Capacity Performance of Tensor Multi-Domain Communication Systems With Discrete Signalling Constellations

Abstract

Modern communication systems employ multi-domain modulation and coding techniques for effectively exploiting all available resources. Hence in such systems the transmit and receive signals have an inherent multi-domain structure which can be represented using tensors. This work considers the capacity of higher order tensor channels associated with such multi-domain communication systems when the elements of the input tensor are constrained to be drawn from discrete signalling constellations. We establish a relationship between the tensor gradient of the mutual information and the error covariance tensor associated with the minimum mean squared error estimator at the receiver. This relation is used to iteratively find a multi-linear precoder at the input which achieves capacity of the tensor channel under the signalling constellation constraints. Through numerical examples, we show the convergence behavior of the proposed precoder, and compare the capacity achieved under different constellations with the capacity when the input is Gaussian. Further, we exploit the tensor formulation of the problem to find the channel capacity under a variety of different power constraints spanning across several domains. At high SNR, the constellation constraints saturate the capacity while at low SNRs, the constellation constraints are not too relevant, and the power constraints dominate and limit the performance. The capacity saturation level depends on the input order and distribution.

Keywords

• Tensor channel
• capacity
• I-MMSE relation
• MMSE tensor estimation
• multi-linear precoder