MINIMUM PHASE SYSTEMS
A causal, rational system is stable if all its poles are inside the unit
circle.
Such a system is called called
minimum phase
if, in
addition, all its
zeros
are inside the unit circle.
Equivalently it has
no zeros outside the unit circle, including no zeros at
∞
.
We now
show that any causal, stable, real rational system,
)
(
z
H
, can be
factored as
)
(
)
(
)
(
z
H
z
H
z
H
a
m
=
(6.1)
where
)
(
z
H
m
is minimum phase and
)
(
z
H
a
is allpass.
Before starting the proof, some discussion is in order.
Suppose
that
)
(
1
z
H
and
)
(
2
z
H
are two causal, stable, real rational systems
such that
)
(
)
(
)
(
2
1
z
H
z
H
z
H
a
=
.
(6.2)
This relation can be thought of in the following way.
The allpass,
)
(
z
H
a
, is
factored out
of
)
(
1
z
H
, leaving the quotient
)
(
2
z
H
.
Note
that
)
(
)
(
)
(
)
(
2
2
1
w
w
w
w
j
j
a
j
j
e
H
e
H
e
H
e
H
=
=
so that
)
(
1
z
H
and
)
(
2
z
H
have same magnitude response.
With
regard to the phase response, let
)
(
),
(
),
(
2
1
w
j
w
j
w
j
a
and
denote
the respective phase response functions.
Then we have
)
(
)
(
)
(
2
1
w
j
w
j
w
j
a
+
=
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[ ]
)
(
)
(
)
(
2
1
w
j
w
j
w
j
a
′

+
′

=
′

.
Recalling that
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 Spring '07
 ee
 Unit Circle, zi, minimum phase

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