signal and system Lecture 8
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Transcript of signal and system Lecture 8
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Fouriers Derivation of the CT Fourier Transform
x(t) - an aperiodic signal
- view it as the limit of a periodic signal as T
For a periodic signal, the harmonic components are
spaced 0 = 2/T apart ...
As T , 0 0, and harmonic components are spacedcloser and closer in frequency
Fourier series Fourier integral
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Discrete
frequencypoints
become
denser in
as Tincreases
Motivating Example: Square wave
increases
kept fixed
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So, on with the derivation ...
For simplicity, assume
x(t) has a finite duration.
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Derivation (continued)
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Derivation (continued)
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Example #1
(a)
(b)
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Example #2: Exponential function
Even symmetry Odd symmetry
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Example #3: A square pulse in the time-domain
Useful facts about CTFTs
Note the inverse relation between the two widths Uncertainty principle
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Example #4: x(t) = eat2
A Gaussian, important in
probability, optics, etc.
Also a Gaussian! Uncertainty Principle! Cannot make
both tand arbitrarily small.
(Pulse width in t)(Pulse width in )
t~ (1/a1/2)(a1/2) = 1
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CT Fourier Transforms of Periodic Signals
periodic in twith
frequency o
All the energy is
concentrated in one
frequency o
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Example #4:
Line spectrum
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Sampling functionExample #5:
Same function in
the frequency-domain!
Note: (period int) T
(period in ) 2/T
Inverse relationship again!
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Properties of the CT Fourier Transform
1) Linearity
2) Time Shifting
FTmagnitude unchanged
Linear change inFTphase
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Properties (continued)
3) Conjugate Symmetry
Even
Odd
Even
Odd
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The Properties Keep on Coming ...
4) Time-Scaling
a) x(t) real and even
b) x(t) real and odd
c)
