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

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 © 2010-2011 Today’s menu Sampling/Anti-Aliasing Communications (intro)

4. Summary: Sampling Leo Lam © 2010-2012 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

5. Quick Recap: Would these alias? Leo Lam © 2010-2012 Remember, no aliasing if How about: 0 1 013-3 NO ALIASING!

6. Would these alias? Leo Lam © 2010-2012 Remember, no aliasing if How about: (hint: what’s the bandwidth?) Definitely ALIASING! Y has infinite bandwidth!

7. Would these alias? Leo Lam © 2010-2012 Remember, no aliasing if How about: (hint: what’s the bandwidth?) -.5 0.5 Copies every.7 -1.5 -.5.5 1.5 ALIASED!

8. How to avoid aliasing? Leo Lam © 2010-2012 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)

9. How bad is anti-aliasing? Leo Lam © 2010-2012 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

10. How bad is anti-aliasing? Leo Lam © 2010-2012 Not bad at all. Check: Energy in the signal (with example) Energy of x(t)? sampler lowpass anti-aliasing filter

11. How bad is anti-aliasing? Leo Lam © 2010-2012 Not bad at all. Check: Energy in the signal (with example) Energy of filtered x(t)? sampler lowpass anti-aliasing filter ~0.455

12. Bandwidth Practice Leo Lam © 2010-2012 Find the Nyquist frequency for: -100 0 100

13. Bandwidth Practice Leo Lam © 2010-2012 Find the Nyquist frequency for: const[rect(  /200)*rect(  /200)] = -200 200

14. Bandwidth Practice Leo Lam © 2010-2012 Find the Nyquist frequency for: (bandwidth = 100) + (bandwidth = 50)