Sorting continuous-time signals and the analog median filter

Abstract

Abstract-There are two natural orderings in signals: temporal order and rank order. There is no compelling reason to explore only one of these orderings, either in the discrete-time or in the continuous-time case. Nevertheless, the concept of rank order for continuous-time signals remains virtually unstudied, which is in striking contrast to the discrete-time case: ranked order discrete-time filters, of which the running median is the most common example, have been intensively studied for three decades. The dependence of these nonlinear systems on the order statistics of the input samples stands in contrast with the tapped delay line filter, which depends on temporal order only. However, continuous-time signals can also be meaningfully sorted: a fact that is explored in this paper to define and study the analog median filter and other ranked-order filters. The paper introduces the basic tools needed to analyze and understand these continuous-time nonlinear filters (the distribution function and the sorting) and presents some of their properties in a tutorial way. The analog median filter is defined in terms of the (unique) nonincreasing left-continuous sorting. More general filters can also be defined, including filters similar to -trimmed mean filters and filters. These include filters that depend on one parameter and contain the running average and running median as special cases. The rate of convergence of the digital median filter to the analog median filter is discussed and related to the signal sampling period, the duration of the filter window, and the smoothness of the input signal. The paper introduces the concept of noise width and studies the effect of additive and multiplicative noise at the output of the analog median filter in terms of the noise width and the smoothness of the input signal.

Sorting continuous-time signals: analog median and median-type filters

“…Longbotham and Bovik [2] show by means of a counterexample that some properties of the filters defined in [1] are more subtle than they initially seem. The issues raised in [2] that are related to the nature and existence of root signals of the median filter as defined in [1] were not addressed until recently [3]. The lack of theoretical results on the analog median filter was noted, for example, in [4].…”

“…Because the sortings may differ only at countably many points of discontinuity, we might be tempted to think that the responses of the two corresponding median filters to the same input signal would differ only at countably many points (thus, they would be equal almost everywhere), but this is false: The outputs may differ not only at isolated points but throughout intervals. For example, it can be verified that the input signal described in [2] does in fact lead to such a situation, which can be properly understood in terms of the continuity properties of the sorting [3].…”

“…The nonequivalent definitions of the analog median filter in [1], and the questions raised in [2] concerning the nature of its root signals, are examples of issues that can be understood in terms of the framework described in this paper. They are related to the left or right continuity of the sorting of the windowed signal at one half the window width [3].…”

A general solution to the continuous-time estimation problem under widely linear processing

“…The study of continuous-time estimation problems is also interesting because it provides precise information on some structural properties of the system under study [8,9]. For instance, an explicit expression of the MS error associated with the optimal estimator can be derived in this approach (e.g., see [12,13]).…”

“…Moreover, the analytical tools developed in the continuous-time case might bring new insights to the analysis which are not possible in their discrete-time counterparts. In particular, [8] illustrates this fact in the problem of sorting continuous-time signals, [9] in the problem of nonfragile H ∞ filtering for a class of continuous-time fuzzy systems, and [10] in the study of the behavior of the continuous-time spectrogram.…”