SPEECH ENHANCEMENT BASED ON F-NORM CONSTRAINED TRUNCATED SVD ALGORITHM - DSP for In-Vehicle and Mobile Systems - page 174

according to the F-norm definition, we have

As X is a

wide-sense stationary stochastic process in the short-time period [0, L - 1],
the autocorrelation of X, does not depend on the placement of the time

origin, i.e., Hence, it is clear that

and that

Similarly, we also have

Correspondingly, it can be shown that

Meanwhile, we also know from the above discussion that

attains its minimum at s = P, and consequently the SNR

of the reconstructed signal attains its maximum.

4. THE FCTSVD ALGORITHM

Based on the above discussion, we can formulate the FCTSVD algorithm
for the case of white noise in the following steps.
(1). Estimate noise vectors W from silence periods in the observed

speech signal.
Form the Hankel matrix

(2).
(3).
(4).
(5).

from the observed signal.

Compute the SVD of
Initialize the order of retained singular values of i.e., s = 0.
Let s = s + 1 and reconstruct the estimated matrix of
using the first s singular values of
Compute the FCF. If s = 1 then return to Step 5 . (6).