Fast Signal Processing Algorithms Week 5

1. Fast Signal Processing AlgorithmsWeek 5 Polyphase Implementation and Filter Banks

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.

44. Two-Channel Filter Banks • It is known that when the signal bandwidth is p (or half of the sampling frequency), there is no room for decimation operation. • In this case filter banks can be used to decompose signals into subband components which has narrower band and decimation can be done for each subband component; • A parallel processing method

45. Two-channel filter banks • 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

46. Two-Channel Filter Banks • Two-channel filter bank --- the simplest filter bank. Hk(z) are analysis filters and Gk(z) are synthesis filters 2 Processing 2 + output 2 Processing 2

47. Two-Channel Filter Banks • In order that the decomposition does not involve loss of information, the following system should meet perfect reconstruction (PR) condition. That is 2 2 + 2 2

48. Two-Channel Filter Banks • Usually H0(z) is a low-pass filter and H1(z) a high pass filter, by which x(n) are decomposed into low and high frequency components respectively 2 2 + output 2 2

49. Two-channel filter bank --- lower channel

50. Two-channel filter bank --- upper channel