Impulse Response and Polyphase Filters in Signal Processing

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Discover the application of impulse response and polyphase filters in signal processing, including M-to-1 and 1-to-M processing, narrow bandwidth filtering, cascade filters, and prototype filters. Explore the design considerations, frequency responses, and spectral masks for optimal signal manipulation.

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Green Filters Green Filters Cascade Polyphase M-to-1 Down Sample Filter, Inner Filter, and Polyphase 1-to-M Up Sample Filter fred harris M-to-1 1-to-M 0 1 0 1 0 1 0 1 M Path Filter M Path Filter .... .... .... .... g(n) M-2 M-2 M-2 M-2 M-1 M-1 M-1 M-1 SDSU College of Engineering

Narrow Bandwidth Filter with Large Ratio Sample Rate to Bandwidth Many Taps N-T ap FIR Filter fs Atten(dB) f N TAPS 22 Reduce Bandwidth and Reduce Sample Rate M-to-1 Increase Sample Rate 1-to-M 0 1 0 1 0 1 0 1 Narrow BW Filter at Reduced Sample Rate M P ath Filter M P ath Filter .... .... .... .... g(n) N/M T ap M-2 M-2 M-2 M-2 M-1 M-1 M-1 M-1 1-Mth Fewer Taps 1-Mth Sample Rate

361-Sample, Impulse Response Standard Design 1 fs Atten(dB) f 100 80 1 22 Amplitude = = N 364 0.5 TAPS 22 0 0 50 100 150 200 250 300 350 Time Index Frequency Response and Spectral Masks 0 Passband: 0-1 kHz Stopband: 2-50 kHz In Band Ripple: 0.1 dB Stop Band Attenuation: 80 dB Log Mag (dB) -20 -40 -60 -80 -100 -50 -40 -30 -20 -10 0 10 20 30 40 50 Frequency (kHz) Zoom to Pass Band Ripple Zoom to Pass Band Ripple 0.2 0 Log Mag (dB) Log Mag (dB) 0.1 -20 Attn (dB) f -40 0 -60 -0.1 -80 -0.2 0 0.5 1 1.5 2 2.5 3 -1.5 -1 -0.5 0 0.5 1 1.5 Frequency (kHz) Frequency (kHz)

Cascade M-to-1 Down Sampling Filter, Inner Filter, and 1-to-M Up Sampling Filter x(n) y(n) N T ap h(n) h (n) 0 h (n) 0 h (n) 1 h (n) 1 x(n) y(n) g(n) h1 h2 0 1 0 1 N/M T ap g(n) H (n) H 20 Path Filter 18 (n) M-1 20 Path Filter M-1 .... .... .... .... H (n) M H (n) M g1 18 19 19

In Particular, Examine a 20 Path Cascade Filter N-T ap FIR Filter h1 h2 0 1 0 1 20 Path Filter 20 Path Filter .... .... .... .... g1 18 18 19 19

120-Sample, Impulse Response, Prototype Filter for 20-to-1 Down Sample Polyphase Filter 1 Amplitude 0.5 0 0 20 40 60 80 100 120 Time Index Frequency Response and Spectral Masks 0 Log Mag (dB) -20 -40 -60 -80 -100 -50 -40 -30 -20 -10 0 10 20 30 40 50 Frequency (kHz) Zoom to Pass Band Ripple Zoom to Transition Bandwidth 0.1 0 Log Mag (dB) Log Mag (dB) 0.05 -20 0 -40 -60 -0.05 -80 -0.1 -1.5 -1 -0.5 0 0.5 1 1.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Frequency (kHz) Frequency (kHz)

20-Sample, Impulse Response, Inner Filter Operating at fs = 5 kHz 1 Amplitude 0.5 0 -2 0 2 4 6 8 10 12 14 16 18 20 Time Index Frequency Response and Spectral Masks 0 Log Mag (dB) -50 -100 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 Frequency (kHz) Zoom to Transition Bandwidth Zoom to Pass Band Ripple 0.1 0 Log Mag (dB) Log Mag (dB) 0.05 -20 -40 0 -60 -0.05 -80 -0.1 0 0.5 1 1.5 2 2.5 -1.5 -1 -0.5 0 0.5 1 1.5 Frequency (kHz) Frequency (kHz)

160-Sample, Impulse Response, Prototype Filter for 1-to-20 Up Sample Polyphase Filter 1 Amplitude 0.5 0 0 20 40 60 80 100 120 140 160 Time Index Frequency Response and Spectral Masks 0 Log Mag (dB) -20 -40 -60 -80 -100 -50 -40 -30 -20 -10 0 10 20 30 40 50 Frequency (kHz) Zoom to Transition Bandwidth Zoom to Pass Band Ripple 0.1 0 Log Mag (dB) Log Mag (dB) 0.05 -20 0 -40 -60 -0.05 -80 -0.1 -1.5 -1 -0.5 0 0.5 1 1.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Frequency (kHz) Frequency (kHz)

621-Sample Impulse Response Cascade 20-to-1 Down Sample Polyphase, 6 wts/path, 20 Tap Inner Filter, and 1-to-20 Up Sample Polyphase 8 ws/path Filter, 15-Ops per Input-Output 1 Amplitude 0.5 0 0 100 200 300 400 500 600 Time Index Frequency Response and Spectral Masks 0 Log Mag (dB) -20 -40 -60 -80 -100 -50 -40 -30 -20 -10 0 10 20 30 40 50 Frequency (kHz) Zoom to Pass Band Ripple Zoom to Transition Bandwidth 0.2 0 Log Mag (dB) Log Mag (dB) 0.1 -20 -40 0 -60 -0.1 -80 -0.2 0 0.5 1 1.5 2 2.5 3 -1.5 -1 -0.5 0 0.5 1 1.5 Frequency (kHz) Frequency (kHz)

Now Consider a 6 Path Cascade Filter N-T ap FIR Filter 0 h1 0 1 h2 0 1 0 0 1 1 0 1 1 6 P ath 6 P ath 20 Path Filter Filter 20 Path Filter Filter .... .... .... .... .... .... .... .... g1 18 4 4 5 4 5 18 4 19 5 19 5

23-Sample, Impulse Response, Prototype Filter for 6-to-1 Down Sample Polyphase Filter 1 Amplitude 0.5 0 0 5 10 15 20 25 Time Index Frequency Response and Spectral Masks 0 Log Mag (dB) -20 -40 -60 -80 -50 -40 -30 -20 -10 0 10 20 30 40 50 Frequency (kHz) Zoom to Pass Band Ripple Zoom to Pass Band Ripple 0.1 0 Log Mag (dB) Log Mag (dB) 0.05 -20 -40 0 -60 -0.05 -80 -0.1 0 2 4 6 8 10 12 14 16 -1.5 -1 -0.5 0 0.5 1 1.5 Frequency (kHz) Frequency (kHz)

63-Sample, Impulse Response, Inner Filter Operating at fs = 16.667 kHz 1 Amplitude 0.5 0 0 10 20 30 40 50 60 Time Index Frequency Response and Spectral Masks 0 Log Mag (dB) -20 -40 -60 -80 -8 -6 -4 -2 0 2 4 6 8 Frequency (kHz) Zoom to Pass Band Ripple Zoom to Pass Band Ripple 0.1 0 Log Mag (dB) Log Mag (dB) 0.05 -20 0 -40 -60 -0.05 -80 -0.1 -1.5 -1 -0.5 0 0.5 1 1.5 0 0.5 1 1.5 2 2.5 3 Frequency (kHz) Frequency (kHz)

19-Sample, Impulse Response, Prototype Filter for 1-to-6 Up Sample Polyphase Filter 1 Amplitude 0.5 0 0 5 10 15 20 Time Index Frequency Response and Spectral Masks 0 Log Mag (dB) -20 -40 -60 -80 -50 -40 -30 -20 -10 0 10 20 30 40 50 Frequency (kHz) Zoom to Pass Band Ripple Zoom to Pass Band Ripple 0.1 0 Log Mag (dB) Log Mag (dB) 0.05 -20 -40 0 -60 -0.05 -80 -0.1 0 2 4 6 8 10 12 14 16 18 -1.5 -1 -0.5 0 0.5 1 1.5 Frequency (kHz) Frequency (kHz)

407-Sample, Impulse Response Cascade 6-to-1 Down Sample, Polyphase, 4-wts/path, 63 Tap Inner Filter and 1-to-6 Up sample Polyphase 4 wts/path Filter, 18.5-Ops per Input-Output 1 Amplitude 0.5 0 0 50 100 150 200 250 300 350 400 Time Index Frequency Response and Spectral Masks 0 Log Mag (dB) -20 -40 -60 -80 -100 -50 -40 -30 -20 -10 0 10 20 30 40 50 Frequency (kHz) Zoom to Pass Band Ripple Zoom to Pass Band Ripple 0.2 0 Log Mag (dB) Log Mag (dB) 0.1 -20 0 -40 -60 -0.1 -80 -0.2 -1.5 -1 -0.5 0 0.5 1 1.5 0 0.5 1 1.5 2 2.5 3 Frequency (kHz) Frequency (kHz)

For Reference, Original Design 361-Sample, Impulse Response Standard Design 1 Amplitude 0.5 0 0 50 100 150 200 250 300 350 Time Index Frequency Response and Spectral Masks 0 Log Mag (dB) -20 -40 -60 -80 -100 -50 -40 -30 -20 -10 0 10 20 30 40 50 Frequency (kHz) Zoom to Pass Band Ripple Zoom to Pass Band Ripple 0.2 0 Log Mag (dB) Log Mag (dB) 0.1 -20 -40 0 -60 -0.1 -80 -0.2 0 0.5 1 1.5 2 2.5 3 -1.5 -1 -0.5 0 0.5 1 1.5 Frequency (kHz) Frequency (kHz)

Workload: Cascade of M-to-1 Down Sample, Inner Filter, and 1-to-M Up Sample 361 Number of Multiplies 99 2 10 50 31 22 19 16 14 12.8 13.4 12.7 13.8 12.9 12.2 11.9 12.4 13.8 IIR 1 10 0 2 4 6 8 10 12 14 16 M Resample Factor for M-to-1 and 1-to-M Delay to Composite Filter Center Sample as Function Of Resample Factor M 280 260 Delay in Samples 240 220 200 180 160 0 2 4 6 8 10 12 14 16 M Resample Factor for M-to-1 and 1-to-M

6 Path Cascade Filter N-T ap FIR Filter 0 h1 0 1 h2 0 1 0 0 1 1 0 1 1 6 P ath 6 P ath 20 Path Filter Filter 20 Path Filter Filter .... .... .... .... .... .... .... .... g1 18 4 4 5 4 5 18 4 19 5 19 5 6-Path Filter with 2- First Order IIR Filters -M -M z z 1 0 M M H (Z ) 0 H (Z ) 1 - - -M -M 0 z z 1

Pole-Zero Diagram 1.5 Don t Care Band 1 0.5 Stop Band Pass Band 0 -0.5 -1 Band Don t care -1.5 -1.5 -1 -0.5 0 0.5 1 1.5

Frequency Response: 6-Path Linear Phase IIR Filter Log Magnitude (dB) 0 -20 -40 -60 -80 -50 -40 -30 -20 -10 0 10 20 30 40 50 Frequency Zoom To Pass Band Response Log Magnitude (dB) 0 -20 -40 -60 -80 -15 -10 -5 0 5 10 15 Frequency Zoom To Pass Band Ripple -8 5x 10 Log Magnitude (dB) 0 -5 -10 -5 -4 -3 -2 -1 0 1 2 3 4 5 Frequency

Phase: 6-Path Linear Phase IIR filter 1 Phase(f)/(2 ) 0 -1 -2 -3 -50 -40 -30 -20 -10 0 10 20 30 40 50 Frequency Group Delay: 6-Path Equal-Ripple Linear Phase IIR filter Input Sample Rate Samples At 20 10 0 -50 -40 -30 -20 -10 0 10 20 30 40 50 Frequency Pass Band Group Delay: 6-Path Equal-Ripple Linear Phase IIR filter 11.8 Input Sample Rate Samples At 11.75 11.7 11.65 -5 -4 -3 -2 -1 0 1 2 3 4 5 Frequency

6-Path Linear Phase Recursive Filter 6-Path IIR Impulse Response, Prototype Filter for 6-to-1 Down Sample Polyphase Filter 1 Amplitude 0.5 0 -0.5 0 10 20 30 40 50 60 70 80 Time Index Frequency Response and Spectral Masks 0 Log Mag (dB) -20 -40 -60 -80 -50 -40 -30 -20 -10 0 10 20 30 40 50 Frequency (kHz) Zoom to Pass Band Ripple Zoom to Pass Band Ripple -7 1x 10 0 Log Mag (dB) Log Mag (dB) 0.5 -20 0 -40 -60 -0.5 -80 -1 -1.5 -1 -0.5 0 0.5 1 1.5 0 2 4 6 8 10 12 14 16 Frequency (kHz) Frequency (kHz)

Cascade 6-Path Chain with Linear Phase Recursive Filter 420-Sample, Impulse Response Cascade 6-to-1 Down Sample, Polyphase IIR, 10/6-wts/path, 63 Tap Inner Filter and 1-to-6 Up sample Polyphase 10/6 wts/path Filter, 13.8-Ops per Input-Output 1 Amplitude 0.5 0 0 50 100 150 200 Time Index 250 300 350 400 Frequency Response and Spectral Masks 0 Log Mag (dB) -50 -100 -50 -40 -30 -20 -10 0 10 20 30 40 50 Frequency (kHz) Zoom to Pass Band Ripple Zoom to Pass Band Ripple 0.2 0 Log Mag (dB) Log Mag (dB) 0.1 -20 0 -40 -60 -0.1 -80 -0.2 -1.5 -1 -0.5 0 0.5 1 1.5 0 0.5 1 1.5 2 2.5 3 Frequency (kHz) Frequency (kHz)

Significant Computational Savings for Narrow Bandwidth Signals with Large Ratio of Sample Rate to BW (Allow Down-Sampling) How do we Access these Savings for Wide Bandwidth Signals? Partition the Wide Bandwidth Signal into Multiple Narrow Bandwidth Segments with each having Large Ratio of Sample Rate to BW! An Input Analysis Channelizer Does the Partition and Down Sampling An Output Synthesis Channelizer Does Reconstruction and Up Sampling 24

Filtering in the Channelizer Domain x x nM/2 P oint S hift M-P oint Circular Buffer nM/2 P oint S hift M-P oint Circular Buffer .. .. M M P oint IFFT M P ath Filter M P oint IFFT .. .. .. P ath Filter .. .. .. .. .. .. .. x x .. .. 25

Impulse Response, 1051 Tap FIR Filter 0.4 Amplitude 0.2 0 -0.2 0 100 200 300 400 Impulse Response, Zoom to Main Lobe 500 600 700 800 900 1000 0.4 Amplitude 0.2 0 -0.2 450 500 550 600 Time Index Frequency response 0 Log Mag (dB) -20 -40 -60 -80 -50 2x 10 -40 -30 -20 -10 0 10 20 30 40 50 Zoom to Pass Band Ripple Zoom to Transition BW -3 Frequency 0 Log Mag (dB) Log Mag (dB) 1 -20 -40 0 -60 -1 -80 -2 -4 -3 -2 -1 0 1 2 3 4 29.5 30 30.5 31 31.5 Frequency Frequency

1151 Point Impulse Response, Prototype Low Pass Synthesis Filter, 9-Samples per Path in 128 Path Filter 1 Amplitude 0.5 0 -600 -400 -200 0 200 400 600 Time Index Frequency Response, Low Pass Analysis Filter (Blue) and Low pass Synthesis Filter (Red) 0 Log Mag (dB) -50 -100 -4 -3 -2 -1 0 1 2 3 4 Frequency Frequency Response, Zoom to Passband Ripple -3 2x 10 Log Mag (dB) 1 0 -1 -2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Frequency

Impulse Response, 61-Bins from 128 Path Channelizer Filter 0.4 Amplitude 76 Ops to Implement 1051 Point FIR Filter 0.2 0 -0.2 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 Impulse Response, Zoom to Main Lobe 0.4 Amplitude 0.2 0 -0.2 1000 1020 1040 1060 1080 1100 1120 1140 1160 1180 Time Index Frequency response 0 Log Mag (dB) -20 -40 -60 -80 -50 2x 10 -40 -30 -20 -10 0 10 20 Zoom to Transition BW 30 40 50 Zoom to Pass Band Ripple -3 Frequency 0 Log Mag (dB) Log Mag (dB) 1 -20 -40 0 -60 -1 -80 -2 29.5 30 30.5 31 31.5 -4 -3 -2 -1 0 1 2 3 4 Frequency Frequency

9-taps/path 1280-Mult/xfrm 9-taps/path 1280-Mult/xfrm x x 64 P oint S hift 128-P nt Circular Buffer 64 P oint S hift 128-P nt Circular Buffer .. .. 128 128 P oint IFFT 128 P ath Filter 128 P oint IFFT .. .. .. P ath Filter .. .. .. .. .. .. .. x x .. .. Input Polyphase Filter 18 Ops/Input, Output Polyphase Filter, 18 Ops/Output 128 point IFFTs, Approximately 1280 Multiplies/Transform, 1280 Ops Amortized over 64 Inputs is 20 Ops/Input Total workload, 18+20+20+18... 76 Ops/Input-Output Direct Filter 1051 Samples... Ratio 1051/76, 13.8 to 1