Least Squares Criterion Adaptive Filtering Algorithms as Tools for Calibration of Arrays with Digital Beamforming

Abstract

This paper explores the application of the adaptive filtering algorithms, based on Least Squares (LS) criterion for the calibration of the receiving arrays with digital beamforming. The digital beamforming assumes the access to the signals in the array channels as a requirement, which has to be provided if the adaptive signal processing is used. In this case, the adaptive filtering algorithms, based on the minimization of the LS error between the desired and array output signals, are able to estimate the channel parameter spread, which values can be stored and used for correction of the array weights during the beam steering. If a set of steered beam positions is known, for example in scanning arrays, the proposed algorithm allows to create a codebook of the array weights, which ensures the random variation of channel’s parameters compensation and correct beam steering. The architecture of the proposed solution and mathematical description of a Recursive Least Squares (RLS) adaptive filtering algorithm, on which the calibration is based on, are presented. The algorithm uses a dynamic regularization of the array input signal correlation matrix. No specific correlation properties are required for the algorithm training signal. The simulation results demonstrate the ability of the calibration algorithm to compensate the channel’s parameters spread. If the array channel Signal-to-Noise Ratio (SNR) is 0 … 30 dB, then the achieved spread of the calibrated channels gain errors is about –20 dB. The norm of the ideal and calibrated array difference is also about –20 dB. The results are obtained with the Phase Shift Keying (PSK) and Quadrature Amplitude Modulated (QAM) training signals. The ability of the proposed calibration algorithm to operate in interference conditions is also demonstrated