Minimum-phase spectrum
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| Series | Geophysical References Series |
|---|---|
| Title | Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing |
| Author | Enders A. Robinson and Sven Treitel |
| Chapter | 6 |
| DOI | http://dx.doi.org/10.1190/1.9781560801610 |
| ISBN | 9781560801481 |
| Store | SEG Online Store |
We saw in the previous section that the transfer function of a digital filter can be expressed conveniently in terms of a magnitude spectrum Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {|}B\left(f\right){|} and a phase-lag spectrum Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \varphi \left(f\right) . It is much easier to visualize the physical significance of the magnitude spectrum of a filter than it is to understand the corresponding phase spectrum. Thus, some people tend to neglect the phase spectra of filters when they are solving actual problems. It turns out, however, that the phase spectra are fundamentally important in classifying filters with identical amplitude characteristics.
This fact can be illustrated best by a simple example. Let us consider two causal FIR filters, one with a Z-transform
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} F\left(Z\right) =l+0.5Z \end{align} ()
and the other with a Z-transform
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} G\left(Z\right) =0.5+Z. \end{align} ()
In other words, the weighting coefficients of filter F(Z) are (1, 0.5), whereas the weighting coefficients of filter G(Z) are (0.5, 1). We recall that the magnitude spectrum of the filter Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): b_0+b_{{1}}Z is
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {|}B\left(f\right){|=}\sqrt{b^{2}_0+2b_0b_{{1}} {\rm \ cos\ 2}\pi f\Delta t+b^{2}_{{1}}}. \end{align} ()
Setting Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): b_0 =1 and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): b_{{\rm l}}=0.5 in this formula, we find that the magnitude spectrum Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): B_F\left(f\right) of the filter F(Z) is
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {|}B_F\left(f\right){|=}\sqrt{{\rm l+\ co}{\rm s\ 2}\pi f\Delta t+0.25}=\sqrt{{125+\ cos\ 2}\pi f\Delta t}. \end{align} ()
Setting Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): b_0={0.5} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): b_{{\rm l}}={1} , we find that the magnitude spectrum Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): B_G\left(f\right) of the G(Z) filter is
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {|}B_G\left(f\right){|=}\sqrt{{\rm 025+\ cos\ 2}\pi f\Delta t+{\rm l}} =\sqrt{{\rm l25+\ cos\ 2}\pi f\Delta t}, \end{align} ()
which is the same as expression 27 for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {|}B_F(f) {|} . Thus, the two filters have the same magnitude spectrum.
The question now arises: What is the relation between the phase spectra of the two filters F(Z) and G(Z)? We recall that the phase spectrum of the filter Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): b_0+b_{{1}}Z is given by
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} \varphi \left(f\right)={{\rm tan}} ^{-{ 1}}\left[\frac{b_{{1}}{\rm \ sin\ 2}\pi f\Delta t}{b_0+b_{{1}}{\rm \ cos\ 2}\pi f\Delta t}\right]. \end{align} ()
Thus, for the F(Z) filter, we have, with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): b_0={1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): b_{{1}}=0.5 ,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\varphi }_F\left(f\right)={{\rm tan}} ^{-{1}}\left[\frac{{\rm 0.5.\ sin\ 2}\pi f\Delta t}{{1+05\ cos\ 2}\pi f\Delta t}\right]. \end{align} ()
On the other hand, letting Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): b_0=0.5 and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): b_{{1}}={1} , we have for the G(Z) filter
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \begin{align} {\varphi }_G\left(f\right)={{\rm tan}} ^{-{l}}\left[\frac{{\rm \ sin\ 2}\pi f\Delta t}{{\rm 0.5+\ cos\ 2}\pi f\Delta t}\right]. \end{align} ()
In summary, the F and G filters have the same magnitude spectra but different phase spectra. The phase spectra for the two filters are plotted in Figure 7a and 7b, respectively.
For a real signal, the phase spectrum is an odd function of frequency f (i.e., the phase spectrum is antisymmetric about the origin of the frequency axis). As a result, we have to plot only Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \varphi _F (f) and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \varphi _G (f) for positive frequencies f. Let us plot both phase curves on the same graph, as we do in Figure 7c.
We see that the phase spectrum Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \varphi _F (f) lies below the phase spectrum Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \varphi _G (f) . We can state now that the filter F(Z) has a phase spectrum that is less than the phase spectrum of the filter G(Z) for the range of positive frequencies. At f = 0, the phase spectra of both filters are zero.
Let us restrict ourselves to real weighting coefficients, and let us consider only causal digital filters, for which the output can never precede the input in time. Then we see that the pair of causal FIR filters F(Z) and G(Z) represents a set of digital causal filters, each of which has the same amplitude spectrum. This set of causal filters is exhaustive in the sense that the two real weighting coefficients 1 and 0.5 can occur only in the sequence (1, 0.5) or in the sequence (0.5, 1). Now we can say that the filter F(Z) has the minimum-phase spectrum of the filter set {F(Z), G(Z)}.
The concept of minimum-phase filters is quite general and can be extended to other sets of causal filters, with each filter in the set having the same amplitude spectrum. In each such set, there is one filter whose phase spectrum is minimum with respect to the phase spectra of all other members of that set, and that filter is the minimum-phase filter. All filters within the given set have the same amplitude spectrum by definition.
The phase-lag jump is defined as the total shift in phase lag over the Nyquist range; that is, the phase jump is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): \phi (\pi ) - \phi ( - \pi ) . A causal wavelet is a minimum-phase-lag wavelet (which simply is called minimum-phase) if and only if its phase-lag jump is zero. An (N + 1)-length causal wavelet is maximum phase if and only if its phase-lag jump is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): 2N\pi . If the phase-lag jump of an (N + 1)-length causal wavelet is between the limits of 0 and $ 2N\pi $, the wavelet is mixed phase. The terms minimum-phase, mixed-phase, and maximum-phase wavelet only apply to causal wavelets.
The phase spectrum of a minimum-phase wavelet is uniquely derivable from its amplitude spectrum. This result means that the coefficients of a minimum-phase wavelet can be determined uniquely from knowledge of its amplitude spectrum. Equivalently, the coefficients of a minimum-phase wavelet can be determined uniquely from knowledge of its autocorrelation.
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| Fourier transform | Inverse Fourier transform |
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| Filtering | Wavelets |
Also in this chapter
- Frequency spectrum
- Magnitude spectrum and phase spectrum
- Fourier transform
- Inverse Fourier transform
- Appendix F: Exercises
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