@ARTICLE\{IMM2005-03093,
    author       = "P. C. Hansen and S. H. Jensen",
    title        = "Prewhitening for Rank-Deficient Noise in Subspace Methods for Noise Reduction",
    year         = "2005",
    pages        = "3718-3726",
    journal      = "{IEEE} Transactions on Signal Processing",
    volume       = "53",
    editor       = "",
    number       = "10/1",
    publisher    = "",
    url          = "http://www2.compute.dtu.dk/pubdb/pubs/3093-full.html",
    abstract     = "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 narrow-band
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 {SVD,} 
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 narrow-band noise."
}