# Zero-phase filtering of a minimum-phase wavelet

Series Geophysical References Series Problems in Exploration Seismology and their Solutions Lloyd P. Geldart and Robert E. Sheriff 9 295 - 366 http://dx.doi.org/10.1190/1.9781560801733 ISBN 9781560801153 SEG Online Store

## Problem 9.16

Show that the result of passing a minimum-phase signal through a zero-phase filter is mixed phase.

### Background

Zero-phase signals are discussed in Sheriff and Geldart, 1995, section 15.5.6d, where it is shown that the spectrum of a zero-phase signal comprises products of pairs of factors of the form $(az-1)(az^{-1}-1)=(1+a^{2})-a(z+z^{-1})=(1+a^{2})-2a{\rm {\;cos\;}}(\omega \Delta )$ , where $a$ can be complex. Because the imaginary part is zero, the phase is zero.

### Solution

Let $\omega _{t}$ be the minimum-phase signal and $f_{t}$ the zero-phase filter. Time-domain filtering is accomplished by convolution, $\omega _{t}*f_{t}$ ; in the frequency domain the result is $W(z)F(z)$ . The factors of $F(z)$ occur in pairs of the form $(az-1)(az^{-1}-1)$ , and each pair has roots $z=a$ , $1/a$ . If $|a|>1$ , then $|1/a|<1$ . Thus, one member of each pair of roots is not minimum-phase and consequently the filtered signal is mixed-phase.