Minimax Design of Nonlinear Phase FIR Filters with Optimality   Certificates
  The Parks-McClellan algorithm provides an efficient method for designing a linear phase FIR filter with a pre-specified weight function on the approximation error. For the given filter order and the specified weight function, the filter designed with this algorithm will have the unique optimal frequency response that approximates a desired filter response as certified by the alternation theorem. In this paper, a nonlinear phase FIR filter design algorithm is provided that allows the specification of a piecewise constant weight function on the approximation error in an analogous manner to linear phase FIR filters. For the given filter order and weight function, the resulting filter will provably have the unique optimal magnitude response that approximates a desired filter response, where the certification of optimality is given and is also based on the alternations that the weighted error function exhibits. Furthermore, the method is applicable to designing filters with both real- and complex-valued coefficients, which in turn determines the number of required alternations. 
Sefa Demirtas
cs.SY