The receiver would be much, much bigger than a square kilometer by many (>6) orders of magnitude. Any civilization capable of interstellar travel would be very capable of setting up a planetary or solar system sized astronomical interferometer (aka hypertelescope [1]) using a large fleet of synchronized satellites. These satellites can be prototyped with today's technology and any interstellar civilization would be capable of both receivers and transmitters with far more power than we have attempted today.

[1] https://en.wikipedia.org/wiki/Astronomical_interferometer

Yes, but isn't interferometry only useful in increasing the angular resolution? It doesn't necessarily boost the signal.

That's if you're trying to reconstruct a "picture" of what the interferometer is pointed at. If you control both the transmitter and receiver, you can use many different techniques to pack more information into the same amount of power. 2G cellular radios, for example, were 2 watts max power while 3G and 4G radios are 500 milliwatts max power despite being orders of magnitude faster thanks to better encoding due to radio sensitivity and spatial resolution due to MIMO - the cellular equivalent of an astronomical interferometer.

The technical term is space-time block coding [1]. The speed negotiation would take at least 3x the distance between the two start systems though, until which time they'd be limited to a conservative estimate based on the channel impairment between the star systems, aka max expected noise between the two astronomical phased arrays.

[1] https://en.wikipedia.org/wiki/Space%E2%80%93time_block_code

It's nowhere near that bad. Arecibo could communicate with a similar installation out to 21,000 LY, at least.[1] That's with a megawatt of power, less than many analog TV stations.

[1] https://www.seti.org/arecibo-message

Seth Shostak, head astronomer for the SETI institute, thinks the actual number is 400 light years, not 21000 light years. And James Benford, President and Founder of Microwave Sciences (and twin brother of science fiction author Gregory Benford), thinks that 400 light years would be the range for detection, not for communication, pointing out that the ground-based receivers wouldn't be able to integrate the signal over enough time due to planetary rotation.

If we're still factoring in ground based solutions for interstellar communication I think we're assuming the wrong level of technological development. I'd like to see how it works out for a few dozen receivers positioned as far apart as possible in Solar orbit.

Keep in mind you can increase the energy per bit by increasing symbol duration.

I had a problem like this on my wireless systems exam recently, where the setup was two Arecibo style antennas with a reasonable 20ly distance, a low 48bps, an Eb/N0 of at least 10dB, and a few other reasonable assumptions. The transmit power needed turned out to be less than 250kW.

Interesting! Clearly something is wrong with my calculations. Given the effects of the inverse square law over 20ly, how is a antenna of any size able to collect a reasonable amount of photons from a 250kW source? Even a beam that is parallel for all terrestrial purposes will spread out quite a bit over such a huge distance.

Would be really grateful if you could link me to some learning resources!

I'd be interested to see your calculations, because I'm just doing it according to the book, blindly following Friis transmission equation. It sounds like you are applying more physics based thinking to the problem. I have no clue how many photons the antennas are spitting out or receiving, just the gains and losses associated with the antenna size and distance.

I'm writing out the basics of the solution below, but the physical reason for the non-impossibility is that the problem statement assumes the lowest possible amount of noise to overcome, long symbol durations, and high gain antennas.

For learning resources to recommend I'm lacking a little, since I mainly relied on lectures etc, but if you're interested in terrestrial wireless communications you could check out Stanford's class by Andrea Goldsmith. [1] There's very little about antennas, satellites, long distance communications and all that, but a draft of her book is available and all the lectures, so it's very comprehensive even though it's only tangentially related. You can also check out the link budget Wikipedia page [2] for an overview of things affecting these kinds of transmissions, as well as many links to interesting related topics, like the voyager program and their unique struggles (300+dB path loss and still hanging in there!)

The path loss is L = (4 * pi * d/lambda)^2, where lambda is wavelength. At 1.5GHz carrier frequency this results in L = 1.41e38 = 382dB, which definitely is a monstrous number, and with no antenna gain and terrestrial noise powers this would be practically impossible to overcome.

But with an equivalent receiver noise temperature of 5K (meaning the noise we are receiving is only the faint radio noise from space [3]) and a bandwidth of 100Hz the signal is only competing with noise with power N_0 = kBT = 6.90e-21W = -202dBW giving us some hope of establishing communications.

The gain of the 300m diameter antenna is G = 4 * pi * Area/lambda^2 = 2.2e7 = 74dB, and we have one transmitting and one receiving so we can double that dB gain.

Given the Eb/N0 = 10dB requirement and the rate of R = 48bps we can figure out the necessary received power after the antenna gain. Energy per bit is N_0 * 10, so the received power has to be Pr = N_0 * 10 * R = 3.3e-18W = -175dBW.

The transmit power plus gains minus loss has to be greater than the receive power limit, so it should be Pt >= Pr - 2G + L = -175dBW -(2 * 74)dB + 382dB = 59dBW = 800kW

I'm getting a different answer here because the exam had some additional pointing/polarization loss, which I apparently flipped the sign on in my solution. So the correct answer for my problem should be in the low MW, but still not outlandish. I have not received the corrected exam yet, so there might be other errors too.

[1] https://web.stanford.edu/class/ee359/courseinfo.html

[2] https://en.wikipedia.org/wiki/Link_budget

[3] I'm a little unclear on how this would be achieved, since the earth antenna is, well, earth based and at some 290K. This is however what the problem assumed, so maybe they're counting on cooled space based antennas.

Incredible! Thanks for putting this together. Yes, I'm using a more first principle based approach (merely applying the inverse square law to the transmission power). Clearly your approach is better (because it accounts for background noise). I shall have a look at all this!

Even worse, what sort of conversation are you contemplating with a round trip on the order of 10K years?

Giant sun ray reflector.

Can the Sun’s power be modulated to produce a signal?

The sun can be used as a gravitational lens.

How would you send out a morse-code like signal?

like -.-- -.-- --..

You arrange satellites behind two suns like this:

    Sat (500AU) sun (N light years) sun (500AU) sat

Where do you get 2 suns from?

Maybe radio is still better than all that stuff.