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FREQUENCY RESPONSE
Consider a discrete time LTI system with a pure complex sinusoid
of frequency
ϖ
as input, as shown below.
The output is given by convolution
∑
∑
∞
∞
=

∞
∞
=

=
=
k
k
j
n
j
k
k
n
j
e
k
h
e
e
k
h
n
y
w
w
w
)
(
)
(
)
(
)
(
or,
n
j
j
e
e
H
n
y
w
w
)
(
)
(
=
.
(4.1)
Thus, the output of an LTI system to a complex sinusoidal input is
again a complex sinusoidal input with exactly the same frequency
as the input.
The only effect of an LTI system on a complex sinusoidal input is to
multiply the input by a complex number,
)
(
w
j
e
H
.
The function ,
)
(
w
j
e
H
, is called the
frequency response
of the
system and it determines the amplitude and phase shifts introduced
by the system in passing a complex sinusoid from input to output.
Indeed, it is conventional to express
)
(
w
j
e
H
in terms of its
magnitude and phase as
)
(
)
(
n
h
e
n
y
n
j
=
w
H(z)
n
j
e
w
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(
)
(
)
(
w
j
w
w
j
j
e
A
e
H
=
(4.2)
where
)
(
)
(
w
w
j
e
H
A
=
(4.3)
and
{ } { }
)
(
)
(
arg
)
(
w
w
w
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This note was uploaded on 10/18/2009 for the course EE ee332 taught by Professor Ee during the Spring '07 term at Istanbul Technical University.
 Spring '07
 ee

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