1. The Echo in the Canyon: Range from Time of Flight

1.1 The one-line physics

You are standing on the rim of a canyon. You shout. A second later you hear your own voice come back. How far away is the canyon wall?

Sound travels at about 343 meters per second, so in one second it covered 343 meters. But your voice did the trip twice, once to the wall and once back, so the wall is at 343/2 = 171 meters. Easy.

Replace sound with a radio pulse and replace the speed of sound with the speed of light, and you have the entire physical principle of radar:

$R = \frac{c \tau}{2}$

where $\tau$ is the round-trip delay, $c \approx 3 \times 10^8$ m/s is the speed of light, and the factor of 2 is there because the signal traveled out and back.

The numbers feel wrong at first. A target ten kilometers away gives a round-trip delay of $\tau = 2R/c = 2 \times 10^4/3\times 10^8 \approx 67$ microseconds. A target two hundred kilometers away, the kind of range an air-traffic-control radar wants to cover, gives a round-trip delay of about 1.33 milliseconds. Sixty-seven microseconds and 1.33 milliseconds are not exotic times; they are easy times to measure with cheap modern electronics. The radar age started in the 1930s precisely when electronics fast enough to measure them became practical.

Why exactly half the speed of light? People sometimes forget the factor of 2 and report twice the range. The factor is physics, not a convention. Light travels at $c$ in vacuum, and approximately $c$ in dry air at the relevant frequencies. The pulse leaves the antenna, takes $R/c$ seconds to reach the target, scatters, and takes another $R/c$ seconds to return. Round-trip time is $2R/c$, so range is $c\tau/2$. Forget the 2 and you double-count the path; you would say a 100 km target is at 200 km.

1.2 A short tour of the early history

The earliest patent for what we would call radar was issued to Christian Hülsmeyer in Germany in 1904. His "Telemobiloskop" bounced spark-gap radio waves off ships and rang a bell when the echo came back. Demonstrated on the Rhine, then forgotten. Nobody bought one. Hülsmeyer was forty years too early; the world did not yet need to detect aircraft.

The world started needing it in the 1930s. By 1935, Britain's Robert Watson-Watt and Arnold Wilkins had built the first operational aircraft-detection radar, the Chain Home system, using long wavelengths around 20 to 50 MHz. Ugly antennas, modest range, but it could see Luftwaffe formations crossing the channel. The Battle of Britain in 1940 was won partly because the RAF had radar and the Luftwaffe did not. A few years later, the British cavity magnetron, developed at Birmingham, made centimeter-wavelength radar practical, and the U.S. MIT Radiation Laboratory turned that one device into a hundred different operational radar systems by 1945.

After WWII, radar fanned out: air traffic control, weather, ship navigation, satellite tracking, ground-penetrating prospecting, traffic enforcement, and finally mass-market automotive collision avoidance. The most recent radar in your daily life might be a 60-GHz chip inside a Pixel phone tracking hand gestures, or a 77-GHz radar in your car's bumper deciding whether to slam on the brakes. The basic principle, send a wave and time the echo, is unchanged from 1904.

1.3 The block diagram, simplified

Every radar in existence, from spark-gap to active electronically scanned array, has a recognizably similar architecture.

              ┌────────────┐
              │   Antenna  │
              └─────┬──────┘
                    │
              ┌─────┴──────┐
              │  Duplexer  │  TX/RX switch (T/R)
              │  (T/R)     │
              └─┬────────┬─┘
       transmit │        │ receive
                │        │
       ┌────────┴──┐  ┌──┴──────────┐
       │ Power     │  │ LNA + mixer │
       │ amplifier │  │             │
       │ (klystron,│  └──────┬──────┘
       │  TWT,     │         │
       │  GaN)     │  ┌──────┴──────┐
       └────┬──────┘  │ IF amp /    │
            │         │ matched     │
       ┌────┴────┐    │ filter      │
       │ Signal  │    └──────┬──────┘
       │ source  │           │
       │ (pulse, │    ┌──────┴──────┐
       │ FMCW)   │    │ Detector    │
       └────┬────┘    │ (envelope   │
            │         │  or I/Q)    │
       ┌────┴────┐    └──────┬──────┘
       │ Master  │           │
       │ clock / │    ┌──────┴──────┐
       │ trigger ├────┤ Display /   │
       └─────────┘    │ DSP /       │
                      │ tracker     │
                      └─────────────┘

Read it from left and top: a master clock triggers the signal source (a pulse or a chirp), which feeds a high-power amplifier (a klystron, a magnetron, a traveling-wave tube, or in modern radars a solid-state GaN or GaAs power module). The amplified signal goes through a duplexer, a switch or circulator that connects the antenna to the transmitter while the pulse is going out and to the receiver during the listening period. The same antenna both transmits and receives. After the antenna, the echo is amplified by a low-noise amplifier (LNA), down-converted to an intermediate frequency by a mixer, passed through the IF amplifier and matched filter, detected, and sent to a display or signal processor.

The duplexer matters because the transmit pulse can be a megawatt and the echo a femtowatt; they share the same antenna and they had better not share the same wire to the receiver. Old radars used a gas-filled spark-gap "TR tube" that ionized and shorted out during the high-power pulse, then de-ionized in time for the echo. Modern radars use ferrite circulators that route signals one way around a three-port loop. We met circulators in Chapter 18.

1.4 Range resolution: how thin a pulse must be

Distinguishing two close targets in range is a different problem from finding any one of them. If the radar transmits a pulse of width $\tau_p$ seconds, the pulse occupies a length $c\tau_p$ in space, and round-trip resolution is half that. Two targets within $\Delta R = c\tau_p/2$ produce overlapping echoes:

$\Delta R = \frac{c \tau_p}{2}$

For a 1-μs pulse, $\Delta R = 150$ m. For a 10-ns pulse, 1.5 m. For a 100-ps effective pulse (the kind automotive FMCW radars achieve via 1-GHz chirp bandwidth), 15 mm.

Why does pulse width set resolution? Repeat the canyon-echo experiment, but sing a long sustained note instead of a clap. As long as you are still singing, the outgoing voice covers any returning voice; you cannot tell when the reflection ended. Stop singing and you hear the trailing edge of the echo. Two walls at different distances produce echoes that overlap in time so long as your note is long. Sing a clap, and the two echoes separate. Pulse width is the radar's "clap duration."

A tension: short pulses have low energy, so they detect more poorly. If transmit power is fixed, shortening the pulse drops its energy proportionally and detection range falls off. Pulse compression resolves this: transmit a long pulse with a wide-bandwidth chirp inside it, then matched-filter to compress the long pulse back down to a short apparent duration, recovering both range resolution and energy. More in Section 5.

1.5 The maximum unambiguous range

There is a different, less obvious limit on range. After you fire a pulse, you have to listen long enough for echoes from the farthest target to return before you fire the next pulse. If the next pulse goes out before all echoes from the previous one have come back, the receiver cannot tell which pulse a late echo belonged to.

For a pulse repetition frequency PRF, the time between pulses is $T_{PRI} = 1/\text{PRF}$. The maximum unambiguous range is

$R_{unamb} = \frac{c}{2\,\text{PRF}}$

A 1-kHz PRF gives 150 km of unambiguous range. A 10-kHz PRF gives only 15 km. A 100-kHz PRF, which automotive radars sometimes use, gives only 1.5 km, which is fine because cars are not interested in airplanes.

The trade-off is brutal. High PRF improves Doppler unambiguity (more on that later) and integration time, but truncates the unambiguous range. Low PRF gives long range but ambiguous Doppler. Modern radars solve this with multiple PRFs, alternating between them and resolving ambiguities in software.

   Transmit pulses (PRI = 1/PRF):
   ┃                    ┃                    ┃                    ┃
   ┃         T_PRI      ┃         T_PRI      ┃         T_PRI      ┃
   ───────────────────────────────────────────────────────────────────  t
 
   Echo from target at range R, delay τ = 2R/c:
 
   ┃                    ┃                    ┃                    ┃
   ┃        ┊  ┃        ┊  ┃        ┊  ┃        ┊  ┃
   ┃        ⤺ τ ⤻       ┊  ┃        ⤺ τ ⤻      ┊  ┃
   ───────────────────────────────────────────────────────────────────  t
 
   If τ > T_PRI, the echo arrives after the next pulse went out.
   The radar misattributes the echo to the new pulse, getting wrong R.

This is sometimes called the second-time-around echo problem. A target at 200 km observed by a 1-kHz PRF radar generates an echo with $\tau = 1.33$ ms; the next transmit pulse goes out at 1.0 ms, so the echo arrives 0.33 ms after the new pulse. The radar interprets that as a target at $0.33\text{ ms} \times c/2 = 50$ km. The target is reported at completely wrong range. Long-range air-defense radars use PRF staggering specifically to detect this and unfold the ambiguity.

1.6 Worked example: a typical air traffic control radar

A primary surveillance radar at L-band (1.3 GHz) with a 1-μs pulse and a 1-kHz PRF:

• Range resolution: $\Delta R = c\tau_p/2 = 150$ m.
• Unambiguous range: $R_{unamb} = c/(2 \cdot 1000) = 150$ km (~80 nautical miles).
• Duty cycle: $\tau_p/T_{PRI} = 10^{-6}/10^{-3} = 0.1$%.
• 1 MW peak power gives 1 kW average. The average is what determines AC mains feed and cooling.

The FAA's ASR-9 airport surveillance radar runs in roughly this regime.