Digital Signal Processing Reference
In-Depth Information
topic. During the past three decades, many single channel speech
enhancement algorithms have been proposed. More recent published
algorithms include variants of spectral subtraction [1] and amplitude
estimation methods [2], methods based on all-pole modeling [3],
enhancement using discrete cosine transformation (DCT)[4] or two-
dimensional Fourier transform [5], schemes based on constructive-
destructive additive noise[6], and signal subspace methods [7]. In this paper,
we propose a new algorithm, called F-norm constrained truncated SVD
(FCTSVD) algorithm, to solve the problem of how to automatically choose
the appropriate order of retained singular values in an SVD scheme.
2.
THE SVD AND SIGNAL SUBSPACE
ESTIMATION
Let be an observed noisy signal vector with
L
samples and we assume that the noise is additive and uncorrelated with
the signal, i.e.
Y = X + W ,
where
X =
[
x
(0),
x
(1)
, ยทยทยท x
(
L-
1)] and
W
represent the original and noise signal, respectively. We can construct the
following MxN
M
Hankel matrix
from
Y
as done in [7], where
L = M + N -
1
and
M > N.
Correspondingly,
can also be written as
where
and
represent the Hankel matrices derived from
X
and
W ,
respectively.
According to the SVD theory, there exist orthogonal matrices
and
such that
where
with
The
nonnegative diagonal elements of are called singular values of
Usually, it is convenient to partition the SVD of
using the first
P
singular values of
as follows:
