1. Leo Lam © 2010-2012 Signals and Systems EE235 Lecture 26

2. Leo Lam © 2010-2012 Today’s menu Fourier Transform

3. Convolution/Multiplication Example Leo Lam © 2010-2012 3 Given f(t)=cos(t)e –t u(t) what is F()

4. More Fourier Transform Properties Leo Lam © 2010-2012 4 Duality Time-scaling Multiplication Differentiation Integration Conjugation time domain Fourier transform Dual of convolution 4

5. Fourier Transform Pairs (Recap) Leo Lam © 2010-2012 5 5 1 Review:

6. Fourier Transform and LTI System Leo Lam © 2010-2012 6 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() X()H() X() Frequency domain input signal’s Fourier transform output signal’s Fourier transform

7. Fourier Transform and LTI (Example) Leo Lam © 2010-2012 Delay: LTI h(t) Time domain:Frequency domain (FT): Shift in time  Add linear phase in frequency 7

8. Fourier Transform and LTI (Example) Leo Lam © 2010-2012 Delay: Exponential response LTI h(t) 8 Delay 3 Using Convolution Properties Using FT Duality

9. Fourier Transform and LTI (Example) Leo Lam © 2010-2012 Delay: Exponential response Responding to Fourier Series LTI h(t) 9 Delay 3

10. Another LTI (Example) Leo Lam © 2010-2012 Given Exponential response What does this system do? What is h(t)? And y(t) if Echo with amplification 10 LTI

11. Another angle of LTI (Example) Leo Lam © 2010-2012 Given graphical H(), find h(t) What does this system do? What is h(t)? Linear phase  constant delay 11 magnitude   phase 0 0 1 Slope=-5

12. Another angle of LTI (Example) Leo Lam © 2010-2012 Given graphical H(), find h(t) What does this system do (qualitatively Low-pass filter. No delay. 12 magnitude   phase 0 0 1

13. Another angle of LTI (Example) Leo Lam © 2010-2012 Given graphical H(), find h(t) What does this system do qualitatively? Bandpass filter. Slight delay. 13 magnitude   phase 0 1

14. Another angle of LTI (Example) Leo Lam © 2010-2012 Given graphical H(), find h(t) What does this system do qualitatively? Bandpass filter. Slight delay. 14 magnitude   phase 0 1

15. Leo Lam © 2010-2012 Summary Fourier Transforms and examples

16. Low Pass Filter (extra examples) Leo Lam © 2010-2011 16 Consider an ideal low-pass filter with frequency response w 0 H() What is h(t)? (Impulse response) Looks like an octopus centered around time t = 0 Not causal…can’t build a circuit.

17. Low Pass Filter Leo Lam © 2010-2011 17 Consider an ideal low-pass filter with frequency response w 0 H() What is y(t) if input is: Ideal filter, so everything above is gone: y(t)

18. Output determination Example Leo Lam © 2010-2011 18 Solve for y(t) Convert input and impulse function to Fourier domain: Invert Fourier using known transform:

19. Output determination Example Leo Lam © 2010-2011 19 Solve for y(t) Recall that: Partial fraction: Invert:

20. Describing Signals (just a summary) Leo Lam © 2010-2011 20 C k and X() tell us the CE’s (or cosines) that are needed to build a time signal x(t) –CE with frequency  (or k 0 ) has magnitude |C k | or |X()| and phase shift <C k and <X() –FS and FT difference is in whether an uncountably infinite number of CEs are needed to build the signal. -B-BB  t x(t) X()

21. Describing Signals (just a summary) Leo Lam © 2010-2011 H(w) = frequency response –Magnitude |H(w)| tells us how to scale cos amplitude –Phase <H(w) tells us the phase shift magnitude phase /2 -2 H() cos(20t) Acos(20t+f) A f 20

22. Example (Fourier Transform problem) Leo Lam © 2010-2011 Solve for y(t) But does it make sense if it was done with convolution? 22 05 -5  F() transfer function H() 01  05 -5  = Z() =0 everywhere 05 -5 w Z() = F() H()

23. Example (Circuit design with FT!) Leo Lam © 2010-2011 Goal: Build a circuit to give v(t) with an input current i(t) Find H() Convert to differential equation (Caveat: only causal systems can be physically built) 23 ???

24. Example (Circuit design with FT!) Leo Lam © 2010-2011 Goal: Build a circuit to give v(t) with an input current i(t) Transfer function: 24 ??? Inverse transform!

25. Example (Circuit design with FT!) Leo Lam © 2010-2011 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? 25 ??? Capacitor Resistor

26. Fourier Transform: Big picture Leo Lam © 2010-2011 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()H() –Multiplication: x(t)m(t)  X()*H()/2 26

27. Fourier Transform: Wrap-up! Leo Lam © 2010-2011 We have done: –Solving the Fourier Integral and Inverse –Fourier Transform Properties –Built-up Time-Frequency pairs –Using all of the above 27

28. Bridge to the next class Leo Lam © 2010-2011 Next class: EE341: Discrete Time Linear Sys Analog to Digital Sampling 28 t continuous in time continuous in amplitude n discrete in time SAMPLING discrete in amplitude QUANTIZATION

29. Leo Lam © 2010-2011 Summary Fourier Transforms and examples Next, and last: Sampling!