Nonlinear phase FIR filter design with minimum LS error and additional constraints

Abstract

W e ezamine the problem of approzrmating a complex frequency response b y a real-valued FIR filter according to the L2 norm subjeci to additional inequality constraints for the complez error function. Starting with the Kuhn-Tucker optimalaty conditions which specialize t o a system of nonlinear equations we deduce an iterative algorithm. These equations are solved by Newton's method in every iteration step. The algorilhm allows arbitrary compromises between an L2 and an L, design. The L2 and the L, solution result as special cases. I IntroductionAlthough FIR filter design has been treated in the past by many publications only few consider the case of a non symmetric impulse response or an arbitrary nonlinear phase complex desired function (NLPFIR filter), which we will do in this paper. The design of FIR filters according to the L, norm with an (anti) symmetric impulse response (LPFIR filters) can be done very efficiently by using either Remez' algorithm [6] or Linear Programming [8]. If the Lz norm is used the problem is reduced to solving a system of linear equations. The latter approach can also be applied to arbitrary complex desired functions. The generalization of the algorithms minimizing the L, norm to this case is more complicated. There are three approaches: A generalization of Remez' algorithm [7, 91 which is fast but not guaranteed to converge, a Linear Programming approach [4, 51 yielding suboptimal results with a high demand of computation time and computer storage and an algorithm of Tang [2, 111 which is not guaranked to converge but does so in nearly all practical applications [9]. As can be seen the optimum filter can be designed if just the L2 or the L, norm is used. With the methods mentioned above the minimization of the error energy (L2 norm) keeping a prescribed maximum error at the same time is not possible. Adams [l] brought up this problem and gave an approach for the solution for LPFIR filters. Considering a discrete frequency grid we generalize his method to the case of NLPFIR. filters and give a simple possibility to essentially improve the convergence behaviour of Adams' approach. Although we do not prove convergence our algorithm converges in most practical applications and we give a short discussion of convergence problems, which are also valid for Adam's method. Convergence of the new algorithm is tantamount to optimality of the solution because of our approach. I1 Description of the ProcedureIn this section we consider the problem of determining a real-valued NLPFIR filter approximating a desired function, which keeps a prescribed error tolerance 6 and minimizes the absolute quadratic error. Both the L , and the L2 solution result as specializations if the minimax error is prescribed for 6 on the one hand and a very large value for 6 on the other hand. By continuously increasing 6 beginning from the minimax error arbitrary compromises between an L, and an L2 solution are possible. We formulate the filter design problem as the problem of minimizing a quadratic function subject to ...