my goal is to scale desired, interfering signal at the receiver in order to achieve desired SIR (signal to interference ratio) for beamforming (source separation) application.
Let be:
- $s(t)$ a known speech signal with zero mean $\mu_s = 0$ and known variance $\sigma^2_s$.
- $q(t)$ a known speech signal with variance $\sigma^2_q$.
- $h_s(t)$ and $h_q(t)$ two known deterministic acoustic impulse responses.
- $y(t) = (h_s \ast s)(t) + (h_q \ast q)(t)$ the reverberant speech signal.
Suppose we have access only to the premix and the mix, that means to $x_s(t) = (h_s \ast s)(t)$, $x_q(t) = (h_q \ast q)(t)$ and $y(t)$.
To achieve desired SIR at the receiver, I can simply make $x_s$ of unit variance, and scale $x_q$ and then mix again the two quantity accordingly. However now, how do I scale $s(t)$ and $q(t)$ accordingly.
I think a more general approach is then, given the deterministic LTI filter $h(t)$, what is the variance of $y(t) = (h \ast s)(t)$?
$$ \sigma^2_y = \mathbb{E}[ (y - \mathbb{E}[y])^2 ] = \mathbb{E}[(h \ast s - \mathbb{E}[h \ast s])^2]$$
Since $s$ is zero-mean periodic speech signal and $h$ is a deterministic filter, then $\mathbb{E}[h \ast s] = 0$.
It follows that $$ \sigma^2_y = \mathbb{E}[ (h \ast s)^2 ]$$ Using the Parceval Theorem and the convolution theorem, I can write $$ \sigma^2_y = \mathbb{E}[ (H X)^2 ]$$
However if I write everything directly in the frequency domain, as $$\mathcal{P}(Y) = \mathbb{E}[ H^2 X^2 ] = | H |^2 \mathbb{E}[X^2] = | H |^2 \mathcal{P}(X)$$ where $H$ and $X$ are the DFT of $h$ and $s$ respectively, while $\mathcal{P}(\cdot)$ is the PSD of operator.
And here I am not sure how to continue, since the PSD is defined over frequencies, while the variance of the signal is a scalar.
Thanks
