# Minimum-phase spectrum

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Series Geophysical References Series Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing Enders A. Robinson and Sven Treitel 6 http://dx.doi.org/10.1190/1.9781560801610 9781560801481 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 ${\displaystyle {|}B\left(f\right){|}}$ and a phase-lag spectrum ${\displaystyle \varphi \left(f\right)}$. 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

 {\displaystyle {\begin{aligned}F\left(Z\right)=l+0.5Z\end{aligned}}} (24)

and the other with a Z-transform

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

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

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

 {\displaystyle {\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 ${\displaystyle {|}B_{F}(f){|}}$. 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 ${\displaystyle b_{0}+b_{1}Z}$ is given by

 {\displaystyle {\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 ${\displaystyle b_{0}={1}}$ and ${\displaystyle b_{1}=0.5}$,

 {\displaystyle {\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 ${\displaystyle b_{0}=0.5}$ and ${\displaystyle b_{1}={1}}$, we have for the G(Z) filter

 {\displaystyle {\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 ${\displaystyle \varphi _{F}(f)}$ and ${\displaystyle \varphi _{G}(f)}$ 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 ${\displaystyle b_{0}={1}}$ and ${\displaystyle b_{1}=0.5}$. (b) The phase spectrum for the maximum-phase G filter ${\displaystyle b_{0}=0.5}$ and ${\displaystyle b_{1}={1.}}$. (c) The phase spectrum for the F filter ${\displaystyle b_{0}={1}}$ and ${\displaystyle b_{1}=0.5}$ lies below the phase spectrum for the G filter ${\displaystyle b_{0}=0.5}$ and ${\displaystyle b_{1}=1}$.

We see that the phase spectrum ${\displaystyle \varphi _{F}(f)}$ lies below the phase spectrum ${\displaystyle \varphi _{G}(f)}$. 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 ${\displaystyle \phi (\pi )-\phi (-\pi )}$. 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 ${\displaystyle 2N\pi }$. If the phase-lag jump of an (N + 1)-length causal wavelet is between the limits of 0 and ${\displaystyle 2N\pi }$, 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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