FIR adaptive filters based on the Hirschman optimal transform

In this paper, we derive a "convolution theorem" suitable for the Hirschman optimal transform (HOT), a unitary transform derived from a discrete-time, discrete-frequency version of the entropy-based uncertainty measure first described by Hirschman. We use the result to develop transform domain adaptive filters. First, we show how our method can be used to implement a fast block-LMS adaptive filter that we call the HOT block-LMS adaptive filter. This filter requires slightly less than half of the computations that are required in an FFT-based block-LMS adaptive filter, We also develop another transform-based adaptive filter algorithm that uses a sliding window instead of a block of data. The HOT version of these sliding algorithms is also significantly computationally more efficient (by root N, where N is the filter order) than the sliding DFT version. Because our work is at an early stage, we develop simulations that explore basic convergence characteristics.