1. Signals and Systems
EE235
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2. Today’s menu
Fourier Transform
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3.  x squared equals 9 x squared plus 1 equals y Find value of y
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4. Fourier Transform: Fourier Transform Inverse Fourier Transform: 4
Weak Dirichlet: Otherwise you can’t solve for the coefficients! 4
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5. 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 5
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6. 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 6
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7. 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 7
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8. 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 8
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9. 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)
??? 9
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10. Example (Circuit design with FT!)
Goal: Build a circuit to give v(t) with an input current i(t)
Transfer function:
???
Inverse transform! 10
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11. 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 11
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12. 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 12
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13. 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 13
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14. 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 14
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15. Summary Fourier Transforms and examples Next, and last: Sampling!
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