# Variance of zero-mean signal after convolution for SIR computation

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

• is there any relation between $x$ and $s$? Commented Dec 16, 2019 at 12:42
• edited, sorry... notation overlap Commented Dec 16, 2019 at 12:57
• Have you checked this answer Commented Dec 16, 2019 at 13:08
• You have to let us know where exactly you're stuck, because this is a classic homework assignment, and we don't provide answers to homework type question unless the OP shows some efforts of their own. Commented Dec 16, 2019 at 13:26