1. Leo Lam © 2010-2012 Signals and Systems EE235 Lecture 27

2. Leo Lam © 2010-2011 Futile Q: What did the monsterous voltage source say to the chunk of wire? A: "YOUR RESISTANCE IS FUTILE!"

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

4. Output determination Example Leo Lam © 2010-2011 4 Solve for y(t) Convert input and impulse function to Fourier domain: Invert Fourier using known transform:

5. Output determination Example Leo Lam © 2010-2011 5 Solve for y(t) Recall that: Partial fraction: Invert:

6. Describing Signals (just a summary) Leo Lam © 2010-2011 6 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()

7. 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

8. Example (Fourier Transform problem) Leo Lam © 2010-2011 Solve for y(t) But does it make sense if it was done with convolution? 8 05 -5  F() transfer function H() 01  05 -5  = Z() =0 everywhere 05 -5 w Z() = F() H()

9. 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) 9 ???

10. 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: 10 ??? Inverse transform!

11. 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? 11 ??? Capacitor Resistor

12. 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 12

13. 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 13

14. Bridge to the next class Leo Lam © 2010-2011 Next class: EE341: Discrete Time Linear Sys Analog to Digital Sampling 14 t continuous in time continuous in amplitude n discrete in time SAMPLING discrete in amplitude QUANTIZATION

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