Zero-phase filtering of a minimum-phase wavelet

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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 , where can be complex. Because the imaginary part is zero, the phase is zero.

Solution

Let be the minimum-phase signal and the zero-phase filter. Time-domain filtering is accomplished by convolution, ; in the frequency domain the result is . The factors of occur in pairs of the form , and each pair has roots , . If , then . Thus, one member of each pair of roots is not minimum-phase and consequently the filtered signal is mixed-phase.

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