A Subspace Approach for Extracting Signals Highly Corrupted by Colored Noise

Estimating a signal which is buried inside colored noise is challenging since significant amount of the noise frequencies with considerable or higher power (signal-to-noise ratio, SNR, being less than 0 dB) reside in the same band as that of the desired waveform. An optimization and eigen-decomposition-based subspace approach has been investigated and tested to estimate signals which are highly corrupted by colored noise; Hu and Loizou [Y. Hu and P. C. Loizou, “A Generalized Subspace Approach for Enhancing Speech Corrupted by Colored Noise,” IEEE Transactions on Speech and Audio Processing, vol. 11, no. 4, pp. 334-341, July 2003] introduced a non-symmetric basis matrix to be eigen-decomposed into its corresponding eigenvalue and eigenvector matrices; the generated eigenvector matrix is supposed to simultaneously diagonalize both the clean speech and noise covariance matrices. They also reported that the utilization of the eigenvector and eigenvalue matrices in the time-domain constrained estimator would result in the optimal estimation of speech corrupted by colored noise. Here we critically examine these matrices and contend that the presented eigen-based equations are mathematically incorrect. The eigenvectors of the proposed basis matrix produce perfectly diagonal eigenvalues for the noise covariance matrix; however, the generated eigenvalues are not the degenerate identity matrix as claimed by the authors. An alternative solution by means of a modified gain matrix is proposed to rectify the mathematical inconsistencies. For validation purposes, the pre- and post-modified algorithms have been assessed in their abilities to extract visual evoked potentials (VEPs) that are corrupted by colored electroencephalogram (EEG) noise. SNR values can be as low as -10 dB in real clinical environments. The simulation results produced by the post-modified SSA2 algorithm, show a higher degree of consistencies in detecting the VEP's P100, P200, and P300 peaks, in comparisons to the pre-modified SSA1 method. Moreover, the results of the real patient data confirm the superiority of SSA2 over SSA1 in estimating VEP's P100 latencies, which are used by doctors to assess the conduction of electrical signals from the subjects' retinas to the visual cortex parts of their brains.