A carrier that costs nothing until the link has nothing left

A deep-space link divides one signal between two jobs. The data travel as sidebands of a radio carrier, and some of the power is left in the carrier itself as a pure tone, so that a tracking loop on the ground can lock to it and measure its frequency to a fraction of a millihertz. That frequency is the spacecraft’s Doppler shift, and a position measured from a frequency is how almost every spacecraft beyond the Moon has been navigated. The tone’s share is set by the modulation index, and the essay on that division found that for a distant spacecraft the binding limit is not the precision of the Doppler but the loop’s ability to stay locked at all: about fifteen decibel-hertz of carrier power over the noise density, below which the loop slips and the Doppler is gone.

There is a way to have the tone for nothing. A receiver can recover the carrier’s phase from the data themselves — by multiplying the signal by the data it carries, which strips the modulation and leaves a tone made of the whole signal’s power. The obvious version uses each symbol’s sign as it arrives, and pays a squaring loss that grows as the symbols weaken. The better version waits for the error-correcting decoder to work out what the bits really were, then uses those decoded bits to strip the modulation, and pays almost no loss at all. Receivers that do this exist and work on links near the Earth. The question the previous essay left was whether they can work at the far end of a deep-space link, where the residual carrier now persists — and whether, if they could, it would matter.

Both halves of the answer turn out to be about time. The carrier costs almost nothing until the link is at its very edge; and at the edge, the decoded bits that would replace it arrive minutes after the phase they describe.

What the tone costs, and where #

The carrier’s cost is not a fixed share of the power. A loop needs a fixed amount of carrier power — fifteen decibel-hertz over the noise — and the rest can go to data. When the link is strong, that fixed amount is a small fraction. At Voyager’s present distance of about 165 astronomical units, the whole signal arrives at about thirty-two decibel-hertz on a seventy-metre dish, and the carrier’s fifteen are two per cent of it. Giving that two per cent to the data would raise the data rate by a tenth of a decibel — two per cent, from 160 bits a second to about 163. It would be invisible in the link margin, which is chosen with several decibels to spare against the weather.

The inverse square changes this only slowly. At six hundred astronomical units the signal is down by another eleven decibels and the carrier’s share has grown to a quarter; dropping it would gain 1.4 decibels. At a thousand astronomical units it is three-quarters, and the gain is six decibels — a factor of four in data rate. Near 1,160 astronomical units the whole signal reaches fifteen decibel-hertz and the residual carrier would take all of it, leaving nothing for data. There, and only there, the gain from dropping the tone is unbounded.

So the residual carrier is cheap insurance for every spacecraft that has ever flown. It becomes expensive only for a mission sent deliberately to many hundreds of astronomical units — the solar gravitational lens at 550 and beyond, or a probe to the heliosphere’s far side — and only in the last factor of two in distance before the link fails.

Three ways to keep the Doppler #

The three schemes can be laid side by side on the same link carrying the same data, protected by a rate-1/6 turbo code, the strongest in routine deep-space use. With a residual carrier, the data rate follows the inverse square until near the end, then falls away steeply as the fixed carrier eats the remaining power; the last bit a second is sent from about 1,120 astronomical units. With a decoder-aided loop, which uses all the power for both jobs, the data rate follows the inverse square until the whole signal reaches the loop’s threshold, at about 1,150 astronomical units. The difference between them is real but confined to the last few tens of astronomical units.

The surprise is the third scheme. A Costas loop — the classical suppressed-carrier receiver, which multiplies the signal by its own symbol-by-symbol decisions — would seem to sit between them, since it also uses the whole signal. It does far worse than either. Its last bit is sent from about 720 astronomical units, four hundred short of the residual carrier, because the turbo code that makes the data so cheap makes the carrier expensive to regenerate.

Why a better code makes the carrier harder to find #

An error-correcting code saves energy per information bit by spreading each bit over several transmitted symbols and letting the decoder pool the evidence. A rate-1/6 code sends six symbols for every bit, so each symbol carries a sixth of the bit’s energy. The turbo code decodes reliably when the energy per information bit is about equal to the noise density, which puts each symbol about eight decibels below it. That is the turbo code’s triumph: it extracts bits from symbols that are individually almost pure noise.

A Costas loop sees only the symbols. It regenerates the carrier by multiplying each received symbol by the sign it appears to have, and when the symbol is buried in noise the sign is a coin toss and the product is mostly noise. The squaring loss is the fraction of power that survives that multiplication, and it depends on the energy per symbol, not per bit. Uncoded data, with strong symbols, keep ninety-six per cent. The convolutional codes of the 1970s and 1980s, with symbols near two decibels, keep three-quarters. The turbo codes of the present keep between a half and a quarter. Every advance in coding since the Voyagers were launched has made the data cheaper and the self-recovered carrier dearer, which is one reason the residual carrier has survived on distant links long after suppressed carriers became standard near the Earth.

A decoder-aided loop escapes this because it does not use the symbols’ individual signs; it uses the decoder’s verdict, which pools the evidence of a whole codeword. Above the decoder’s threshold that verdict is almost always right, and multiplying the signal by the right symbols strips the modulation completely. The cost is that the verdict arrives only after the whole codeword has been received.

A codeword that outlasts the sky’s phase #

A turbo codeword is long by design — the longer the word, the closer such a code comes to the theoretical limit — and the standard deep-space lengths hold 1,784 or 8,920 information bits. At a high data rate that is a moment. At Voyager’s 160 bits a second the longer word takes fifty-six seconds to arrive. At the ten bits a second that the edge of the link supports, it takes fifteen minutes, and even the short word takes three.

A loop steered by decoded bits learns what each symbol was only when its codeword has been decoded. To use that knowledge it must hold the carrier’s phase in the meantime, predicting it across the codeword from what came before, so that the decoded symbols can be applied to samples whose phase is still known. If the phase wanders by more than about a third of a radian over the codeword, the symbols are being stripped against the wrong reference and the regenerated tone degrades.

How long the phase stays predictable is set by the same media between the antenna and the spacecraft that limit the Doppler’s precision. The solar wind’s electrons and the troposphere’s water vapour change the path length by fractions of a millimetre over a minute in quiet conditions, which at X band is a third of a radian of phase after about a minute. A one-way link referred to the spacecraft’s own crystal oscillator adds a comparable term; a two-way link referred to a hydrogen maser on the ground removes it but leaves the sky’s. Near solar conjunction the plasma’s contribution is ten times larger and the phase stays predictable for well under a second.

The two time scales cross at exactly the wrong place. At high data rates, where a codeword is shorter than a minute, decoder-aided tracking works — and there it saves only the residual carrier’s two per cent. At low data rates, where dropping the residual carrier would gain several decibels, a codeword lasts several minutes and the phase has wandered by radians before its symbols are known. The receiver cannot simply wait for the decoder.

Below a few bits a second the loop decides #

There is a deeper limit underneath. However the carrier is recovered — as a residual tone, by squaring, or with decoded bits — some loop must hold its phase, and a loop needs its fifteen decibel-hertz. The data, with a strong code and a few decibels of margin, need about five decibels of signal-to-noise density for each doubling of the data rate, and at about nine bits a second the two requirements are equal. Above that rate the data’s need dominates, a residual carrier is a small surcharge, and decoder-aided tracking saves the surcharge. Below it the loop’s need dominates, and no improvement in coding changes it. A link sending one bit a second still needs fifteen decibel-hertz for its Doppler.

That is why the gain from dropping the tone runs to infinity at the edge but buys so little distance: the decoder-aided link and the residual-carrier link both fail at the loop’s threshold, within a few per cent of the same distance. What the decoder-aided loop saves is the data rate in the last stretch before that threshold, not the threshold itself.

What a receiver at the edge would actually do #

None of this means the data cannot carry the Doppler at the edge. It means a loop running in real time cannot, and the way round it is not to run in real time. The Deep Space Network’s radio-science receivers record the raw signal at a stable sampling rate, open-loop, without trying to lock to anything, and the recording can be processed afterwards as many times as needed. A receiver working on a recording can decode a codeword with an initial guess at the phase, use the decoded symbols to estimate the phase across the codeword, decode again with the better phase, and iterate — estimating the carrier and the data jointly rather than one before the other. Methods of this kind have been demonstrated on weak signals, and they turn the latency problem into a computational one.

What they cannot escape is the phase coherence. A joint estimate across a codeword still needs the phase to vary in a way that can be modelled over its duration, and if the plasma moves it by radians in a few minutes, the model has to follow those movements using only the signal itself — at a signal-to-noise ratio low enough that the movements are hard to see. The sky’s coherence time sets how much signal can be pooled into each phase estimate, and at the edge of the link that pool is not much bigger than what a residual carrier provides. The navigation the Doppler is wanted for does not need it in real time, so the open-loop approach is available. But the gain over a residual carrier, once the coherence limit is respected, is a few decibels at most, at distances no spacecraft has yet reached.

The other two measurements share the same edge #

Doppler is one of three radiometric measurements, and the other two meet the same wall from different sides. Range is measured by timing a code or a set of tones sent up to the spacecraft and turned round, and the ranging signal is modulated onto the same carrier as the data, taking its own share of the power. A ranging measurement integrates for minutes to build up enough signal to resolve the round-trip delay, and it too needs the carrier’s phase to be followed while it integrates; at the edge it competes for the same fifteen decibel-hertz and the same minute of coherence. The newer regenerative and telemetry-based ranging schemes, which extract the range from the timing of the data frames themselves, are the ranging counterpart of decoder-aided carrier recovery, and they share its dependence on the decoder working.

The third measurement, the angle on the sky, is made by comparing the spacecraft’s signal with a quasar’s at two widely separated antennas. It needs tones or wideband signal spread across a bandwidth, recorded open-loop and correlated afterwards, and it is limited by the same media — the troposphere over each station, which is a few metres of delay overhead and tens at the horizon, and the plasma along each line of sight. Its edge is set by how much of the spacecraft’s signal survives correlation in the few minutes before the quasar has to be observed again.

All three measurements, then, are bounded by the coherence of the path and by the power left after the data are paid for. A spacecraft far enough away that its data rate is a few bits a second is a spacecraft whose position is known mostly from its trajectory, propagated from a time when it was nearer, and only weakly from anything measured now. The residual carrier keeps the one measurement — Doppler — that degrades most gracefully.

The tone as an instrument #

The arithmetic leaves the residual carrier with a better case than it seemed to have. There is also a reason beyond arithmetic. A pure tone is not only a navigation aid but a measurement in its own right: radio-science experiments at the planets measure gravity fields, atmospheric profiles and ring structure from the carrier’s Doppler and amplitude, and for those the cleanest possible tone, uncontaminated by data, is the whole point. A tone recovered from data inherits whatever errors the decoder makes, and a decoder at its threshold makes them in bursts. For a spacecraft whose science is its carrier, the residual tone is the instrument.

And there is a reason in the history of the most distant link. The Voyagers carry convolutional codes and a residual carrier, designed in the 1970s for a link that was expected to last a decade. That they are still navigated and still return data at 165 astronomical units is due in part to a design that kept the tone simple: a ground station can find a pure carrier with no knowledge of the data’s format or rate, and can measure its Doppler while the data are failing. A spacecraft whose carrier existed only inside its data would be lost at the same moment as its data. The residual carrier is what lets a dying link still be found.

What the model leaves out #

The link is the idealised Voyager-like X-band link used for dividing the signal between carrier and data, held at fixed transmitter power, with a single required energy per bit for each code and a fixed three decibels of implementation margin. The codes’ operating points are typical values for a frame error rate near one in ten thousand, not specifications. The squaring loss is the textbook expression for a Costas loop on binary phase-shift keying with hard decisions; real receivers use soft decisions and can do somewhat better. The phase-coherence estimate treats the plasma and troposphere as white frequency noise at a single quiet-condition level, which ignores their real spectra and their dependence on the angle to the Sun. Arraying several ground antennas, which adds directly to the received signal, moves every curve outward together without changing their relationships.

Still open: a link too weak for any loop #

Every scheme here ends at the same threshold, the loop’s fifteen decibel-hertz, and it is natural to ask whether that threshold is fundamental. It is not, quite: it comes from how a loop’s bandwidth must be chosen to follow the spacecraft’s dynamics and the medium’s phase wander, and a spacecraft coasting in the outer heliosphere has almost no dynamics to follow. A narrower loop needs less power. What limits the narrowing is the phase wander of the medium itself, and the same coherence time that defeats the decoder-aided loop sets the narrowest bandwidth any loop can usefully have.

So the edge of the link is set, in the end, by how long the interplanetary plasma leaves a radio wave’s phase alone. That varies with the solar cycle, with the angle to the Sun, and with distance along the path — and for a link reaching beyond the heliopause, part of the path runs through the interstellar medium, whose density fluctuations are measured from pulsars to be far smaller. Whether the coherence time of a signal from many hundreds of astronomical units is set by the near-Sun solar wind, by the heliosheath, or by the receiving station’s own atmosphere, and therefore how much of a carrier the most distant spacecraft will need to be found, has not been measured, because no signal has yet come from far enough away to ask.