Digital Signal Processing Reference
In-Depth Information
•
Rewrite
H
(
z
) as follows, by replacing every root outside the unit
circle with its conjugate reciprocal root (i.e., replace
z = c
with
z =
1
/c*
, where the symbol * denotes complex conjugate):
⎛
⎜
1
2
⎞
⎟
⎛
⎜
1
2
⎞
⎟
z
+
z
−
(
z
+
2
1
2
)
Hz
()
=
⎛
⎜
⎞
⎟
⎛
⎜
⎞
⎟
1
3
z
+
z
+
which can be separated as
H
(
z
) =
H
min
(
z
)
×
H
ap
(
z
).
•
, we have to determine the
magnitude of the all-pass system response as follows:
In order to ensure
H
(
e
j
ω
)
=
H
min
(
e
j
ω
)
(
z
+
2
1
2
)
Hz
ap
()
=
⎛
⎜
⎞
⎟
z
+
•
Hence, the frequency response of the all-pass system is
(
)
j
ω
e
+
2
1
2
()
=
j
ω
He
ap
⎛
⎜
⎞
⎟
j
ω
e
+
and the magnitude response is
(
since the system is all-
pass, the magnitude response is independent of frequency
)
H
(
e
j
ω
)
=
H
(
e
j
0
)
(
)
j
0
e
+
2
1
2
()
=
(
12
+
)
j
ω
He
=
=
2
ap
⎛
⎜
⎞
⎟
⎛
⎜
⎞
⎠
1
2
j
0
e
+
1
+
⎟
•
Hence, if we redefine
⎛
⎜
1
2
⎞
⎟
⎛
⎜
1
2
⎞
⎟
z
+
z
−
Hz
min
()
=
2
⎛
⎜
⎞
⎟
1
3
z
+
and
