1. Leo Lam © 2010-2011 Signals and Systems EE235 Leo Lam

2. Leo Lam © 2010-2011 Today’s menu Fourier Transform – Last two examples Sampling And we are set!

3. Fourier Transform: Leo Lam © 2010-2011 3 Fourier Transform Inverse Fourier Transform:

4. Example (Fourier Transform problem) Leo Lam © 2010-2011 Solve for y(t) But does it make sense if it was done with convolution? 4 05 -5  F() transfer function H() 01  05 -5  = Z() =0 everywhere 05 -5 w Z() = F() H()

5. 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) 5 ???

6. 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: 6 ??? Inverse transform!

7. 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? 7 ??? Capacitor Resistor

8. 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 8

9. 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 9

10. Bridge to the next class Leo Lam © 2010-2011 Next class: EE341: Discrete Time Linear Sys Analog to Digital Sampling 10 t continuous in time continuous in amplitude n discrete in time SAMPLING discrete in amplitude QUANTIZATION

11. Sampling Leo Lam © 2010-2011 Convert a continuous time signal into a series of regularly spaced samples, a discrete-time signal. Sampling is multiplying with an impulse train 11 t t t multiply = 0 TSTS

12. Sampling Leo Lam © 2010-2011 Sampling signal with sampling period T s is: Note that Sampling is NOT LTI 12 sampler

13. Sampling Leo Lam © 2010-2011 Sampling effect in frequency domain: Need to find: X s () First recall: 13 timeT Fourier spectra 0 1/T

14. Sampling Leo Lam © 2010-2011 Sampling effect in frequency domain: In Fourier domain: 14 distributive property Impulse train in time  impulse train in frequency, dk=1/Ts What does this mean?

15. Sampling Leo Lam © 2010-2011 Graphically: In Fourier domain: No info loss if no overlap (fully reconstructible) Reconstruction = Ideal low pass filter 0 X() bandwidth

16. Sampling Leo Lam © 2010-2011 Graphically: In Fourier domain: Overlap = Aliasing if To avoid: Equivalently: 0 Shannon’s Sampling Theorem Nyquist Frequency (min. lossless)

17. Sampling (in time) Leo Lam © 2010-2011 Time domain representation cos(2  100t) 100 Hz Fs=1000 Fs=500 Fs=250 Fs=125 < 2*100 cos(2  25t) Aliasing Frequency wraparound, sounds like Fs=25 (Works in spatial frequency, too!)

18. Summary: Sampling Leo Lam © 2010-2011 Review: –Sampling in time = replication in frequency domain –Safe sampling rate (Nyquist Rate), Shannon theorem –Aliasing –Reconstruction (via low-pass filter) More topics: –Practical issues: –Reconstruction with non-ideal filters –sampling signals that are not band-limited (infinite bandwidth) Reconstruction viewed in time domain: interpolate with sinc function

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