i.e. the output of a real LTI system is a cosine at the same
frequency as the input (
ω
0
), with scaling and phase shift
determined by the magnitude and phase of the frequency
response.
EE 224, #14
7
Input Consisting of Multiple Complex Sinusoids
Suppose that we can express a CT signal
x
(
t
)
as
x
(
t
) =
X
k
a
k
e
j ω
k
t
ω
k
in rad/s
.
Then, the output
y
(
t
)
can be
easily
obtained as
y
(
t
) =
T {
x
(
t
)
}
=
X
k
a
k
H
F
(
ω
k
)
e
j ω
k
t
.
EE 224, #14
8
Example: RC Circuit
Calculate the frequency response of an RC circuit.
We obtained the impulse response of this RC circuit in handout
#12 [see (8) in handout #12]:
h
(
t
) =
1
R C
e

t/
(
R C
)
u
(
t
) =
α e

α t
u
(
t
)
where
α
= 1
/
(
R C
)
. The circuit’s frequency response is then
H
F
(
ω
)
=
Z
+
∞
∞
e

j ω τ
h
(
τ
)
dτ
=
Z
+
∞
∞
e

j ω τ
α e

α τ
u
(
τ
)
dτ
=
α
Z
+
∞
0
e

(
α
+
j ω
)
τ
dτ
=
α
e

(
α
+
j ω
)
τ
α
+
j ω
0
τ
=+
∞
=
α
α
+
j ω
.
EE 224, #14
9
An
alternative
approach
for
computing
the
frequency
response
H
F
(
ω
)
.
By Kirchoff’s voltage law,
x
(
t
) =
R i
(
t
) +
y
(
t
)
.
Using
i
(
t
) =
C y
0
(
t
)
, we have
y
0
(
t
) +
α y
(
t
) =
α x
(
t
)
(4)
which is an LCCD equation. Now, consider the input
x
(
t
) =
e
j ω t
.
Since the system is linear,
y
(
t
)
=
H
F
(
ω
)
e
j ω t
y
0
(
t
)
=
H
F
(
ω
)
j ω e
j ω t
.
Substitute the above into (4):
H
F
(
ω
)
j ω e
j ω t
+
α H
F
(
ω
)
e
j ω t
=
α e
j ω t
cancel
e
j ω t
and solve for
H
F
(
ω
)
:
H
F
(
ω
) =
α
α
+
j ω
.
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 Fall '09
 Digital Signal Processing, Frequency, LTI system theory, Impulse response