[karl-j]

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.

[hliyan]

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!

[karl-j]

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.

[hliyan]

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!