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Sampling Theorem

Sampling Theorem. 主講者：虞台文. 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

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Sampling Theorem

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

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

  6. xc(t) xc(t) t t T 2T 3T 2T 0 T 2T 3T 4T 8T 4T 0 2T 4T 8T 10T x(n) x(n) n n 1 2 3 2 0 1 2 3 4 6 4 0 2 4 6 8 Sampling with Periodic Impulse train

  7. xc(t) xc(t) t t T 2T 3T 2T 0 T 2T 3T 4T 8T 4T 0 2T 4T 8T 10T x(n) x(n) n n 1 2 3 2 0 1 2 3 4 6 4 0 2 4 6 8 We want to restore xc(t) from x(n). Sampling with Periodic Impulse train What condition has to be placed on the sampling rate?

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

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

  10. C/D System s: Sampling Frequency

  11. C/D System

  12. Sampling Theorem Sampling of Band-Limited Signals

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

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

  15. Sampling Theorem Aliasing --- Nyquist Rate

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

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

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

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

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

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

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

  23. Sampling Theorem CFT vs. DFT

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

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

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

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

  28. CFT vs. DFT

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

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

  31. Sampling Theorem Reconstruction of Band-limited Signals

  32. xc(t) Xc(j)  t T 3T 2T 0 T 2T 3T 4T /T /T X(ej) x(n)    n 1 3 2 0 1 2 3 4 Key Concepts CFT ICFT Sampling C/D Retrieve One period FT IFT

  33. Interpolation

  34. Interpolation n(t) x(n)

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

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

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

  38. xc(t) x(n) xr(t) C/D D/C T T Ideal D/C Reconstruction System In what condition xr(t) = xc(t)?

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

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

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

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

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

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

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

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

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

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

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

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

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