Prewhitening for rank-deficient noise in subspace methods for noise reduction

A fundamental issue in connection with subspace methods for noise reduction is that the covariance matrix for the noise is required to have full rank in order for the prewhitening step to be defined. However, there are important cases where this requirement is not fulfilled, e.g., when the noise has narrowband characteristics or in the case of tonal noise. We extend the concept of prewhitening to include the case when the noise covariance matrix is rank deficient, using a weighted pseudoinverse and the quotient singular value decomposition, and we show how to formulate a general rank-reduction algorithm that works also for rank-deficient noise. We also demonstrate how to formulate this algorithm by means of a quotient ULV decomposition, which allows for faster computation and updating. Finally, we apply our algorithm to a problem involving a speech signal contaminated by narrowband noise.