1. 1 Iterative Reweighted Least-Squares Algorithm for 2-D IIR Filters Design Bogdan Dumitrescu, Riitta Niemistö Institute of Signal Processing Tampere University of Technology, Finland

2. 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. 3 2-D IIR filters Transfer function Degrees m 1, m 2, n 1, n 2 are given Coefficients are optimized Denominator can be separable or not

4. 4 Optimization criterion p -norm error w.r.t. desired frequency response Special case: p large (approx. Chebyshev) The error is computed on a grid of frequencies

5. 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. 6 convex domain around current denominator Iteration structure set of stable denominators descent direction - current denominator - next denominator

7. 7 2-D convex stability domain Based on the positive realness condition Described by a linear matrix inequality (LMI) Using a parameterization of sum-of-squares multivariable polynomials Pole radius bound possible

8. 8 Gauss-Newton descent direction In each iteration, the descent direction is found by a convexification of the criterion Semidefinite programming (SDP) problem

9. 9 IRLS - IIR filters with fixed denominator Start with Increase exponent with Compute new weights LS optimize: Update numerator Repeat until convergence

10. 10 GN_IRLS Algorithm 1. Set 2. Set 3. Compute new weights 4. Compute GN direction with new weights 5. Find optimal step by line search 6. Compute new filter 7. With i=i+1, repeat from 2 until convergence

11. 11 GN_IRLS+ Design IIR filter using GN_IRLS (with trivial initialization) Then, keeping fixed the denominator, reoptimize the numerator using IRLS

12. 12 Design example Desired response: ideal lowpass filter with linear phase in passband

13. 13 Design details Design data (as in [1]):  Degrees:  Separable denominator  Group delays:  Stop- and pass-band:  Pole radius:  Norm: Implementation: Matlab + SeDuMi

14. 14 Example, magnitude

15. 15 Example, group delay

16. 16 Comparison with [1] This paper[1] Stopband attenuation42.5 dB39.4 dB Passband deviation0.00740.0081 Max. group delay error0.526- Execution time6 min27 min

17. 17 How to choose  ? GN_IRLS only: variations with  GN_IRLS+: many values of  give similar results

18. 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.