Polynomial Approximation

The goal is to find the distribution (mean and variance) of noisy speech parameter $y$, given the distributions for noise parameters $n$ and $h$, and clean speech parameter $x$. First, mean, i.e. expectation value, of $y$ is given as:

$$E(y) = E(x)+E(h)+E[ln(1+e^{n-x-h})]$$

$$= E(x)+E(h)+E[g(x,n,h)]$$ (5)

Provided $x$, $n$ and $h$ have Gaussian distributions, $E[g(x,n,h)]$ does not have any closed-form expressions.

Now, the task here is to approximate the function $g(x,n,h)$ 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 $g(x,n,h)$ to univariate one, by taking $z=n-x-h$, where $z\sim\mathcal{N}(\mu_{n}-\mu_{x}-\mu_{h}, \sigma_{n}^{2}+\sigma_{x}^2+\sigma_{h}^2)$.

We choose here 2nd-order Lagrange polynomial to approximate the function $g(z)$, as given by:

$$g(z)\approx P_2(z)=\sum_{k=0}^2 g(z_k) \prod_{i=0, i\neq k}^2 \frac{(z-z_i)}{(z_k-z_i)}$$ (6)

The points $z_0$, $z_1$ and $z_2$ can be specified manually (one point at $z=\mu_z$ 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 $g(z)=az^2+bz+c$ form, where $a$, $b$ and $c$ are constants. Therefore:

$$E[g(z)]=a(\sigma_{z}^2+\mu_{z}^2)+b\mu_z+c$$ (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 $z$, and diagonal variances can be adapted as well.