Tensor Recursive Least Squares Filters for Multichannel Interrelational Signals

It is well known that recursive least squares (RLS) filter is a prevalent adaptive filter that recursively updates the filter-coefficients to minimize a weighted cost function related to an input signal. Conventionally, such an input signal is a sequence of scaler numbers. Multichannel signals generated from one or more sources are often encountered nowadays. Therefore, there emerges a urgent demand to generalize the conventional RLS filter to accommodate multichannel signals with implicit interrelations. The key issue of this new generalization is to convert the correlation matrix involved in the conventional RLS filter for one-dimensional input to the correlation tensor required by the new RLS filter for multi-dimensional input. Due to lack of the sufficient mathematical framework of the required tensor operations, here we make the first attempt to established the new tensor recursive least squares filter (TRLS) theory based on our newly derived Woodbury tensor identity. We apply the Woodbury tensor identity to design a recursive algorithm for updating the weight-tensor. Furthermore, the performance analysis of the proposed new TRLS algorithm, including mean-squared deviation and misadjustment, are facilitated using the established tensor calculus theory. Numerical experiments are also presented to investigate the convergence speed, mean-squared deviation, and misadjustment for input signals with various statistical characteristics and different dimensions.