The speeds a pulsed radar folds into one

The shift a mirror gives twice followed a radar that measures speed by mixing its echo with its own transmission and listening to the beat: a continuous tone reflected from a moving target comes back shifted by twice the speed over the wavelength, and the beat frequency is the speed. That is how a police speed gun works, and it works because the gun transmits continuously and listens continuously. A weather radar cannot. It has to know where each echo came from as well as how fast it was moving, and to time the echoes it transmits in short pulses and listens in between.

A pulsed radar measures speed in a way that sounds almost the same and behaves quite differently. It does not hear the Doppler shift. It samples the echo once per pulse, and from one sample to the next it notes how far the echo’s phase has turned. A phase is known only to within a whole turn, and that small fact folds every speed the radar can see into a narrow window, and ties that window to how far the radar can see.

A phase that turns between pulses #

Each pulse that reflects from a raindrop at range $R$ comes back with a phase set by the round-trip path, $4\pi R/\lambda$. If the drop is moving at radial speed $v$, between one pulse and the next — an interval $T = 1/\text{PRF}$ set by the pulse repetition frequency — the round trip changes by $2vT$ and the phase by $4\pi vT/\lambda$. The radar compares each echo with the last and reads that step. It is the Doppler shift, measured as a phase change over a fixed interval rather than as a frequency.

The trouble is that a phase step of $\phi$ and a phase step of $\phi + 2\pi$ produce the same sample. A target closing at 60 metres a second advances the echo’s phase by one whole turn more, between pulses, than a target at 10 — and the extra turn leaves no trace, because the radar is not looking while it happens. Speeds that differ by $\lambda\cdot\text{PRF}/2$ give identical sequences of echoes. The radar must choose one of them, and it chooses the one whose phase step is smallest, which puts every reported speed in a window of half that width either side of zero:

$$v_N = \frac{\lambda\cdot\text{PRF}}{4},$$

the Nyquist velocity. For a ten-centimetre radar pulsing a thousand times a second it is 25 metres a second.

How a radar reads a phase #

To read a phase at all, the radar has to remember the phase of what it sent. A coherent radar keeps a stable reference oscillator running between pulses and mixes each echo against it, producing two numbers per echo — the in-phase and quadrature parts, the cosine and sine of the echo’s phase relative to the reference. A string of those pairs, one per pulse, is a sampled version of the Doppler signal, and the standard estimate of the mean speed, the pulse-pair method, multiplies each pair by the complex conjugate of the one before and averages: the angle of the average is the mean phase step, and the speed follows. The same average, through its magnitude, gives the spread of speeds in the volume, because a wide spread decorrelates successive echoes. The fringe and the spectrum are one measurement found the general fact underneath: the correlation of a signal with itself at a delay and its spectrum are a Fourier pair, so a radar that measures the correlation at one delay, the pulse interval, has measured one coefficient of the Doppler spectrum — enough for its centre and width, and blind to everything the delay cannot resolve.

That blindness is the fold. A correlation at a single delay $T$ cannot distinguish a spectrum centred at a frequency $f$ from one centred at $f + 1/T$, because $e^{2\pi i f T}$ and $e^{2\pi i (f + 1/T)T}$ are equal. The radar’s measurement is periodic in frequency with period equal to the pulse rate, and every Doppler spectrum is wrapped onto one period of width PRF. The ripple that counts the neighbours used the same counting — one independent sample per Nyquist interval — in a quite different setting.

Every speed folded into one window #

Within the window the radar is right. Outside it, the reported speed folds back.

The sawtooth is aliasing, the same effect as the wagon wheel in a film that seems to turn slowly backwards when it is turning fast forwards: a rotation sampled less often than twice per turn cannot be told from a slower one, in either direction. The sampling theorem says it in general: a signal can be recovered from samples taken at a rate $f_s$ only if it contains no frequency above $f_s/2$. The radar’s Doppler signal is sampled at the pulse rate, and its frequency, $2v/\lambda$, must stay below half that rate — which is exactly the condition $|v| < v_N$.

It might seem that the cure is to pulse faster. A radar at ten thousand pulses a second would have a Nyquist velocity of 250 metres a second, enough for any wind on Earth. The reason weather radars do not do that is the other thing pulses are for.

Range and speed tied together #

A pulse sent at one moment returns from a drop at range $R$ after $2R/c$. If that is longer than the interval between pulses, the echo arrives after the next pulse has gone out, and the radar, timing it from the most recent pulse, places it at the wrong range — far rain appearing close, a second-trip echo. Range is unambiguous only out to

$$R_{\max} = \frac{c}{2\cdot\text{PRF}}.$$

Pulsing faster shrinks it. Second-trip echoes are not merely misplaced; they are recognisable. They appear as thin, radially stretched wedges of weak echo close to the radar, with no storm structure to them, because they belong to rain hundreds of kilometres further out, compressed into the first interval. An operator who sees one knows that the pulse rate is too high for the weather present. And the product of the two limits contains no pulse rate at all:

$$R_{\max}\,v_N = \frac{c\lambda}{8}.$$

This is the Doppler dilemma, and only the wavelength moves it. A longer wavelength raises the product, which is why the radars of the United States weather network work in the S band at about ten centimetres, despite the larger antennas the wavelength demands; shorter-wavelength C- and X-band radars, cheaper and more compact, live with a smaller product. None of them can see 200 kilometres and measure 50 metres a second in the same scan. The constraint is not engineering. It is two consequences of the same sampling — one interval between pulses, which must be short to catch the phase and long to catch the echo.

Why ten centimetres #

The wavelength enters the dilemma’s constant directly, and the choice of wavelength is a negotiation with several other parts of physics. A longer wavelength widens the velocity window at a given range. It also weakens the echo: raindrops are small compared with radar wavelengths and scatter in the Rayleigh regime, with a cross-section falling as the inverse fourth power of the wavelength — the dependence the cross-section that forgets the colour set beside the scattering of free electrons, which has none — so a ten-centimetre radar sees the same rain about a hundred times more faintly than a three-centimetre one. Against that, shorter wavelengths are absorbed by the rain they look through, and an X-band radar looking through a heavy storm can lose the far side of it altogether, while S band passes through almost unattenuated. And the angular resolution of a dish is the wavelength over its diameter, as how far apart two things have to be found for any aperture, so a beam a degree wide at ten centimetres needs a dish eight or nine metres across.

National networks choose the long wavelength and the big dish, buying a larger $c\lambda/8$ and freedom from attenuation at the cost of size; local and mobile radars choose the short wavelength and live with folding and attenuation. Neither escapes the dilemma; they choose where to sit in it.

Two pulse rates that disagree usefully #

If a single pulse rate cannot have both, two can be combined. Scan the same volume at two pulse rates whose Nyquist velocities differ, and each folds the true speed into its own window, at different places. A true speed of 40 metres a second appears as −10 at a thousand pulses a second and as 2.5 at seven hundred and fifty; a true speed of −10 appears as −10 at both. The pair of readings distinguishes them where neither reading alone could.

For two rates whose Nyquist velocities are $v_1$ and $v_2$, the pair of folded readings repeats only after $v_1v_2/|v_1 - v_2|$, so for rates in the ratio four to three the recoverable window is three times the faster rate’s. Weather radars use exactly this, alternating pulse rates from one radial to the next or one scan to the next. The method needs both readings to be accurate: a wind noisy enough that one of them is off by a few metres a second can be assigned the wrong fold, and dual-rate velocity images show occasional pixels of wildly wrong speed — misfolded points — that a cleaning algorithm has to recognise as isolated jumps and repair. Other schemes vary the phase of each pulse in a coded sequence, so that second-trip echoes, which carry the wrong code, can be separated from first-trip ones; each buys back part of what the dilemma takes.

A tornado turned inside out #

Folding matters most where the winds are fastest, and the fastest winds a weather radar sees are in tornadoes.

A tornado seen by a radar beam crossing it appears as a velocity couplet: strong winds towards the radar on one side of the core and away on the other, side by side within a few hundred metres. It is the signature forecasters look for. When the winds exceed the Nyquist velocity, the couplet folds: the strongest inbound winds near the core are reported as outbound, and the outbound as inbound, so the couplet appears inside out, with sharp jumps between neighbouring range gates where the reported speed leaps from one edge of the window to the other. Read naively, the fold hides the very feature being looked for. Read correctly — a jump of nearly twice the Nyquist velocity between adjacent gates is almost never a real wind, which would have to change by fifty metres a second across a few hundred metres of air — the fold is unfolded by continuity from the slower winds around it, and the couplet reappears at its true strength.

A wind seen from one side #

Folding is not the radar’s only blind spot about speed, and the other one shapes how tornado signatures are read. A radar measures only the component of motion along its beam. Wind blowing across the beam produces no first-order Doppler shift and is invisible; the Doppler shift with nothing coming closer found frequency shifts produced by rotation rather than approach, but a radar beam crossing a moving parcel of air meets neither kind. So a single radar maps one component of a three-dimensional wind, and a vortex appears as a couplet only because, on either side of its core, the circulating wind happens to blow along the beam in opposite directions. Two radars viewing the same storm from different directions combine their radial components into horizontal winds — dual-Doppler analysis — and each of the two must first be unfolded, so that an error in either propagates into both components of the wind. The fold and the radial view compound: the radar sees one component, wrapped.

The same arithmetic everywhere sampling meets phase #

The dilemma is not peculiar to weather. Every instrument that measures a phase by sampling meets the same fold. Medical ultrasound imagers that colour blood flow by its Doppler shift alias in fast jets, through narrowed heart valves, and the cardiologist’s colour image shows the jet’s centre in the wrong colour; the same trade between depth and velocity limits how deep the imager can look at a given speed. Airborne radars that look down at the ground for moving targets meet the dilemma as blind speeds — speeds at which a target’s phase step is a whole number of turns and it is indistinguishable from the stationary ground. The note that changes on approach began with the Doppler shift as a continuous frequency change; pulsed instruments never see that frequency, only its samples, and everything they report is that frequency’s alias.

Everything from an exchange of pulses built relativity out of pulses sent and received at known intervals, measuring times and distances by echo. The pulsed radar is the same arrangement used for speed, and it inherits the same reliance on the interval: a measurement made at discrete times knows nothing about what happened between them, and anything that could have happened in a whole number of turns between samples is invisible.

What the drawings leave out #

The figures treat a single target with a single speed and the radar’s samples as noiseless. A real radar volume holds a distribution of drops with a spread of speeds, and the radar estimates the mean speed from the correlation of many echoes, with a statistical uncertainty that grows as the spread approaches the window’s width; at that point the estimate becomes unreliable even without folding, which sets a practical limit tighter than the Nyquist velocity itself. The vortex is a Rankine vortex with no inflow or updraft, crossed exactly through its centre by a beam far narrower than it, where a real radar beam at range is often as wide as the tornado and averages its winds. The radar bands are representative wavelengths; real weather radars run at specific frequencies within them. And the figures ignore the ground: every radar receives strong echoes from hills and buildings at zero speed, and removes them with a filter that discards echoes near zero Doppler. Folding makes that filter dangerous, because weather moving at exactly twice the Nyquist velocity folds to zero and is removed along with the ground — a blind speed for rain — so the choice of pulse rate also decides which real winds the clutter filter will erase. The domain of the drawings is single-target, noiseless sampling at pulse rates of a hundred to five thousand per second.

Still open: how much of the dilemma can be coded away #

Modern radars encode each pulse with a phase pattern so that echoes from different pulses can be separated in processing, and transmit staggered sequences of pulse intervals that fold speeds differently from pulse to pulse, and some phased-array weather radars now steer their beams electronically and revisit each volume many times a minute. Together these recover much of what the dilemma forbids a single uniform pulse train. Whether coding and staggering can, in the presence of real noise, ground clutter and overlapping storms, deliver unambiguous range and velocity at the extremes forecasters want — two hundred kilometres and eighty metres a second in one scan — or whether the constant $c\lambda/8$ always reappears as a noise penalty somewhere, is a question being worked out in the design of the next generation of national radar networks.

The arithmetic at the bottom of it is short. A pulsed radar measures a target’s speed by the turn of its echo’s phase between pulses, 4πv/λ·PRF, known only modulo 2π, so every speed folds into ±λ·PRF/4 — ±25 m/s for a 10 cm radar at 1000 pulses a second — while echoes are placed correctly only out to c/2·PRF, and the two limits multiply to cλ/8 whatever the pulse rate: 3.75 million square metres per second in the S band, never enough for 50 m/s winds at 200 km. A radar that listens in pulses hears every speed only up to a whole number of turns.