Hierarchical Filters in a Collaborative Filtering Framework for System Identification and Knowledge Retrieval

This chapter provides a critical review of hierarchical filters and the associated adaptive learning algorithms. Hierarchical filters are collaborative adaptive filtering architectures where short-length adaptive transversal filters are combined into layers, which are then combined into a multilayered structures. These structures offer potentially faster speed of convergence compared to the standard finite impulse response (FIR) filters, which is due to the small order of their constituting sub-filters. Several approaches can be used to adapt the coefficients of hierarchical filters. These include the use of the standard least mean square (LMS) algorithm for every sub-filter, via a variant of linear backpropagation, through to using a different algorithm for every layer and every sub-filter within the layer. Unless the input signal is white or the unknown channel is sparse, hierarchical filters converge to biased solutions. We make use of this property to propose a collaborative approach to the identification of sparse channels. The performances of these algorithms are evaluated for a variety of applications, including system identification and sparsity detection. The benefits and limitations of hierarchical adaptive filtering in this context are highlighted.