ESE 531: Digital Signal Processing Lec 10: February 15th, 2018 Practical and Non-integer Sampling, Multirate Sampling
Signals and Systems Review 3
Lecture Outline! Review: Downsampling/Upsampling! Non-integer Resampling! Multi-Rate Processing " Interchanging Operations! Polyphase Decomposition! Multi-Rate Filter Banks 4
Downsampling! Definition: Reducing the sampling rate by an integer number 5
Example i=1 2π i=0 4π 6
Example i=1 2π i=2 4π i=0 6π 7
Example 8
Upsampling! Definition: Increasing the sampling rate by an integer number x[n] = x c (nt ) x i [n] = x c (nt ') 9
Upsampling x i [n] 10
Frequency Domain Interpretation 11
Frequency Domain Interpretation 12
Example 13
Non-integer Resampling
Non-integer Resampling! T =TM/L 15
Non-integer Resampling! T =TM/L " Upsample by L, then downsample by M interpolator decimator 16
Non-integer Resampling! T =TM/L " Upsample by L, then downsample by M interpolator decimator 17
Example! T =3/2T # L=2, M=3 18
Example! T =3/2T # L=2, M=3 19
Non-integer Sampling! T =TM/L " Downsample by M, then upsample by L? interpolator decimator 20
Multi-Rate Signal Processing! What if we want to resample by 1.01T? " Expand by L=100 " Filter π/101 ($$$$$) " Downsample by M=101! Fortunately there are ways around it! " Called multi-rate " Uses compressors, expanders and filtering 21
Interchanging Operations Upsampling -expanding in time -compressing in frequency Downsampling -compressing in time -expanding in frequency 22
Interchanging Operations - Expander Upsampling -expanding in time -compressing in frequency? 23
Interchanging Operations - Expander Upsampling -expanding in time -compressing in frequency? 24
Interchanging Operations - Expander Upsampling -expanding in time -compressing in frequency 25
Interchanging Operations - Expander Upsampling -expanding in time -compressing in frequency = 26
Interchanging Operations - Compressor Downsampling -compressing in time -expanding in frequency = 27
Interchanging Operations - Compressor = 28
Interchanging Operations - Compressor = = 29
Interchanging Operations - Compressor = = 30
Interchanging Operations - Compressor = 31
Interchanging Operations - Compressor = ~ 32
Interchanging Operations - Compressor = ~ = 33
Interchanging Operations - Summary Filter and expander Expander and expanded filter* Compressor and filter Expanded filter* and compressor *Expanded filter = expanded impulse response, compressed freq response 34
Multi-Rate Signal Processing! What if we want to resample by 1.01T? " Expand by L=100 " Filter π/101 ($$$$$) " Downsample by M=101! Fortunately there are ways around it! " Called multi-rate " Uses compressors, expanders and filtering 35
Polyphase Decomposition! We can decompose an impulse response (of our filter) to: 36
Polyphase Decomposition! We can decompose an impulse response (of our filter) to: 37
Polyphase Decomposition 38
Polyphase Decomposition 39
Polyphase Decomposition 40
Polyphase Decomposition 41
Polyphase Decomposition 42
Polyphase Implementation of Decimation! Problem: " Compute all y[n] and then throw away -- wasted computation! " For FIR length N # N mults/unit time 43
Polyphase Implementation of Decimation 44
Polyphase Implementation of Decimation 45
Interchanging Operations - Summary Filter and expander Expander and expanded filter Compressor and filter Expanded filter and compressor 46
Polyphase Implementation of Decimation 47
Polyphase Implementation of Decimation 48
Polyphase Implementation of Decimation Each filter computation: -N/M multiplications -1/M rate per sample #N/M*(1/M) mults/unit time Total computation: -M filters #N/M mults/unit time 49
Multi-Rate Signal Processing! What if we want to resample by 1.01T? " Expand by L=100 " Filter π/101 ($$$$$) " Downsample by M=101! Fortunately there are ways around it! " Called multi-rate " Uses compressors, expanders and filtering 50
Polyphase Implementation of Decimator interpolator decimator 51
Polyphase Implementation of Interpolation interpolator decimator E 0 (z) E 1 (z) 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
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
Multi-Rate Filter Banks! Assume h 0, h 1 are ideal low/high pass with ω C =π/2 55
Multi-Rate Filter Banks! Assume h 0, h 1 are ideal low/high pass with ω C =π/2 56
Multi-Rate Filter Banks! Assume h 0, h 1 are ideal low/high pass with ω C =π/2 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
Multi-Rate Filter Banks! Assume h 0, h 1 are ideal low/high pass with ω C =π/2 59
Multi-Rate Filter Banks! Assume h 0, h 1 are ideal low/high pass with ω C =π/2 60
Multi-Rate Filter Banks! Assume h 0, h 1 are ideal low/high pass with ω C =π/2 61
Multi-Rate Filter Banks! h 0, h 1 are NOT ideal low/high pass 62
Non Ideal Filters! h 0, h 1 are NOT ideal low/high pass 63
Non Ideal Filters 64
Perfect Reconstruction non-ideal Filters 65
Quadrature Mirror Filters Quadrature mirror filters 66
Perfect Reconstruction non-ideal Filters 67
Big Ideas! Downsampling/Upsampling! Practical Interpolation! Non-integer Resampling! Multi-Rate Processing " Interchanging Operations! Polyphase Decomposition! Multi-Rate Filter Banks 68
Admin! HW 4 due Sunday 69