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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