Decolored Ll filter lattice: a subclass of Ll filters useful in practice

The Ll filters constitute a wide class of filters including both linear and L filters, and it has been shown that they outperform linear and L filters. A close examination of the optimization of the Ll filters in real applications, however, casts a problem of importance that the Ll filters can be degraded from insufficient signal statistics for optimization since a sufficiently long training data sequence is not tractable in practice for the Ll filters, especially, when the the number of input taps is large. We formally propose a new class of order-statistic filters referred to as the decolored Ll filter lattice, which exploits a decolored rank permutation space. By decoloring the rank permutation space, having sufficient training data for optimization in practice is feasible. Various filters are obtained in the class of decolored Ll filters for different orders of decoloring. It is shown that the set of decolored Ll filters of all orders form a lattice structure. At the simplest level in the structure, the decolored Ll filters reduce to a simple linear (FIR) filter, and we obtain the Ll filter at the highest level in the structure.