The long pulse that arrives as a spike

The packet that will not keep its shape followed a pulse spreading as it travels through a dispersive medium, its different frequencies moving at different speeds until the packet is long and chirped — its low frequencies at one end and its high at the other. It ended by noting that the spreading can be run backwards: send a deliberately chirped pulse into a medium whose dispersion is opposite, and it arrives shorter than it left. That is how the shortest laser pulses are made. It is also, in a different guise, how every modern radar sees.

The shift a mirror gives twice found the trade at the heart of radar design: a short pulse resolves range finely but carries little energy, a long one carries energy but blurs range, and a radar must choose. Pulse compression is the escape from the choice, and the escape works because resolution was never really a property of the pulse’s length.

A pulse that sings a rising note #

A transmitter is limited in its peak power — by the tube or transistor, by the voltage the waveguide will stand before it arcs. The energy in a pulse is the peak power times its duration, and the faintest echo a radar can detect is set by the energy that comes back, not the power. So a radar that must see far, through a small target’s tiny echo, needs long pulses. A pulse of fifty microseconds carries fifty times the energy of a pulse of one.

A plain pulse of fifty microseconds blurs every target over fifty microseconds of delay, seven and a half kilometres of range. The radar cannot tell two aircraft apart unless they are that far apart. The way out is to make the long pulse distinguishable along its length: to give each part of it a different frequency, so that the echo carries in its pitch a label saying which part of the pulse it is.

The simplest labelling is a linear sweep: the frequency rises steadily from one edge of a band $B$ to the other over the pulse’s duration $T$. The waveform is $e^{i\pi(B/T)t^2}$, a chirp, after the sound such a signal makes when played as audio — a bird’s rising whistle. Bats that hunt by echolocation use exactly such sweeps, and so do the radars of aircraft, ships and weather services.

A filter that waits for the low notes #

To compress the echo, the receiver passes it through a filter that delays each frequency by a different amount: the low frequencies, which left first, by more; the high frequencies, which left last, by less. Matched to the sweep, the filter delays every frequency by exactly enough that all of them emerge at the same instant. Fifty microseconds of echo arrive as one spike.

The filter that does this best has a name that says why. The matched filter — the filter whose impulse response is the transmitted pulse reversed in time — is the filter that maximises the ratio of signal to noise at its output for a known pulse buried in white noise; what the instrument actually hears met it as the way a gravitational-wave detector finds a chirp a tenth the size of its noise. Its output is the correlation of the echo with the transmitted pulse, and for a chirp that correlation collapses to a narrow peak.

The width of the spike is about $1/B$, set by the band the chirp sweeps and nothing else, because the correlation of any waveform with itself is the Fourier transform of its power spectrum, and a flat band $B$ wide transforms into a peak $1/B$ wide. The fringe and the spectrum are one measurement found the same pairing in interferometry. The depth a broad spectrum can see is the same rule in medicine: an optical scanner resolves layers inside a retina to a few micrometres because its light spans a broad spectrum, not because any pulse of it is short — most such instruments use continuous light and no pulse at all. The duration of the pulse does not enter. A pulse of fifty microseconds sweeping five megahertz compresses to about two hundred nanoseconds, a range resolution of thirty metres, while carrying the energy of fifty microseconds. The compression ratio is the time-bandwidth product $TB$, and pulse-compression radars run at products of a hundred to many thousands.

The uncertainty that is not violated #

It can look as though the chirp has beaten the trade sharpness has to be paid for described: a signal cannot be both short in time and narrow in frequency. It has not. The compressed spike is short because the pulse has a wide band; a short spike needs a wide band, and the chirp has one. What the chirp adds is length — a long pulse with a wide band, which a plain pulse cannot be, since a plain pulse’s band is fixed by its length at $1/T$. The time-bandwidth product of a plain pulse is about one; a chirp’s can be as large as the transmitter can sweep and sustain. The uncertainty relation sets a minimum for the product; nothing sets a maximum.

That is the general fact underneath pulse compression. Resolution in time belongs to bandwidth. Energy belongs to duration. A waveform whose time-bandwidth product is large separates the two, and the receiver’s matched filter collects the energy of the whole duration into a peak whose width is set by the band.

Why the noise does not compress with it #

The spike is fifty times taller than the long echo was, relative to its width, but that alone would prove nothing: a filter that sharpened everything would sharpen the noise as well. What makes compression worth doing is that the noise is not sharpened. Receiver noise is incoherent — its value at one instant says nothing about its value a microsecond later — so when the filter adds up fifty microseconds of input with the delays matched to the chirp, the echo’s pieces arrive in step and add as amplitudes, while the noise’s pieces arrive with random phases and add only as powers. The echo’s peak power grows as the square of the number of independent pieces, the noise’s power only in proportion to it, and the ratio between them rises by the time-bandwidth product.

That gain is exactly what a short pulse of the same peak power would have lacked. A one-microsecond pulse and a fifty-microsecond chirp, both at a megawatt, give the same peak power on the target; after the matched filter the chirp’s spike stands fifty times further above the noise, because fifty times as much energy went into it. The filter has not created signal out of nothing. It has gathered energy that was spread along the pulse into one instant, and gathered the noise only as fast as noise can be gathered. The noise every amplifier must add set a floor under any receiver’s noise; compression does not lower that floor, it lifts the signal further above it.

Nature found the same arrangement first. Bats hunting by echolocation emit sweeps lasting a few milliseconds that fall through tens of kilohertz, and the sweep’s bandwidth gives them range resolution of a centimetre or so while its duration gives them the energy to hear a moth several metres away. Whether a bat’s auditory system performs anything like a matched filter, or reads the echo some other way, has been argued over for decades; that the sweeps let it resolve far better than their length suggests is measured. The ionosphere makes chirps without trying: the whistle that arrives sorted followed a lightning crack spread by a dispersive plasma into a falling tone seconds long, the same stretching the radar’s compressor undoes, done by a medium rather than a design.

Sidelobes, and the price of hiding them #

The compressed spike is not clean. A flat band with sharp edges transforms into a $\sin x/x$ shape, with sidelobes on either side of the peak that fall off slowly — the first at about thirteen decibels below the peak, a twentieth of its power. A target twenty times weaker than a neighbour, in the right place, sits inside that neighbour’s sidelobe and is invisible; a ship next to a small boat, a storm beside a light shower.

The cure is to soften the band’s edges by weighting the filter, emphasising the middle of the sweep and de-emphasising its ends. A Hamming weighting brings the sidelobes down by more than twenty decibels. It costs two things. The spike widens, because a tapered band is effectively narrower, by about one and a half times. And the filter no longer exactly matches the echo, so the peak’s signal-to-noise ratio falls by a decibel or two. Every pulse-compression radar sits at a chosen point on that trade between resolution, sensitivity and the ability to see a weak target beside a strong one, and more elaborate waveforms — nonlinear sweeps that spend more time at the band’s centre, or phase codes whose correlations have uniformly low sidelobes — move the trade without abolishing it.

A speed that looks like a range #

A chirp has one further peculiarity, which is useful or harmful depending on what the radar is for. Along a linear sweep, a shift of frequency and a shift of time are interchangeable: the sweep at frequency $f$ is reached at time $fT/B$ later than at zero. An echo whose frequency has been raised by a Doppler shift $f$ matches the transmitted sweep best when it is slid back in time by $fT/B$, and the matched filter’s spike moves by that amount.

A moving target therefore appears at the wrong range, displaced by an amount proportional to its speed, and the spike shrinks a little because part of the echo’s band has moved out of the filter’s. For most targets the displacement is small — a Doppler shift of a few kilohertz on a sweep of megahertz moves the spike a fraction of its width — and the robustness of the chirp against Doppler is one of its virtues: a spike that moves a little is still a spike, while a phase-coded pulse may lose its compression altogether. Where the displacement matters, the radar sweeps alternately up and down; the two displacements are opposite, their average is the true range, and their difference measures the speed.

The speeds a pulsed radar folds into one found range and speed tied together through the pulse rate. Here they are tied through the pulse’s own shape. The general statement, worked out for radar in the 1950s by Philip Woodward, is the ambiguity function: a map of the matched filter’s response over every combination of delay and Doppler shift, whose total volume is fixed for any waveform. A waveform can be designed to put that volume where the radar can tolerate it — a narrow ridge along a diagonal for a chirp, a thumbtack spike surrounded by a low plateau for a well-chosen code — but it cannot be made to vanish.

Two targets the long pulse merges #

The payoff is in what the radar can see.

Two targets whose echoes overlap almost completely in the raw signal — the second arriving three units of $1/B$ after the first, inside a pulse fifty units long — come out of the matched filter as two clean spikes, each about one unit wide. A plain pulse of the same length would have needed the targets fifty units apart to separate them; a plain pulse short enough to separate them would have carried a fiftieth of the energy and seen a fiftieth as far, or rather a little over a third as far, since a radar’s range grows only as the fourth root of its pulse energy. The chirp sees as far as the long pulse and as sharply as the short one. In numbers: the five-megahertz chirp of fifty microseconds resolves two aircraft thirty metres apart at whatever range its energy reaches, where a plain pulse with the same thirty-metre resolution would have to last two hundred nanoseconds and would carry a two-hundred-and-fiftieth of the energy and reach, with the same transmitter, only a quarter as far.

The pair in the figure is not chosen to flatter the method. The weaker target is at seven-tenths of the stronger one’s amplitude, three units away — far enough that the stronger spike’s mainlobe has fallen to nearly nothing there, close enough that its first sidelobes sit nearby. Bring a much weaker target to that spot and it begins to compete with the sidelobe rather than with the mainlobe, which is the case the weighting of the earlier figure exists to handle. Resolution and dynamic range are separate properties of a compressed pulse, and a radar specification names both: how close two equal targets can be, and how weak a target can be next to a strong one.

The same trick in light #

Optical pulse compression runs the same idea in reverse for a different reason. The most intense laser pulses cannot be amplified directly, because a short pulse at high power damages the amplifier. In chirped-pulse amplification, a short pulse is first stretched into a long chirp with a dispersive element, amplified while it is long and its peak power low, and then compressed back with the opposite dispersion — a pair of diffraction gratings that delay the long wavelengths more than the short. The compressor is the optical matched filter, and the compressed pulse is as short as the band of the original allows, now carrying the energy of the amplifier. Donna Strickland and Gérard Mourou demonstrated it in 1985, and it earned them a share of the 2018 Nobel Prize in physics; every petawatt laser in the world uses it.

The two versions differ in one respect that matters. A radar’s compressor works on the echo, with noise mixed in, and its job is to find a weak signal; an optical compressor works on a strong pulse and its job is to shorten it. But both rely on the same fact: a chirp is a waveform whose duration and bandwidth are independent, and a dispersive delay that runs the chirp backwards concentrates the duration’s energy into the bandwidth’s sharpness. The speed that depends on the length found the dispersion that makes the delay possible; here it is engineered rather than suffered.

What the drawings leave out #

The figures use an ideal linear chirp with an exactly flat band and a rectangular envelope, sampled finely, and a matched filter computed as a direct correlation. Real transmitters have bands that are not flat, envelopes that rise and fall over finite times, and nonlinearities that distort the sweep, all of which raise the sidelobes; the correlation is computed digitally after the echo is sampled, which introduces its own small errors. The targets are points with no extent, no fluctuation and no noise. The Doppler shifts are drawn large, as fractions of the band, to make the coupling visible; real targets give shifts thousands of times smaller relative to the band. The domain of the drawings is a time-bandwidth product of fifty, weightings of none and Hamming, and Doppler shifts up to 0.15 of the band.

Still open: how low sidelobes can be pushed #

For radars that must see weak targets beside strong ones — weather radars measuring light rain beside a thunderstorm core, or radars looking for small boats in heavy sea clutter — the sidelobes of pulse compression are the limiting problem, and the best waveforms and filters now reach sidelobes fifty or sixty decibels below the peak. How far they can go with real transmitters, whose distortions put a floor under every design, and whether adaptive filters that change from one echo to the next can push below that floor without losing sensitivity, are active questions in radar engineering. For a known waveform in white noise the matched filter is optimal; the question is what to do when neither the waveform nor the noise is quite what the theory assumes.

The arithmetic underneath is short. A chirp lasting T and sweeping a band B, passed through the filter matched to it, collapses to a spike about 1/B wide — 0.878/B at half power, compressed by the time-bandwidth product, 50 here — so resolution belongs to bandwidth and energy to duration; the spike’s −13.6 dB sidelobes can be pushed below −37.5 dB at the cost of a spike 1.49 times wider, and a Doppler shift f moves it by fT/B, misplacing a moving target in range. A radar can send a long pulse and still see sharply, because what it measures is not how long the pulse is but how much of the spectrum it covers.