1. Signals and Systems
EE235
Lecture 23
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2. Today’s menu Fourier Series Example Fourier Transform
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3. Motivation
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4. Fourier Series: Quick exercise
Given:
Find its exponential Fourier Series:
(Find the coefficients dn and w0) 4
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5. Fourier Series: Fun examples
Rectified sinusoids
Find its exponential Fourier Series: t
f(t) =|sin(t)|
Expand as exp., combine, integrate 5
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6. Fourier Series: Circuit Application
Rectified sinusoids
Now we know:
Circuit is an LTI system:
Find y(t)
Remember: + -
sin(t)
full
wave
rectifier
y(t)
f(t)
Where did this come from?
Find H(s)! S 6
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7. Fourier Series: Circuit Application
Finding H(s) for the LTI system:
est is an eigenfunction, so
Therefore:
So:
Shows how much an exponential gets amplified at different frequency s 7
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8. Fourier Series: Circuit Application
Rectified sinusoids
Now we know:
LTI system:
Transfer function:
To frequency: + -
sin(t)
full
wave
rectifier
y(t)
f(t) 8
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9. Fourier Series: Circuit Application
Rectified sinusoids
Now we know:
LTI system:
Transfer function:
System response: + -
sin(t)
full
wave
rectifier
y(t)
f(t) 9
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10. Summary
Fourier Series circuit example
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11. Fourier Series: Dirichlet Conditon
Condition for periodic signal f(t) to exist has exponential series:
Weak Dirichlet:
Strong Dirichlet (converging series): f(t) must have finite maxima, minima, and discontinuities in a period
All physical periodic signals converge
Weak Dirichlet: Otherwise you can’t solve for the coefficients! 11
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12. End of Fourier Series We have accomplished: Next: Fourier Transform 12
Introduced signal orthogonality
Fourier Series derivation
Approx. periodic signals:
Fourier Series Properties
Next: Fourier Transform 12
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13. Fourier Transform: Introduction
Fourier Series: Periodic Signal
Fourier Transform: extends to all signals
Recall time-scaling: 13
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14. Fourier Transform: Recall time-scaling: 14 Fourier Spectra for T,
Fourier Spectra
for T,
for 2T, 14
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15. Fourier Transform: Non-periodic signal: infinite period T 15
Fourier Spectra
for T,
for 2T, 15
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16. Fourier Transform: Fourier Formulas:
For any arbitrary practical signal
And its “coefficients” (Fourier Transform):
F(w) is complex
Rigorous math derivation in Ch. 4 (not required reading, but recommended)
Time domain to
Frequency domain
Weak Dirichlet: Otherwise you can’t solve for the coefficients! 16
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17. Fourier Transform: Fourier Formulas compared: 17 Fourier transform
Fourier transform coefficients:
Fourier transform
(arbitrary signals)
Fourier series (Periodic signals):
Fourier series coefficients:
and 17
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18. Fourier Transform (example):
Find the Fourier Transform of
What does it look like?
If a <0, blows up
phase
varies with 
magnitude
varies with  18
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