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(5) |
Now, the task here is to approximate the function by some polynomial
that can closely approximate it within a given range, with its low order
as far as possible. The polynomial approximation can be simplified by
reducing function
to univariate one, by taking
, where
.
We choose here 2nd-order Lagrange polynomial to approximate the function
, as given by:
The points ,
and
can be specified manually (one point
at
and other two chosen to minimize error in the required range),
or instead, Chebyshev-Lagrange polynomial can be used that specifies the
points itself.
Finally, Eq.(6) reduces to
form, where
,
and
are constants. Therefore:
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(7) |
The accurate value of mean is more important than that of
variance. Therefore, the covariance matrix can be retained as it is for
the clean speech. However, expression for adapting diagonal variance
can be derived from above approximation, in terms of higher-order
moments (up to 4th moment) of , and diagonal variances can be adapted
as well.