Paulo J. S. G. Ferreira

## Sorting and the analog median filter

The articles in this section explore the possibility of "sorting a continuous-time signal". My original motivation was to define and study an analog version of the well-known digital median filter. The digital median filter is a moving window, nonlinear filter: the value of the output at a certain time sample is the median of the input samples inside the observation window. The median filter is able to remove impulsive noise without smoothing edges, as a linear filter would do. I wanted to define and study a continuous-time, single-input, single-output median filter, and guessed that it would lead to more attractive implementations (there is no need for A/D and D/A converters, a good thing when power and circuit area are severely limited resources).

The tapped delay line filter depends on temporal order only. The nonlinear systems based on sorting depend on rank order. These are the two natural orderings in signals: temporal order, and rank order.

I see 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, in striking contrast to the discrete-time case. These publications are part of an effort to fill in this gap.

For a different application see the section on Uncertainty-based and fuzzy information.

P. J. S. G. Ferreira and M. J. C. S. Reis. Impulsive Noise, Fuzzy Uncertainty, and the Analog Median Filter. In: Systematic Organisation of Information in Fuzzy Systems, P. Melo-Pinto and H.-N. Teodorescu (Eds.). IOS Press, pp. 373-392, 2002.

Part of the material in this book chapter was presented at the NATO Advanced Research Workshop Systematic Organisation of Information in Fuzzy Systems, Oct. 2001.

P. J. S. G. Ferreira. Sorting Continuous-Time Signals: Analog Median and Median-Type Filters. IEEE Transactions on Signal Processing, vol. 49, no. 11, pp. 2734-2744, Nov. 2001.

This paper studies the analog median filter and other ranked order filters. It 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.

Main results:
• 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 alpha-trimmed mean filters, and L-filters. These include filters that depend upon 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.

P. J. S. G. Ferreira. Sorting Continuous-Time Signals and the Analog Median Filter. IEEE Signal Processing Letters, vol. 7, no. 10, pp. 281-283, Oct. 2000.

Continuous-time signals can be meaningfully "sorted", or rearranged, leading to precise formulations of analog median filters and other variants of ranked order filters. This paper shows how this new interpretation leads to a broader view of the analog median and related filters, and settles certain issues regarding continuity and root signals.