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

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.

Fuzzy Information On Discrete And Continuous Domains: Approximation Results

“…The result now follows from the uniqueness of the rearrangement of f. More precisely, f is the only non-increasing left-continuous function equidistributed with f (for details, see Ferreira, 2001). Therefore, the non-increasing step function that has been built is in fact the rearrangement of f.…”

“…See Ferreira (2000;2001) for details. The only difficulty is that M f , which for discontinuous functions can be constant over certain intervals, does not in general have an inverse.…”

“…For an application of these concepts in the context of analog median filtering see Ferreira (2000;2001).…”

Sensor Signal Processing for Autonomous Wireless Sensors

“…For example, median hybrid filters suggested by Heinonen, and Neuvo [6,7], running medians with robust regression, analized by Davies et al [8], Qc-method-based filters proposed by Croux and Rousseeuw [9,10]. For audio signal elaboration and image processing, many realizations of efficient median filters have been proposed [11][12][13][14], even if usually in this field fast and powerful processors or DSPs are employed. Particularly interesting are the so called trimmed or winsorized mean filter, better described in the next section [15].…”

Signal analysis through analog representation

“…However, while analog implementation of the mean is trivial, median estimators are much harder to implement in analog form (Jarske & Vainio 1993), since, traditionally, their determination has involved 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 (Bottema 1991;Ferreira 2000Ferreira , 2001.…”