Adaptive approximation of feedback rank filters for continuous signals

Abstract

Rank-based nonlinear filtering techniques are steadily gaining in popularity due to their robustness. However, the analog implementation of these techniques meets with considerable conceptual and practical difficulties. Here we describe an adaptive approximation for a rank filter of a continuous signal expressed in terms of a system of differential equations easily implementable in an analog circuit. The design is based on consideration of the finite precision of physical measurements, which leads to simple and efficient implementation of many traditionally digital analysis tools. We also illustrate the performance of the adaptive approximation filter in comparison with the respective ‘exact’ rank filter in a boxcar moving window.

Introduction

The main advantage of analog signal processing with respect to digital processing is simple implementation and efficient handling of nonlinearities. However, there are many signal processing tasks for which digital algorithms are well known, but corresponding analog operations are hard to reproduce. One example which falls within this category is related to the use of signal processing techniques based on order statistics,¹ such as implementing median and other order statistic filtering [16]. Order statistic filters are gaining wider recognition for their ability to provide more robust estimators of signal properties. For example, the median value of a set of measurements usually represents the general trend in a signal better than the mean value, since the latter is more sensitive to outliers. However, while analog implementation of the mean is trivial, median estimators are much harder to implement in analog form [5], [6], [7], since, traditionally, their determination involves the operation of sorting or ordering a set of measurements. Indeed, there is no conceptual difficulty in sorting a set of discrete measurements, but it is much less obvious how to perform similar operations for continuous signals [3], [4].

As pointed out by some authors [14], the major problem in analog rank processing is the lack of an appropriate differential equation for ‘analog sorting’. There have been many attempts to implement such sorting and build continuous-time rank filters. Examples of these efforts include optical rank filters [12], analog sorting networks [13], [14], and analog rank selectors based on minimization of a nonlinear objective function [17]. However, the term ‘analog’ is often perceived as only ‘continuous-time’, and these efforts fall short of considering the threshold continuity, which is necessary for a truly analog representation of differential sorting operators. Even though the recent work by Ferreira [4] extensively discusses threshold distributions, these distributions are only piecewise-continuous and thus do not allow straightforward introduction of differential operations with respect to threshold.

Recently, we have proposed a new approach to constructing analog devices for performing traditionally digital signal processing tasks [9], [10], [11]. This approach is based on the consideration of the finite precision of real measurements, with the resulting modification of the definitions of various signal properties and underlying mathematical equations. Since analog systems are implemented using physical components, the mathematical description of such systems must take into account their limited precision and inertial characteristics. Therefore, the output of an analog device typically represents a weighted average over a nonzero time and threshold intervals. Realization of this fact enables us to rewrite many problems of signal analysis in the form readily addressed by methods of differential calculus, which are suitable for analog implementation, rather than by the algebraic or logic operations of the digital approach. In [9], we have outlined the general principles of this approach and suggested several applications. In the present article we apply these principles to develop a simple and accurate approximation for a rank filter of a continuous signal in a boxcar moving window.