Norsig 2004, Espoo1 Least Squares Optimization of 2-D IIR Filters Bogdan Dumitrescu Tampere Int. Center for Signal Processing Tampere University of Technology, Finland

Norsig 2004, Espoo 2 Summary 2-D IIR filters: least-squares optimization problem 2-D convex stability domain Gauss-Newton algorithm Experimental results

Norsig 2004, Espoo 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

Norsig 2004, Espoo 4 Optimization criterion Least-squares error with respect to a desired frequency response The error is computed on a grid of frequencies

Norsig 2004, Espoo 5 Optimization difficulties The set of stable IIR filters is not convex The optimization criterion is not convex SOLUTIONS Iterative optimization Convex stability domain around current denominator Gauss-Newton descent technique

Norsig 2004, Espoo 6 convex domain around current denominator Iteration structure set of stable denominators descent direction - current denominator - next denominator

Norsig 2004, Espoo 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

Norsig 2004, Espoo 8 Gauss-Newton descent direction In each iteration, the descent direction is found by a convexification of the criterion This is a semidefinite programming (SDP) problem

Norsig 2004, Espoo 9 Algorithm 1. Set 2. Set 3. Compute GN direction 4. Find optimal step by line search 5. Compute new filter 6. With i=i+1, repeat from 2 until convergence

Norsig 2004, Espoo 10 Design problems Desired response: ideal lowpass filter with linear phase in passband

Norsig 2004, Espoo 11 Passband and stopband shapes circularrhomboidalelliptic

Norsig 2004, Espoo 12 Experiments details Numerator degree: 12 Denominator degree: 2 to 10 Pole radius: 0.9 Implementation: Matlab + SeDuMi Execution time: 3-10 minutes on PC at 1GHz

Norsig 2004, Espoo 13 Results nonseparableseparable circular2.07e-58.76e-6 rhomboidal5.98e-47.05e-4 elliptic9.15e-59.18e-5

Norsig 2004, Espoo 14 Example, magnitude

Norsig 2004, Espoo 15 Example, group delay