Identification of Channels with Single and Multiple Inputs and Outputs under Linear Constraints

The channel identification problem for multiple-input multiple-output (MIMO) channels under linear constraints can be formulated as solving a linear system which involves finite-dimensional Gabor matrices and a pre-determined unstructured matrix that represents the linear constraints. While matrices of the latter type are fixed a priori by the constraints, Gabor matrices depend on the choice of their generating windows which is often chosen by the user. This is important since even if the matrix associated with the linear constraints is ill-conditioned, the full system may be solvable if the windows are designed appropriately.We prove that linear constraints consisting of a single equation always remove a single degree of freedom in the channel identification problem, in the sense that preknowledge of such constraints allows identification of MIMO channels with support size one greater than the fundamental limit. However, we give an explicit example showing that this statement does not generalize to the case of multiple constraints. In the single-input single-output (SISO) case, we provide some sufficient conditions on the linear side constraints under which the corresponding SISO channels are identifiable.