1. Notice  HW problems for Z-transform at available on the course website  due this Friday (9/26/2014)  http://sist.shanghaitech.edu.cn/faculty/luoxl/class/2014Fall_DSP/DSPclass.ht m http://sist.shanghaitech.edu.cn/faculty/luoxl/class/2014Fall_DSP/DSPclass.ht m

2. Lecture 3: Sampling XILIANG LUO 2014/9

3. Periodic Sampling  A continuous time signal is sampled periodically to obtain a discrete- time signal as: Ideal C/D converter

4. C/D : Not Invertible in General However, it is possible to reconstruct original signal by restricting the frequency content of the input signal!

5. Ideal Sampling  Impulse train modulator

6. Ideal Sampling

8. Fourier Transform of Ideal Sampling Fourier transform of the ideal sampled signal is the convolution of the FT of original continuous signal and the impulse train

9. Fourier Transform of Ideal Sampling Fourier Transform of periodic impulse train is an impulse train:

10. Ideal Sampling  Fourier Transform of the ideal sampled signal consists of periodically repeated copies of the Fourier Transform of the original continuous time signal

11. Ideal Sampling

14. Observation  When Sampling Frequency satisfies the following condition, the replicas of Xc(j Ω ) do not overlap:  When replicas of Xc(j Ω ) do not overlap, ideal lowpass filter will be able to recover original continuous time signal from the ideal sampled signal with the impulse train

15. Exact Recovery

17. Aliasing  A simple cosine signal:

18. Aliasing

19. Nyquist-Shannon Sampling

20. What about DTFT

21. This is the general relationship between the periodically sampled sequence and the underlying continuous time signal

22. Reconstruction  If we are given a sequence, we can formulate an impulse train:  Let this impulse train be the input to a reconstruction filter

23. Reconstruction System

24. Reconstruction Filter

25. Ideal Low Pass Filter

26. Ideal Band-limited Interpolation

27. Ideal Reconstruction System Note: output is always bandwidth limited to the cutoff frequency of the lowpass filter

28. Process Cont. Signal  A main application of discrete-time systems is to process continuous- time signal in discrete-time domain

29. Sampling Process

30. Ideal D/C

31. Discrete-Time System Role LTI

32. Discrete-Time System Role

33. Band-limited Signal

34. Observations  For band-limited signal, we are processing continuous time signal using discrete-time signal processing  For band-limited signal, the overall system behaves like a linear time- invariant continuous-time system with the following frequency domain relationship:

35. Question  In order for the following equation to hold, can we allow some aliasing to happen, i.e., the sampling rate is less than the Nyquist rate?

36. Impulse Invariance

37. Example: Low-Pass Filter

38. Process Discrete-Time Signal

40. Example: Non-Integer Delay

41. Next 1. Change sampling rate 2. Multi-rate signal processing 3. Quantization 4. Noise shaping

42. HW Due on 10/10 4.31 4.34 4.53 4.60 4.61 4.21 4.54 need multi-rate signal processing knowledge