One signal that carries the bits and the Doppler

Everything known about where a deep-space probe is comes from its radio signal: a frequency shift and a round-trip delay, measured on the ground against atomic clocks and turned into a position by the equations of motion. Everything the probe has learned about the place it is in comes home on the same signal. There is one transmitter on board, one antenna pointed at the Earth, and one stream of watts leaving it. The navigation and the science are paid for from the same budget.

The two jobs want different things from that signal. Navigation wants a pure, steady tone — a carrier — whose frequency and phase the ground can follow to a fraction of a cycle. Data want the signal to change, so that each change can carry a bit. A signal that is perfectly steady carries no information, and a signal that is changing all the time has no steady tone left to follow. The compromise used across deep-space communication is to modulate the carrier only partly, leaving some of the power in a residual tone and putting the rest into data sidebands on either side of it. How much goes where is set by a single angle, the modulation index.

This essay follows that compromise from a spacecraft near the Earth, where it hardly matters, out to the most distant spacecraft ever built, where it decides whether the probe can be heard at all. The answer at the far end is not what precision would suggest. The carrier is never starved of power because the Doppler would become too noisy; it is starved because, below a certain power, the ground can no longer stay locked to it, and then there is no Doppler at all.

A ledger in decibels #

The power that reaches the ground is set by the Friis transmission equation, which multiplies four factors and divides by a fifth: the transmitter’s power, the gain of the spacecraft’s antenna, the gain of the ground antenna, the square of the wavelength over the circumference of a sphere of radius equal to the distance, and — to turn power into a signal-to-noise ratio — the noise of the receiver. Multiplying is awkward across thirty orders of magnitude, so the factors are written in decibels and added.

The ledger for a Voyager-like downlink at 165 astronomical units is dominated by one entry. The transmitter supplies about twenty watts, thirteen decibels above a watt. The spacecraft’s dish, 3.66 metres across, concentrates it into a beam about half a degree wide, a gain of forty-eight decibels. Then the beam spreads over a sphere twenty-five billion kilometres in radius, and that spreading costs 319 decibels — a factor of $10^{31.9}$ in power. The seventy-metre dish on the ground buys back seventy-four decibels by collecting over a large area, and the receiver, cooled to twenty kelvin, contributes a noise density of $kT$, about $3 \times 10^{-22}$ watts in each hertz of bandwidth.

What arrives is about $4 \times 10^{-19}$ watts. The useful measure is the ratio of that power to the noise density, which has units of hertz: 32 decibel-hertz, or about 1,600 hertz. It means the whole signal carries as much power as the receiver’s noise does in a band 1,600 hertz wide. Every job the signal does — locking a tracking loop, carrying a bit — has to be paid for out of that number, and the number falls as the square of the distance.

One dial between two jobs #

For a residual-carrier signal with a data subcarrier, the modulation index $\beta$ divides the power cleanly: the carrier keeps $\cos^2\beta$ of it and the data sidebands get $\sin^2\beta$.

At small indices the signal is mostly carrier: an excellent navigation beacon carrying a trickle of data. At ninety degrees the carrier is suppressed completely and every watt is data. A mission chooses the index for each phase of flight, and the choice is a trade between the two people waiting for the signal — the navigator, who wants the carrier, and the scientist, who wants the bits.

The data side of the trade is simple. The data rate a signal can support is the power in the data, divided by the noise density, divided by the energy each bit needs relative to the noise for it to be decoded reliably. Modern error-correcting codes need about one to two decibels of energy per bit over the noise density; with losses in the hardware and a margin for the weather, a working figure is a few times that. Halve the data power and the rate halves. The carrier side is where the trade becomes interesting, because how much carrier the navigator needs is not obvious.

Precision that the medium will not deliver #

A tracking receiver on the ground follows the carrier with a phase-locked loop: an oscillator steered continuously to stay in step with the incoming tone. The loop’s phase error has a random part set by the carrier’s power over the noise density and by the loop’s bandwidth, about a hertz for deep-space tracking:

$$\sigma_\phi^2 = \frac{B_L}{P_c/N_0} .$$

Doppler is the rate of change of that phase, counted over an interval — typically a minute — and a phase error of $\sigma_\phi$ radians at each end of a sixty-second count becomes a velocity error of about $\lambda\sigma_\phi\sqrt{2}/(4\pi T)$ for a two-way link at wavelength $\lambda$.

For the Voyager-like link the thermal Doppler noise is a few thousandths of a millimetre a second at moderate indices, and it rises steeply only near the end. That has to be compared with the noise that the signal picks up on its way, which no amount of carrier power removes: the fluctuating electron density of the solar wind and the ionosphere, and the water vapour of the troposphere, which together add a phase scintillation equivalent to a few hundredths of a millimetre a second over a minute at X band. The thermal noise stays below that floor right up to the point where the carrier becomes too weak for the loop to hold.

That is the shape of the trade, and it is not a smooth exchange of precision for bits. Over most of the range of the modulation index, more carrier buys Doppler precision the medium will not let the navigator use. Then, at a definite index, the carrier power drops below what the loop needs to stay in step — the loop begins to slip cycles, losing count of whole turns of phase, and the Doppler becomes unusable rather than noisy. The threshold depends on the loop bandwidth and the margin a mission insists on; taken here as fifteen decibel-hertz, it is reached at an index of about eighty-two degrees, with ninety-eight per cent of the power in the data. The navigator’s real requirement is not “as much carrier as possible” but “enough carrier to stay locked, plus a margin”, and everything above that belongs to the data.

The inverse square takes the margin #

The margin is a fixed number of decibel-hertz, and the signal that pays for it shrinks with distance. At Mars, with a 34-metre station, the whole signal is millions of hertz and the carrier’s few dozen are a rounding error; the data can have almost everything at any reasonable index. At Jupiter the signal is a hundred times weaker. At the edge of the heliosphere it is a further factor of a thousand down, and the carrier’s fixed minimum is no longer small.

The best data rate falls as the inverse square of the distance over most of the range, a straight line on logarithmic axes, and then bends down where the carrier’s minimum becomes a large fraction of the total. The line for a fixed index runs parallel to it and then simply stops: at some distance the fixed share of the carrier drops below the lock threshold, and the link is lost even though there is still enough power for data. A mission that kept its index fixed would lose its navigation and its data together, at a distance set by the index rather than by the power. Changing the index as the spacecraft recedes — giving the carrier an increasing share of a shrinking total — moves that edge out by a factor of more than two in distance.

This model is an idealisation of a real link, with an assumed loop threshold, coding requirement and loss allowance. It gives a ceiling of a few hundred bits a second at 165 astronomical units with a seventy-metre dish, and Voyager 1’s science data come home at 160 bits a second, with the difference spent on margin. The order of magnitude is the point. The same seventy-metre dish that receives 160 bits a second from beyond the heliopause could receive millions from Mars, and the ratio is exactly the square of the ratio of the distances.

What the navigator actually needs from the carrier #

It is worth asking what a navigator does with the lock once it has it, because that explains why the floor of the trade sits where it does. A single Doppler count measures one component of velocity, along the line of sight, to a few hundredths of a millimetre a second. The other two components, and the position across the sky, are inferred over hours from the way the Earth’s rotation sweeps the station back and forth across the signal’s direction, and over weeks from the way the trajectory curves. A Doppler record with gaps in it — passes lost because the loop could not hold — loses exactly that sweep, and the orbit fit loses its leverage on the two directions the line of sight does not see.

The other tools a deep-space navigator uses fill the same gaps by different routes. Measuring the signal’s arrival at two stations a continent apart against a quasar in nearly the same direction gives the angular position directly, in an hour rather than a week, and it uses the data sidebands or dedicated tones rather than the carrier. Ranging gives the distance from a round-trip time. None of them substitutes for continuous Doppler on the approach to a target, where an error of a millimetre a second at departure has grown to tens of kilometres at arrival and the correction has to be computed from data only days old. That is the navigator’s case for a margin above the lock threshold: not precision in any one count, but the guarantee of an unbroken record when it matters most.

Why a near spacecraft can do without the carrier #

There is a way to have both. A suppressed-carrier signal — modulation index ninety degrees, every watt in the data — still contains the carrier’s frequency, folded into the data. Squaring the signal, or multiplying it by a copy of itself with the data’s sign flips removed, regenerates a tone at the carrier frequency that a loop can track. A receiver doing this, a Costas loop, uses all of the signal’s power for tracking and all of it for data at once.

It pays a price, the squaring loss. The regenerated tone is made by multiplying noisy symbols by noisy decisions about them, and when the symbols are weak the decisions are often wrong and the product is mostly noise.

The loss depends on the energy in each symbol, not on the total power. A spacecraft near the Earth sending high-rate data with strong symbols loses only a few per cent and gains everything the residual carrier would have cost; Earth-orbiting satellites and high-rate links from Mars routinely run this way. A distant spacecraft sending a few hundred bits a second through a powerful error-correcting code transmits symbols whose individual energy is near or below the noise — the coding works by spreading the information across many weak symbols — and there the squaring loss takes most of the power. Far from the Earth the residual carrier wins, because a modest share of the power in a clean tone is worth more to the tracking loop than all of it in symbols too weak to be trusted one at a time.

So the choice between residual and suppressed carrier is set by the same inverse square that sets the data rate. The architecture of a deep-space link changes with distance not because the engineering fashion changed but because the energy per symbol did.

A shorter wavelength squeezes the same dishes #

The ledger has one more lever. The gain of a dish of fixed diameter rises as the inverse square of the wavelength, and both ends of the link have one, so moving from X band at 8.4 gigahertz to Ka band at 32 gigahertz multiplies the received power by the square of the frequency ratio twice and divides it by that square once for the spreading — a net factor of about fourteen.

The real gain is less than fourteen. A Ka-band ground antenna has to be shaped and pointed four times more accurately to realise its theoretical gain, so its efficiency is lower; the sky is noisier at the shorter wavelength; and water in the atmosphere absorbs it. In a clear sky the net gain is a factor of six or seven, which is why the newer deep-space missions carry Ka-band downlinks for their high-rate science. In heavy rain the absorption and the added noise can take the whole gain away. The answer missions have adopted is to schedule Ka-band passes with a forecast and to fall back to X band when needed — the same two-job compromise appearing again, this time between the bits and the weather.

Ka band also helps the navigator, by a different route. The plasma along the path delays a radio signal by an amount inversely proportional to the square of its frequency, so the plasma noise in Ka-band Doppler is fourteen times smaller than at X band. Two frequencies sent together let the plasma’s effect be measured and removed entirely, and the Doppler residuals that revealed an ocean in a moon’s gravity field were measured on links built that way. The floor in the earlier figure is not a fixed property of nature; it is a property of the wavelength and the calibration, and lowering it changes how much carrier a mission can usefully spend on precision.

The surprising connection: the carrier is the experiment #

The modulation index looks like an engineering parameter, but the residual carrier is often the main scientific instrument. A gravity experiment at a planet or a moon measures nothing but the Doppler shift of the carrier as the spacecraft is accelerated by the body it passes; an atmospheric occultation measures the carrier’s phase and amplitude as it grazes through the atmosphere on its way to the Earth; a test of relativity during a solar conjunction measures the carrier’s delay as it passes close to the Sun. In all three the data stream is irrelevant, and missions switch their links to give the carrier as much power as possible for the experiment’s few hours.

This turns the trade on its head at those moments. The navigator’s “enough to stay locked” is a floor set by the medium and the loop; the radio scientist’s requirement is a ceiling set by the physics being measured, and the small residuals that have turned out to matter — the slight extra acceleration of a spacecraft from its own anisotropic heat, a moon’s tidal flexing, the curvature of spacetime near the Sun — were all found in carrier phase measured with more power than navigation alone would have spent. The same watts that would have carried a few thousand more bits of images carried, instead, the measurement.

What the model leaves out #

The link is idealised throughout. Real tracking loops have bandwidths chosen for the spacecraft’s dynamics and a lock threshold that depends on how the loop is designed and how much cycle slipping a mission will tolerate; the fifteen decibel-hertz used here is representative, not universal. The data-rate model uses a single required energy per bit and a fixed loss, where a real link budget carries a dozen loss terms and a margin chosen statistically against the weather. The Doppler floor of 0.03 millimetres a second at X band over a minute is typical of quiet conditions away from the Sun and can be ten times worse near solar conjunction.

The spacecraft’s transmitter power is held fixed with distance. Voyager’s power, from radioisotope power sources, has fallen steadily through the mission, and its instruments have been switched off one by one to keep the transmitter and the heaters running; at the far end of the curves its link is limited as much by the declining supply as by the distance. Ranging — the other half of radiometric navigation — also takes power from the same budget, through a ranging signal modulated onto the carrier alongside the data, and that three-way division is omitted here for clarity. And the ground antennas can be arrayed, combining several dishes to synthesise a larger collecting area; arraying adds directly to the ground gain and is how the Deep Space Network has repeatedly extended the reach of its most distant links.

Still open: navigating with the data alone #

The residual carrier persists at large distances because a suppressed-carrier loop needs strong symbols, and strong symbols are what a distant, heavily coded link does not have. That is a statement about the loop, not about information. The data stream itself carries the carrier’s phase, and a receiver that decoded the data first and then used the decoded bits to strip the modulation could recover a clean tone from the whole signal without the squaring loss — tracking on what the decoder knows rather than on what each symbol says. Receivers of this kind exist, and they are used on Earth-orbiting and lunar links.

Whether they can work at the edge of the inverse square, where the decoder itself is close to failing and its errors would feed back into the loop, has not been settled. If they can, the most distant missions of the coming decades — probes sent deliberately to hundreds of astronomical units — could give their entire signal to the data and still be located from it, recovering the factor that the residual carrier now spends. If they cannot, the carrier’s fixed minimum will remain the quantity that sets how far away a spacecraft can be and still be both heard and found.