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Hassan Bhatti

Digital Signal Processing Lecture-4 Fall 2009 Polyphase Decomposition and Mutirate Signal Processing Chap.-4. Hassan Bhatti. Motivation. Up- and down sampling combined with filtering are the usual operations in multirate systems. Polypahse approach will yield simple implementations.

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Hassan Bhatti

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  1. Digital Signal Processing Lecture-4 Fall 2009Polyphase Decomposition and Mutirate Signal Processing Chap.-4 Hassan Bhatti

  2. Motivation • Up- and down sampling combined with filtering are the usual operations in multirate systems. • Polypahse approach will yield simple implementations

  3. Outlines • Two basic multirate operations • Polyphase interpolator – upsampling followed by a filter • Polyphase decimatator – a filter followed by a decimator • Two-channel filter banks • Perfect reconstruction condition • Quadrature mirror filter (QMF) filter banks • Design of two-channel filter banks with PR • Multiple-channel filter banks • Tree- structured filter banks • Octave-band filter banks

  4. Basic Multirate Operations • Decimation and interpolation • Z-domain and Frequency domain analysis of up-and downsampled version of a signal • Polyphase decomposition • Noble Identities

  5. Decimation and Interpolation • Decimation---down-sampling N x(n)

  6. Decimation and Interpolation • Decimation---down-sampling N

  7. Decimation and Interpolation • Decimation---down-sampling N y(m)

  8. Decimation and Interpolation • Interpolation --- up-sampling N

  9. Decimation and Interpolation • Interpolation --- up-sampling N

  10. Decimation and Interpolation • Interpolation --- up-sampling N

  11. H(z) N N G(z) Decimation and Interpolation A typical building block of multirate filter bank We want to know the relationships between the above signals

  12. Decimation and Interpolation Up-sampling N

  13. Decimation and Interpolation Upsampling when N=2

  14. Decimation and Interpolation Downsampling followed by upsampling N N as hence

  15. Decimation and Interpolation Downsampling followed by upsampling N N

  16. Decimation and Interpolation Downsampling followed by upsampling N N image spectra original spectrum

  17. Decimation and Interpolation Downsampling followed by upsampling image spectra original spectrum

  18. Decimation and Interpolation Downsampling followed by upsampling when N=2 Image spectra

  19. Decimation and Interpolation Downsampling N We know upsampling We know downsampling +upsampling We can get downsampling

  20. Decimation and Interpolation Downsampling N Example, N=2

  21. Decimation and Interpolation Downsampling when N=2 Image spectra

  22. Decimation and Interpolation Downsampling when N=2 Image spectra

  23. Decimation and Interpolation Downsampling when N=2 Image spectra

  24. H(z) N N G(z) Decimation and Interpolation A typical building block of multirate filter bank We want to know the relationships between the above signals

  25. Decimation and Interpolation– overall system

  26. Polyphase Decomposition • Polyphase decomposition is the decomposition of a sequence x(n) into sub-sequences x(mN+i) • There are four types of polyphase decomposition. Type-1 [ 0,1,2,3,4,5,6,7,8,9,10,11] M=3 [ 0,3,6,9] [ 1,4,7,10] [ 2,5,8,11]

  27. Polyphase Decomposition Type-1 M M + T 2T M

  28. Polyphase Decomposition Type-2 [ 0,1,2,3,4,5,6,7,8,9,10,11] M=3 [ 2,5,8,11] [ 1,4,7,10] [ 0,3,6,9]

  29. Polyphase Decomposition Type-3: we want to have hence Type-3 is not very straightforward as there is a casualty problem.

  30. Nobel Identities Identity I N G(z) G(zN) N

  31. Nobel Identities Identity II N G(zN) G(z) N

  32. H(z) N N G(z) Decimation and Interpolation—polyphase implementation A typical building block of multirate filter bank We want to know if there is an efficient way to implement the above system

  33. Polyphase Interpolator N G(z) x(n) y(m) v(n) Multiplications with zeros are involved

  34. Polyphase Intepolator • We decompose the filter into polyphase components (Type-1): • In z-domain:

  35. Polyphase Decomposition （from the notes last week) Type-1 M M + T 2T M

  36. N G(z) x(n) y(m) v(n) Polyphase Intepolator N x(n) y(m) v(n)

  37. N G(z) x(n) y(m) v(n) N N N Polyphase Intepolator Using the second Noble identity: x(n) y(m)

  38. N G(z) x(n) y(m) v(n) N x(n) N y(m) N Polyphase Intepolator For input signal of length M, and G(z) of length L, convolution of v(n) (of length NM) and g(n) (of length L) requires NML mutiplications

  39. N x(n) N y(m) N Polyphase Intepolator • Each branch has a convolution of y(m) (of length M) with Gl(z) (of length L/N). ML/N multiplications are required; • Hence for N branches, ML multiplications are required in total; Computation is greatly reduced

  40. Polyphase Intepolator • example: M=1024, N=2,L=64, MNL=128k multiplications are required for convolution • when using polyphase approach, only ML=64k multiplications are required.

  41. Polyphase Decimator – polyphase implementation H(z) N y(m) x(n) v(n) The number of multiplications is: ML

  42. Polyphase Decimator Let i=jN+k let

  43. Polyphase Decimator N y(m) x(n) N N The number of multiplications is: N (M/N)(L/N) =ML/N. Also reduced a lot.

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