1 / 43

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.

evelyn-savage
Télécharger la présentation

Hassan Bhatti

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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.

More Related