1

Iterative Reweighted Least-Squares Algorithm for 2-D IIR Filters Design

Bogdan Dumitrescu, Riitta NiemistöInstitute of Signal Processing

Tampere University of Technology, Finland

2

Summary

Problem: Chebyshev design of 2-D IIR filters Algorithm: combination of

iterative reweighted least squares (IRLS) Gauss-Newton convexification convex 1-D and 2-D stability domains

Optimization tool: semidefinite programming

3

2-D IIR filters

Transfer function

1

1

2

2

21

21

1

1

2

2

21

21

0 0 21,

0 0 21,

21

2121 ),(

),(),( n

k

n

k

kkkk

m

k

m

k

kkkk

zza

zzb

zzA

zzBzzH

Degrees m1, m2, n1, n2 are given Coefficients are optimized Denominator can be separable or not

2121 ,, , kkkk ab

4

Optimization criterion

p-norm error w.r.t. desired frequency response

1

1

2

2

2211

21211 1

)()(,, ),(),,(

L Lpjj eeHDpBAJ

Special case: p large (approx. Chebyshev) The error is computed on a grid of frequencies

5

Optimization difficulties

The set of stable IIR filters is not convex The optimization criterion is not convex

SOLUTIONS Iterative reweighed LS (IRLS) optimization Convex stability domain around current

denominator Gauss-Newton descent technique

6

convex domain around

current denominator

Iteration structure

)(iA

)1( iA

set of stable denominators

descent direction

)(iA - current denominator)1( iA - next denominator

)(iA

)(iD

7

2-D convex stability domain

Based on the positive realness condition

0),(

),(1Re

21

21

)(

)(

jji

jjiA

eeA

ee

Described by a linear matrix inequality (LMI) Using a parameterization of sum-of-squares

multivariable polynomials Pole radius bound possible

8

Gauss-Newton descent direction In each iteration, the descent direction is found by

a convexification of the criterion

(i)D

)()(

2

1 1

)()(,

)(,,,

..

min 1

1

2

2 21212121

iA

i

L L iiTi

Ats

HHD

Semidefinite programming (SDP) problem

)(

)(

iB

iA

9

IRLS - IIR filters with fixed denominator Start with Increase exponent with Compute new weights

),,min( 1 ppp ii 1

2)(,,,, 21212121

~ ipiHD

LS optimize: Update numerator Repeat until convergence

),(,2 )1(0 ABBp LS

)~,( ABB LS

)()1(

1

2

1

1 i

i

i

i

i Bp

pB

pB

10

GN_IRLS Algorithm

1. Set 2. Set3. Compute new weights4. Compute GN direction with new weights5. Find optimal step by line search

6. Compute new filter

7. With i=i+1, repeat from 2 until convergence

)(,1,2,1 )1()1()1(0 ABBApi LS

)()( , iB

iA

),(min )()()()(10

* iB

iiA

i BAJ

)(*)()1()(*)()1( , iB

iiiA

ii BBAA

),min( 1 ppp ii

11

GN_IRLS+

Design IIR filter using GN_IRLS (with trivial initialization)

Then, keeping fixed the denominator, reoptimize the numerator using IRLS

12

Design example

Desired response: ideal lowpass filter with linear phase in passband

)(2121

2211),(),( jeDD

s

pD

22

21

22

21

21if,0

if,1),(

13

Design details

Design data (as in [1]): Degrees: Separable denominator Group delays: Stop- and pass-band: Pole radius: Norm:

Implementation: Matlab + SeDuMi

8,12 2121 nnmm

821

9.0 7.0,5.0 sp

120p

14

Example, magnitude

15

Example, group delay

16

Comparison with [1]

This paper [1]

Stopband attenuation 42.5 dB 39.4 dB

Passband deviation 0.0074 0.0081

Max. group delay error 0.526 -

Execution time 6 min 27 min

17

How to choose ?

GN_IRLS only: variations with GN_IRLS+: many values of give similar results

18

References

[1] W.S.Lu, T.Hinamoto. Optimal Design of IIR Digital Filters with Robust Stability Using Conic-Quadratic Programming Updates. IEEE Trans. Signal Proc., 51(6):1581-1592, June 2003.

[2] B.Dumitrescu, R.Niemistö. Multistage IIR Filter Design Using Convex Stability Domains Defined by Positive Realness. IEEE Trans. Signal Proc., 52(4):962-974, April 2004.

[3] C.S.Burrus, J.A.Barreto, I.W.Selesnick. Iterative Reweighted Least-Squares Design of FIR Filters. IEEE Trans. Signal Proc., 42(11):2926-2936, Nov. 1994.