Lec 11: February 20, 2020 Polyphase Decomposition and ...ese531/spring2020/handouts/lec11.pdf ·...
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ESE 531: Digital Signal Processing
Lec 11: February 20, 2020 Polyphase Decomposition and Multi-rate
Filter Banks
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Lecture Outline
! Review: Interchanging Operations ! Polyphase Decomposition ! Multi-Rate Filter Banks
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Expander and Compressor
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Upsampling -expanding in time -compressing in frequency
Downsampling -compressing in time -expanding in frequency
Interchanging Operations - Summary
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Filter and expander Expander and expanded filter*
Compressor and filter Expanded filter* and compressor
*Expanded filter = expanded impulse response, compressed freq response
Polyphase Decomposition
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! The polyphase decomposition of a sequence is obtained by representing it as a superposition of M subsequences, each consisting of every Mth value of successively delayed versions of the sequence.
! When this decomposition is applied to a filter impulse response, it can lead to efficient implementation structures for linear filters in several contexts.
Polyphase Decomposition
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! We can decompose an impulse response (of our filter) to:
Polyphase Decomposition
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! We can decompose an impulse response (of our filter) to:
M=2
Polyphase Decomposition
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! We can decompose an impulse response (of our filter) to:
M=2
Polyphase Decomposition
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M=2
Polyphase Decomposition
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M=2
Polyphase Decomposition
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Polyphase Decomposition
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Polyphase Decomposition
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Polyphase Implementation of Decimation
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! Problem: " Compute all y[n] and then throw away -- wasted
computation!
Polyphase Implementation of Decimation
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! Problem: " Compute all y[n] and then throw away -- wasted
computation! " For FIR length N # N multiplications/unit time
Polyphase Implementation of Decimation
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Polyphase Implementation of Decimation
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Polyphase Implementation of Decimation
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Polyphase Implementation of Decimation
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Polyphase Implementation of Decimation
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Polyphase Implementation of Decimation
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Polyphase Implementation of Decimation
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Each filter computation: -N/M multiplications
Total computation: -M filters #N mults??
Polyphase Implementation of Decimation
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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
Polyphase Implementation of Decimator
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interpolator decimator
Polyphase Implementation of Interpolation
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interpolator decimator
E0(z)
E1(z)
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
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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
! h0[n] is low-pass, h1[n] is high-pass " Often h1[n]=ejπnh0[n] $ shift freq resp by π
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Multi-Rate Filter Banks
! Assume h0, h1 are ideal low/high pass with ωC=π/2
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Multi-Rate Filter Banks
! Assume h0, h1 are ideal low/high pass with ωC=π/2
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Multi-Rate Filter Banks
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! Assume h0, h1 are ideal low/high pass with ωC=π/2
Multi-Rate Filter Banks
! Assume h0, h1 are ideal low/high pass with ωC=π/2
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Multi-Rate Filter Banks
! Assume h0, h1 are ideal low/high pass with ωC=π/2
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Have to be careful with
order!
Multi-Rate Filter Banks
! Assume h0, h1 are ideal low/high pass with ωC=π/2
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Multi-Rate Filter Banks
! Assume h0, h1 are ideal low/high pass with ωC=π/2
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Multi-Rate Filter Banks
! Assume h0, h1 are ideal low/high pass with ωC=π/2
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Multi-Rate Filter Banks
! Assume h0, h1 are ideal low/high pass with ωC=π/2
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Multi-Rate Filter Banks
! Assume h0, h1 are ideal low/high pass with ωC=π/2
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Multi-Rate Filter Banks
! Assume h0, h1 are ideal low/high pass with ωC=π/2
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Multi-Rate Filter Banks
! h0, h1 are NOT ideal low/high pass
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Non Ideal Filters
! h0, h1 are NOT ideal low/high pass
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Non Ideal Filters
! h0, h1 are NOT ideal low/high pass
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Non Ideal Filters
! h0, h1 are NOT ideal low/high pass
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Non Ideal Filters
! h0, h1 are NOT ideal low/high pass
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Non Ideal Filters
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Non Ideal Filters
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Non Ideal Filters
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Perfect Reconstruction non-Ideal Filters
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Quadrature Mirror Filters
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Quadrature mirror filters
Perfect Reconstruction non-Ideal Filters
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Haar Filter Example
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Polyphase Filter Bank
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p[n]
q[n]
p[n]
q[n]
Polyphase Decomposition
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Polyphase Filter Bank
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p[n]
q[n]
p[n]
q[n]
Polyphase Filter Bank
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-
p[n]
q[n]
ADC
Analog to Digital Converter
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Anti-Aliasing Filter with ADC
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Aliasing
! If ΩN>Ωs/2, xr(t) an aliased version of xc(t)
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Anti-Aliasing Filter with ADC
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1
-ΩN ΩN
XC ( jΩ)X LP ( jΩ)
ΩN
ΩS/2
XS ( jΩ)1/T
1
-ΩN ΩN
XC ( jΩ)
ΩS/2
ΩN
ΩS/2
XS ( jΩ)1/T
Non-Ideal Anti-Aliasing Filter
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Non-Ideal Anti-Aliasing Filter
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Non-Ideal Anti-Aliasing Filter
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! Problem: Hard to implement sharp analog filter ! Consequence: Crop part of the signal and suffer
from noise and interference
Oversampled ADC
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Oversampled ADC – Simple filter
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Oversampled ADC – M=2
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Oversampled ADC
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Oversampled ADC – Sharp digital filter/Downsample
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Oversampled ADC
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Oversampled ADC – Sharp digital filter/Downsample
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Oversampled ADC
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Big Ideas
! Interchanging Operations ! Polyphase Decomposition ! Multi-Rate Filter Banks
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Admin
! HW 4 due Sunday
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