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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 pathloss 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 eachother. 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 farfield equation so for simplicity I choose this, the Friis formula can be used.

So the pathloss equation reveals that this communication channel has a ~ -14 dB pathloss. 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 sensivity 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.


Therefor 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 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.

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    \$\begingroup\$ I'm amazed that no one says anything about it, but yes you can by using coded bits. In other words, instead of sending 8 bits that you want to send, you use some other longer sequence that translates to those 8 bits. And the sequence you choose is not just any sequence, it utilizes the Hamming distance.. Just click the video-link if you want to "read" up on it. Here's a video about it and video2 \$\endgroup\$ – Harry Svensson Jul 29 '17 at 22:50
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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:

  • Range (distance) of a radar target (not my choice of words, it's simply called "target" in radar)
  • Relative speed of a target
  • number of targets
  • Size of targets
  • Material/shape properties of targets

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

Power Received > Noise Floor , then communication is possible, otherwise it's not

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:

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.

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:

  • What a "working communication" is, and what isn't, is up to the definition of yourself, and
  • Receiver systems simply aren't as sensitive to noise as they are to the signal they want to see – and thus, there's systems that can even work with Noise > Signal energy
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    \$\begingroup\$ This has just the right mix of practical grounding in reality and actual math and theory that makes for a truly excellent answer in my opinion. 👍 \$\endgroup\$ – metacollin Jul 29 '17 at 13:48
  • \$\begingroup\$ Reality get in the way far too often for my liking. :) +1 \$\endgroup\$ – Wossname Jul 31 '17 at 13:23
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Fundamentaly, we have the Shannon-Hartley formula for the communication capacity of a channel:

$$C = B \log_2\left(1+{\rm SNR}\right).$$

\$C\$ is the error-free channel capacity in bits per second of information transferred. \$B\$ is the bandwidth of the channel in Hz. \$\rm SNR\$ is the signal to noise ratio of the channel.

There is no stipulation that \$\rm SNR\$ must be greater than 1.

With an appropriate coding scheme you can communicate over a channel with \${\rm SNR} < 1\$, but you'll never achieve an error-free bit rate better than what's given by the Shannon-Hartley formula. And this limit approaches 0 as SNR approaches 0.

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  • \$\begingroup\$ How to describe this in decibels? In my question I have used a 3dB value, is it possible to translate this formula into dB? \$\endgroup\$ – David K. Jul 30 '17 at 1:30
  • \$\begingroup\$ Yes, just use the usual formula for converting dB to linear power ratio. (3 dB = 2x ratio). \$\endgroup\$ – The Photon Jul 30 '17 at 1:32
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    \$\begingroup\$ I am not sure I follow, in my example is the SNR = 1.9952 or ~ 2, based on the 3dB value? So the bit rate at 1 Hz of my example would be 1.58 bits/s. \$\endgroup\$ – David K. Jul 30 '17 at 4:28
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    \$\begingroup\$ @DavidK. wikipedia decibel; \$x_{linear}= 10^{\frac{x_{dB}}{10}}\$. \$\endgroup\$ – Marcus Müller Jul 30 '17 at 11:47
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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.

DSSS (direct sequence spread spectrum) radio can have a power level below the prevailing noise level and still work: -

enter image description here

It relies on "process gain".

A simplified example of process gain would sum many, many versions of the signal and each signal is selected from different points in the spectrum to achieve an enhanced SNR. Each addition doubles the signal amplitude (an increase of 6 dB) but the noise is only raised by 3 dB. Thus, with two carriers you get a 3 dB increase in SNR. With 4 carriers you get another 3 dB etc etc.. So 4 carriers improves SNR by 6 dB. 16 carriers would get 12 dB improvement. 64 carriers gets an 18 dB improvement.

Its origins were originally military because it made it hard to eavesdrop on secret communications.

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    \$\begingroup\$ The principle of processing gain is correct, but this isn't a particularly accurate description of how DSSS is demodulated; see this answer on Signal Processing.SE for more details on what DSSS buys you. The key is that the information-carrying portion of the signal has a much narrower bandwidth than the spread-spectrum waveform; there is proportionally less noise power in that smaller bandwidth, thus the processing gain. \$\endgroup\$ – Jason R Jul 29 '17 at 15:51
  • \$\begingroup\$ @JasonR I wasn't attempting to provide an accurate description of how DSSS can get below the noise floor. I'll make that a tad clearer in my answer. \$\endgroup\$ – Andy aka Jul 31 '17 at 12:37
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the power received by the antenna in dBW has to be higher than the noise floor in dBW

"noise floor" as most people would understand it is not measured in dBW, or any other unit of power. Rather, the noise floor is defined by the noise spectral density, which is measured in watts per hertz, or equivalently watt-seconds.

The noise floor can be measured with a spectrum analyzer:

SpectrumAnalyzerDisplay.png
CC BY-SA 3.0, Link

Here, the noise floor appears to be around -97 on the Y axis. Assuming this analyzer is calibrated and appropriately normalized, that's -97 dBm per Hz.

"Below the noise floor" would then mean a signal so weak that it doesn't register visually on the spectrum analyzer. Alternately, you might define "below the noise floor" as so weak it can't be heard: it sounds indistinguishable from noise.

So then, are communications possible when the signal is below the noise floor? Yes they are.

Let's say we are transmitting just an unmodulated carrier, so weak it's not audible or visible on a typical spectrum analyzer. How can we detect it?

A carrier is just one frequency. That is, it's infinitely narrow. So if noise spectral density is defined in power per hertz, the narrower we can make a filter, the less noise there will be. Since the carrier has zero width in frequency, the filter can be arbitrarily narrow, and thus the noise can be made arbitrarily small.

However there's a catch: the more certain we are about a thing in frequency, the less certain we can be in frequency. For a time uncertainty of \$\Delta t\$ in seconds and a frequency uncertainty of \$\Delta \nu\$ in Hz, this relation must hold:

$$ \Delta t \Delta \nu \ge {1\over 4}\pi $$

Consequently, if we want to limit our measurement to an extremely narrow bandwidth (thus minimizing noise power), we must observe for an extremely long time.

One way to do this is to take the FFT of the signal, like the spectrum analyzer does. But rather than displaying one FFT after another, average them together. The noise, being random, will average out. But the extremely weak carrier introduces a constant bias at one point, which will eventually win over the averaged random noise. Some spectrum analyzers have an "average" mode which does precisely this.

Another way is to record the signal for a very long time, then take a very long FFT. The longer (in time) the input to the FFT, the higher frequency resolution it has. With increasing length in time, the width of each frequency bin becomes smaller, as does the noise power in each bin. At some point the noise power becomes small enough that the weak carrier can be resolved.

Although given enough time any simple carrier can be detected, if we wish to transmit any information the carrier can't go on forever. It must be modulated somehow: perhaps switched on and off, shifted in phase, or in frequency, etc. This puts a limit on how fast information can be transmitted. The ultimate limit is given by the Shannon-Hartley theorem:

$$ C = B \log_2\left( 1 + {S \over N} \right) $$

  • \$C\$ is the channel capacity, in bits per second
  • \$B\$ is the channel bandwidth, in Hz
  • \$S\$ and \$N\$ are the signal and noise powers respectively, in watts

From this you can see that communication never becomes impossible with a poor signal to noise ratio (\$S/N\$), although there is an upper bound on the rate of information that can be transmitted.

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  • \$\begingroup\$ Great answer, thanks it cleared up some confusions in my mind. \$\endgroup\$ – David K. Aug 5 '17 at 14:37
  • \$\begingroup\$ Here, the noise floor appears to be around -97 on the Y axis. Assuming this analyzer is calibrated and appropriately normalized, that's -97 dBm per Hz. I disagree: It is 97 dBm per /110 kHz. Your RBW is 110 kHz. \$\endgroup\$ – user94729 Dec 12 '17 at 12:39
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As a practical adjunct to Marcus Müller's excellent answer...

Ham radio has a number of digital modes suitable for successful signal reception below the noise floor. These numbers have a caveat, which I explain afterwards.

The above are all examples of leveraging processing gain. However, the oldest amateur radio digital mode, CW (Morse code, typically) can be properly copied by ear at 18 dB below the noise floor.

Note that the numbers above compute SNR relative to a 2500 Hz bandwidth. This allows apples-to-apples comparisons of modes, but can be misleading for very wide or very narrow signals (for which filtering will necessary include or exclude, respectively, more noise). The last link explains that E_b/N_0, where E_b is the energy per bit and N_0 is the noise power in 1 Hz is a better scoring metric (and provides more direct coupling to the theoretical numbers you are generating). Happily, Shannon has shown that there is an absolute lower bound on E_b/N_0 of -1.59 dB, so any mode that gets close to this is very good. As the table at that link shows, "Coherent BPSK on VLF" has E_b/N_0 of -1 dB ("-57 dB below the noise floor" relative to 2.5 kHz, as a comparison with the above numbers). That BPSK result is more of a heroic experiment, repeated for transatlantic VLF transmissions (with more details at the link above and at its references).

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  • \$\begingroup\$ Interesting, so in my calculation I assumed a 3 dB S/N ratio, should I use instead -57 dB since according to the link provided, this has also been tested, and proven to work. \$\endgroup\$ – David K. Jul 30 '17 at 1:27
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    \$\begingroup\$ No. These far-below-zero numbers are result of filtering, discarding almost all of the bandwidth. This filtering may be with correlation or de-spreading, used by GPS and other systems. The chip-rate of GPS is 2million chips per second; the useful bitrate is much lower and the noise bandwidth is therefore much lower than 2MHz. \$\endgroup\$ – analogsystemsrf Jul 30 '17 at 2:53
  • \$\begingroup\$ A decodable PSK31 signal is clearly audible or visible on a spectogram. That is not "below the noise floor" in my book. The mistake you are making is "noise floor" is not the same thing as "noise power in a 2500 Hz bandwidth". \$\endgroup\$ – Phil Frost Jul 31 '17 at 14:10
  • \$\begingroup\$ @PhilFrost : Take it up with David Farrell, as cited for PSK31. "PSK31 signals can be recovered from 7 dB below the noise floor." I have observed PSK31 signals recovered that do not clearly stand out in a waterfall, so his claim accords with my observations. \$\endgroup\$ – Eric Towers Aug 1 '17 at 0:07
  • \$\begingroup\$ In my book, if you can see it on the waterfall, or hear it, it's not "below the noise floor". \$\endgroup\$ – Phil Frost Aug 1 '17 at 1:17
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Any communications medium will attempt to distinguish among various possible states, e.g.

  • The remote device is trying to transmit a "zero".
  • The remote device is trying to transmit a "one".
  • The remote device is not trying to transmit a "zero" or a "one".

A receiver can't be 100% certain about the actual state of the transmitter. Any means the receiver uses to ascertain the sender's state will have a non-zero probability of misjudging at least some such states (a receiver that unconditionally decides the transmitter isn't sending anything would misjudge that state 0% of the time, but misjudge other states 100% of the time).

As signals approach or fall below the noise floor, the probability of misjudging states will go up. This will in many cases limit the usefulness of communication that can be performed. On the other hand, if a channel which is only 51% reliable is used to send the same bit three times, it would have a 13.27% chance of reporting the correct value all three times, a 38.2% chance of reporting the correct value twice, and a 36.7% chance of reporting the wrong value twice, and an 11.7% chance of reporting the wrong value all three times. Not great odds, but the probability of reporting the correct value would increase from 51.0% to just under 51.5%. That might not seem like much, but if data is sent enough times and failures are independent, the probability of the majority being correct may be brought arbitrarily close to one.

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In RADAR, the false-alarm detectors are adjustable; those are down at the 3dB region; at 10dB SNR, the BER (false alarms) occur 0.1% of the time; note the 10dB depends on how the bandwidth is defined --- some use 1/2 bitrate, some use bitrate, causing a 7dB SNR for 1/2 bitrate. Various modulation methods has different spectral masks and thus use differing ratios of bandwidth to bitrate, thus SNR varies.

Key: classical communications [before bit-error-correction methods arrived] need 20dB SNR for clean digital data to be communicated; ditto for FM music; clean video needs 50 or 60dB SNR, to avoid irksome beatnotes of chroma crawling up the screen; MorseCode sometimes works below the noise floor, because the human ear is extracting the beep---beep---beeeeeeeep---beep out of the noise.

Here is a BER curve from Wikipedia

enter image description here

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You can detect and communicate with signals below the noise level by exploiting differences between the noise and the signal frequency distributions and by exploiting known timing characteristics of the signal that the noise does not share. Or the transmitter can run at very high power for brief instants, so that the average power level is low. That means filtering and gating at the receiving end. Error correcting codes can be used for further gain.

An example of an extreme case is the SETI effort to detect signals from extraterrestrial sources. (Of course they haven't found anything yet, but if a signal were there, they would find it.) SETI uses extremely narrow band filters to cut out the noise. There is a proposal for an optical SETI that will look everywhere at once and look for bright flashes.

In ham radio we have a mode called JT6M that makes the most of very low power transmissions by combining extreme narrow bandwidth with known timing of the signal bits and an error correcting code. Check it out.

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