Optimal design of nonuniform FIR transmultiplexer using semi-infinite programming

Abstract

Abstract-This paper considers an optimum nonuniform FIR transmultiplexer design problem subject to specifications in the frequency domain. Our objective is to minimize the sum of the ripple energy for all the individual filters, subject to the specifications on amplitude and aliasing distortions, and to the passband and stopband specifications for the individual filters. This optimum nonuniform transmultiplexer design problem can be formulated as a quadratic semi-infinite programming problem. The dual parametrization algorithm is extended to this nonuniform transmultiplexer design problem. If the lengths of the filters are sufficiently long and the set of decimation integers is compatible, then a solution exists. Since the problem is formulated as a convex problem, if a solution exists, then the solution obtained is unique and the local solution is a global minimum.Index Terms-Nonuniform transmultiplexer, semi-infinite programming, dual parameterization.

Efficient Algorithm for Solving Semi-Infinite Programming Problems and Their Applications to Nonuniform Filter Bank Designs

“…(19) Now we see that (16), (18), and (19) contradict the fact that ) (x J is strictly convex. Therefore, x is feasible to problem P. …”

“…The formulation of this optimum nonuniform symmetric/antisymmetric linear phase FIR filter bank design problem is similar to that of the nonuniform transmultiplexer design problem [18], so only the summary of the formulation is stated in this paper. For interested readers, we recommend them to study [18].…”

“…Define the cost function of an optimization problem as the sum of the ripple energy for all the individual filters in the passband and stopband subject to the specifications on the passband and stopband ripple magnitudes, as well as on the amplitude and aliasing distortions. Using similar approach in [18], the nonuniform filter bank design problem can be formulated as the following semi-infinite programming problem:…”

“…In order to solve these problems, numerically efficient simplex extension algorithms were employed in [6] and [16], conventional dual parameterization approaches were employed in [5], [12], [17], [18] and [27], an extended active set strategy was employed in [4], an extended version of Remez algorithm was employed in [7] to generate a feasible starting condition for the dual method, discretization methods were employed in [9], [15] and [16], outer-approximation cutting plane algorithms were employed in [10], [15] and [30], penalty function approaches were employed in [11] and [26], and constraint transcription approaches were employed in [13] and [25]. Among them, the constraint transcription methods, the penalty function approaches, the discretization methods, the cutting plane methods and the conventional dual parameterization methods are the most common methods employed for solving engineering problems.…”

“…The rest of this paper is organized as follows. In Section II, we summarize the formulation of the optimum nonuniform symmetric/anti-symmetric linear phase FIR filter bank design problem as a quadratic semi-infinite programming problem [18]. In Section III, an improved dual parameterization algorithm for solving the semi-infinite programming problem is proposed.…”