Matrix and tensor decompositions for identification of block-structured nonlinear channels in digital transmission systems

In this paper, we consider the problem of identification of nonlinear communication channels using input-output measurements. The nonlinear channel is structured as a LTI-ZMNL-LTI one, i.e. a zero-memory nonlinearity (ZMNL) sandwiched between two linear time-invariant (LTI) subchannels. Considering Volterra kernels of order higher than two as tensors, we show that such a kernel associated with a LTI-ZMNL-LTI admits a PARAFAC decomposition with matrix factors in Toeplitz form. From a third-order Volterra kernel, we show that the PARAFAC decomposition allows estimating directly the linear subchannels. In the case of a LTI-ZMNL channel, such a task is achieved by considering an eigenvalue decomposition of a given slice of such a tensor. Then, the nonlinear subsystem is estimated in the least squares sense. The proposed identification method is illustrated by means of simulation results.