Digital Signal Processing Reference
In-Depth Information
of
P
results in information loss, while a big value leads to a matrix not
absolutely noise free.
3.
F-NORM CONSTRAINED CRITERION
In order to obtain the appropriate order for the best reconstruction, an F-
norm constrained function used in an analysis-by-synthesis procedure is
proposed. Let
represent the estimated matrix of
from the first
s
singular values of
i.e.
where is the
sxs
matrix filter as described in (5). We define an F-
norm constrained function (FCF), denoted by
as follows:
It can be proved that the FCF is a monotonically decreasing function with
increasing value of the variable
s.
According to the F-norm definition and
the assumptions, we have
Obviously, we also have,
due to the use of a new singular value in the reconstruction process of the
estimated Hankel matrix with additional power. From (8) and (9), it is clear
that the FCF is a monotonically decreasing function.
Equation (8) states
that the F-norm of the reconstructed
with
s <P,
is less than that of
coming from the clean signal. This is because
