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On the Optimization of the Tay-Kingsbury 2-D Filterbank

Bogdan Dumitrescu

Tampere Int. Center for Signal Processing

Tampere University of Technology, Finland

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Summary

Problem: design of 2-channel 2-D FIR filterbank Idea: 1-D to 2-D McClellan transformation Math tool for optimization:

sum-of-squares polynomials (on the unit circle) Optimization tool: semidefinite programming

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2-Channel 2-D Filterbank

Quincunx sampling Non-separable filters FIR

H0(z) 2 2 F0(z)

H1(z) 2 2 F1(z)

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Tay-Kingsbury idea

),(),(

),(),(

21021211

21021211

21

21

zzHzzzzF

zzFzzzzHKK

KK

PR condition

Take

),(),(),( 21021021 zzFzzHzzD Define

2),(),( 2121 zzDzzD

or

0,,evenif,0

0if,1

2121

21, 21 nnnn

nnd nn

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Tay-Kingsbury transformation

1-D halfband filter

factorized

2)()( ZDZD pp

)()()( ZFZHZD ppp

2-D transformation G(z1,z2) with

evenif,0 21, 21 nng nn

PR filterbank:)),((),(

)),((),(

21210

21210

zzGFzzF

zzGHzzH

p

p

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Transformation properties

Ideal frequency response

otherwise,1

||||if,1),( 2121

G

Denote ),(1),(~

2121 zzGzzG

Property: 2),(~

0then,0),(~

if 2121 GG

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Optimization of the transformation Minimize stopband energy

gΦg ~~),(~

21

2

21T

s

s

ddGE

where is the vector of coefficients and

is a positive definite matrix

g~

Constraint: !!!0),(~

21 G

Φ

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Stopband shape

1

2

s

),(

),(

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Sum-of-squares polynomials

A symmetric polynomial is sum-of-squares on the unit circle if

1

2

2121

1

12

112121

),(),(~

),(),(),(~

FG

zzFzzFzzG

A sos polynomial is nonnegative on the unit circle

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Positive polynomials

Basic result: all polynomials positive on the unit circle can be expressed as sum-of-squares

However, theoretically it is possible that

),(~

deg),(deg 2121 zzGzzF

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Parameterization of sos polynomials A symmetric polynomial is sos if and only if there

exists a positive semidefinite matrix Q such that

where1221 , kkkk TTT

0

0elementary Toeplitz

Gram matrix QT

2121 ,,~

kkkk traceg

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Resulting optimization problem

Semidefinite programming (SDP) problem

0

~evenif,~

~~

..

min

2121

2121

,,

21,,

,~

Q

QT

gQgQg

kkkk

nnnn

T

traceg

nngts

Unique solution, reliable algorithms

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Example of design

1-D halfband prototype

)8)(1()(

),3)(1()(2

12

14

1

ZZZZF

ZZZH

p

p

Transformation degree: 3 (symmetric polynomial)

Execution time: 1.2 seconds

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Frequency response H0

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Frequency response F0

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Improvement of synthesis filter

Since usually degF0>degH0, a new synthesis filter of same degree can be obtained via lifting

evenif,0 21, 21 nna nn

),(),(),(),(~

21021210210 zzHzzAzzFzzF

Optimization of a quadratic with linear constraints

A is obtained by solving a linear system.

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Frequency response of improved F0