Optimization of two-dimensional IIR filters with nonseparable and separable denominator

We present algorithms for the optimization of two-dimensional (2-D) infinite impulse response (IIR) filters with separable or nonseparable denominator, for least squares or Chebyshev criteria. The algorithms are iterative, and each iteration consists of solving a semidefinite programming problem. For least squares designs, we adapt the Gauss-Newton idea, which outcomes to a convex approximation of the optimization criterion. For Chebyshev designs, we adapt the iterative reweighted least squares (IRLS) algorithm; in each iteration, a least squares Gauss-Newton step is performed, while the weights are changed as in the basic IRLS algorithm. The stability of the 2-D IIR filters is ensured by keeping the denominator inside convex stability domains, which are defined by linear matrix inequalities. For the 2-D (nonseparable) case, this is a new contribution, based on the parameterization of 2-D polynomials that are positive on the unit bicircle. In the experimental section, 2-D IIR filters with separable and nonseparable denominators are designed and compared. We show that each type may be better than the other, depending on the design specification. We also give an example of filter that is clearly better than a recent very good design.