z''<sup>4</sup>. (<b>d</b>) <b>Maximum phase</b>: (0.8–''z'')<sup>2</sup>(0.5+''z'')<sup>2</sup>=0.16+0.24''z''–0.71''z''<sup>2</sup>–0.6''z''<sup>3</sup>+''z''<sup>4</sup>. (e) <b>Zero phase</b>: 0.4''z''<sup>–2</sup>+0.18''z''<sup>–1</sup>–1.25+0.18''z''+0.4''z''<sup>2</sup>. 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: ''<sub>''xy''</sub>(''z'')=(1–0.8''z'')<sup>2</sup>(0.8–''z'')<sup>2</sup>(1+0.5''z'')<sup>2</sup>(0.5+''z'')<sup>2</sup>. In a more general case roots may be complex.]] | [[File:Segp2.jpg|thumb|FIG. P-2. (a) <b>Phase characterization of wavelets</b> having the same amplitude spectrum. (b) <b>Minimum-phase</b> wavelet and its phase spectrum: (1–0.8''z'')<sup>2</sup>(1+0.5''z'')<sup>2</sup>=1–0.6''z''–0.71''z''<sup>2</sup>+0.24''z''<sup>3</sup>+0.16''z''<sup>4</sup>. (c) <b>Linear phase</b>: (1–0.8''z'')(0.8–''z'')((1+0.5''z'')(0.5+''z'')=0.4+0.18''z''–1.25''z''<sup>2</sup>+0.18''z''<sup>3</sup>+0.4''z''<sup>4</sup>. (<b>d</b>) <b>Maximum phase</b>: (0.8–''z'')<sup>2</sup>(0.5+''z'')<sup>2</sup>=0.16+0.24''z''–0.71''z''<sup>2</sup>–0.6''z''<sup>3</sup>+''z''<sup>4</sup>. (e) <b>Zero phase</b>: 0.4''z''<sup>–2</sup>+0.18''z''<sup>–1</sup>–1.25+0.18''z''+0.4''z''<sup>2</sup>. 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: ''<sub>''xy''</sub>(''z'')=(1–0.8''z'')<sup>2</sup>(0.8–''z'')<sup>2</sup>(1+0.5''z'')<sup>2</sup>(0.5+''z'')<sup>2</sup>. In a more general case roots may be complex.]] | ||

<b>1</b>. 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 [[Special:MyLanguage/Dictionary:Fig_P-2|P-2]].) The principal feature of <b>minimum phase</b> 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 <b>minimum delay</b>) than for any other causal wavelet with the same amplitude spectrum (or same autocorrelation). A two-term wavelet (or <b>doublet</b>) [''a,b''] is minimum phase (minimum delay) if |''a''|>|''b''|. | <b>1</b>. 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 [[Special:MyLanguage/Dictionary:Fig_P-2|P-2]].) The principal feature of <b>minimum phase</b> 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 <b>minimum delay</b>) than for any other causal wavelet with the same amplitude spectrum (or same autocorrelation). A two-term wavelet (or <b>doublet</b>) [''a,b''] is minimum phase (minimum delay) if |''a''|>|''b''|. | ||

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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''<sup>2</sup>), 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 <b>maximum-phase</b> or <b>maximum-delay</b> doublet [''a,b''] has |''a''|<|''b''|. Maximum-phase wavelets have all their roots inside the unit circle in the ''z''-plane. For a <b>linear-phase</b> wavelet, the phase-frequency plot is linear. If its intercept is ''n''π (where ''n'' is any integer), such a wavelet is symmetrical. | 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''<sup>2</sup>), 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 <b>maximum-phase</b> or <b>maximum-delay</b> doublet [''a,b''] has |''a''|<|''b''|. Maximum-phase wavelets have all their roots inside the unit circle in the ''z''-plane. For a <b>linear-phase</b> wavelet, the phase-frequency plot is linear. If its intercept is ''n''π (where ''n'' is any integer), such a wavelet is symmetrical. | ||

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A <b>zero-phase</b> wavelet has phase identically zero; it is symmetrical about zero but is not causal. | A <b>zero-phase</b> wavelet has phase identically zero; it is symmetrical about zero but is not causal. | ||

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## Latest revision as of 10:55, 9 March 2018

**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.