Using Bayesian inference for the design of FIR filters with signed power-of-two coefficients

Abstract

The design approach presented in this paper applies Bayesian inference to the design of finite impulse response (FIR) filters with signed power-of-two (SPoT) coefficients. Given a desired frequency magnitude response specified by upper and lower bounds in decibels, Bayesian parameter estimation and model selection are adapted to produce a distribution of potential designs, all of which perform at or close to the specified standard. In the process, having incorporated prior information such as the maximum acceptable number of SPoT terms and filter length, and the practical design requirement to use the fewest bits possible, the total number of bits, filter taps and SPoT terms, and the filter length required in a design are automatically determined. The produced design candidates have design complexity appropriate to the design specifications and requirements, as designs with higher design complexity than required are rendered less probable by the embedded Ockham's razor. This innate ability is a prominent advantage that the newly developed framework possesses over many optimization based techniques as it leads to designs that require fewer SPoT terms and filter taps. Most importantly, it avoids the intricacy, arduousness and rigorousness involved in devising an appropriate scheme for balancing design performance against design complexity.

Introduction

A design process involves determining appropriate values for the design parameters based on prescribed design specifications and requirements. For a given design problem, the existence and uniqueness of solutions are not guaranteed. A design problem may have more than one solution such that different designs are capable of meeting the same design specifications and requirements. Alternatively, a design problem may have no solution, meaning that no design can be produced to meet the specified standard. A design problem shares these characteristics with inverse problems, and can be treated as a generalized inverse problem. In particular, this notion is applicable to the design of a linear phase finite impulse response (FIR) filter with signed power-of-two (SPoT) coefficients. Given a desired frequency magnitude response, which is characterized by design specifications such as the passband and stopband edge frequencies, the maximum passband ripple, and the minimum stopband attenuation, the goal is to produce a design that realizes the desired frequency response. To obtain a design, the values of all design parameters must be determined, including the total required number of SPoT terms and filter taps, the index of the filter taps, the filter length and the SPoT coefficient values. Depending on the design criteria and complexity, a filter design problem may have several sets of appropriate design parameter values, or none.

Bayesian inference has been used extensively to solve generalized inverse problems in a wide variety of applications, and the tools and methods developed for Bayesian parameter estimation and model comparison can be adapted to solve filter design problems. The Bayesian inference framework for design has already been applied to two design applications: linear antenna arrays [1], [2] and linear phase FIR filters with continuous coefficients [3]. Below, the developed framework is extended to design linear phase FIR filters with SPoT coefficients.

The design of a linear phase finite impulse response (FIR) filter with signed power-of-two (SPoT) coefficients is commonly formulated as an optimization problem. The goal of the optimization approach is, in general, to minimize the difference between the achieved and desired frequency magnitude response. Many optimization techniques, such as hybrid genetic algorithm [4], polynomial-time algorithm [5], simulated annealing [6], discrete filled function [7], mixed integer linear programming [8] and others [9] have been successful; however, the optimization approach has the drawback that one or more design parameters, such as the number of SPoT terms and filter taps, and the filter length, are fixed in the design process. This limitation leads to filter designs with greater design complexity than necessary. The methods in [5], [6], [8] use a prescribed fixed filter length and prescribed number of SPoT terms per coefficient or entire filter, and consequently generate designs that have more SPoT terms than required. The method in [4] aims to minimize the required number of SPoT terms and filter taps by including these factors in the fitness function. Even though this author establishes a reasonable fitness function through a simulation study of a large number of filter designs, there is no fundamental principle for the fitness function. Consequently—as will emerge below—the resulting designs still involve more SPoT terms than necessary even though some improvement is attained over the methods in [5], [6], [8]. These comments show the enormous complexity involved in devising a method for designing filters with design complexity appropriate to the design requirements. This capability is innate in the Bayesian inference framework because it contains a principle of parsimony and quantitatively implements the Ockham's razor principle, rendering filter designs with greater design complexity than required less probable. The embedded Ockham's razor principle makes it superfluous to devise complicated and ultimately arbitrary schemes for balancing design performance against design complexity. Automatic determination of the design complexity, done properly, is equivalent to Bayesian model selection. The mechanism by which Ockham's razor operates in Bayesian model selection is presented at length in, among other places, [10, Chapter 3; 11, Chapter 4; 12, Chapter 28].

In the framework of Bayesian inference, the solution to a filter design problem is encapsulated in the posterior probability distribution, which is a function of the design parameters. Bayes' theorem, which is an immediate consequence of the sum and product rules of probability, states that the posterior is proportional to the product of the likelihood and the priors. The priors are probability distribution functions which encapsulate prior information about constraints on the design parameters, and the likelihood is obtained by assigning a probability distribution function to the error. In an inverse problem, the error is defined as the difference between the parametric model and the measured data. There are no data or measurements in a filter design problem, however. By exploiting the parallelism between inference and design problems, the error in a filter design problem can be identified as the difference between the desired and achieved frequency response. The posterior is approximated by a Monte Carlo method, as a closed-form solution cannot be obtained. In this method, a reasonable number of samples are drawn from the posterior using a Markov chain Monte Carlo method [13], [14]. From the point of view of design, each posterior sample represents a potential design solution having specific values for the design parameters. As a result, the solution to a design problem consists of a number of design candidates rather than a single final design. Note that unlike many techniques in the optimization framework, the potential designs are all achieved without the aid of a set of continuous filter coefficients that has been predetermined to meet the desired frequency response. To obtain the final design, a designer has to select a single design candidate based on additional design criteria.

This paper is organized as follows. Starting with a description of the parametric model, Section 2 presents the basic concepts of the Bayesian inference framework for design. The application of the Bayesian inference framework to two representative design examples is discussed in Section 3, along with a comparison of results. Section 4 concludes with some final reflections.