Maximum-likelihood binary shift-register synthesis from noisy observations

We consider the problem of estimating the feedback coefficients of a linear feedback shift register based on noisy observations. The problem of determining feedback coefficients in the absence of noise is now classical (Massey's (1969) algorithm). In the current approach to the problem of estimation with noisy observations, the coefficients are endowed with a probabilistic model. Gradient ascent updates to coefficient probabilities are computed using recursions developed by means of the expectation-maximization (EM) algorithm. Reduced-complexity approximations are also developed by reducing the number forward probability terms propagated at each stage. While suffering from a local-maximum problem typical of many maximum-likelihood (ML) procedures, the method does exhibit convergence.