A unified derivation of square-root multichannel least-squares filtering algorithms

The authors present a new derivation of exact least-squares multichannel and multidimensional adaptive algorithms, based on explicitly formulating the problem as a state-space estimation problem and then using different square-root versions of the Kalman, Chandrasekhar, and information algorithms. The amount of data to be processed here is usually significantly higher than in the single-channel case, and reducing the computational complexity of the standard multichannel RLS (recursive least square) algorithm is thus of major importance. This reduction is usually achieved by invoking the existing shift structure in the input data. For this purpose, it is shown how to apply the extended Chandrasekhar recursions, with an appropriate choice of the initial covariance matrix, to reduce the computations by an order of magnitude. In multichannel filters, the number of weights in different channels is not necessarily the same. This is illustrated with two examples: a nonlinear Volterra-series filter and a two-dimensional filter. In the former case the number of weights varies among the channels, but in the latter case all channels have the same number of weights.<>