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Suppose we have some physical variable $y$ that is changing in some way and we want to detect this change in the presence of noise (e.g. white noise) in that same physical quantity. For example $y$ is an electric field amplitude and there is black body radiation or something like that. If we know of some pattern to be expected in the change of $y$ then this helps to detect it against the noise. For example $y$ could be an oscillation at a very well-defined frequency and we know the phase, so we can use a lock-in amplifier (also known as phase-sensitive detection). Or, $y$ could be a binary message encoded using an error-correcting code (e.g. transmission from space probes). By such ingenuity one can extract very low-level signals against very noisy backgrounds. What is the best anyone has ever done? What limits have been encountered?

Added remark

I was aware when asking that this is a somewhat ill-defined question because for many types of noise you can do better by averaging if the signal lasts long enough. But typically the signal is not permanent. I am asking about what really happens, especially in the presence of thermal noise, when people want to communicate or measure. For example, what is the noise at LIGO (in the relevant frequency band) before signal processing? Here for example I mean the noise in the photocurrent at the light detectors (converted into equivalent strain noise).

I see there is already a helpful answer from one user so please don't close this question! I think further helpful answers will emerge.

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    $\begingroup$ Doesn't this depend on how you define signal and noise? In a one second exposure with a telescope, you might have a signal to noise ratio of $0.1$ for some star. In a $10^4$ second exposure this rises to a signal to noise ratio of $10$. So does this count as an observation with an SNR of $10$ or $0.1$? $\endgroup$
    – knzhou
    Commented Dec 8, 2022 at 22:59
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    $\begingroup$ If it's the latter, then I can get an arbitrarily low SNR by dividing an observation into many tiny pieces. If it's the former (the final SNR in the processed combined signal), then essentially by definition you can never see a signal with SNR below order one. $\endgroup$
    – knzhou
    Commented Dec 8, 2022 at 22:59
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    $\begingroup$ I'm not sure this is a well-defined question, since the signal-to-noise ratio needs to be defined on a case-by-case basis. It depends on the search algorithm being used, among other things. For example, gravitational waves from the binary neutron star GW170817 were detected by matched filter searches with a very large SNR, but not detected at all by a search for generic transient signals. Generally, the SNR for any clear signal should be large enough that the probability noise would generate that SNR is small. $\endgroup$
    – Andrew
    Commented Dec 8, 2022 at 23:19
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    $\begingroup$ I have detected/decoded binary phase shift coded signals at less than $CNR=-30dB$ where $CNR = 10log\frac{A^2}{k_BTW}$ and the signal is $\pm Acos(\omega t)$ and $W=1/T_b$, $T_b$ is the length of each $+$ or $-$ pseudo-random phase jump. In addition to thermal noise of relative intensity -30dB the signal was also covered by much worse narrow band "jamming", too. I know it can be done at $CNR=-40dB$, but much lower than that is probably classified. $\endgroup$
    – hyportnex
    Commented Dec 9, 2022 at 3:19
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    $\begingroup$ I just started writing an answer about how we communicate with the Voyager spacecrafts, but then I realized that this might not be what you're after. Are answers about how we narrow the received noise with antenna design interesting to you? $\endgroup$
    – DanielSank
    Commented Dec 9, 2022 at 6:07

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WSPR, a "ham radio" digital radio transmission protocol with redundancy and error-correction, can be routinely decoded at a level 29dB below the noise floor. This is good enough to permit error-free radio communication between any two points on earth on much less than one watt of radiated power, as long as ionospheric bounce conditions prevail.

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  • $\begingroup$ Yes typically 2 sensors at different frequency are used then the difference if used to capture the signal. Sometimes 2 transmitters are used in conjunction $\endgroup$
    – ChemEng
    Commented Dec 10, 2022 at 14:06

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