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The Frequency Domain Representation
of LTI Systems 75
In general, the frequency response H(e jω )isacomplex function of
ω. The magnitude |H(e jω )| of H(e jω )iscalled the magnitude (or gain)
response function, and the angle ̸ H(e jω )iscalled the phase response
function as we shall see below.
3.3.2 RESPONSE TO SINUSOIDAL SEQUENCES
Let x(n) =A cos(ω 0 n + θ 0 )beaninput to an LTI system h(n). Then
from (3.17) we can show that the response y(n) isanother sinusoid of the
same frequency ω 0 , with amplitude gained by |H(e jω0 )| and phase shifted
by ̸ H(e jω0 ), that is,
y(n) =A|H(e jω0 )| cos(ω 0 n + θ 0 + ̸ H(e jω0 )) (3.18)
This response is called the steady-state response, denoted by y ss (n). It
can be extended to a linear combination of sinusoidal sequences.
∑
A k cos(ω k n + θ k ) −→ H(e jω ) −→ ∑ k A k|H(e jω k
)|
k
cos(ω k n + θ k + ̸ H(e jω k
))
3.3.3 RESPONSE TO ARBITRARY SEQUENCES
Finally, (3.17) can be generalized to arbitrary absolutely summable sequences.
Let X(e jω )=F[x(n)] and Y (e jω )=F[y(n)]; then using the
convolution property (3.11), we have
Y (e jω )=H(e jω ) X(e jω ) (3.19)
Therefore an LTI system can be represented in the frequency domain by
X(e jω ) −→ H(e jω ) −→ Y (e jω )=H(e jω ) X(e jω )
The output y(n) isthen computed from Y (e jω ) using the inverse
discrete-time Fourier transform (3.2). This requires an integral operation,
which is not a convenient operation in MATLAB. As we shall see in
Chapter 4, there is an alternate approach to the computation of output to
arbitrary inputs using the z-transform and partial fraction expansion. In
this chapter we will concentrate on computing the steady-state response.
□ EXAMPLE 3.13 Determine the frequency response H(e jω )ofasystem characterized by h(n) =
(0.9) n u(n). Plot the magnitude and the phase responses.
Solution Using (3.16),
∞∑
∞∑
H(e jω )= h(n)e −jωn = (0.9) n e −jωn
=
−∞
∞∑
(0.9e −jω ) n 1
=
1 − 0.9e −jω
0
0
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