Robust frequency-selective filtering using weighted myriad filters admitting real-valued weights

Weighted myriad smoothers have been proposed as a class of nonlinear filters for robust non-Gaussian signal processing in impulsive noise environments. However, weighted myriad smoothers are severely limited since their weights are restricted to be non-negative. This constraint makes them unusable in bandpass or highpass filtering applications that require negative filter weights. Further, they are incapable of amplifying selected frequency components of an input signal. In this paper, we generalize the weighted myriad smoother to a richer structure: a weighted myriad filter admitting real-valued weights. This involves assigning a pair of filter weights (one positive and the other negative) to each of the input samples. Equivalently, the filter can be described as a weighted myriad smoother applied to a transformed set of samples that includes the original input samples as well as their negatives. The weighted myriad filter is analogous to a normalized linear FIR filter with real-valued weights whose absolute values sum to unity. By suitably scaling the output of the weighted myriad filter, we extend it to yield the so-called scaled weighted myriad filter, which includes (but is more powerful than) the traditional unconstrained linear FIR filter. Finally we derive stochastic gradient-based nonlinear adaptive algorithms for the optimization of these novel myriad filters under the mean square error criterion.