Multi-rate adaptive filters have numerous advantages such as low computational load, fast convergence and parallelism in the adaptation. Drawbacks when using multi-rate processing are mainly related to aliasing and reconstruction effects. These effects can be minimized by introducing appropriate problem formulation and employing sophisticated optimization techniques. In this paper, we propose a formulation for the design of filter bank which controls the distortion level for each frequency component directly and minimizes the inband aliasing and the residual aliasing between different subbands. The advantage of this problem formulation is that the distortion level can be weighted for each frequency depending on the particular practical application. A new iterative algorithm is proposed to optimize simultaneously over both the analysis and the synthesis filter banks. This algorithm is shown to have a unique solution for each iteration. For a fixed distortion level, the proposed algorithm yields a significant reduction in both the inband aliasing and the residual aliasing levels compared to existing methods applied to the numerical examples.

The weighted Chebyshev design of two-dimensional FIR filters is in general not unique since the Haar condition is not generally satisfied. However, fo r a design on a discrete frequency domain, the Haar condition might be fulf illed. The question of uniqueness is, however, rather extensive to investig ate. It is therefore desirable to define some simple additional constraints to the Chebyshev design in order to obtain a unique solution. The weighted Chebyshev solution of minimum Euclidean filter weight norm is always uniqu e, and represents a sensible additional constraint since it implies minimum white noise amplification. This unique Chebyshev solution can always be ob tained by using an efficient quadratic programming formulation with a stric tly convex objective function and linear constraints. An example where a co nventional Chebyshev solution is nonunique is discussed in the brief.

Using the recently developed semi-infinite linear programming techniques and Caratheodory's dimensionality theory, we present a unified approach to digital filter design with time and/or frequency-domain specifications. Through systematic analysis and detailed numerical design examples, we demonstrate that the proposed approach exhibits several salient features compared to traditional methods: 1) using the unified approach, complex responses can be handled conveniently without resorting to discretization; 2) time-domain constraints can be included easily; and 3) any filter structure, recursive or nonrecursive, can be employed, provided that the frequency response can be represented by a finite-complex basis. More importantly, the solution procedure is based on the numerically efficient simplex extension algorithms. As numerical examples, a discrete-time Laguerre network is used in a frequency-domain design with additional group-delay specifications, and in a H-infinity-optimal envelope constrained filter design problem. Finally, a finite impulse response phase equalizer is designed with additional frequency domain H-infinity, robustness constraints.

The Multi-Mode Mean Field Annealing (MM-MFA) approach to combinatorial optimization is introduced as a tool to design recursive (IIR) digital filters with discrete coefficients. As an application example demonstrating the potential of the method we consider the design of structurally passive IIR digital filters realized as the sum of two allpass functions. The new design technique facilitates the solution of non-trivial filter design problems such as satisfying a general frequency specification by solving a combinatorial optimization problem over discrete coefficients and a max-norm cost. The final solution is not guaranteed to be a globally optimal solution but the convergence time is short enough to allow interactive design even for large problems.

The multi-mode mean field annealing (MM-MFA) approach to combinatorial optimization is introduced as a tool to design recursive infinite-impulse response (IIR) digital filters with discrete coefficients. As an application example demonstrating the potential of the method we consider the design of structurally passive IIR digital filters realized as the sum of two all-pass functions. The new design technique facilitates the solution of nontrivial filter design problems such as satisfying a general frequency specification by solving a combinatorial optimization problem over discrete coefficients and a max-norm cost. The final solution is not guaranteed to be a globally optimal solution but the convergence time is short enough to allow interactive design even for large problems.