R24 - ENEE 241 02 READING ASSIGNMENT 24 Tue 05/12 Lecture...

ENEE 241 02
*
READING ASSIGNMENT 24 Tue 05/12 Lecture
Topics: convolution in the z -domain; frequency-selective 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 time-invariant system
are related by
Y ( z ) = H ( z ) X ( z )
Thus convolution in the time domain amounts to multiplication in the (so-called) 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 finite-duration 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 two-sided exponential input to the cascade.
1

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)