When do radio and tv signals become indistinguishable from background noise of the universe?

Question

I understand the rate an EM signal broadcast uniformly from the Earth will decrease in its power is governed by the inverse square law. How far from Earth will radio and tv signals become indistinguishable from background noise from varied sources in the galaxy, even CMB.

I know nothing about the type of equipment that would be used to gather and analyze such a signal. I don't know what the geometry of the observation means. Assuming the optimal equipment and conditions we currently have and no other line-of-sight sources.

Answer 1

If an extraterrestrial civilization has a SETI project similar to our own, could they detect signals from Earth?
In general, no. Most earthly transmissions are too weak to be found by equipment similar to ours at the distance of even the nearest star. But there are some important exceptions. High-powered radars and the Arecibo broadcast of 1974 (which lasted for only three minutes) could be detected at distances of tens to hundreds of light-years with a setup similar to our best SETI experiments.

SETI FAQ

Answer 2

If a signal of power $P$ is spread isotropically it will have power density $P_s = P/4\pi r^2$ at distance $r$. If there is a noise power density $P_n$ then the signal to noise ratio will be $$\text{SNR} = \frac{P_s}{P_n} = \frac{P}{4\pi r^2 P_n}.$$ Conversely, signals can be detected when the SNR>1, or $$ r < \sqrt{\frac{P}{ 4\pi P_n}}.$$

Now we need to find $P$ and $P_n$. Full-power UHF TV stations have a few hundred kilowatt to a megawatt ERP (that is, they transmit as if they had that power in normally used directions - they are not isotropic). The carrier signal is about 0.1 Hz (the actual signal is about 6 MHz, too broad to be easy to detect over the backround). In that range the background astronomical flux density is a few Janskys, $\sim 3 \times 10^{-26}$ watt per square meter per Hz (see fig 2.1 here). So if we assume that the TV signal is aimed right at the receiver I get $r<\sqrt{10^6/4\pi (0.1\times 3\times 10^{-26})}=0.1669$ parsec. Not too impressive.

We can of course assume that we should be summing over all TV stations, which presumably multiplies this by a factor of perhaps 1000 to 10000 (let's be generous). That just boosts the detectability distance up to a factor of 100, to 16.6911 parsec. Now, this is not much by astronomical standards, but it is a few hundred stars.