On the dimension of the Krylov subspace in low complexity wireless communications linear receivers

In this paper, we analyse the dimension of the Krylov subspace obtained in Krylov solvers applied to signal detection in low complexity communication receivers. These receivers are based on the Wiener filter as a pre-processing step for signal detection, requiring the computation of a matrix inverse, which is computationally demanding for large systems. When applying Krylov solvers to the computation of the Wiener filter, a prescribed number of iterations is applied to solve the associated linear system. This allows for obtaining a reduced number of floating point operations. We base our analysis on relating the Krylov subspace with the eigenvalues and the eigenvectors of the received covariance matrix, and on the the cross-covariance matrix between the received and the transmitted signals. In our analysis, we show that the particular structure of communication systems can yield a unique Krylov subspace for several right hand sides. Based on the latter, we further extend our findings by solving the multivariate version of the Wiener filter utilising Galerking projections.