Properties of minimum-phase wavelets

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Problem 9.12a

Of the four wavelets given in problem 9.8, which are minimum-phase?

Background

Minimum-phase wavelets are discussed briefly in problem 9.11 and in more detail in Sheriff and Geldart, 1995, section 9.4 and section 15.5.6); -transforms are discussed in Sheriff and Geldart, 1995, section 15.5.3.

Solution

The four wavelets are: , , , and . The roots of the wavelets and are 2 and , so both are minimum-phase because the magnitudes of the roots are greater than unity. Since , the roots are 3/2 and 2; is therefore also minimum-phase. Finally, , the roots being and . Because , is mixed-phase.

Problem 9.12b

Find and by calculating in the time domain.

Solution

Problem 9.12c

Repeat part (b) except using transforms.

Solution

Problem 9.12d

Find .

Solution

Problem 9.12e

Does the largest value of a minimum-phase wavelet have to come at ?

Solution

The wavelet is minimum-phase if and . The ratio of the first two terms is and it has its minimum absolute value when and have the same signs. When and have the same sign and are both slightly larger than unity, the ratio is close to 1/2 and the second term is larger than the first. As and/or increase, the ratio increases; the first and second terms are equal when . If and differ significantly in magnitude, the second term can be larger than the first for large values of or ; e.g., if , the ratio is when .

If and have opposite signs, the ratio cannot be smaller than 1 since the two terms in the denominator have opposite signs and the denominator cannot exceed the numerator.

When the wavelet has three factors, the ratio of the first to second term takes the form . When , , and are all close to unity and of the same polarity, the magnitude of the ratio is and the second term is larger than the first. Generalizing to terms the ratio can be .

Problem 9.12f

Can a minimum-phase wavelet be zero at ?

Solution

If a wavelet is zero at , it is of the form (}, so

Since one of the roots is , the wavelet is not minimum-phase.

When we deal with an individual wavelet, we avoid the root by taking as the time when the first nonzero value occurs.

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