Introduction to Signal Processing
1
Copyright © 2005-2009 – Hayder Radha
D.
Signal Transmission Through
Linear Time-Invariant Systems
If
( )
x t
and
( )
y t
are the input and output of an LTIC
system with impulse response
( )
h t
then recall that the
output of the system (
zero-state response
) can be
evaluated as the convolution of the input
( )
x t
and the
impulse response
( )
h t
:
Introduction to Signal Processing
2
Copyright © 2005-2009 – Hayder Radha
( )
( )
( )
*
y t
x t
h t
=
.
where
( )
( ) (
)
y t
x
h t
d
τ
τ
τ
∞
−∞
=
−
∫
( )
( )
( )
*
y t
x t
h t
=
( )
h t
( )
x t
Introduction to Signal Processing
3
Copyright © 2005-2009 – Hayder Radha
Let
( )
x t
,
( )
y t
, and
( )
h t
have corresponding Fourier
transform functions
(
)
X
ω
,
(
)
Y
ω
, and
(
)
H
ω
respectively:
( )
(
)
x t
X
ω
⇔
( )
(
)
y t
Y
ω
⇔
( )
(
)
h t
H
ω
⇔
Introduction to Signal Processing
4
Copyright © 2005-2009 – Hayder Radha
Now, by using the time-convolution property, we have
the following:
(
)
(
)
(
)
Y
H
X
ω
ω
ω
=
.
(
)
(
)
(
)
Y
X
H
ω
ω
ω
=
(
)
H
ω
(
)
X
ω

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