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Lesson Title: Impulse Invariant FIR
Lesson Number: 13 (Section 61 to 62)
Background:
In Chapter 5, elementary FIR filters were discussed. The Chapter began by establishing
the rules and purpose of convolution and ended with a discussion of some of the basic
properties of FIRs supported with selected demonstrations.
In Chapter 6, the behavior
of such systems are explored in the frequency domain.
DiscreteTime System Frequency Response
It has been established that the response of an LTI consists of transient and steady
state responses.
The focus of initial Chapter 6 studies is the analysis of the steady state
responses of LTI systems in the frequency domain. This analysis gives rise to the
concept of the
steadystate
frequency
response of the LTI system.
The development of a steady state frequency response framework can begin with the
now familiar convolution sum:
[
]
[
]
k
n
x
h
n
y
N
k
k

=
∑
=
0
1.
Consider the special case where the input is the discretetime signal x[n] is given by:
[
]
(
29
π
ω
ϖ
φ
<
≤

=
=
+
:
/
;
s
n
j
T
Ae
n
x
2.
which represents a complex exponential signal of magnitude A,
normalize frequency
ω
,
and phase offset
φ
.
In continuous time, the signal is assumed to satisfy:
(
29
(
29
+
=
t
j
Ae
t
x
3.
Upon substituting Equation 2 into Equation 1, the response of an (N+1)order FIR to a
complex exponential signal is given by:
[
]
(
29
(
29
(
29
(
29
(
29
(
29
(
29
(
29
(
29
(
29
n
j
j
n
j
j
k
j
N
k
k
k
j
n
j
j
N
k
k
k
n
j
N
k
k
e
Ae
e
Ae
e
h
e
e
Ae
h
Ae
h
n
y
ℑ
=
=
=
=

=

=
+

=
∑
∑
∑
0
0
0
4.
where the operator
ℑ
is called the
frequency response function
or simply
frequency
response
.
Specifically, the frequency response can be defined to be:
(
29
(
29
(
29
C
∈
=
=
ℑ

=
∑
k
j
N
k
k
j
e
h
e
H
0
5
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View Full Document which is a complex function of the normalized frequency
ω
.
It should be noted that
Equation 4 is restricted to the case where the input is a complex exponential
[
]
(
29
φ
ω
+
=
n
j
Ae
n
x
.
The exponential form of the input also suggests (using Euler’s
equation) that Equation 5 can expand the support the study of a system’s steadystate
sinusoidal response.
Whether the input is an exponential or sinusoid, the frequency
response function of the filter can be represented in magnitude/phase form or real and
imaginary terms as suggested in Equation 6, namely:
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This note was uploaded on 08/21/2010 for the course EEL 3135 taught by Professor ? during the Spring '08 term at University of Florida.
 Spring '08
 ?

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