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On blind channel identification and equalization over Galois fields

We consider the problem of blind identification and equalization of a Linear, Time-Invariant (LTI) system, where the input and output signals, as well as the linear operations, all reside in a finite (Galois) field. We point out some fundamental differences from the classical version of this problem. We show that if the input process is a sequence of independent, identically distributed random variables, the system is identifiable if and only if the (marginal) distribution of the input is non-uniform. For an autoregressive (AR) channel a finite impulse response equalizer can be found by minimizing the marginal entropy of its output signal. However, an exhaustive search for the minimizing equalizer, although theoretically possible, is not necessary: Based on somewhat surprising properties of the AR channel's output (not shared by the classical case), we show that the equalizer can be found directly from the empirical characteristic tensor of this output. We demonstrate the success rate of the proposed methods in simulation.