Digital Signal Processing Reference
In-Depth Information
does not contain all the harmonic information of the clean signal
x.
On the other hand, when
s > P,
the F-norm of the reconstructed is
larger than that of due to noise. This implies that crosses zero at
s = P.
However, the non-correlation assumption is not a true condition in
practice, especially in the case of short data records. Thus a small bias may
exist in the values of This slight deviation can be solved by
investigating the change of the difference values of the FCF as follows.
Based on the definition of
we have
Note that consists of only the noise component when
This implies that the difference value of the FCF converges to 0, i.e.
In true condition, the value of
converges to a small value. Thus, we can use a threshold value If the
value of is initially less than with the increase of
s,
we
can then obtain the wanted order of retained singular values as follows:
Note that the selected
P
by F-norm constrained criterion leads to the
best approximation of original speech in terms of signal-to-noise ratio
(SNR). Let be a frame of the reconstructed signal and it consists of the
original signal
X
and error signal i.e. Assuming the means
of
X,
and are zero, and that the error signal is statistically
independent of the clean signal
X,
the SNR of the reconstructed signal is
then given as follows:
where
E
is the statistical expectation operator. Next, we will prove that the
SNR
becomes maximum when the FCF reaches the point
s = P.
Let
