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R to the fourth: why doubling radar range costs sixteen times the power

Doubling a radar’s range takes sixteen times the transmit power, because the signal spreads out twice. The pulse thins with the square of the distance on its way to the target, and the echo thins with the square again on its way back. Received power therefore falls with R⁴, and every doubling of range costs a factor of 16, or 12 dB.

The numbers in this post come from the radar calculators, each with its formula and sources:

• Radar range equation calculator maximum range from power, gain, RCS and receiver sensitivity
• Radar horizon calculator how far a radar sees over the curved earth
• PRF calculator: unambiguous range, blind speeds and duty cycle unambiguous range and velocity, blind speeds, duty cycle
• Radar resolution calculator range resolution from pulse or bandwidth, cross-range from the beam
• FMCW radar calculator chirp bandwidth, beat frequency and maximum range of mmWave sensors
• Radar cross section (RCS) calculator: m² ↔ dBsm m² to dBsm, reference targets and simple shapes
• Weather radar dBZ to rain rate converter weather radar reflectivity to rain rate and back
• Doppler shift calculator one-way, and the two-way mode for radar echoes and speed guns
• Radar and radio frequency band designations IEEE letter bands, ITU band numbers and NATO letters

Where does the fourth power come from?

From two trips through the inverse-square law. A transmitter’s power spreads over a sphere, so the power density at the target falls with R². The target intercepts a slice of it, set by its radar cross-section, and re-radiates it in all directions. That echo spreads over a new sphere on the way back and loses R² a second time. The radar range equation multiplies the two: P_r = P_t·G²·λ²·σ / ((4π)³·R⁴).

Take a typical marine X-band radar: 12 kW peak at 9.41 GHz, a 31 dBi antenna, and a 10 m² cabin cruiser as the target. With a receiver that needs −92.0 dBm, the boat shows up to 12.5 km. The chart plots its echo against range. Moving the boat from 12 km to 6 km makes the echo 12.0 dB stronger, which is the same factor of 16 seen from the other side.

What does doubling the power actually buy?

About 19 % more range. Twice the power stretches the marine radar from 12.5 to 14.8 km, and ten times the power multiplies the range by only 1.78. Power is the most expensive lever there is. The others act on the same equation and are often cheaper.

• Antenna gain counts twice. A monostatic radar transmits and receives with the same antenna, so 3 dB more gain is worth 6 dB, the same as four times the power. The marine radar would reach 17.6 km.
• Target size counts once, as a fourth root. At 1 m², about the size of a person, the same radar reaches 7.0 km. A 100 m² target shows up at 22.2 km, one of 0.01 m² at 2.2 km. Every factor of ten in RCS is 10 dBsm and a factor of 1.78 in range.
• Sensitivity counts like power. Every decibel the receiver gains, through a lower noise figure, a narrower bandwidth or by integrating several pulses, is worth exactly one decibel of transmit power.

Can more power beat the horizon?

No. A radar sees along straight lines, bent slightly by the atmosphere, and the earth curves away beneath them. With the usual 4/3 effective earth radius, a radar 15 m above the sea sees a boat whose top is 3 m high out to 23.1 km. Our 192 kW radar would detect that boat at 25.0 km, so nearly 2 km of the sixteenfold power are spent behind the horizon.

Height works where power cannot. An air-surveillance radar 30 m up sees an aircraft at 10,000 ft (3,048 m) out to 250.1 km. At 200 km a target must fly above 1,853 m to be seen at all, however strong the transmitter.

Why must a long-range radar pulse slowly?

Because each echo has to come home before the next pulse leaves. Otherwise a distant target is read as a near echo of the following pulse. The maximum unambiguous range is c/(2·PRF). The US NEXRAD weather radars at 2.8 GHz measure without ambiguity out to 471.4 km at a PRF of 318 Hz, and only to 115.3 km at 1300 Hz.

The slow pulse rate has a price. A pulse-Doppler radar samples the phase of each echo once per pulse, so the radial speed it can read is limited to ±λ·PRF/4. That is ±8.51 m/s at 318 Hz and ±34.80 m/s at 1300 Hz. The product of the two limits, c·λ/8, does not depend on the PRF at all. That is why weather radars such as NEXRAD combine scans at a low and a high PRF.

The calculator uses the coherent ±λ·PRF/4 convention. Texts that quote λ·PRF/2 mean the full span from −v to +v. The physics is the same.

Modern radars beat the fourth power with signal processing rather than raw power. Pulse compression codes a long pulse with a linear FM chirp or a phase code, and a matched filter squeezes the echo into a short spike. Resolution then follows the bandwidth, c/(2B), while the long pulse carries the energy. Against a plain pulse with the same 15.0 m resolution, a 10 µs pulse with 10 MHz gains B·τ = 100, or 20 dB. Under R⁴ those 20 dB buy a factor of (B·τ)^1/4 = 3.16 in range. Its blind range of c·τ/2 = 1.5 km is covered by a short fill pulse. Coherent integration of N pulses raises the signal-to-noise ratio up to N times, so 16 pulses add 10·log₁₀(16) = 12.0 dB, exactly what doubling the range costs. Doppler and MTI processing separate moving targets from sea and ground clutter. Electronically scanned arrays dwell longer where a target is suspected, and low-noise receivers lower the detection threshold decibel for decibel. All of these move the left-hand side of the budget. R⁴ still sets the slope.

The radar resolution calculator gives c/(2B) for any bandwidth, and c·τ/2 for a plain pulse.

Does a longer range cost resolution as well?

Across the beam, yes. Range resolution depends only on the pulse length τ and equals c·τ/2, 149.9 m for a 1 µs pulse at any distance. Cross-range resolution is the width of the beam at the target, so it grows with range. A 1.23° beam is 215 m wide at 10 km and 429 m at 20 km. At 20 km, two boats side by side merge into one echo if they are closer than that.

Long range wants long pulses, because detection depends on the energy in each pulse, and long pulses smear range. Pulse compression and FMCW escape this by setting resolution through bandwidth, c/(2B). An automotive 77 GHz sensor with a 4 GHz chirp resolves 3.75 cm. With a 10 Msps ADC and a 40 µs chirp, its maximum range is 13.5 m. Spreading the same chirp over 1 GHz stretches that to 54.0 m at 15.0 cm resolution. 77 GHz lies in the IEEE W band, 75 to 110 GHz.

R⁴ still applies. With the calculator’s typical automotive preset, a car of 10 m² is detectable out to 99 m and a 1 m² pedestrian to 56 m. In the short-range chirp above, the ADC limit of 13.5 m bites long before R⁴ does.

What do weather radar and speed guns add to the picture?

Weather radar turns the echo into a rain rate. It measures the reflectivity factor Z on a decibel scale, dBZ, and a Z-R relation converts it. With the Marshall–Palmer default Z = 200·R^1.6, 20 dBZ is 0.65 mm/h, 40 dBZ is 11.5 mm/h and 50 dBZ is 48.6 mm/h. The NEXRAD default, 300·R^1.4, reads 40 dBZ as 12.2 mm/h. The relation describes a typical drop-size distribution, so the choice matters more than the decimals.

A speed gun is the two-way case of the Doppler shift. The car receives a shifted wave and sends it back as a moving source, so the shift is applied twice, 2·v·f/c instead of v·f/c. At 24.15 GHz in K band, 100 km/h gives 4,475 Hz, twice the one-way 2,238 Hz. X-band guns at 10.525 GHz see 1,950 Hz. Measured 10° off the line of travel, the cosine effect makes the gun read 98.5 km/h, so the error falls in the driver’s favour.

Where does real terrain come in?

The radar horizon assumes a smooth earth. Real coastlines, ridges and buildings cut the view much earlier, and the shadow they cast is exactly what a line-of-sight viewshed shows. In the coverage simulator you can set a mast height, drop the radar on the map and see where a target of a given height is visible over real terrain. For the arithmetic, all radar calculators are on the tools page.