Blind multi-channel system identification and deconvolution: performance bounds

We consider the problem of estimating the parameters of an unknown multi-input multi-output linear system, and the related problem of deconvolving and recovering its inputs, using only observations of the system outputs. We derive simple closed-form asymptotic expressions for the Cramer-Rao lower bound (CRLB) for the system parameters, as well as lower bounds on the signal reconstruction performance. These show that the identification/deconvolution performance depend on the accuracy with which the scale and the location parameters of the input probability density-functions can be identified from observation of the input signals. It is also shown that the CRLB possesses a block diagonal structure, indicating that the general multichannel deconvolution problem is decoupled into two independent simpler sub-problems. The signal separation problem where the unknown system is deconvolved to a diagonal one, and the remaining independent single-channel-deconvolution problems associated with the equalization of each of its diagonal elements.