Not my field, but assuming transmitting hardware (including beam forming) is constant and that atmosphere can mostly be ignored (see comments about it usually being a non-impact in the transmission frequencies), two approaches would suggest:
1. Increase the effective receiving dish size, to capture more of the signal. Essentially, this would be effective in direct proportion to beam spread (the more beam spread, the bigger dish you can use to capture signal).
In practice, this would use multiple geographically-displaced dishes to construct a virtually-larger dish, to allow for better noise-cancellation magic (and at lower cost than one huge dish). I believe the deep space network (DSN) already does this? Edit: It certainly has arrayed antennae [0], though not sure how many are Voyager-tasked.
2. Increase the resilience of the signal, via encoding. The math is talking about bits and photons, but not encoded information. By trading lower bit-efficiency for increased error tolerance (i.e. including redundant information) we can extract a coherent signal even accounting for losses.
Someone please point out if I'm wrong, but afaik the Shannon–Hartley limit speaks to "lower" in the physical stack than error coding. I.e. one can layer arbitrary error coding on top of it to push limits (at the expense of rate)?
If the above understanding is correct, is there a way to calculate maximum signal distance assuming a theoretically maximally efficient error coding (is that a thing?) ? Or is that distance effectively infinite, assuming you're willing to accept an increasingly slow bit receiving rate?
> The uplink carrier frequency of Voyager 1 is 2114.676697 MHz and 2113.312500 MHz for Voyager 2. The uplink carrier can be modulated with command and/or ranging data. Commands are 16-bps, Manchester-encoded, biphase-modulated onto a 512 HZ square wave subcarrier.
Note that "16 bps" while the system runs at 160 bps. This suggests that the data is repeated ten times and xor'ed with a clock running at 10 HZ.
While there's no VOY set up now, https://eyes.nasa.gov/dsn/dsn.html will occasionally show it. When that happens, you will likely see two set up for it. I've not seen them set up across multiple facilities - the facilities are 120° apart and only one has a spacecraft above the horizon for any given length of time.
At the Moon's surface, the beam is about 6.5 kilometers (4.0 mi) wide[24][i] and scientists liken the task of aiming the beam to using a rifle to hit a moving dime 3 kilometers (1.9 mi) away. The reflected light is too weak to see with the human eye. Out of a pulse of 3×10^17 photons aimed at the reflector, only about 1–5 are received back on Earth, even under good conditions. They can be identified as originating from the laser because the laser is highly monochromatic.
While there's no signal there, we're still looking at very sensitive equipment.
There's another major factor. I suppose it falls under the encoding. Frequency stability. In a certain sense, having an extremely precise oscillator at both the receiver and the transmitter, is the same thing as just having a better more frequency-stable antenna, or less noise in the channel (because you know what the signal you're listening for should look like).
I'm no physicist here so take this with a major grain of salt. I think the limit might ultimately arise from the uncertainty principle? Eventually the signal becomes so weak that measuring it, overwhelms the signal. This is why the receiver of space telescopes is cooled down with liquid helium. The thermally-generated background RF noise (black bodies radiate right down into the radio spectrum) would drown everything else out otherwise.
Along those lines, while I'm still not quite sure where the limit is, things become discrete at the micro level, and the smallest possible physical state change appears to be discrete in nature: https://en.wikipedia.org/wiki/Landauer%27s_principle Enough work physically must occur to induce a state change of some kind at the receiver, or no communication can occur. (But this interpretation is disputed!)
Coding counters the uncertainty principle by allowing multiple measurements, which can then be averaged. Thant counters the contribution of random noise.
There are practical signals we use every day that are "below the noise floor" before we decode them.
So while there is an ultimate limit of the maximum coding rate for a given signal-to-noise ratio, this is expressed in terms of a data rate (i.e. bits per second). If you're fine with lowering your data rate, there is no fundamental theoretical limit, as far as I understand.
TIL from another comment that the Shannon limit assumes Gaussian noise so it's not actually always the theoretical limit.
You can't work around the Shannon limit by using encoding. It's the theoretical information content limit. But you can keep reducing the bandwidth and one way of doing that is adding error correction. So intuitively I'd say yes to your question, the distance can go to infinity as long as you're willing to accept an increasingly low receive bit rate. What's less clear to me is whether error correction on its own can be used to approach the Shannon limit for a given S/N ratio - I think the answer is no because you're not able to use the entire underlying bandwidth. But you can still extract a digital signal from noise given enough of a signal...
Of course that would mean sending (a) giant receiver dish(es) in the general direction a probe is sent. On the flip side, if using a single relay it could travel at roughly 1/2 the speed of the probe.
Note that signal strength weakens with distance^2. So if eg. you'd have 2 relays (1/3 and 2/3 between Earth & the probe), each relay would receive 9x stronger signal.
No doubt the 'logistics' (trajectory, gravity assist options, mission cost etc) make this impractical. But it is an option.
That'd be really cool! And definitely helped by the fact that you'd only need to head out at a fraction of Voyagers speeds to get the benefit.
There's some details on the Voyager gravity-assist mechanics here [0], but you'd also need the escape trajectory to be pointed in the Voyager direction which would further constrain...
That said, Earth-Jupiter-Saturn alignments don't seem that rare (on a decades scale).
If the current number of photons per bit is 1500 and the effective limit at 8.3GHz is 25, that is a factor of 60.
With each doubling of distance the number decreases by the inverse square law, so with the current setup we'd have a maximum distance of log2(60) = 5.9 times the current distance (about 163 astronomical units (AU)) which is 961 AU.
In comparison, the closest star to our sun, Proxima Centauri, is 268774 AU away!
Which means we would need something 268774/961 ~= 280 times SQUARED = 78222 times more sensitive than the current setup at the Shannon limit to communicate with it if it managed to get that far.
Maybe multiple receiving antennas could be used, spaced widely apart (e.g. one on Earth, one on the moon, one in orbit somewhere); each would receive some signal affected by noise, but the noise would be different for each (such as the atmosphere affecting only the on-ground antenna). Also given that we know the precise location of the transmitter, we can work-out the exact phase difference between the antennas. Some digital post-processing would combine the raw signal+noise from the multiple receivers and extract the signal.
In fact, I assume that at this distance, even a very narrow signal would spread wide enough to illuminate more than just the Earth diameter.
> So what does that conclusion mean about when the probe will be so far away that we are below the Shannon limit?
I think the practical limit right now is that the Voyagers are losing power.
"The radioisotope thermoelectric generator on each spacecraft puts out 4 watts less each year. [...] The two Voyager spacecraft could remain in the range of the Deep Space Network through about 2036, depending on how much power the spacecraft still have to transmit a signal back to Earth."
I guess we could just program Voyager 1 to lower the data bitrate, adding more redundancy / error correction. I think there's no real limit to how far it could get if we keep doing that.
> And can we beat the Shannon limit somehow, eg collect for longer
If you turn on a faucet for longer, you're not beating the "gallon limit" of the system. The limit is not a fixed number, it's directly based on how much you do to improve the signal.
And they have already slowed down the transmission speed repeatedly.
There's not a lot of appetite for that, though. the 70m receiver is already one of the largest ever built, and for most use cases it's looking better to use an array of smaller ones. Which works fine for receiving but not so much for transmitting (while there are multiple sites in the world capable of receiving from voyager 2, there's only one dish which can actually transmit to it)
(I recall seeing a video on that dish, and the director seemed confident there was enough noise margin left that voyager's power would fail before they lost contact with it)
And can we beat the Shannon limit somehow, eg collect for longer, put the dish outside the atmosphere, and so on?