Leo Lam © 2010-2011 Signals and Systems EE235. Leo Lam © 2010-2011 Fourier Transform Q: What did the Fourier transform of the arbitrary signal say to.

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Presentation transcript:

Leo Lam © Signals and Systems EE235

Leo Lam © Fourier Transform Q: What did the Fourier transform of the arbitrary signal say to the Fourier transform of the sinc function? A: "You're such a square!"

Leo Lam © Today’s menu Sampling Laplace Transform

Sampling Leo Lam © Convert a continuous time signal into a series of regularly spaced samples, a discrete-time signal. Sampling is multiplying with an impulse train 4 t t t multiply = 0 TSTS

Sampling Leo Lam © Sampling signal with sampling period T s is: Note that Sampling is NOT LTI 5 sampler

Sampling Leo Lam © Sampling effect in frequency domain: Need to find: X s () First recall: 6 timeT Fourier spectra 0 1/T

Sampling Leo Lam © Sampling effect in frequency domain: In Fourier domain: 7 distributive property Impulse train in time  impulse train in frequency, dk=1/Ts What does this mean?

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

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

Sampling (in time) Leo Lam © 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!)

Summary: Sampling Leo Lam © 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

Leo Lam © Summary Sampling and the frequency domain representations Sampling frequency conditions

Laplace Transform Leo Lam © Focus on: –Doing (Definitions and properties) –Understanding its possibilities (ROC) –Poles and zeroes (overlap with EE233)