1. Signals and Systems
EE235
Lecture 26
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2. Today’s menu
Fourier Transform
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3. Convolution/Multiplication Example
Given f(t)=cos(t)e–tu(t) what is F()
Not a good Duality Example, but a great proof for the Multiplication properties! 3
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4. More Fourier Transform Properties
time domain Fourier transform
Duality
Time-scaling
Multiplication
Differentiation
Integration
Conjugation
Dual of convolution 4 4
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5. Fourier Transform Pairs (Recap)
Review: 1 5 5
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6. Fourier Transform and LTI System
Back to the Convolution Duality:
And remember:
And in frequency domain
Convolution in time
h(t)
x(t)*h(t)
x(t)
Time domain
Multiplication in frequency
H(w)
X(w)H(w)
X(w)
Frequency domain
input signal’s
Fourier transform
output signal’s
Fourier transform 6
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7. Fourier Transform and LTI (Example)
Delay:
h(t)
LTI
Time domain:
Frequency domain (FT):
Shift in time 
Add linear phase in frequency 7
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8. Fourier Transform and LTI (Example)
Delay:
Exponential response
h(t)
LTI
Delay 3
Using Convolution Properties
Using FT Duality 8
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9. Fourier Transform and LTI (Example)
Delay:
Exponential response
Responding to Fourier Series
h(t)
LTI
Delay 3
Delay 3 9
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10. Another LTI (Example) Given Exponential response
What does this system do? What is h(t)?
And y(t) if
Echo with amplification
LTI 10
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11. Another angle of LTI (Example)
Given graphical H(w), find h(t)
What does this system do? What is h(t)?
Linear phase  constant delay
magnitude w
phase 1
Slope=-5 11
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12. Another angle of LTI (Example)
Given graphical H(w), find h(t)
What does this system do (qualitatively
Low-pass filter. No delay.
magnitude w
phase 1 12
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13. Another angle of LTI (Example)
Given graphical H(w), find h(t)
What does this system do qualitatively?
Bandpass filter. Slight delay.
magnitude w
phase 1 13
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14. Another angle of LTI (Example)
Given graphical H(w), find h(t)
What does this system do qualitatively?
Bandpass filter. Slight delay.
magnitude w
phase 1 14
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15. Summary
Fourier Transforms and examples
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16. Low Pass Filter (extra examples)
What is h(t)? (Impulse response)
Consider an ideal low-pass filter with frequency response
H(w) w
Looks like an octopus centered
around time t = 0 
Not causal…can’t build a circuit. 16
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17. Low Pass Filter What is y(t) if input is:
Ideal filter, so everything above is gone:
y(t)
Consider an ideal low-pass filter with frequency response
H(w) w 17
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18. Output determination Example
Solve for y(t)
Convert input and impulse function to Fourier domain:
Invert Fourier using known transform:
Use the e-(jw0t)  2*pi*delta(w-w0) pair 18
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19. Output determination Example
Solve for y(t)
Recall that:
Partial fraction:
Invert: 19
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20. Describing Signals (just a summary)
Ck and X(w) tell us the CE’s (or cosines) that are needed to build a time signal x(t)
CE with frequency w (or kw0) has magnitude |Ck| or |X(w)| and phase shift <Ck and <X(w)
FS and FT difference is in whether an uncountably infinite number of CEs are needed to build the signal. -B B w t
x(t)
X(w) 20
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21. Describing Signals (just a summary)
H(w) = frequency response
Magnitude |H(w)| tells us how to scale cos amplitude
Phase <H(w) tells us the phase shift
H(w)
cos(20t)
Acos(20t+f)
p/2
magnitude phase A
-p/2 f 20 20
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22. Example (Fourier Transform problem)
Solve for y(t)
But does it make sense if it was done with convolution? 5 -5 w
F(w)
transfer function H(w) 1 -1 w 5 -5 w
Z(w) = F(w) H(w) 5 -5 w =
Z(w) =0 everywhere 22
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23. Example (Circuit design with FT!)
Goal: Build a circuit to give v(t) with an input current i(t)
Find H(w)
Convert to differential equation
(Caveat: only causal systems can be physically built)
??? 23
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24. Example (Circuit design with FT!)
Goal: Build a circuit to give v(t) with an input current i(t)
Transfer function:
???
Inverse transform! 24
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25. Example (Circuit design with FT!)
Goal: Build a circuit to give v(t) with an input current i(t)
From:
The system:
Inverse transform:
KCL: What does it look like?
???
Capacitor
Resistor 25
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26. Fourier Transform: Big picture
With Fourier Series and Transform:
Intuitive way to describe signals & systems
Provides a way to build signals
Generate sinusoids, do weighted combination
Easy ways to modify signals
LTI systems: x(t)*h(t)  X(w)H(w)
Multiplication: x(t)m(t)  X(w)*H(w)/2p 26
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27. Fourier Transform: Wrap-up!
We have done:
Solving the Fourier Integral and Inverse
Fourier Transform Properties
Built-up Time-Frequency pairs
Using all of the above 27
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28. Bridge to the next class
Next class: EE341: Discrete Time Linear Sys
Analog to Digital
Sampling t
continuous in time
continuous in amplitude n
discrete in time
SAMPLING
discrete in amplitude
QUANTIZATION 28
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29. Summary Fourier Transforms and examples Next, and last: Sampling!
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