Understanding Sampling and Signal Processing Fundamentals

Sampling plays a crucial role in converting continuous-time signals into discrete-time signals for processing. This lecture covers periodic sampling, ideal sampling, Fourier transforms, Nyquist-Shannon sampling, and the processing of band-limited signals. It delves into the relationship between periodically sampled sequences and continuous-time signals, emphasizing the importance of multi-rate signal processing knowledge. The content also touches on discrete-time system applications and examples of processing signals with non-integer delays.

• Sampling
• Signal Processing
• Fourier Transform
• Nyquist-Shannon
• Band-limited

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1. Lecture 4: Sampling [2] XILIANG LUO 2014/10

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

3. Ideal Sampling Impulse train modulator

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

5. What about DTFT This is the general relationship between the periodically sampled sequence and the underlying continuous time signal

6. Nyquist-Shannon Sampling Let ??(?) be a band-limited signal with Then ??(?) is uniquely determined by its samples if: The frequency ? is referred to as the Nyquiest frequency The frequency 2 ? is called Nyquist rate

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

8. Band-limited Signal

9. 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:

10. Process Discrete-Time Signal

11. Process Discrete-Time Signal

12. Example: Non-Integer Delay

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

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

15. Change Sampling Rate Conceptually, we can do this by reconstruct the continuous time signal first, then resample the reconstructed continuous signal

16. Sampling Rate Reduction Down-sampling

17. Downsampling

18. Downsampling

19. Anti-Aliasing Filter

20. Aliasing Example

21. Upsampling

22. Upsampling

23. Frequency Domain

24. Upsampling

25. Filtering Compressor

26. Filtering Expander

27. Polyphase Decomposition Goal: efficient implementation structure k=0,1, ,M-1

28. Polyphase Decomposition

29. Polyphase in Freq Domain Polyphase component filters

30. Polyphase Filters y[n]=x[n]*h[n]

31. Polyphase + Decimation Filter

32. Polyphase + Decimation Filter

33. Polyphase + Decimation Filter

34. Polyphase + Interp Filter

35. Polyphase + Interp Filter

36. Polyphase + Interp Filter

37. Ideal

38. Practical

39. Avoid Aliasing

40. Simple Anti-Aliasing Filter

41. Oversampling C/D

42. Oversampling C/D

43. Oversampling C/D Advantages nominal analog filter exact linear phase

44. A/D Conversion

45. Zero-order Hold System

46. Quantization

47. a Typical Quantizer

48. Quantization Error

49. Quantization Error Assumptions:

50. Quantization Error

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