Matrix Approximation Based Design of a Combined Analog and Digital Beamformer in Frequency-Selective Fading Channels

In this letter, a combined analog and digital beamforming system is considered in frequency-selective fading channels. Analog beamforming matrices (mapping between RF chains and antennas) should be constant over frequency, while digital beamforming matrices (mapping between spatial data streams and RF chains) can be adapted to frequency. More RF chains than the number of spatial data streams are employed to enable a combined analog and digital beamformer to be adapted to frequency. A novel method of designing analog beamforming matrices is proposed by using the Eckart-Young-Mirsky matrix approximation theorem. The proposed method is optimal in a sense that for a given number of RF chains, the weighted sum of the squared distances between the combined analog and digital beamforming vectors (comprising a beamforming matrix) derived by the proposed method and the ideal beamforming vectors (singular vectors of the channel matrix at each frequency) is minimized, where the weights are the singular values corresponding to the singular vectors. The proposed beamformer is shown to significantly outperform a conventional one.