On the algebraic identifiability of finite impulse response channels driven by linearly precoded signals

It is common in wireless communications to perform some form of linear precoding operation on the source signal prior to transmission over a channel. Although the traditional reason for precoding is to improve the performance of the communication system, this paper draws attention to the fact that the receiver can identify the impulse response of the channel without any prior knowledge of the transmitted signal simply by solving a system of polynomial equations. Since different precoders lead to different systems of equations, this paper addresses the fundamental issue of determining which classes of linear precoders lead to a system of equations having a unique solution. In doing so, basic properties of polynomial equations which are useful for studying other identifiability issues commonly encountered in engineering and the applied sciences are presented.