Modeling, Estimation and Optimal Filtration in Signal Processing

Appendix C

Subspace Decomposition Applied to Speech Enhancement

The subspace method can potentially be used in the field of speech enhancement using a single microphone. In this context, our purpose is to estimate the signal s(k) using observations disturbed by a white additive noise b(k). Instead of directly operating on the correlation matrix, an alternative procedure is to carry out the singular value decomposotion of the noisy observations' Hankel matrix. This algorithm consists of three steps:

1) First we construct the L×M Hankel matrix H_y using the noisy data as follows:

All the elements of the anti-diagonal in the Hankel matrix are equal to one another. L and M are such that L + M = N + 1. Moreover, we choose L >> M.

2) Then, the least squares estimate of the signal subspace, i.e. H_s^LS, can be obtained only by considering the K dominant singular values of the observation Hankel matrix H_y. The criterion to be considered for this step is:

where ||H||_F is the Frobenius¹ norm of matrix H.

Given that:

where U R^L×M, Σ R^M×M and V R^M×M, and where U_1,K R^L×K and V_K+1,M R^M×M–K and: ...