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The Frequency Domain Representation
of LTI Systems 77
10
Magnitude Response
8
|H|
6
4
2
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
frequency in π units
Phase Response
0
Phase in π Radians
−0.1
−0.2
−0.3
−0.4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
frequency in π units
FIGURE 3.7 Frequency response plots in Example 3.13
then to evaluate its frequency response from (3.16), we would need the impulse
response h(n). However, using (3.17), we can easily obtain H(e jω ).
We know that when x(n) =e jωn , then y(n) must be H(e jω )e jωn . Substituting
in (3.20), we have
H(e jω )e jωn +
N∑
a l H(e jω )e jω(n−l) =
l=1
M∑
b m e jω(n−m)
m=0
or
∑ M
H(e jω m=0
)=
b m e −jωm
1+ ∑ N
l=1 a (3.21)
l e −jωl
after canceling the common factor e jωn term and rearranging. This equation
can easily be implemented in MATLAB, given the difference equation
parameters.
□ EXAMPLE 3.15 An LTI system is specified by the difference equation
y(n) =0.8y(n − 1) + x(n)
a. Determine H(e jω ).
b. Calculate and plot the steady-state response y ss(n) to
x(n) =cos(0.05πn)u(n)
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