Introduction to Signal Processing
1
Copyright © 2005-2009 – Hayder Radha
IV.
Continuous-Time Signal Analysis: The
Fourier Transform
As we have seen, the Fourier series provided a useful
frequency-domain analytical tool for the representation
of periodic signals (that have finite power).
We now focus on the frequency-domain representation of
aperioidc signals, which are much more common in
many practical systems.
Introduction to Signal Processing
2
Copyright © 2005-2009 – Hayder Radha
A.
Aperiodic Signal Representation by the
Fourier Integral
The various forms of the
Fourier series
we studied earlier
provided us with the means for representing
periodic
signals
with period
0
T
as the sum of sinusoidal functions.
Here, we will build on the Fourier series by studying an
equivalent method for the representation of a general
aperiodic signal
( )
x t
.
Introduction to Signal Processing
3
Copyright © 2005-2009 – Hayder Radha
We can utilize the Fourier series by modifying an
aperiodic signal into a periodic one. Hence, let us
construct a new signal
0
( )
T
x
t
by repeating
( )
x t
with a
time period
0
T
.
Introduction to Signal Processing
4
Copyright © 2005-2009 – Hayder Radha
If we let
0
T
→ ∞
and, hence, make the interval duration
infinite, we get back the original signal
( )
x t
.
0
0
lim
( )
( )
T
T
x
t
x t
→∞
=

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