iterative reweighted least-squares algorithm for 2-d iir filters design
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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
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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
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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
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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
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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
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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
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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
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GN_IRLS+
Design IIR filter using GN_IRLS (with trivial initialization)
Then, keeping fixed the denominator, reoptimize the numerator using IRLS
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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),(
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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
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Example, magnitude
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Example, group delay
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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
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How to choose ?
GN_IRLS only: variations with GN_IRLS+: many values of give similar results
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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.
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