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Well the concept is similar to a radio telescope, but with a twist.

Assuming I have a directional antenna, and I point it to a weak transmitter, and the two are separated by great distance. Is there a limit to the distance that a communication from a weak receiver, can be achieved that is only dependent to the weaker transmitter, as influenced by the conditions on earth?

Thinking about this on my own, I can identify that the noise caused by the atmosphere would have some self-imposed distance limit. And this is why radio telescopes are usually build in high altitude and where the atmosphere is thin. I am just not able to identify any formulas. To make things easier, picture a radio telescope and a wifi signal coming from within earth.

Is there a formula that takes into account both the antenna gain of the receiver and transmitter and determines the maximum distance based on atmospheric pressure and such weather conditions?

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    $\begingroup$ Are you trying to stealing wifi from the mole people? Where does the atmosphere coming into it for a signal within the earth to a receiver on the surface? en.wikipedia.org/wiki/Minimum_detectable_signal may be of interest. $\endgroup$ – user118047 Dec 9 '16 at 23:36
  • $\begingroup$ I think the question is too open-ended stated like this. There are too many parameters and a successful communication is not well defined here (error rate, minimum bit rate, etc.). You can communicate on an extremely noisy line if your ready to have a very slow chat. The limits are fundamentally engineering trade-offs and cannot be put in a formula. $\endgroup$ – G. Bergeron Dec 10 '16 at 1:35
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    $\begingroup$ Actually it's relatively easy and within some statistical probabilities one can get rough answers. Look it up in any introductory book on communications systems engineering. Or maybe Wikipedia. Then think and ask here what you really want. Can one receive a signal from a well transmitter? Definitely, depends on the numbers but we receive radio signals from relatively weak transmitters on spacecraft probes that are further away that Saturn, and I believe Pluto. The main equation if $\endgroup$ – Bob Bee Dec 10 '16 at 7:17
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Actually it's relatively easy and within some statistical probabilities one can get rough answers. Look it up in any introductory book on communications systems engineering. Or maybe Wikipedia. Then think and ask here what you really want. Rough answer to can one receive a signal from a well transmitter:

Definitely, depends on the numbers but we receive radio signals from relatively weak transmitters on spacecraft probes that are further away that Saturn, and I believe Pluto. The main equation is

Pr = Pt X Gt x Gr/4x pi x $R^n$, where The first two terms are power received and transmitted, the next two are antenna gains for receiver and transmitter, R is distance, and n is an exponent, which in free space is 2. In the atmosphere or water or any propagation media n will typically be larger and should be thought of as a model of the propagation loss in the media. It clearly depends on frequency. You can find models and parameters and get close enough unless you are trying to get real scientific or engineering numbers.

So yes, Pt can be small and if you make the G's large enough, or even just the GR, you can get enough Pr to detect it. How much you need depends on another equation, derived from SNR = Pr/kTB, B the bandwidth, k Boltzmann constant and T the noise temperature.

Again it depends on the numbers which depends on what the design and operation provides for those numbers.

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  • $\begingroup$ Thank you, I think you have steered me in the right direction. I will keep researching. $\endgroup$ – Iordanis Dec 10 '16 at 13:22
  • $\begingroup$ You are quite welcome. There is history there for another factor . The error correction codes were designed and initially implemented, particularly the tree search decoding (there is a official name with the inventor, famous but doesn't come to mind), for long and far NASA missions and the weak signals it was going to receive. Very low data rates, and long and good codes to fix errors. That would be part of generically the processing gain, and for error correcting gains, if you break up SNR above in the equation it is = EbC/NoB, with C the data rate and B bandwidth. You can manipulate R/B $\endgroup$ – Bob Bee Dec 10 '16 at 19:54

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