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

2. Leo Lam © 2010-2011 Today’s menu Sampling – Examples and practices And we are done!

3. Real-life LPF Ideal LPF problems: –Non-causal –An infinite impulse response –What’s the problem? Can’t build it! Real-life cc  c mostly  0 Leo Lam © 2010-2011

4. Real-life LPF Practical LPF –Causal –Gradual cutoff, not flat in pass-band, Stop band may not be zero, phase varies Need to sample faster: Give the filter “room” 0 (signal bandwidth) Leo Lam © 2010-2011

5. Real-life LPF Practical LPF…is tough Lucky for you, not in the scope of this class! 0 (signal bandwidth) Leo Lam © 2010-2011

6. Avoiding aliasing Pick a “reasonable” sampling frequency Band-limit signal to  c < s /2 with LPF Then sample Sample A/D Interpolate D/A B w s > 2w c time signal x(t) X(w) Anti-aliasing filter w c < B Z(w) z(t) z(n) Leo Lam © 2010-2011

7. Reconstruction in time-domain See what happens in time t Recall: convolving a signal with a shifted delta just shifts the signal Leo Lam © 2010-2011

8. Reconstruction in time-domain Some math manipulations: Tada! Sinc function is the ideal “interpolator” sum of shifted/scaled sinc functions t n=0n=1Sum to n=∞ Leo Lam © 2010-2011

9. Reconstruction in time-domain Sampling: –Determined “Safe” sampling frequencies –What signals can be sampled (finite region in frequency domain) –Sampling in time and effect in frequency domain –Reconstruction in frequency and effect in time domain –Practical problems with reconstruction And with this… Leo Lam © 2010-2011

10. Woohoo!!! Leo Lam © 2010-2011