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74 Chapter 3 THE DISCRETE-TIME FOURIER ANALYSIS
3.3 THE FREQUENCY DOMAIN REPRESENTATION
OF LTI SYSTEMS
We earlier stated that the Fourier transform representation is the most
useful signal representation for LTI systems. It is due to the following
result.
3.3.1 RESPONSE TO A COMPLEX EXPONENTIAL e jω0n
Let x(n) =e jω0n be the input to an LTI system represented by the impulse
response h(n).
e jω0n −→ h(n) −→ h(n) ∗ e jω0n
Then
y(n) =h(n) ∗ e jω0n =
∞∑
−∞
h(k)e jω0(n−k)
[ ∞
]
∑
= h(k)e −jω0k e jω0n (3.15)
−∞
=[F[h(n)]| ω=ω0 ] e jω0n
DEFINITION 1
[Frequency Response] The discrete-time Fourier transform of an impulse
response is called the frequency response (or transfer function)ofanLTI
system and is denoted by
∞∑
H(e jωn ) =
△ h(n)e −jωn (3.16)
−∞
Then from (3.15) we can represent the system by
x(n) =e jω0n −→ H(e jω ) −→ y(n) =H(e jω0 ) × e jω0n (3.17)
Hence the output sequence is the input exponential sequence modified by
the response of the system at frequency ω 0 . This justifies the definition
of H(e jω )asafrequency response because it is what the complex exponential
is multiplied by to obtain the output y(n). This powerful result
can be extended to a linear combination of complex exponentials using
the linearity of LTI systems.
∑
A k e jωkn −→ h(n) −→ ∑ A k H(e jω k
) e jω kn
k
k
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