DiscreteTime Signals and Systems
Dr. Fred J. Taylor, Professor
Lesson Title: FIR Frequency Response
Lesson Number: 14 (Section 63 to 64)
L
Background:
In Chapter 6, the concept of frequency response for FIRs was introduced.
Sections 63 and 64
examines this concept in increasing detail.
DiscreteTime System Frequency Response
If an LTI FIR is presented with periodic input data, then using an exponential or trigonometric
Fourier series, the input can be partitioned into a collection of discrete input signals at discrete
frequencies.
The output consists of a like collection of exponential or trigonometric signals at
the frequencies of the input signal, but with altered amplitudes and phases.
This analysis gives
rise to the concept of the LTI’s frequency response.
In Chapter 6, the steadystate frequency response of a discretetime LTI FIR system is
determined by evaluating the convolution sum at specific frequency locations. In particular, the
frequency response of an
N
th
order FIR, having an impulse response given by
h
[
k
]={
h
0
,
h
1
, … ,
h
N1
}, and sampled at a rate
f
s
, can be expressed as:
(
29
(
29
(
29
ω
ω
ω
ω
φ
j
j
N
m
m
j
i
j
e
e
H
e
h
e
H
∠
=
=
∑

=

1
0
1
where
ω
=
ϖ
T
s
is the normalized discretetime baseband frequency ranging over
ω∈
[

π
,
π
].
Periodicity
Observe that over an extended range
ϖ
=
ω
+(2
π
)s, for s an integer, that:
(
29
(
29
(
29
s
j
j
e
H
e
H
π
ω
ω
2
+
=
.
2
That is, the FIR frequency response replicates itself on integer multiples of the normalized
baseband frequency as motivated in Figure 1. This is a fundamentally important observation in
that it characterizes the frequency domain signature of periodically sampled signal processes is
likewise periodic.
It is implicitly assumed, however, that the sample rate f
s
was chosen with
knowledge that the highest frequency in the signal being sampled and processed is bounded by
f
s
/2.
Therefore, the periodically extended frequency responses are devoid of actual signal
energy in the ideal case.
Figure 1: Baseband and full spectrum of a discretetime filter.
1

π
0
π
f
s
/2
0
f
s
/2
Baseband
4
π
2
π

π
0
π
2
π
4
π
2f
s
f
s
0
f
s
2f
s
…
…
Actual Spectrum
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DiscreteTime Signals and Systems
Dr. Fred J. Taylor, Professor
The magnitude frequency response is called the filter's
gain
(defined at frequency
ω
) and
provides information about how the FIR amplifies or attenuates an input signal.
The
phase
response
(defined at frequency
ω
) establishes the amount of steadystate phase shift imparted
to the input signal.
There are a number of techniques that can be used to mathematically or experimentally
produce an FIR's frequency response. These concepts will only be superficially motivated at this
time. The first method presumes that an FIR's impulse response,
h
[
k
], has been measured or
derived from its underlying equations.
The analysis of the impulse response can be
accomplished using one or more of the many Fourierbased techniques (e.g., Fourier Series),
as suggested in Figure 2. The accuracy of each method is based on the quality of the impulse
response definition and the selected Fourier tool. The top method shown in Figure 2, uses a tool
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 Spring '08
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 Digital Signal Processing, Frequency, Signal Processing, Fir, Dr. Fred J., Fred J. Taylor

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