Minimum-phase spectrum

Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing

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 ${|}B\left(f\right){|}$ and a phase-lag spectrum $\varphi \left(f\right)$. 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

${\begin{aligned}F\left(Z\right)=l+0.5Z\end{aligned}}$

(24)

and the other with a Z-transform

${\begin{aligned}G\left(Z\right)=0.5+Z.\end{aligned}}$

(25)

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 $b_{0}+b_{1}Z$ is

${\begin{aligned}{|}B\left(f\right){|=}{\sqrt {b_{0}^{2}+2b_{0}b_{1}{\rm {\ cos\ 2}}\pi f\Delta t+b_{1}^{2}}}.\end{aligned}}$

(26)

Setting $b_{0}=1$ and $b_{\rm {l}}=0.5$ in this formula, we find that the magnitude spectrum $B_{F}\left(f\right)$ of the filter F(Z) is

${\begin{aligned}{|}B_{F}\left(f\right){|=}{\sqrt {{\rm {l+\ co}}{\rm {s\ 2}}\pi f\Delta t+0.25}}={\sqrt {{125+\ cos\ 2}\pi f\Delta t}}.\end{aligned}}$

(27)

Setting $b_{0}={0.5}$ and $b_{\rm {l}}={1}$, we find that the magnitude spectrum $B_{G}\left(f\right)$ of the G(Z) filter is

${\begin{aligned}{|}B_{G}\left(f\right){|=}{\sqrt {{\rm {025+\ cos\ 2}}\pi f\Delta t+{\rm {l}}}}={\sqrt {{\rm {l25+\ cos\ 2}}\pi f\Delta t}},\end{aligned}}$

(28)

which is the same as expression 27 for ${|}B_{F}(f){|}$. 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 $b_{0}+b_{1}Z$ is given by

${\begin{aligned}\varphi \left(f\right)={\rm {tan}}^{-{1}}\left[{\frac {b_{1}{\rm {\ sin\ 2}}\pi f\Delta t}{b_{0}+b_{1}{\rm {\ cos\ 2}}\pi f\Delta t}}\right].\end{aligned}}$

(29)

Thus, for the F(Z) filter, we have, with $b_{0}={1}$ and $b_{1}=0.5$,

${\begin{aligned}{\varphi }_{F}\left(f\right)={\rm {tan}}^{-{1}}\left[{\frac {{\rm {0.5.\ sin\ 2}}\pi f\Delta t}{{1+05\ cos\ 2}\pi f\Delta t}}\right].\end{aligned}}$

(30)

On the other hand, letting $b_{0}=0.5$ and $b_{1}={1}$, we have for the G(Z) filter

${\begin{aligned}{\varphi }_{G}\left(f\right)={\rm {tan}}^{-{l}}\left[{\frac {{\rm {\ sin\ 2}}\pi f\Delta t}{{\rm {0.5+\ cos\ 2}}\pi f\Delta t}}\right].\end{aligned}}$

(31)

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 $\varphi _{F}(f)$ and $\varphi _{G}(f)$ for positive frequencies f. Let us plot both phase curves on the same graph, as we do in Figure 7c.

Figure 7.  (a) The phase spectrum for the minimum-phase F filter $b_{0}={1}$ and $b_{1}=0.5$. (b) The phase spectrum for the maximum-phase G filter $b_{0}=0.5$ and $b_{1}={1.}$. (c) The phase spectrum for the F filter $b_{0}={1}$ and $b_{1}=0.5$ lies below the phase spectrum for the G filter $b_{0}=0.5$ and $b_{1}=1$.

We see that the phase spectrum $\varphi _{F}(f)$ lies below the phase spectrum $\varphi _{G}(f)$. 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 $\phi (\pi )-\phi (-\pi )$. 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 $2N\pi$. 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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Also in this chapter

• Frequency spectrum
• Magnitude spectrum and phase spectrum
• Fourier transform
• Inverse Fourier transform
• Appendix F: Exercises

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