d

1. 1
“Figures and images used in these lecture notes by permission,
copyright 1997 by Alan V. Oppenheim and Alan S. Willsky”
Signals and Systems
Spring 2003
Lecture #8
Jacob White
(Slides thanks to A. Willsky, T. Weiss,
Q. Hu, and D. Boning)

2. 2
Fourier Transform
• 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

3. 3
The Frequency Response of an LTI System

4. 4

5. 5
First Order CT Low Pass Filter
Direct Solution of Differential Equation

6. 6
Using Impulse Response
Note map from
unit sample
response to
frequency
response

7. 7
Fourier’s Derivation of the CT Fourier
Transform
• 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 w0 = 2p/T apart ...
as T  and wo  0, then w = kw0 becomes continuous

Fourier series  Fourier integral



8. 8

9. 9
Discrete
frequency
points
become
denser in
w as T
increases
Square Wave Example

10. 10
“Periodify” a non-periodic signal
For simplicity, assume
x(t) has a finite duration.

11. 11
Fourier Series For Periodified x(t)

12. 12

13. 13
Limit of Large Period

14. 14
a) Finite energy
In this case, there is zero energy in the error
What Signals have Fourier Transforms?
(1) x(t) can be of infinite duration, but must satisfy:
c) By allowing impulses in x(t) or in X(jw), we can represent
even more signals
b) Dirichlet conditions (including )

15. 15
Fourier Transform Examples
(a)
(b)
Impulses

16. 16

17. 17
Fourier Transform of Right-Sided Exponential
Even symmetry Odd symmetry

18. 18
Fourier Transform of square pulse
Useful facts about CTFT’s
Note the inverse relation between the two widths  Uncertainty principle

19. 19
Fourier Transform of a Gaussian
x(t)  eat2
— A Gaussian, important in
probability, optics, etc.
(Pulse width in t)•(Pulse width in w)
∆t•∆w ~ (1/a1/2)•(a1/2) = 1

20. 20

21. 21
CT Fourier Transforms of Periodic Signals

22. 22
Fourier Transform of Cosine

23. 23
Note: (period in t) T
(period in w) 2p/T
Impulse Train (Sampling Function)

24. 24

25. 25
Properties of the CT Fourier Transform
FT magnitude unchanged
Linear change in FT phase
1) Linearity
2) Time Shifting

26. 26
Properties (continued)
3) Conjugate Symmetry
Or
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: w ≥ 0 will be sufficient.
Even
Odd
Even
Odd

27. 27
More Properties
4) Time-Scaling
a) x(t) real and even
b) x(t) real and odd
c)

28. 28

29. 29
Conclusions
• 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