Blind deconvolution of discrete-valued signals

The paper shows that when the input signal to a linear system is discrete-valued the blind deconvolution problem of simultaneously estimating the system and recovering the input can be solved more efficiently by taking into account the discreteness of the input signal. Two situations are considered. One deals with noiseless data by an inverse-filtering procedure which minimizes a cost function that measures the discreteness of the output of an inverse filter. For noisy data, observed from FIR systems, the Gibbs sampling approach is employed to simulate the posteriors of the unknowns under the assumption that the input signal is a Markov chain. It is shown that in the noiseless case the method leads to a highly efficient estimator for parametric systems so that the estimation error decays exponentially as the sample size grows. The Gibbs sampling approach also provides rather precise results for noisy data, even if the initial and transition probabilities of the input signal and the variance of the noise are completely unknown