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Multi-Rate Digital Signal Processing

Multi-Rate Digital Signal Processing. Y. C. Jenq, Ph.D. Department of Electrical & Computer Engineering Portland State University Portland, Oregon 97207 jenq@ece.pdx.edu. x[n]. M. x d [n]. Decimation System (Down Sampling). M=3. Decimation System (Down Sampling).

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Multi-Rate Digital Signal Processing

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  1. Multi-RateDigital Signal Processing Y. C. Jenq, Ph.D. Department of Electrical & Computer Engineering Portland State University Portland, Oregon 97207 jenq@ece.pdx.edu

  2. x[n] M xd[n] Decimation System (Down Sampling) M=3

  3. Decimation System (Down Sampling) xd[n] = x[nM], where M is an integer Xd(z) =Sn xd[n]z-n =(1/M)Sm=0,(M-1) X(z(1/M)e-jm2p/M) Xd(ejw) =(1/M)Sm=0,(M-1) X(ej(w-m2p)/M)

  4. Decimation System (Down Sampling) Xd(ejw) =(1/M)Sm=0,(M-1) X(ej(w-m2p)/M) X(ejw/M) X(ejw) M=3 0 p 2p 4p 6p 8p X(ej(w-2p)/M) 0 2p X(ej(w-4p)/M) 0 4p

  5. M Sampling Rate Reduction System Low-pass filter with cutoff at p/M yd[n] x[n] y[n]

  6. L x[n] xu[n] Interpolation System (Up Sampling) L=3

  7. Interpolation System (Up Sampling) xu[n] = x[n/L], n = 0, ±L, ±2L, … 0, otherwise Xu(z) =Sn xu[n]z-n = X(zL) Xu(ejw) = X(ejwL)

  8. Interpolation System (Up Sampling) Xu(ejw) = X(ejwL) L=3 X(ejw) X(ejwL) 0 2p/3 4p/3 2p

  9. Sampling Rate Increase System Low-pass filter with cutoff at p/L xu[n] L y[n] x[n]

  10. Decimation and Interpolation M M x[n] y[n] M=3

  11. Decimation and Interpolation M M x[n] y[n] Let WM = e-j2p/M Y(z) = (1/M)Sm=0,(M-1) X(ze-jm2p/M) = (1/M)Sm=0,(M-1) X(zWMm) Y(ejw)= (1/M)Sm=0,(M-1) X(ej(w-m2p/M))

  12. Decimation and Interpolation Y(ejw)= (1/M)Sm=0,(M-1) X(ej(w-jm2p/M)) X(ejw) M=3 0 2p/3 2p X(ej(w-2p/M)) 0 2p/3 X(ej(w-4p/M)) 0 4p/3

  13. L M Fractional Sampling Rate Change Low-pass filter with cutoff at min(p/L,p/M) y[n] x[n]

  14. M M M M M Block Interconnection Identities C C  Multiply by a Constant 

  15. M M M Block Interconnection Identities 

  16. L C C L L  L L Block Interconnection Identities

  17. L  L L Block Interconnection Identities

  18. M M Multi-Rate Identities H(zM)  H(z)  L H(zL) H(z) L

  19. 3 3 3 Multi-rate Switch Models x[n] n = 0, 3, 6,… Z-1  n = -1, 2, 5,… Z-1 n = -2, 1, 4,…

  20. 3 3 3 Multi-rate Switch Models Serial to Parallel Converter x[3n] x[n] Z-1  x[3n-1] x[n] Z-1 x[3n-2]

  21. Multi-rate Switch Models 3 x1[n] n = 2, 5, 8,… Z-1 3  x2[n] n = 1, 4, 7,… Z-1 3 n = 0, 3, 6,… x3[n]

  22. Multi-rate Switch Models 3 Parallel to Serial Converter x1[n] x[3n+2] Z-1 3  x[3n+1] x2[n] x[n] Z-1 3 x[3n] x3[n]

  23. Poly-phase Structure of Decimation Filter y[n] x[n] 3 H(z) y[3n] H(z) = D0(z3)+ z-1D1(z3)+z-2D2(z3) D0(z3) x[n] Z-1 y[n] 3 y[3n] D1(z3) Z-1 D2(z3)

  24. 3 3 3 y[3n] Poly-phase Structure of Decimation Filter H(z) = D0(z3)+ z-1D1(z3)+z-2D2(z3) D0(z3) x[n] Z-1 D1(z3) Z-1 D2(z3)

  25. 3 3 3 y[3n] Poly-phase Structure of Decimation Filter H(z) = D0(z3)+ z-1D1(z3)+z-2D2(z3) D0(z) x[n] Z-1 D1(z) Z-1 D2(z)

  26. y[3n] Poly-phase Structure of Decimation Filter H(z) = D0(z3)+ z-1D1(z3)+z-2D2(z3) Serial to Parallel Converter D0(z) x[3n] x[n] D1(z) x[3n-1] D2(z) x[3n-2]

  27. Poly-phase Structure of Interpolation Filter 3 x[n] H(z) y[n] H(z) = z-2I0(z3)+ z-1I1(z3)+I2(z3) 3 I0(z3) Z-2 x[n] I1(z3) Z-1 y[n] I2(z3)

  28. Poly-phase Structure of Interpolation Filter H(z) = z-2I0(z3)+ z-1I1(z3)+I2(z3) 3 x[n] I0(z3) Z-1 3 I1(z3) Z-1 3 I2(z3) y[n]

  29. Poly-phase Structure of Interpolation Filter H(z) = z-2I0(z3)+ z-1I1(z3)+I2(z3) 3 x[n] I0(z) Z-1 3 I1(z) Z-1 3 I2(z) y[n]

  30. Poly-phase Structure of Interpolation Filter H(z) = z-2I0(z3)+ z-1I1(z3)+I2(z3) Parallel to Serial Converter x[n] I0(z) y[3n+2] I1(z) y[3n+1] y[n] I2(z) y[3n]

  31. Poly-phase Structure of Fractional Sampling Rate Filter H(z) = z-2F0(z3)+ z-1F1(z3)+F2(z3) (M=4, L=3) 3 F0(z3) Z-2 x[n] 4 F1(z3) Z-1 y[n] F2(z3)

  32. Poly-phase Structure of Fractional Sampling Rate Filter H(z) = z-2F0(z3)+ z-1F1(z3)+F2(z3) (M=4, L=3) 3 F0(z) Z-2 x[n] 4 3 F1(z) Z-1 y[n] 3 F2(z)

  33. Poly-phase Structure of Fractional Sampling Rate Filter H(z) = z-2F0(z3)+ z-1F1(z3)+F2(z3) (M=4, L=3) 3 F0(z) Z6 Z-8 x[n] y[n] 4 3 F1(z) Z3 Z-4 3 F2(z)

  34. Poly-phase Structure of Fractional Sampling Rate Filter H(z) = z-2F0(z3)+ z-1F1(z3)+F2(z3) (M=4, L=3) 3 4 F0(z) Z2 Z-2 x[n] y[n] 3 4 F1(z) Z1 Z-1 3 4 F2(z)

  35. Poly-phase Structure of Fractional Sampling Rate Filter H(z) = z-2F0(z3)+ z-1F1(z3)+F2(z3) (M=4, L=3) 3 4 Z2 F0(z) Z-2 x[n] y[n] 3 4 Z1 F1(z) Z-1 3 4 F2(z)

  36. Poly-phase Structure of Fractional Sampling Rate Filter H(z) = z-2F0(z3)+ z-1F1(z3)+F2(z3) (M=4, L=3) 4 3 Z2 F0(z) Z-2 x[n] y[n] 4 3 Z1 F1(z) Z-1 4 3 F2(z)

  37. Poly-phase Structure of Fractional Sampling Rate Filter H(z) = z-2F0(z3)+ z-1F1(z3)+F2(z3) (M=4, L=3) 4 Parallel to Serial Converter x[n] Z2 F0(z) 4 y[n] Z1 F1(z) 4 F2(z)

  38. Poly-phase Structure of Fractional Sampling Rate Filter H(z) = z-2F0(z3)+ z-1F1(z3)+F2(z3) Fk(z) = Fk0(z4)+ z-1Fk1(z4)+z-2Fk2(z4)+z-3Fk3(z4) (M=4, L=3) 4 Parallel to Serial Converter x[n] Z2 F0(z) 4 y[n] Z1 F1(z) 4 F2(z)

  39. Poly-phase Structure of Fractional Sampling Rate Filter H(z) = z-2F0(z3)+ z-1F1(z3)+F2(z3) Fk(z) = Fk0(z4)+ z-1Fk1(z4)+z-2Fk2(z4)+z-3Fk3(z4) (M=4, L=3) Fk0 Serial to Parallel Converter Fk1 Parallel to Serial Converter x[n] Z2 Fk2 Fk3 y[n] 4 Z1 F1(z) 4 F2(z)

  40. Poly-phase Structure of Fractional Sampling Rate Filter H(z) = z-2F0(z3)+ z-1F1(z3)+F2(z3) Fk(z) = Fk0(z4)+ z-1Fk1(z4)+z-2Fk2(z4)+z-3Fk3(z4) (M=4, L=3) Fk0 Serial to Parallel Converter x[n] Fk1 Parallel to Serial Converter Z2 Fk2 Fk3 y[n] 4 Z-1 F1(z) 4 Z-1 F2(z)

  41. Efficient Design for Very Narrow-band Filters x[n] H(z) y[n] x[n] F(z) G(zM) y[n]

  42. 0 0 0 0 p p p p Efficient Design for Very Narrow-band Filters H(z) wp ws G(z) Desired passband Mws Mwp Images G(zM) wp ws 2p/M F(z) wp 2p/M-ws

  43. z-1 z-1 Efficient Design for Very Narrow-band Filters M x[n] F(z) G(zM) y[n] M x[n] F(z) G(z) y[n] x[n] F0(z) M G(z) y[n] F1(z) + M

  44. Efficient Design for Very Narrow-band Filters F0(z) Serial to Parallel Converter G(z) y[n] F1(z) + x[n] FM-1(z)

  45. Multi-stage Decimation System M x[n] H(z) y[n] M1M2 x[n] F(z) G(zM1) y[n] M2 M1 y[n] x[n] F(z) G(z)

  46. Multi-stage Decimation System F0(z) G0(z) Serial to Parallel Converter Serial to Parallel Converter F1(z) G1(z) x[n] + + y[n] GM2 -1(z) FM1 -1(z)

  47. F(z) F(z) Multi-stage Interpolation System L x[n] H(z) y[n] L1L2 x[n] G(zL1) y[n] L2 L1 y[n] x[n] G(z)

  48. Multi-stage Interpolation System G0(z) Parallel to Serial Converter Parallel to Serial Converter F0(z) G1(z) F1(z) x[n] y[n] GL2-1(z) FL1-1(z)

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