ENEE 241 02
*
READING ASSIGNMENT 24
Tue 05/12 Lecture
Topics:
convolution in the
z
domain; frequencyselective filters
Textbook References:
sections 4.5, 4.6.5
Key Points:
•
If
X
(
z
)
def
=
∞
X
n
=
∞
x
[
n
]
z

n
,
then the input
x
[
·
], impulse response
h
[
·
] and output
y
[
·
] of a linear timeinvariant system
are related by
Y
(
z
) =
H
(
z
)
X
(
z
)
Thus convolution in the time domain amounts to multiplication in the (socalled)
z
domain.
•
Ideal filters have amplitude responses which are piecewise constant functions of
ω
.
These
responses are unattainable in practice.
•
Ideal filters can be approximated by FIR filters. A longer coefficient vector affords a better
approximation to an ideal amplitude response, at the expense of a longer delay between input
and output.
Theory and Examples:
1
. The
z
transform of a sequence
x
=
x
[
·
] is the power series defined by
X
(
z
) =
∞
X
n
=
∞
x
[
n
]
z

n
Convergence may be an issue if
x
[
·
] has infnite duration.
Notably, exponential sequences
such as
x
[
n
] =
z
n
0
(for all
n
) do not have a
z
transform except in the trivial case
z
0
= 0.
2
. The
z
transform of a finiteduration sequence is always well defined; it was introduced earlier
as the system function of a FIR filter:
H
(
z
) =
M
X
n
=0
b
n
z

n
=
∞
X
n
=
∞
h
[
n
]
z

n
We have also seen that the system function
H
(
z
) of the cascade of two FIR filters (labeled 1
and 2) satisfies
H
(
z
) =
H
1
(
z
)
H
2
(
z
)
This result was obtained by applying a twosided exponential input to the cascade.
1
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δ
H
1
h
*
h
(2)
(1)
h
(1)
H
2
3
. By applying an input
δ
=
δ
[
·
] to the same cascade, we see (figure above) that the impulse
response of the cascade is given by
h
=
h
(1)
*
h
(2)
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 Spring '08
 staff
 Digital Signal Processing, Frequency, Signal Processing, LTI system theory, Impulse response, amplitude response

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