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We have seen that in a cascade of filters
the Ztransform polynomials are multiplied together.
For
filters in parallel
the polynomials add.
See Figure 21.
parallel
Figure 21
Filters operating in parallel.

 
We have seen also that a cascade of filters is minimumphase if,
and only if,
each element of the product is minimumphase.
Now we will find a condition that
is sufficient (but not necessary) for
a sum A(Z) + G(Z) to be minimumphase.
First, assume that A(Z) is minimumphase.
Then write
 
(40) 
The question as to whether A(Z) + G(Z) is minimumphase
is now reduced to determining
whether A(Z) and 1 + G(Z)/A(Z) are both minimumphase.
We have assumed that A(Z) is minimumphase.
Before we ask whether 1 + G(Z)/A(Z) is minimumphase,
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 minimumphase is that
the spectrum of A exceed that of G at all frequencies.
In other words, for any real , A > G .
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, obtaining 1 + G(Z)/A(Z),
then the curve will always have a positive real part as in
Figure 22.
garbage
Figure 22
A phase trajectory as in Figure 18
left, but more complicated.

 
Since the curve cannot enclose the origin,
the phase must be that of a minimumphase function.
You can add garbage to a minimumphase wavelet
if you do not add too much.

This abstract theorem has an immediate physical consequence.
Suppose a wave characterized
by a minimumphase 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 minimumphase,
we note that the magnitude of Z^{n}
is unity and the reflection coefficient must be less than unity
(to avoid perpetual motion),
so that takes the role of G(Z).
Thus, a minimumphase wave along with its echo is minimumphase.
We will later consider wave propagation with echoes of echoes
ad infinitum.
EXERCISES:

Find two nonminimumphase wavelets whose sum is minimumphase.

Let A(Z) be a minimumphase 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 that A(Z) is maximumphase and that the degree of G(Z) is less
than or equal to the degree of A(Z). Assume A>G.
Show that A(Z) + G(Z) is maximumphase.

Let A(Z) be minimumphase.
Where are the roots of
in the three cases
 c  < 1,  c  > 1,  c  = 1?
(HINT: The roots of a polynomial are continuous functions of the
polynomial coefficients.)
Next: About this document ...
Up: Spectrum and phase
Previous: ROBINSON'S ENERGYDELAY THEOREM
Stanford Exploration Project
10/21/1998