Optimization-based design and implementation of multidimensional zero-phase IIR filters

This paper considers multidimensional infinite-impulse response (IIR) filters that are iteratively implemented. The focus is on zero-phase filters with symmetric polynomials in the numerator and denominator of the multivariable transfer function. A rigorous optimization-based design of the filter is considered. Transfer function magnitude specifications, convergence speed requirements for the iterative implementation, and spatial decay of the filter impulse response (which defines the boundary condition influence in the spatial domain of the filtered signal) are all formulated as optimization constraints. When the denominator of the zero-phase IIR filter is strictly positive, these frequency domain specifications can be cast as a linear program and then efficiently solved. The method is illustrated with two two-dimensional IIR filter design examples.