ch4 The Continuous-Time Fourier Transform.ppt

INTRODUCTION
Represent continuous-time aperiodic signals as binations plex exponentials.
Fourier transform and inverse Fourier transform.
(傅立叶变换) (傅立叶逆变换)
Use Fourier methods to analyze and understand signals and LTI systems.
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1. Representation of Aperiodic Signals: Continuous-Time Fourier Transform
1) Development of the Fourier transform representation of an aperiodic signal
(1) Example
( Go from Fourier series to Fourier transform )
Over one period of the continuous-time periodic square wave:
We can regard Tak as samples of an envelope function, specifically
---continuous variable
---the envelope of Tak
---equally spaced samples
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As T increases with T1 fixed, we have two observations:
A. The harmonics are packed more and more closer to each other;
B. However, the shape (the envelope) of the spectrum remains the same (and it is determined by the pulse shape of the signal).
What do we know from the above graph?
T=4T1
T=8T1
T=16T1
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(2) Go from periodic to aperiodic
Construct a periodic signal from the aperiodic signal x(t); later, let
For periodic signal , we have its Fourier serires:
For aperiodic signal x(t) :
When
How to represent the aperiodic signal x(t) ?
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As ,
Use X(jω) to denote this integral, then we have:
Since
spectrum of x(t)
Although X(jω) is often abbreviated as “spectrum”, it is different from ak, which is the spectrum of periodic signals.
Fourier transform of x(t)
X(jω) is in fact spectrum-density function(频谱密度函数)
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Thus, we obtain
Inverse Fourier transform
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Fourier transform pair :
or
Important:
The aperiodic signals can still be represented as a bination plex exponentials.
The magnitude ponent with frequence ω is
An useful relationship:
where X(jω) is the Fourier transform of x(t), ak is the Fourier coefficients of . x(t) is one period of the periodic signal
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or
Example:
periodic square wave:
So aperiodic signal x(t):
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2) Convergence of Fourier transform
Dirichlet conditions:
(1) x(t) is ab