# metric for signal to noise ratio in communication systems

I'm not quite sure about how to define a good measure of the quality of a communication channel with fading and interference. Let us assume the simplest case, where a node in a network receives the following quantity:
$$y = s + w$$
where $s$ is the amplitude (positive real) of the signal and $w$ the noise (gaussian with zero mean and variance $\sigma^2$). A simple measure of the channel quality is simply given by ${\rm SNR} = s^2 / \sigma^2$, i.e., the ratio of the power of the signal vs the power of the noise.
Now, suppose the received signal is $$y = h_s s + w$$
where $h_s$ and $w$ are complex gaussian with zero mean and unitary variance. A measure of the SNR that makes sense could be derived by looking at the power of the received signal $$|y|^2 = y y' = |h_s|^2 s^2 + 2{\rm Re}(h_ss~w') + |w|^2$$
now, since for practical applications the noise process is much faster than the fading (represented by $h_s$) what people normally do is to average over $w$, obtaining
$$E[|y|^2]_w = |h_s|^2 s^2 + \sigma^2$$
hence, a measure of the signal strength that makes sense would be ${\rm SNR} = |h_s|^2 s^2 / \sigma^2$.
What I am interested about is the case with interference. Suppose there is an extra term that takes into account for interfering signals
$$y = h_s s + h_{\rm I} s_{\rm I} + w$$
where $h_I$ is complex with zero mean and unitary variance. How could I possibly define the SNR in this case? By taking $|y|^2$ there are cross terms, i.e., products of $h_s$ and $h_{\rm I}$ that do not disappear even if I average over the noise $w$. So it seems not obvious to me how to decouple $|y|^2$ into two different pieces, one for the signal and the other for the noise. The final goal would be determining a good measure of the SNR in order to be able to compute probabilities like

$$P({\rm SNR} \geq \gamma_0).$$

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if one can assume that $h_I$ is uncorrelated with $w$ and $h_s$, then you could just average $∣y∣^2$ over both $h_I$ and $w$; cross terms would still vanish; this is not what you want? –  Carlo Beenakker Aug 16 '11 at 13:47
yes I know, but typically, the $h$ terms vary more slowly than the noise $w$. So averaging over $h$ is not that practical. –  Bob Aug 16 '11 at 22:37

The general case is not known. If you find the correct information theoretic metric for the interference channel, you can find the capacity of interference channels (an open problem since the 1960s).

In your case, it looks like a MAC channel. Decode the stronger signal first and then decode the weaker one. Assuming you have a powerful code underneath, Euclidean distance as metric suffices.

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