In this paper, we consider a blind channel estimation and equalization for single input multiple output (SIMO) channels. It is based on the one-step forward multichannel linear prediction error method. The derivation of the existing method is based on the noiseless assumption, however, we analyze the effects of additive noise at the output of the one-step forward multichannel linear prediction error filters. Moreover, we derive analytical results for the error in the blind channel estimation and equalization using linear prediction.

In this study, a fourth-order cumulants based iterative algorithm for blind channel equalization is introduced, which is robust with respect to the existence of heavy Gaussian noise in a channel and does not require the minimum phase characteristic of the channel. The transmitted signals at the receiver are over-sampled to ensure the channel described by a full-column rank matrix. It changes a single-input/single-output (SISO) finite-impulse response (FIR) channel to a single-input/multi-output (SIMO) channel. Based on the properties of the fourth-order cumulants of the over-sampled channel inputs, the iterative algorithm is derived to estimate the deconvolution matrix which makes the overall transfer matrix transparent, i.e., it can be reduced to the identity matrix by simple reordering and scaling. In simulation studies, both a closed-form and a stochastic version of the proposed algorithm are tested with three-ray multi-path channels, and their performances are compared with the methods based on conventional second-order statistics and higher-order statistics (HOS) as well. Relatively good results with fast convergence speed are achieved, even when the transmitted symbols are significantly corrupted with Gaussian noise.

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.