# How to define SNR for multiple signals in noise?

I have to estimate the number of signals present in a measurement contaminated by additive noise given $n$-dimensional snapshot vectors $\bf x$, modeled as $\bf x = \bf A \bf s + \bf z$ where $\bf s$ is a $k \times 1$ vector representing $k$ different signals. $\bf A$ is an $n \times k$ non-random matrix and $\bf z$ is an $n \times 1$ noise vector. (This model is common in array processing problems.)

How should I define the SNR—per signal, averaged or otherwise?

• What algorithm are using to estimate $k$? – Emre Apr 15 '12 at 1:19
• I'm looking at AIC, MDL based estimators developed by Wax-Kailath. – sauravrt Apr 15 '12 at 2:50
• In that case I assume you've read this paper. Why don't you use the same definition as them and papers that cite them do? – Emre Apr 15 '12 at 7:47
• @Emre I hadn't read that paper by Kundu. Thanks for pointing it out. I read Wax-Kailath and specifically I'm interested in this paper by Rao-Edelman. Here the authors define a term Eigen-SNR for which I didn't find a clear definition in the papers. – sauravrt Apr 15 '12 at 10:29
• @Emre I skimmed through Kundu's paper. On page 64 he computes SNR for different values of $\sigma$. I couldn't figure out how he computed them. Could you explain them to me? I assume the total signal power is $Tr(\Psi)$. – sauravrt Apr 15 '12 at 14:35

The measurement that is important to you will depend on the application. If you are looking for an over all measurement of all signal power to noise power then you define signal to be the power in all signal bands and the noise all of the powers in the noise bands.

However, if you are trying to use an sub-band adaptive filter to correct for some sort of distortion then the SNR in the particular band that you care about would be important not an overall SNR.

In the same application both measurements could be important at different stages of the system. If you are working in wireless communications and the frequency range that is being digitized contains multiple signals then the over all SNR into the receiver needs to be high enough to limit the noise introduced by amplifiers before digitization. But if after I digitize the band it is then split into multiple sub sections for demodulating the signals I would only care about the SNR of the signal currently being demodulated.

For your application it sounds like an over all measure is important because you are treating the range of frequencies as a whole.

In the application of mismatch correction algorithms for time-interleaved ADCs (my topic) we sometimes use multiple sinusoids input to the interleaved converters to measure the performance increase the correction algorithms. This makes it easier to calculate and visualize things like SNR and SFDR when the desired signal locations are known.

I hope this helps,

Charna

• So in my case of estimating number of signals present in the measurement, all of the arriving signals are equally important. The SNR should be like $\sum s_i^2 / \sigma_n^2$ ? – sauravrt Apr 15 '12 at 2:59
• Yes, I believe so. You may want to try as Jim mentioned: try to isolate one signal at a time, determine that it is a valid signal before you count it and then move to the next determine it is valid etc. – Charna Apr 16 '12 at 21:31

As Charna said, it depends on what you are trying to do, but usually when you are interested in multiple signals you have to deal with those signals individually.

For instance, if I were creating a W-CDMA phone I would always have to deal with multiple signals at the same frequencies in the presence of noise. It does not matter, in this situation, what the total signal power is. To recover any useful information I need to recover individual signals, so I need to have each individual SNR be high enough to demodulate the signal. In fact, when dealing with each signal the other signals act as interferers, which is why they use SINR to measure how recoverable the signal is, not SNR.

Anyway, long story short, it depends on what you are trying to do, but normally you only care about the SNR for individual signals, and in that case the other signals might even be interferers.

• I understand that when dealing with individual signals treating others at inteferers, SNR per signal is useful. But in my case when I'm trying to estimate the total number of signals ( or sources ) present in the measurement, each one is equally important for me. As charna mentioned, in that case it will have to be overall signal power against nosie power. – sauravrt Apr 15 '12 at 2:56