This relates to my previous question, which I think I have asked in the wrong way:
I wasn't really interested in detectability of the signal, and I have phrased that question very ambiguously, so let me ask what I would really like to know.
Question:
What I would really like to know is that is it possible to establish a communication channel (sending information) if the received power level of the signal, received by the receiver antenna is below the noise floor.
Let me explain:
I did more research on this and the power level is usually expressed in dBm or dBW. In this question I will be expressing it in dBW.
Then we have the power inserted into the transmitter antenna, and we have the path loss equation to determine how much of that is attenuated by the time the signal reaches the receiver antenna.
So we have two dBW values, and my theory is that the power received by the antenna in dBW has to be higher than the noise floor in dBW.
1)
For the sake of this argument let's use a transmitter/receiver antenna 20 cm long, at 5 Ghz frequency at 1 meter from each other. Again I am using the maximum gain fundamentally possible, because I am also looking whether the communication channel can be established at all, so I have to insert the most extreme values in order to determine the fundamental limit. In this case both antennas have a gain of 16.219 dB which is the maximum gain they can have at this frequency, and by maximum I mean a gain higher than this would violate the laws of energy conservation. So these antennas are in theory perfect lossless antennas. This is a far field equation so for simplicity I choose to use the Friis formula.
So the path loss equation reveals that this communication channel has a ~ -14 dB path loss. So if we are inserting 1 Watt of power, the receiver antenna should receive no more than -14dBW.
2)
I've stumbled across a paper:
It claims the minimum sensitivity for a receiver antenna is this:
$$ S_{min} = 10* \log_{10}( (S/N)*k*T_0*f*N_f ) $$
$$where$$
-
S/N= Signal to noise rate
-
k = Boltzmann constant
-
T0 = Temperature of the receiver antenna
-
f = frequency
-
Nf= noise factor of the antenna
And this is also a dBW unit. This formula would describe the noise floor at that frequency.
Going back to our calculation, the paper recommends, in best case scenario, when a skilled manual operator is involved a 3 dB S/N ratio (max), we will use 290 Kelvin for room temperature, the frequency 5 Ghz as above, and the noise factor I will ignore since we assumed a perfect antenna earlier.
This would give us -104 dBW noise floor.
Therefore since the received power level is -14 dBW and the noise floor is considerably lower at -104 dBW, and this assumes a best case scenario with generous estimates, as in the best case scenario.
So in this example, communication is possible, very much. However if the received power level would be lower than the noise floor, then it would not be.
So my hypothesis is that if:
Power Received > Noise Floor , then communication is possible, otherwise it's not
Since the power received is way higher than the noise received, it means that communication at this frequency is theoretically possible.
Practically speaking of course issues could arise as the gain would be lower, and the antenna operator would receive too many false positives at such strict S/N rate (3 db), so in reality the noise floor would probably be 50-60 dB higher. I haven't calculated that.
Best Answer
Short answer: yes, possible. GPS does that (nearly) all of the time.
Long answer:
The SNR your receiver system needs depends on the type of signal you're considering. For example, good old analog color TV needs, depending on standard, some 40 dB SNR to be "viewable".
Now, any receiver is, mathematically, an estimator. An estimator is a function that maps an observation that usually includes a random variable to a underlying value that led to the observed quantity. So, that TV receiver is an estimator for the picture that the station meant to send. The performance of that estimator is basically, how "closely" you can get back to the original information that was transmitted. "Closely" is a term that needs definition – in the analog TV sense, one receiver might be a really good estimator in terms of variance (from the "real" value) of the image brightness, but terrible for color. Another one might be so-and-so for both aspects. The quality of a receiver thus depends on what you need to optimize.
For radar, things are a bit clearer. You use radar to detect only a very limited set of things; amongst these, we can pick out a few from the following things, which we can simply represent as real numbers:
If you restrict yourself to one thing, let's say range, then your radar estimator can get something like a "range variance over SNR" curve.
Just a quick reminder: Variance of an estimator \$R\$ is defined as the expectation value of
$$\text{Var}(R) = E{(R-\mu)^2}$$
with \$\mu\$ being the expectation value of the "actual" phenomenon (in this case, the actual distance, assuming we've got an unbiased estimator).
So, one person might say "OK, it's not really a usable estimate for distance of cars unless range variance drops below 20 m², so we need at least an SNR of \$x\$ so that we get a variance below \$y\$", whilst another person, who might be detecting a different kind of thing (let's say planets), can live with a much much higher variance, and thus, much much lower SNR. Including SNR where noise is much stronger than signal.
For many things, your combined observation's variance gets better (==lower) the more observations you combine – and combination is a very common way of getting what we call processing gain, ie. an improve of estimator performance equal to improving the SNR by a specific factor.
To come back to my GPS example:
GPS uses a ca 1MHz bandwidth to transmit signals spread out in time – the actual GPS symbol rate is much much lower than the bandwidth. This happens by multiplying a single transmit symbol \$s\$ with a long, long sequence of numbers \$l[n],\,\, n\in[0,1,\ldots,N]\$ that then gets transmitted. In the receiver, you correlate with the same sequence, and sum things up – through linear algebra, noise (which we model as uncorrelated to any signal) doesn't add up constructively, while the energy in the send sequence times receive sequence grows with \$N\$. That's how GPS cannot even be seen in a spectrum plot, but easily received by extremely cheap receivers with inefficient antennas, noise amplifiers, ridiculously low-resolution ADCs and without anyone having to point a large high-gain antenna in the directions of satellites.
Thus, your hypothesis
doesn't stand. "Possible" or "impossible" depends on the error you're willing to accept (and that can be quite a lot!), and even more so on the processing gain between where you look at the receive power–to–noise ratio and the actual estimate.
So, your core question:
Yes, very much so. Global localization systems depend on it, and cellular IoT networks will, probably, too, as transmit power is very expensive to those.
Ultra-Wideband (UWB) is kind of a dead idea in communication designs (mainly due to regulatory problems), but those devices hide e.g. a forwarded USB communication far below the detectable spectral power density level. The fact that radioastronomers are able to tell us about far away stars also backs this.
Same applies to the radar satellite images that are produced using lower-earth-orbit satellites. You'll hardly be able to detect the radar waveformes with which they illuminate the earth – and they're even weaker when their reflection reaches the satellite again. Still, these waves carry information (and that's the same as communicating) about structures much smaller than 1m on earth, at high rates (getting the actual earth shape / property estimates stored or sent back to earth is a very serious problem for these satellites – there's just so much info transferred with signals that are far, far below thermal noise).
So, if you need to remember only two things about this: