Weighted and Unweighted Subspace Fitting without Eigendecomposition

Subspace fitting has become a well known method to identify FIR Single Input Multiple Output (SIMO) systems, only resorting to second-order statistics. The main drawback of this method is its computational cost, due to the eigendecomposition of the sample covariance matrix. We propose a scheme that solves the subspace fitting problem without using the eigendecomposition of the cited matrix. The approach is based on the observation that the signal subspace is also the column space of the noise-free covariance matrix. We suggest a two-step procedure. In the first step, the column space is generated by arbitrary combinations of the columns. In the second step, this column space estimate is refined by optimally combining the columns using the channel estimate resulting from the first step. Using recent results on Weighted Subspace Fitting, we are able to incorporate the optimal weighting in the second step. For the unweighted signal subspace fitting, our method only requires computation of two eigenvectors of a small matrix and of two projection matrices, although yielding the same performance as the usual subspace fitting. Furthermore, a detailed analysis of the displacement structure of the various matrices involved leads to a fast algorithm. Furthermore, detailing the expressions of the criteria leads us to draw equivalences between various blind identifications schemes, namely SRM (Subchannel Response Matching), DML (Deterministic Maximum Likelihood) and linear prediction approaches.