1. Sampling Theorem 主講者：虞台文

2. Content Periodic Sampling Sampling of Band-Limited Signals Aliasing --- Nyquist rate CFT vs. DFT Reconstruction of Band-limited Signals Discrete-Time Processing of Continuous-Time Signals Continuous-Time Processing of Discrete-Time Signals Changing Sampling Rate Realistic Model for Digital Processing

3. Sampling Theorem Periodic Sampling

4. Continuous to Discrete-Time Signal Converter C/D T xc(t)xc(t) x(n)= x c (nT) Sampling rate

5. C/D System Conversion from impulse train to discrete-time sequence xc(t)xc(t) x(n)= x c (nT)  s(t)s(t) xs(t)xs(t)

6. Sampling with Periodic Impulse train t xc(t)xc(t) 0T2T2T3T3T4T4T TT 2T2T 3T3T n x(n)x(n) 01234 11 22 33 t xc(t)xc(t) 02T4T4T8T8T10T 2T2T 4T4T 8T8T n x(n)x(n) 02468 22 44 66

7. Sampling with Periodic Impulse train t xc(t)xc(t) 0T2T2T3T3T4T4T TT 2T2T 3T3T n x(n)x(n) 01234 11 22 33 t xc(t)xc(t) 02T4T4T8T8T10T 2T2T 4T4T 8T8T n x(n)x(n) 02468 22 44 66 What condition has to be placed on the sampling rate? We want to restore x c (t) from x(n).

8. C/D System Conversion from impulse train to discrete-time sequence xc(t)xc(t) x(n)= x c (nT)  s(t)s(t) xs(t)xs(t)

9. C/D System Conversion from impulse train to discrete-time sequence xc(t)xc(t) x(n)= x c (nT)  s(t)s(t) xs(t)xs(t)

10. C/D System  s : Sampling Frequency

11. C/D System

12. Sampling Theorem Sampling of Band-Limited Signals

13.  Yc(j)Yc(j) Band-Limited Band-Unlimited  Xc(j)Xc(j) NN  N 1

14. Sampling of Band-Limited Signals Band-Limited  Xc(j)Xc(j) NN  N 1  ss  s 2s2s 3s3s 2s2s 3s3s S(j)S(j) 2/T2/T  4s4s 4s4s 2s2s 6s6s 2s2s 6s6s S(j)S(j) 2/T2/T Sampling with Higher Frequency Sampling with Lower Frequency

15. Sampling Theorem Aliasing --- Nyquist Rate

16. Recoverability Band-Limited  Xc(j)Xc(j) NN  N 1  ss  s 2s2s 3s3s 2s2s 3s3s S(j)S(j) 2/T2/T  4s4s 4s4s 2s2s 6s6s 2s2s 6s6s S(j)S(j) 2/T2/T Sampling with Higher Frequency Sampling with Lower Frequency  s > 2  N  s < 2  N

17. Case 1:  s > 2  N  Xc(j)Xc(j) NN  N 1  ss  s 2s2s 3s3s 2s2s 3s3s S(j)S(j) 2/T2/T 1/T1/T  ss 2s2s 3s3s 2s2s 3s3s Xs(j)Xs(j)

18. Case 1:  s > 2  N  Xc(j)Xc(j) NN  N 1  ss  s 2s2s 3s3s 2s2s 3s3s S(j)S(j) 2/T2/T 1/T1/T  ss 2s2s 3s3s 2s2s 3s3s Xs(j)Xs(j) Passing X s (j  ) through a low-pass filter with cutoff frequency  N <  c <  s   N, the original signal can be recovered. X s (j  ) is a periodic function with period  s.

19. Case 2:  s < 2  N  Xc(j)Xc(j) NN  N 1 1/T1/T  2s2s 2s2s 4s4s 6s6s 4s4s 6s6s S(j)S(j) 2/T2/T  2s2s 2s2s 4s4s 6s6s 4s4s 6s6s Xs(j)Xs(j)

20. Case 2:  s < 2  N  Xc(j)Xc(j) NN  N 1 1/T1/T  2s2s 2s2s 4s4s 6s6s 4s4s 6s6s S(j)S(j) 2/T2/T  2s2s 2s2s 4s4s 6s6s 4s4s 6s6s Xs(j)Xs(j) Aliasing No way to recover the original signal. X s (j  ) is a periodic function with period  s.

21. Nequist Rate  Xc(j)Xc(j) NN  N 1 Band-Limited Nequist frequency (  N ) The highest frequency of a band-limited signal Nequist rate = 2  N

22. Nequist Sampling Theorem  Xc(j)Xc(j) NN  N 1 Band-Limited  s > 2  N  s < 2  N Recoverable Aliasing

23. Sampling Theorem CFT vs. DFT

24. C/D System Conversion from impulse train to discrete-time sequence xc(t)xc(t) x(n)= x c (nT)  s(t)s(t) xs(t)xs(t)

25. Continuous-Time Fourier Transform Conversion from impulse train to discrete-time sequence xc(t)xc(t) x(n)= x c (nT)  s(t)s(t) xs(t)xs(t)

26. CFT vs. DFT Conversion from impulse train to discrete-time sequence xc(t)xc(t) x(n)= x c (nT)  s(t)s(t) xs(t)xs(t) x(n)x(n)

27. CFT vs. DFT Conversion from impulse train to discrete-time sequence xc(t)xc(t) x(n)= x c (nT)  s(t)s(t) xs(t)xs(t) x(n)x(n)

28. CFT vs. DFT

29.  Xs(j)Xs(j) ss  s 1/T  X(ej)X(ej) 22 22 4444  Xc(j)Xc(j) 1

30. CFT vs. DFT  Xs(j)Xs(j) ss  s 1/T  X(ej)X(ej) 22 22 4444  Xc(j)Xc(j) 1 Amplitude scaling & Repeating Frequency scaling  s  2 

31. Sampling Theorem Reconstruction of Band-limited Signals

32. Key Concepts t xc(t)xc(t) 0T2T2T3T3T4T4T TT 2T2T 3T3T n x(n)x(n) 01234 11 22 33  X(ej)X(ej)  FT IFT  Xc(j)Xc(j) /T/T  /T Sampling C/D Retrieve One period ICFT CFT

33. Interpolation

34. x(n)x(n) n(t)n(t)

35. Ideal D/C Reconstruction System x(n)x(n) xs(t)xs(t) xr(t)xr(t) Covert from sequence to impulse train T Ideal Reconstruction Filter H r (j  ) Ideal Reconstruction Filter H r (j  ) T

36. x(n)x(n) xs(t)xs(t) xr(t)xr(t) Covert from sequence to impulse train T Ideal Reconstruction Filter H r (j  ) Ideal Reconstruction Filter H r (j  ) T Ideal D/C Reconstruction System  /T/T  /T Hr(j)Hr(j) T Obtained from sampling x c (t) using an ideal C/D system.

37. x(n)x(n) xs(t)xs(t) xr(t)xr(t) Covert from sequence to impulse train T Ideal Reconstruction Filter H r (j  ) Ideal Reconstruction Filter H r (j  ) T Ideal D/C Reconstruction System

38. x(n)x(n)xr(t)xr(t) D/C T xc(t)xc(t) C/D T In what condition x r (t) = x c (t)?

39. Sampling Theorem Discrete-Time Processing of Continuous-Time Signals

40. The Model y(n)y(n)yr(t)yr(t) D/C T Discrete-Time System Discrete-Time System T xc(t)xc(t) C/D x(n)x(n) Continuous-Time System Continuous-Time System xc(t)xc(t) yr(t)yr(t)

41. The Model y(n)y(n)yr(t)yr(t) D/C T Discrete-Time System Discrete-Time System T xc(t)xc(t) C/D x(n)x(n) Continuous-Time System Continuous-Time System xc(t)xc(t) yr(t)yr(t) H eff (j  ) H (ej)H (ej) H (ej)H (ej)

42. LTI Discrete-Time Systems y(n)y(n)yr(t)yr(t) D/C T Discrete-Time System Discrete-Time System T xc(t)xc(t) C/D x(n)x(n) H (ej)H (ej) H (ej)H (ej) H r (j  )

43. LTI Discrete-Time Systems y(n)y(n)yr(t)yr(t) D/C T Discrete-Time System Discrete-Time System T xc(t)xc(t) C/D x(n)x(n) H (ej)H (ej) H (ej)H (ej) H r (j  )

44. LTI Discrete-Time Systems Continuous-Time System Continuous-Time System xc(t)xc(t) yr(t)yr(t) H eff (j  )

45. Example:Ideal Lowpass Filter y(n)y(n)yr(t)yr(t) D/C T Discrete-Time System Discrete-Time System T xc(t)xc(t) C/D x(n)x(n) 1  cc  c H(ej)H(ej)

46. Example:Ideal Lowpass Filter Continuous-Time System Continuous-Time System xc(t)xc(t) yr(t)yr(t) 1  cc  c H eff (j  )

47. Example: Ideal Bandlimited Differentiator Continuous-Time System Continuous-Time System xc(t)xc(t)

48. Example: Ideal Bandlimited Differentiator Continuous-Time System Continuous-Time System xc(t)xc(t)  |H eff (j  )|

49. Example: Ideal Bandlimited Differentiator Continuous-Time System Continuous-Time System xc(t)xc(t)  |H eff (j  )|

50. Impulse Invariance Continuous-Time LTI system h c (t), H c (j  ) Continuous-Time LTI system h c (t), H c (j  ) xc(t)xc(t) yc(t)yc(t) y(n)y(n)yc(t)yc(t) D/C T Discrete-Time LTI System h(n) H(e j  ) Discrete-Time LTI System h(n) H(e j  ) T xc(t)xc(t) C/D x(n)x(n) What is the relation between h c (t) and h(n)?

51. Impulse Invariance

53. Continuous-Time LTI system h c (t), H c (j  ) Continuous-Time LTI system h c (t), H c (j  ) xc(t)xc(t) yc(t)yc(t) y(n)y(n)yc(t)yc(t) D/C T Discrete-Time LTI System h(n) H(e j  ) Discrete-Time LTI System h(n) H(e j  ) T xc(t)xc(t) C/D x(n)x(n) What is the relation between h c (t) and h(n)?

54. Sampling Theorem Continuous-Time Processing of Discrete-Time Signals

55. The Model yc(t)yc(t)y(n)y(n) C/D T Continous-Time System Continous-Time System T x(n)x(n) D/C xc(t)xc(t) Discrete-Time System Discrete-Time System x(n)x(n) y(n)y(n)

56. The Model yc(t)yc(t)y(n)y(n) C/D T Continous-Time System Continous-Time System T x(n)x(n) D/C xc(t)xc(t) Discrete-Time System Discrete-Time System x(n)x(n) y(n)y(n) H (ej)H (ej) H (ej)H (ej) Hc(j)Hc(j) Hc(j)Hc(j)

57. The Model yc(t)yc(t)y(n)y(n) C/D T Continous-Time System Continous-Time System T x(n)x(n) D/C xc(t)xc(t) Hc(j)Hc(j) Hc(j)Hc(j)

58. The Model

59. Discrete-Time System Discrete-Time System x(n)x(n) y(n)y(n) H (ej)H (ej) H (ej)H (ej)

60. The Model Discrete-Time System Discrete-Time System x(n)x(n) y(n)y(n) H (ej)H (ej) H (ej)H (ej) yc(t)yc(t)y(n)y(n) C/D T Continous-Time System Continous-Time System T x(n)x(n) D/C xc(t)xc(t) Hc(j)Hc(j) Hc(j)Hc(j)

61. Sampling Theorem Changing Sampling Rate Using Discrete-Time Processing

62. The Goal Down/Up Sampling Down/Up Sampling

63. Sampling Rate Reduction By an Integer Factor Down Sampling Down Sampling

64. Sampling Rate Reduction By an Integer Factor

65. Let r = kM + i

66. Sampling Rate Reduction By an Integer Factor NN  N Xc(j)Xc(j)  NN X s (j  ), X (e j  T )  2/T2/T 2/T2/T 1/T N=NTN=NT  N X (ej)X (ej)  2222 1/T

67. Sampling Rate Reduction By an Integer Factor X d (e j  )  2222 1/MT M=2 X d (e j  T )  1/T’ 2  /T’  2  /T’4  /T’  4  /T’ NN  N Xc(j)Xc(j)  NN X s (j  ), X (e j  T )  2/T2/T 2/T2/T 1/T N=NTN=NT  N X (ej)X (ej)  2222 1/T  N <  : no aliasing

68. Antialiasing NN  N X (ej)X (ej)  2222 1/T M=3  X d (e j  ) 1/MT 2222 Aliasing

69. Antialiasing NN  N X (ej)X (ej)  2222 1/T  2222  /3 H d (e j  )  2222 1  /3  2222  /3  /3 However, x d (n)  x(nT’) M=3

70. Decimator Lowpass filter Gain = 1 Cutoff =  /M Lowpass filter Gain = 1 Cutoff =  /M MM MM

71. Increasing Sampling Rate By an Integer Factor Up Sampling Up Sampling T

72. Increasing Sampling Rate By an Integer Factor Up Sampling Up Sampling X (ej)X (ej)   1/T X’ (e j  )   L/TL/T

73. Interpolator Lowpass filter Gain = L Cutoff =  /L Lowpass filter Gain = L Cutoff =  /L LL LL

74. Interpolator

75. X (ej)X (ej)   1/T Xe(ej)Xe(ej)   1/T Xi(ej)Xi(ej)   L/TL/T L=3  Hi(ej)Hi(ej)  L  /3  /3

76. Changing the Sampling Rate By a Noninteger Factor Resampling

77. Changing the Sampling Rate By a Noninteger Factor Lowpass filter Gain = 1 Cutoff =  /M Lowpass filter Gain = 1 Cutoff =  /M MM MM Lowpass filter Gain = L Cutoff =  /L Lowpass filter Gain = L Cutoff =  /L LL LL Sampling Periods: MM MM Lowpass filter Gain = L Cutoff = min(  /L,  /M) Lowpass filter Gain = L Cutoff = min(  /L,  /M) LL LL

78. Sampling Theorem Realistic Model for Digital Processing

79. Ideal Discrete-Time Signal Processing Model y(n)y(n)yc(t)yc(t) D/C T Discrete-Time LTI System Discrete-Time LTI System T xc(t)xc(t) C/D x(n)x(n) Real world signal usually is not bandlimited Ideal continuous-to- discrete converter is not realizable Ideal discrete-to- continuous converter is not realizable

80. More Realistic Model y(n)y(n)yc(t)yc(t) D/C T Discrete-Time LTI System Discrete-Time LTI System T xc(t)xc(t) C/D x(n)x(n) Anti- aliasing filter Sample and Hold A/D converter Discrete-time system D/A converter Compensated reconstruction filter TTT

81. Analog-to-Digital Conversion T Sample and Hold Sample and Hold A/D converter TT

82. Sample and Hold T t T ho(t)ho(t) t

83. T t T ho(t)ho(t) t xo(t)xo(t)

84. t T ho(t)ho(t) t xo(t)xo(t)

85. Zero-Order Hold h o (t) Zero-Order Hold h o (t) Goal: To hold constant sample value for A/D converter.

86. A/D Converter C/D T Quantizer Coder

87. Typical Quantizer 2Xm2Xm (B+1)-bit Binary code 2’s complement code Offset binary code 011 010 001 000 111 110 101 100 111 110 101 100 011 010 001 000

88. Analysis of Quantization Errors C/D T Quantizer Coder Quantizer Q[ ] Quantizer Q[ ]

89. Analysis of Quantization Errors The error sequence e(n) is a stationary random process. e(n) and x(n) are uncorrelated. The random variables of the error process are uncorrelated, i.e., the error is a white-noise process. e(n) is uniform distributed.

90. SNR (Signal-to-Noise Ratio)

91. 每增加一個 bit ， SNR 增加約 6dB

92. SNR (Signal-to-Noise Ratio)  x 大較有利，但不得過大 ( 為何？ )  x 過小不利   x 每降低一倍 SNR 少 6dB X~N(0,  x 2 )  P(|X|<4  x )  0.00064 Let  x =X m / 4  SNR  6B  1.25 dB