# Properties of minimum-phase wavelets

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Series Geophysical References Series Problems in Exploration Seismology and their Solutions Lloyd P. Geldart and Robert E. Sheriff 9 295 - 294 http://dx.doi.org/10.1190/1.9781560801733 ISBN 9781560801153 SEG Online Store

## 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); ${\displaystyle z}$-transforms are discussed in Sheriff and Geldart, 1995, section 15.5.3.

### Solution

The four wavelets are: ${\displaystyle a_{t}=[2,\;-1]}$, ${\displaystyle b_{t}=[4,1]}$, ${\displaystyle c_{t}=[6,\;-7,\;2]}$, and ${\displaystyle d_{t}=[4,9,2]}$. The roots of the wavelets ${\displaystyle a_{t}}$ and ${\displaystyle b_{t}}$ are 2 and ${\displaystyle -4}$, so both are minimum-phase because the magnitudes of the roots are greater than unity. Since ${\displaystyle c_{t}\leftrightarrow (6-7z+2z^{2})=(3-2z)(2-z)}$, the roots are 3/2 and 2; ${\displaystyle c_{t}}$ is therefore also minimum-phase. Finally, ${\displaystyle d_{t}\leftrightarrow (4+9z+2z^{2})=(4+z)(1+2z)}$, the roots being ${\displaystyle -4}$ and ${\displaystyle -1/2}$. Because ${\displaystyle |-1/2|<1}$, ${\displaystyle d_{t}}$ is mixed-phase.

## Problem 9.12b

Find ${\displaystyle a_{t}*b_{t}}$ and ${\displaystyle a_{t}*c_{t}}$ by calculating in the time domain.

### Solution

{\displaystyle {\begin{aligned}a_{t}*b_{t}=[2,\;-1]*[4,\;1]\;=8,\;-4\\2,\;-1\\=[8,\;-2,\;-1];\end{aligned}}} {\displaystyle {\begin{aligned}a_{t}*c_{t}=[2,\;-1]*[6,\;-7,\;2]\;=12,\;-6\\-14,\;7\\4,\;-2\\=[12,\;-20,\;11,\;2].\end{aligned}}}

## Problem 9.12c

Repeat part (b) except using transforms.

### Solution

{\displaystyle {\begin{aligned}A(z)=(2-z)\;,\;B(z)=(4+z)\;,\;C(z)=(6-7z+2z^{2});\\a_{t}*b_{t}\leftrightarrow (2-z)(4+z)=8-2z-z^{2}\leftrightarrow [8,\;-2,\;-1];\\a_{t}*c_{t}\leftrightarrow (2-z)(6-7z+2z^{2})=12-20z+11z^{2}-2z^{3}\\=[12,\;-20,\;11,\;-2].\end{aligned}}}

## Problem 9.12d

Find ${\displaystyle a_{t}*b_{t}*c_{t}}$.

### Solution

{\displaystyle {\begin{aligned}a_{t}*b_{t}*c_{t}=(a_{t}*b_{t})*c_{t}=[8,\;-2,\;-1]*[6,\;-7,\;2]\\=48,\;-12,\;-6\\-56,\;14,\;7\\16,\;-4,\;-2\\=[48,\;-68,\;24,\;3,\;-2].\\\end{aligned}}}

## Problem 9.12e

Does the largest value of a minimum-phase wavelet have to come at ${\displaystyle t=0}$?

### Solution

The wavelet ${\displaystyle (a+z)(b+z)=[ab+(a+b)z+z^{2}]}$ is minimum-phase if ${\displaystyle |a|>1}$ and ${\displaystyle |b|>1}$. The ratio of the first two terms is ${\displaystyle ab/(a+b)=a/(1+a/b)}$ and it has its minimum absolute value when ${\displaystyle a}$ and ${\displaystyle b}$ have the same signs. When ${\displaystyle a}$ and ${\displaystyle b}$ 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 ${\displaystyle a}$ and/or ${\displaystyle b}$ increase, the ratio increases; the first and second terms are equal when ${\displaystyle a=b=2}$. If ${\displaystyle a}$ and ${\displaystyle b}$ differ significantly in magnitude, the second term can be larger than the first for large values of ${\displaystyle a}$ or ${\displaystyle b}$; e.g., if ${\displaystyle b\approx 1}$, the ratio is ${\displaystyle \approx 0.9}$ when ${\displaystyle a=9}$.

If ${\displaystyle a}$ and ${\displaystyle b}$ 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 ${\displaystyle abc/(ab+ac+bc)}$. When ${\displaystyle a}$, ${\displaystyle b}$, and ${\displaystyle c}$ are all close to unity and of the same polarity, the magnitude of the ratio is ${\displaystyle \approx 1/3}$ and the second term is larger than the first. Generalizing to ${\displaystyle n}$ terms the ratio can be ${\displaystyle \approx 1/n}$.

## Problem 9.12f

Can a minimum-phase wavelet be zero at ${\displaystyle t=0}$?

### Solution

If a wavelet is zero at ${\displaystyle t=0}$, it is of the form (${\displaystyle 0,a,b,c,\ldots }$}, so

{\displaystyle {\begin{aligned}\left[0,\;a,\;b,\;c,\;.\;.\;.\right]\;\leftrightarrow (0+az+bz^{2}+cz^{3}+\cdots \cdot )=z(a+bz+cz^{2}+\cdots ).\end{aligned}}}

Since one of the roots is ${\displaystyle z=0<1}$, the wavelet is not minimum-phase.

When we deal with an individual wavelet, we avoid the root ${\displaystyle z=0}$ by taking ${\displaystyle t=0}$ as the time when the first nonzero value occurs.