ESE 531: Digital Signal Processing

Transcription

1 ESE 531: Digital Signal Processing Lec 10: February 15th, 2018 Practical and Non-integer Sampling, Multirate Sampling

2 Signals and Systems Review 3

3 Lecture Outline! Review: Downsampling/Upsampling! Non-integer Resampling! Multi-Rate Processing " Interchanging Operations! Polyphase Decomposition! Multi-Rate Filter Banks 4

4 Downsampling! Definition: Reducing the sampling rate by an integer number 5

5 Example i=1 2π i=0 4π 6

6 Example i=1 2π i=2 4π i=0 6π 7

7 Example 8

8 Upsampling! Definition: Increasing the sampling rate by an integer number x[n] = x c (nt ) x i [n] = x c (nt ') 9

9 Upsampling x i [n] 10

10 Frequency Domain Interpretation 11

11 Frequency Domain Interpretation 12

12 Example 13

13 Non-integer Resampling

14 Non-integer Resampling! T =TM/L 15

15 Non-integer Resampling! T =TM/L " Upsample by L, then downsample by M interpolator decimator 16

16 Non-integer Resampling! T =TM/L " Upsample by L, then downsample by M interpolator decimator 17

17 Example! T =3/2T # L=2, M=3 18

18 Example! T =3/2T # L=2, M=3 19

19 Non-integer Sampling! T =TM/L " Downsample by M, then upsample by L? interpolator decimator 20

20 Multi-Rate Signal Processing! What if we want to resample by 1.01T? " Expand by L=100 " Filter π/101 ($$$$$) " Downsample by M=101! Fortunately there are ways around it! " Called multi-rate " Uses compressors, expanders and filtering 21

21 Interchanging Operations Upsampling -expanding in time -compressing in frequency Downsampling -compressing in time -expanding in frequency 22

22 Interchanging Operations - Expander Upsampling -expanding in time -compressing in frequency? 23

23 Interchanging Operations - Expander Upsampling -expanding in time -compressing in frequency? 24

24 Interchanging Operations - Expander Upsampling -expanding in time -compressing in frequency 25

25 Interchanging Operations - Expander Upsampling -expanding in time -compressing in frequency = 26

26 Interchanging Operations - Compressor Downsampling -compressing in time -expanding in frequency = 27

27 Interchanging Operations - Compressor = 28

28 Interchanging Operations - Compressor = = 29

29 Interchanging Operations - Compressor = = 30

30 Interchanging Operations - Compressor = 31

31 Interchanging Operations - Compressor = ~ 32

32 Interchanging Operations - Compressor = ~ = 33

33 Interchanging Operations - Summary Filter and expander Expander and expanded filter* Compressor and filter Expanded filter* and compressor *Expanded filter = expanded impulse response, compressed freq response 34

34 Multi-Rate Signal Processing! What if we want to resample by 1.01T? " Expand by L=100 " Filter π/101 ($$$$$) " Downsample by M=101! Fortunately there are ways around it! " Called multi-rate " Uses compressors, expanders and filtering 35

35 Polyphase Decomposition! We can decompose an impulse response (of our filter) to: 36

36 Polyphase Decomposition! We can decompose an impulse response (of our filter) to: 37

37 Polyphase Decomposition 38

38 Polyphase Decomposition 39

39 Polyphase Decomposition 40

40 Polyphase Decomposition 41

41 Polyphase Decomposition 42

42 Polyphase Implementation of Decimation! Problem: " Compute all y[n] and then throw away -- wasted computation! " For FIR length N # N mults/unit time 43

43 Polyphase Implementation of Decimation 44

44 Polyphase Implementation of Decimation 45

45 Interchanging Operations - Summary Filter and expander Expander and expanded filter Compressor and filter Expanded filter and compressor 46

46 Polyphase Implementation of Decimation 47

47 Polyphase Implementation of Decimation 48

48 Polyphase Implementation of Decimation Each filter computation: -N/M multiplications -1/M rate per sample #N/M*(1/M) mults/unit time Total computation: -M filters #N/M mults/unit time 49

49 Multi-Rate Signal Processing! What if we want to resample by 1.01T? " Expand by L=100 " Filter π/101 ($$$$$) " Downsample by M=101! Fortunately there are ways around it! " Called multi-rate " Uses compressors, expanders and filtering 50

50 Polyphase Implementation of Decimator interpolator decimator 51

51 Polyphase Implementation of Interpolation interpolator decimator E 0 (z) E 1 (z) 52

52 Multi-Rate Filter Banks! Use filter banks to operate on a signal differently in different frequency bands " To save computation, reduce the rate after filtering 53

53 Multi-Rate Filter Banks! Use filter banks to operate on a signal differently in different frequency bands " To save computation, reduce the rate after filtering! h 0 [n] is low-pass, h 1 [n] is high-pass " Often h 1 [n]=e jπn h 0 [n] $ shift freq resp by π 54

54 Multi-Rate Filter Banks! Assume h 0, h 1 are ideal low/high pass with ω C =π/2 55

55 Multi-Rate Filter Banks! Assume h 0, h 1 are ideal low/high pass with ω C =π/2 56

56 Multi-Rate Filter Banks! Assume h 0, h 1 are ideal low/high pass with ω C =π/2 57

57 Multi-Rate Filter Banks! Assume h 0, h 1 are ideal low/high pass with ω C =π/2 Have to be careful with order! 58

58 Multi-Rate Filter Banks! Assume h 0, h 1 are ideal low/high pass with ω C =π/2 59

59 Multi-Rate Filter Banks! Assume h 0, h 1 are ideal low/high pass with ω C =π/2 60

60 Multi-Rate Filter Banks! Assume h 0, h 1 are ideal low/high pass with ω C =π/2 61

61 Multi-Rate Filter Banks! h 0, h 1 are NOT ideal low/high pass 62

62 Non Ideal Filters! h 0, h 1 are NOT ideal low/high pass 63

63 Non Ideal Filters 64

64 Perfect Reconstruction non-ideal Filters 65

65 Quadrature Mirror Filters Quadrature mirror filters 66

66 Perfect Reconstruction non-ideal Filters 67

67 Big Ideas! Downsampling/Upsampling! Practical Interpolation! Non-integer Resampling! Multi-Rate Processing " Interchanging Operations! Polyphase Decomposition! Multi-Rate Filter Banks 68

68 Admin! HW 4 due Sunday 69