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Chapter 4 sampling of continous-time signals

4.1 periodic sampling. 4.2 discrete-time processing of continuous-time signals. 4.3 continuous-time processing of discrete-time signal. 4.4 digital processing of analog signals. 4.5 changing the sampling rate using discrete-time processing. Chapter 4 sampling of continous-time signals.

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Chapter 4 sampling of continous-time signals

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  1. 4.1 periodic sampling 4.2 discrete-time processing of continuous-time signals 4.3 continuous-time processing of discrete-time signal 4.4 digital processing of analog signals 4.5 changing the sampling rate using discrete-time processing Chapter 4 sampling of continous-time signals

  2. 4.1 periodic sampling1.ideal sample T:sample period fs=1/T:sample rate Ωs=2π/T:sample rate

  3. Figure 4.1ideal continous-time-to-discrete-time(C/D)converter

  4. time normalization tt/T=n Figure 4.2(a) mathematic model for ideal C/D

  5. Figure 4.3 frequency spectrum change of ideal sample No aliasing aliasing aliasing frequency

  6. Period =2πin time domain: w=2.1πand w=0.1πare the same trigonometric function property

  7. high frequency is changed into low frequency in time domain:w=1.1π and w=0.9πare the same trigonometric function property

  8. 2.ideal reconstruction Figure 4.10(b) ideal D/C converter

  9. ideal reconstruction in frequency domain Figure 4.4

  10. Take sinusoidal signal for example to understand aliasing from frequency domain EXAMPLE Figure 4.5

  11. EXAMPLE Sampling frequency:8Hz Reconstruct frequency:

  12. Figure 4.10(a) mathematic model for ideal D/C

  13. ideal reconstruction in time domain

  14. EXAMPLE Figure 4.9

  15. understand aliasing from time-domain interpolation EXAMPLE

  16. 3.Nyquist sampling theorems

  17. Nyquist sampling theorems:

  18. Figure 4.41 Digital processing of analog signals

  19. examples of sampling theorem(1) The highest frequency of analog signal ,which wav file with sampling rate 16kHz can show , is: 8kHz The higher sampling rate of audio files, the better fidelity.

  20. examples of sampling theorem(2) according to what you know about the sampling rate of MP3 file,judge the sound we can feel frequency range( ) (A)20~44.1kHz (B)20~20kHz (C)20~4kHz (D)20~8kHz B

  21. Matlab codes to realize interpolation EXAMPLE

  22. T=0.1; n=0:10; x=cos(10*pi*n*T); stem(n,x); dt=0.001; t=ones(11,1)* [0:dt:1]; n=n'*ones(1,1/dt+1); y=x*sinc((t-n*T)/T); hold on; plot(t/T,y,'r')

  23. Supplement: band-pass sampling theorem:

  24. 4.1 summary 1.representation in time domain of sampling

  25. 2.changes in frequency domain caused by sampling

  26. 3. understand reconstruction in frequency domain

  27. 4. understand reconstruction in time domain

  28. 5. sampling theorem

  29. Requirements and difficulties: frequency spectrum chart of sampling and reconstruction comprehension and application of sampling theorem

  30. Figure 4.11 4.2 discrete-time processing of continuous-time signals

  31. Figure 4.12 conditions:LTI;no aliasing or aliasing occurred outside the pass band of filters EXAMPLE

  32. EXAMPLE aliasing occurred outside the pass band of digital filters satisfies the equivalent relation of frequency response mentioned before. Figure 4.13

  33. Figure 4.16 4.3 continuous-time processing of discrete-time signal

  34. c Figure 4.12

  35. EXAMPLE Ideal delay system:noninteger delay

  36. quantization and coding Figure 4.41 4.4 digital processing of analog signals

  37. Sampling and holding Figure 4.46(b)

  38. uniform quantization and coding Figure 4.48

  39. Figure 4.51 quantization error of 3BIT quantization error of8BIT

  40. nonuniform quantization

  41. 一维信号例子: index0 x 4 4 3 3 3 3 3 2 2 1 1 d(x,c0)=5 d(x,c1)=11 d(x,c2)=8 d(x,c3)=8 codewordc1 1 codeword c0 x 4 4 3 3 2 2 1 codewordc2 codewordc3 1 码书 vector quantization

  42. Example: image coding • Initial image block(4 gray-levels,dimentions k=4×4=16) x 0 1 2 3 Code book C ={y0, y1 , y2, y3} y0 y1 y2 y3 codeword y1 is the most adjacent to x,so it is coded by the index “01”. d(x,y0)=25 d(x,y1)=5 d(x,y2)=25 d(x,y3)=46 vector quantization

  43. reconstruction Figure 4.53 D/A过程

  44. Figure 4.5

  45. record the digital sound

  46. Influence caused by sampling rate and quantizing bits

  47. Different tones require different sampling rates.

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