The Optimization of a Class of Non-Linear Filters

An optimum linear filter, in the mean-square-error sense, is no better than the optimum attenuator if there is no dissimilarity between the spectral densities of the signal and the noise. However, non-linear filters can use more statistical information about the signal and noise, so that, although they both possess the same spectral densities, a non-linear filter may be able to introduce a significant improvement in the mean-square error (i.e. it can do better than the optimum attenuator). For this reason the use of a non-linear filter in certain circumstances may well justify the greater difficulties encountered in its optimization and physical realization. The class of filters considered in the paper may be defined by a general expression relating output to input: y(t) =¿r=tR¿0¿¿r(¿)¿r[x(t¿¿)]d¿ An almost routine procedure is proposed whereby the optimum set of weighting functions, ¿r(¿), can be determined given either long enough samples of the combined input and the signal or sufficient statistical information about their characteristics. Some worked examples demonstrate that: (a) A significant improvement in mean-square error is possible even under the condition when the signal and noise possess the same spectral densities. (b) The class of non-linear filters under consideration can be optimized unhindered by the need to evaluate difficult integrals. (c) Although the physical complexity of the filter increases rapidly with the value of R, the mean-square error may converge rapidly to an asymptotic value as R is increased; in one example the performance was found to be within about 5% of the asymptotic value with R = 2.