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Leo Lam © 2010-2013 Signals and Systems EE235

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Leo Lam © 2010-2013 Futile Q: What did the monsterous voltage source say to the chunk of wire? A: "YOUR RESISTANCE IS FUTILE!"

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Leo Lam © 2010-2013 Today’s menu Sampling/Anti-Aliasing Communications (intro)

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Sampling Leo Lam © 2010-2013 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

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Sampling Leo Lam © 2010-2013 Sampling signal with sampling period T s is: Note that Sampling is NOT LTI 5 sampler

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Sampling Leo Lam © 2010-2013 Sampling effect in frequency domain: Need to find: X s () First recall: 6 timeT Fourier spectra 0 1/T

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Sampling Leo Lam © 2010-2013 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?

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Sampling Leo Lam © 2010-2013 Graphically: In Fourier domain: No info loss if no overlap (fully reconstructible) Reconstruction = Ideal low pass filter 0 X() bandwidth

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Sampling Leo Lam © 2010-2013 Graphically: In Fourier domain: Overlap = Aliasing if To avoid Alisasing: Equivalently: 0 Shannon’s Sampling Theorem Nyquist Frequency (min. lossless)

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Sampling (in time) Leo Lam © 2010-2013 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!)

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Summary: Sampling Leo Lam © 2010-2013 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

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Would these alias? Leo Lam © 2010-2013 Remember, no aliasing if How about: 0 1 013-3 NO ALIASING!

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Would these alias? Leo Lam © 2010-2013 Remember, no aliasing if How about: (hint: what’s the bandwidth?) Definitely ALIASING! Y has infinite bandwidth!

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Would these alias? Leo Lam © 2010-2013 Remember, no aliasing if How about: (hint: what’s the bandwidth?) -.5 0.5 Copies every.7 -1.5 -.5.5 1.5 ALIASED!

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Would these alias? Leo Lam © 2010-2013 Remember, no aliasing if How about: (hint: what’s the bandwidth?) -.5 0.5 Copies every.7 -1.5 -.5.5 1.5 ALIASED!

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How to avoid aliasing? Leo Lam © 2010-2013 We ANTI-alias. SampleReconstruct B w s > 2w c time signal x(t) X(w) Anti-aliasing filter w c < B Z(w) z(n)

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How bad is anti-aliasing? Leo Lam © 2010-2013 Not bad at all. Check: Energy in the signal (with example) Sampled at Add anti-aliasing (ideal) filter with bandwidth 7 sampler lowpass anti-aliasing filter

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How bad is anti-aliasing? Leo Lam © 2010-2013 Not bad at all. Check: Energy in the signal (with example) Energy of x(t)? sampler lowpass anti-aliasing filter

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How bad is anti-aliasing? Leo Lam © 2010-2013 Not bad at all. Check: Energy in the signal (with example) Energy of filtered x(t)? sampler lowpass anti-aliasing filter ~0.455

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Bandwidth Practice Leo Lam © 2010-2013 Find the Nyquist frequency for: -100 0 100

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Bandwidth Practice Leo Lam © 2010-2013 Find the Nyquist frequency for: const[rect( /200)*rect( /200)] = -200 200

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Bandwidth Practice Leo Lam © 2010-2013 Find the Nyquist frequency for: (bandwidth = 100) + (bandwidth = 50)

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Leo Lam © 2010-2013 Summary Sampling and the frequency domain representations Sampling frequency conditions

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