When radio signals leave earth, they propagate out in a wave form. But how far they can “spread out” over distance until they become indistinguishable from background noise?
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3Related: How big is Earth's sphere of broadcast influence? Also, could you please [edit] to clarify your question? As it is, it's rather broad and open to interpretation. For example, radio signals at different transmit power output, wavelength, even solid angle of its main lobe or where they were transmitted from and in what direction will effectively have different signal to noise ratio at some specific range, location even. It also depends on technological ability to discern signal out of it. – TildalWave Nov 13 '14 at 16:16
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You can always get a bigger dish, so there is no absolute ceiling... – PearsonArtPhoto Nov 13 '14 at 17:38
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1Also depends on how noisy are your neighbors. :) – TildalWave Nov 13 '14 at 18:11
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Modulation will also make a difference ... AM, FM, PM? – Everyone Nov 15 '14 at 15:45
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2@Everyone Actually no, not for attenuation at least which is given as $\alpha [\text{dB}/(\text{MHz} \cdot \text{cm})] \cdot \ell [\text{cm}] \cdot \text{f}[\text{MHz}]$. Actually, the question doesn't even clarify if it's merely about detecting a signal and inferring its artificial nature, or also discerning its meaning. Modulation might have a role with the latter (some are easier to flatten-out with interference of strong natural radio sources), but might not be even relevant. What's odd tho is that nobody voted to close as unclear / too broad. – TildalWave Nov 15 '14 at 16:01
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Also possibly related: What do they exactly look for when searching for extraterrestrial intelligence?, Does mankind send any signals to attract aliens?, What evidence would be needed to determine a signal was artificial in origin?, Has Earth ever sent out a repeating signal into space?, Interstellar eavesdropping,... – TildalWave Nov 15 '14 at 16:11
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There's a lot of factors in this question, but it really comes down to a link budget. With a fixed power, gain (gain = 1 for omnidirectional antennas), wavelength etc. it really just comes down to running the numbers.
The real equation of note here is the free space path loss equation:
http://en.m.wikipedia.org/wiki/Free-space_path_loss#Free-space_path_loss_formula
That equation will give the the loss at a given distance. You can use the temperature of the background radiation in you link budget, and quickly come up with a distance at which the signal strength is less than the noise. Although it's typical when designing communications system to require a S/N ration of greater than 3, so that might be a more practical target.
ThePlanMan
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Perhaps the question breaks down to two parts: the distance beyond which no terrestrial "leakage" could be recovered from the noise as a usable signal, and the distance beyond which no such terrestrial emissions could be recognizable as artificial. – Anthony X Nov 15 '14 at 15:16