Presentation on:
Signals and Systems
By – Aakash Maurya
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Fourier Transform
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System Frequency Response and Unit
Sample Response
Derivation of CT Fourier Transform
pair
Examples of Fourier Transforms
Fourier Transforms of Periodic Signals
Properties of the CT Fourier Transform
2
The Frequency Response of an LTI System
3
First Order CT Low Pass Filter
Direct Solution of Differential Equation
4
Using Impulse Response
Note map from
unit sample
response to
frequency
response
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Fourier’s Derivation of the CT Fourier
Transform
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x(t) - an aperiodic signal
- view it as the limit of a periodic signal as T ! 1
•
For a periodic sign, the harmonic components are
spaced 0 = 2/T apart ...
as T  and o  0, then  = k0 becomes continuous


Fourier series  Fourier integral
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Square Wave Example
Discrete
frequency
points
become
denser in
 as T
increases
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“Periodify” a non-periodic signal
For simplicity, assume
x(t) has a finite duration.
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Fourier Series For Periodified x(t)
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Limit of Large Period
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What Signals have Fourier Transforms?
(1) x(t) can be of infinite duration, but must satisfy:
a) Finite energy
In this case, there is zero energy in the error
b) Dirichlet conditions (including
)
c) By allowing impulses in x(t) or in X(j), we can represent
even more signals
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Fourier Transform Examples
Impulses
(a)
(b)
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Fourier Transform of Right-Sided Exponential
Even symmetry
Odd symmetry
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Fourier Transform of square pulse
Note the inverse relation between the two widths  Uncertainty principle
Useful facts about CTFT’s
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Fourier Transform of a Gaussian
x(t)  e at
2
— A Gaussian, important in
probability, optics, etc.
(Pulse width in t)•(Pulse width in )
∆t•∆ ~ (1/a1/2)•(a1/2) = 1
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CT Fourier Transforms of Periodic Signals
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Fourier Transform of Cosine
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Impulse Train (Sampling Function)
Note: (period in t) T
(period in ) 2/T
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Properties of the CT Fourier Transform
1)
Linearity
2) Time Shifting
FT magnitude unchanged
Linear change in FT phase
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Properties (continued)
3) Conjugate Symmetry
Even
Odd
Even
Or
Odd
When x(t) is real (all the physically measurable signals are real), the
negative frequency components do not carry any additional information
beyond the positive frequency components:  ≥ 0 will be sufficient.
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More Properties
4) Time-Scaling
a)
x(t) real and even
b)
x(t) real and odd
c)
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Conclusions
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System Frequency Response and Unit
Sample Response
Derivation of CT Fourier Transform pair
CT Fourier Transforms of pulses,
exponentials
FT of Periodic Signals  Impulses
Time shift, Scaling, Linearity
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