It may seem that spectrum subtraction noise reduction is only based on an engineering intuition. However, this engineering intuition has a direct mathematics connection to the theoretically optimal Winer filter approach. The basic spectrum subtraction in frequency domain, $\left|\hat{S}\left(f\right)\right|^2=\left|X\left(f\right)\right|^2-\left|N\left(f\right)\right|^2$   (1)

if N(f) is available or can be estimated.

We can write is as the following, $\left|\hat{S}\left(f\right)\right|^2=\left|X\left(f\right)\right|^2-\left|N\left(f\right)\right|^2$   (2) $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\frac{\left|X\left(f\right)\right|^2-\left|N\left(f\right)\right|^2}{\left|X\left(f\right)\right|^2}\left|X\left(f\right)\right|^2\$   (3)

and we can recognize immediately the spectrum subtraction approach is actually equivalent to the Wiener filtering solution. Therefore, spectrum subtraction is mathematically optimum in the same sense as Wiener filtering! $H(f)=\frac{\left|X\left(f\right)\right|^2-\left|N\left(f\right)\right|^2}{\left|X\left(f\right)\right|^2}$