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
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 0 = 2/T apart ...
as T and o 0, then = k0becomes continuous
Fourier series Fourier integral
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(j), we can represent
even more signals
b) Dirichlet conditions (including )
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 )
∆t•∆ ~ (1/a1/2
)•(a1/2
) = 1
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: ≥ 0 will be sufficient.
Even
Odd
Even
Odd
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
