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We have seen that in a cascade of filters
the filter polynomials are multiplied together.
One might conceive of adding two polynomials
*A*(*Z*) and *G*(*Z*) when they correspond to filters which operate
in parallel. See Figure 10.

**2-10
**

Figure 10
Filters operating in parallel.

When filters operate in parallel their *Z* transforms add together.
We have seen that a cascade of filters is minimum phase if,
and only if, each element of the product is minimum phase.
Now we will see a sufficient (but not necessary)
condition that the sum *A*(*Z*) + *G*(*Z*) be minimum phase.
First of all, let us assume that *A*(*Z*) is minimum phase.
Then we may write

| |
(13) |

The question whether *A*(*Z*) + *G*(*Z*) is minimum phase is now reduced
to determining whether *A*(*Z*) and 1 + *G*(*Z*)/*A*(*Z*) are both minimum phase.
We have assumed that *A*(*Z*) is minimum phase.
Before we ask whether 1 + *G*(*Z*)/*A*(*Z*) is minimum phase
we need to be sure that it is causal.
Since 1/*A*(*Z*) is expandable in positive powers of *Z* only,
then *G*(*Z*)/*A*(*Z*) is also causal.
We will next see that a sufficient condition for
1 + *G*(*Z*)/*A*(*Z*) to be minimum phase is that
the spectrum of *A* exceeds that of *G* at all frequencies.
In other words, for any real , .
Thus, if we plot the curve of *G*(*Z*)/*A*(*Z*) in the complex plane,
for real it lies everywhere inside the unit circle.
Now if we add unity --getting 1 + *G*(*Z*)/*A*(*Z*), the curve
will always have a positive real part. See Figure 11.
**2-11
**
Figure 11
Phase of a positive real function lies between . |
| |

Since the curve cannot enclose the origin,
the phase must be that of a minimum-phase function.
In words, ``You can add garbage to a minimum-phase wavelet
if you do not add too much." This somewhat abstract theorem has
an immediate physical consequence. Suppose a wave characterized
by a minimum phase *A*(*Z*) is emitted from a source and
detected at a receiver some time later.
At a still later time an echo bounces off
a nearby object and is also detected at the receiver.
The receiver sees the signal
where *n* measures the delay from
the first arrival to the echo and represents the amplitude
attenuation of the echo. To see that *Y*(*Z*) is minimum phase, we
note that the magnitude of *Z*^{n} is unity and that the reflection
coefficient must be less than unity (to avoid perpetual
motion) so that takes the role of *G*(*Z*). Thus
a minimum-phase wave along with its echo is minimum phase. We will
later consider wave propagation situations with echoes of the echoes
ad infinitum.

## EXERCISES:

- Find two nonminimum-phase wavelets whose sum is minimum phase.
- Let
*A*(*Z*) be a minimum-phase polynomial of degree *N*. Let
. Locate in the complex *Z* plane
the roots of *A*'(*Z*). *A*'(*Z*) is called *maximum phase*.
[HINT: Work the simple case *A*(*Z*) = *a*_{0} + *a*_{1}Z first.]
- Suppose
*A*(*Z*) is maximum phase and that the degree of *G*(*Z*) is less
than or equal to the degree of *A*(*Z*). Assume .Show that *A*(*Z*) + *G*(*Z*) is maximum phase.
- Let
*A*(*Z*) be minimum phase. Where are the roots of
in the three cases
?
(HINT: The roots of a polynomial are continuous functions of the
polynomial coefficients.)

** Next:** POSITIVE REAL FUNCTIONS
** Up:** One-sided functions
** Previous:** MINIMUM PHASE
Stanford Exploration Project

10/30/1997