The joint estimation of two unknowns, i.e. system and input sequence, is overviewed in two methodologies of equalization and identification. Statistical approaches such as optimizing the ensamble average of the cost function at the equalizer output have been widely researched. One is based on the principle of distribution matching that total system must be transparent when the equalizer output has the same distribution as the transmitted sequence. Several generalizations for the cost function to measure mis-matching between distributions have been proposed. The other approach applies the higher order statistics like polyspectrum or cumulant, which possesses the entire information of the system. For example, the total response can be evaluated by the polyspectrum measured at equalizer output, and by zero-forcing both side of the response tail the time dependency in the equalizer output can be eliminated. This is based on the second principle that IID simultaneously at input and at output requires a tranparent system. The recent progress of digital mobile communication gives an incentive to a new approach in the Viterbi algorithm. The Viterbi algorithm coupled with the blind channel identification can be established under a finite alphabet of the transmitted symbols. In the blind algorithm, length of the candidate sequence, which decides the number of trellis states, should be defined as long enough to estimate the current channel response. The channel impairments in mobile communication, null spectrum and rapid time-variance, are solved by fast estimation techniques, for example by Kalman filters or by direct solving the short time least squared error equations. The question of what algorithm has the fastest tracking ability is discussed from algebraic view points.