Digital Signal Processing Reference
In-Depth Information
radians, the frequency
response,
H
(
e
j
ω
), of any discrete-time system is also periodic with period 2
Since the function
e
j
ω
is periodic with period 2
π
π
radians. This is one important distinction between continuous-time and
discrete-time systems. The frequency response is, in general, complex, and
hence we define the
magnitude response
H
(
e
j
ω
)
and phase response/
H
(
e
j
ω
).
Example
An
N
th order
comb filter
has the system function
H
(
z
) = 1 -
z
-N
.
i.
Determine the pole-zero locations of
H
(
z
)
.
ii.
Evaluate the impulse response
h
(
n
)
.
iii.
Determine the frequency response (magnitude and phase) of the
system.
Solution
The system function is
−
N
Hz
()
=−
1
z
N
z
−
1
=
N
z
i.
From the system function, the poles and zeros can be obtained:
Zeros
:
N
zeros at
z
N
==
1
e
j
2
π
k
,
k
=
0 1
,
,
… −
N
1
j
2
π
kN
⇒=
ze
,
k
=
01
,
,
… −
N
1
Poles: N
poles at
z
= 0.
ii.
The impulse response can be obtained by taking the inverse
Z
-trans-
form of the system function
H
(
z
):
h
(
n
) = (
n
)
h
(
n
) =
δ
(
n
) -
δ
(
n
- 1)
iii.
The frequency response can also be obtained from the system func-
tion
H
(
z
) by using Equation 2.10:
()
=
j
ω
−
jN
ω
He
Hz
()
=
1
−
e
j
ω
ze
=
which can be simplified to yield the final result:
()
=
(
)
j
ω
−
j
ω
N
2
sin
He
2
je
ω
N
2
