Everything an EW specialist does — recognising a radar signal, converting a frequency, deriving a distance from a travel time — rests on a small number of arithmetic foundations. This chapter lays those foundations. Anyone who has mastered the concepts quantity, unit, frequency and the powers of 10 can follow every later chapter without stumbling over the maths.
The material is deliberately practical: no derivations, but the formulas and conversions you actually use in the field and in the classroom.
Quantity and unit
A quantity is a property you can measure or calculate: length, time, velocity, frequency, power, temperature. A unit is the measure in which you express that quantity: metre, second, metre per second, hertz, watt, kelvin. A measured value without a unit is meaningless — "the distance is 5" says nothing, "the distance is 5 km" does.
Internationally it has been agreed to work with the SI system (Système International). The quantities most important for EW and their SI units:
| Quantity | Symbol | SI unit | Abbreviation |
|---|---|---|---|
| Length / distance | l, d, R | metre | m |
| Time | t | second | s |
| Frequency | f | hertz | Hz |
| Velocity | v | metre per second | m/s |
| Power | P | watt | W |
| Temperature | T | kelvin | K |
| Wavelength | λ (lambda) | metre | m |
| Angle | θ, φ | degree / radian | ° / rad |
Write the unit down with every calculation. Many calculation errors in practice are really unit errors: confusing kilometres with metres, megahertz with gigahertz.
Frequency
Frequency is how many times per second a wave goes up and down. The higher the frequency, the shorter the wave (the wavelength). Low frequencies have waves several metres long; radar works with waves of a few centimetres.
In practiceAntennas are usually a sizeable fraction of the wavelength. A VHF combat net radio (30–88 MHz) therefore has a whip antenna one to several metres long, while an X-band radar (10 GHz, a wavelength of 3 cm) fits hundreds of small radiating elements into the nose of a fighter. A GPS antenna fits inside a phone.
Frequency (f) is the number of times a periodic phenomenon repeats per second. The unit is the hertz (Hz): 1 Hz = 1 oscillation per second. For EW, frequency is the central quantity: the entire electromagnetic spectrum is divided up by frequency, and radars, radios and jammers are characterised by the frequency at which they operate.
The time one complete oscillation takes is called the period (T). Frequency and period are each other's inverse:
A signal has a period of 1 ms (0.001 s). The frequency is then f = 1 / 0.001 = 1,000 Hz = 1 kHz. Conversely, a radar operating at 10 GHz has a period of T = 1 / 10,000,000,000 = 0.1 nanosecond.
Commonly used derived units: 1 kHz = a thousand Hz, 1 MHz = a million Hz, 1 GHz = a billion Hz, 1 THz = a trillion Hz. Military radars typically operate in the MHz to GHz range; infrared and light sit in the THz range.
Large numbers, number names and powers of 10
In the EM spectrum you work with extremely large numbers (gigahertz: 1,000,000,000 Hz) and extremely small ones (nanoseconds: 0.000000001 s). Writing them out is clumsy and error-prone; that is why we use powers of 10 and SI prefixes.
| Prefix | Symbol | Power of 10 | Written out | Number name |
|---|---|---|---|---|
| tera | T | 1012 | 1,000,000,000,000 | trillion |
| giga | G | 109 | 1,000,000,000 | billion |
| mega | M | 106 | 1,000,000 | million |
| kilo | k | 103 | 1,000 | thousand |
| — | — | 100 | 1 | one |
| milli | m | 10-3 | 0.001 | thousandth |
| micro | µ | 10-6 | 0.000001 | millionth |
| nano | n | 10-9 | 0.000000001 | billionth |
| pico | p | 10-12 | 0.000000000001 | trillionth |
Calculating with powers of 10 follows two simple rules: when multiplying you add the exponents (103 × 106 = 109), when dividing you subtract them (109 / 106 = 103).
A radar transmits at 9.4 GHz. That is 9.4 × 109 Hz = 9,400 MHz = 9,400,000 kHz. A pulse width of 1.5 µs is 1.5 × 10-6 s.
The Dutch miljard (109) is a billion in English, and the Dutch biljoen (1012) is a trillion in English. In international documentation and in a NATO context this is a classic source of errors.
Speed of light
Electromagnetic waves — radio waves, radar, infrared, light, laser — propagate in a vacuum, and to a close approximation also in air, at the speed of light c:
This number is of direct practical importance to the EW specialist, for two reasons:
- Range measurement: radar and laser rangefinders measure the travel time of a signal. Because c is known, travel time translates directly into distance (see Radar).
- The relationship between frequency and wavelength: frequency f and wavelength λ are linked through c.
An X-band radar operates at 10 GHz. The wavelength is λ = (3 × 108) / (10 × 109) = 0.03 m = 3 cm. A VHF radio at 150 MHz has λ = (3 × 108) / (150 × 106) = 2 m.
High frequency = short wavelength; low frequency = long wavelength. A handy mnemonic: 300 MHz ↔ 1 m. From there you scale easily: 3 GHz ↔ 10 cm, 30 GHz ↔ 1 cm.
Temperature
At first sight temperature seems out of place in an EW course, but it is essential for understanding infrared (Infrared): every object with a temperature above absolute zero emits electromagnetic radiation, and its amount and wavelength depend directly on the temperature.
There are three scales you need to know:
| Scale | Symbol | Zero point | Use |
|---|---|---|---|
| Celsius | °C | melting point of ice | everyday use |
| Kelvin | K | absolute zero (-273.15 °C) | science and engineering, IR calculations |
| Fahrenheit | °F | — | American documentation |
Absolute zero (0 K = −273.15 °C) is the lowest possible temperature; there all molecular motion stops and an object emits no thermal radiation at all. Everything above it radiates — a human (~310 K), a vehicle engine (~350–400 K), a jet engine exhaust (~700–1000 K). The hotter the object, the more IR radiation and the shorter the peak wavelength. This is the physical principle behind thermal imaging cameras and IR-guided missiles.
An exhaust plume of 627 °C is 627 + 273 = 900 K. In Infrared you will see that such a source peaks around 3.2 µm — right in the operating band of many heat-seeking missiles.
Azimuth, elevation and range
To fix the position of a target relative to a sensor we use three coordinates — together the spherical coordinate system of the radar world:
- Azimuth (Az): the horizontal direction to the target, measured in degrees clockwise from north (000° = north, 090° = east, 180° = south, 270° = west).
- Elevation (El): the vertical angle to the target, measured from the horizontal plane. 0° = horizon, 90° = directly above the sensor.
- Range (R): the slant distance (slant range) from the sensor to the target, along the straight line of sight.
With these three values the position of a target is fully determined. A 2D radar measures only azimuth and range; a 3D radar also measures elevation and can calculate the target's height from it (see 2D and 3D radar).
In reports, azimuth is always given in three digits: "target at 045, range 12 kilometres". Remember that range is the slant distance: a target at 10 km range and 30° elevation is closer than 10 km measured along the ground.
Radial velocity
Only the part of the speed that is directly towards you or away from you counts as radial velocity. A car overtaking you on the motorway hardly gets any closer at the moment it is right beside you — even though it is doing 130.
In practiceA fighter pilot being tracked by a pulse-Doppler radar can deliberately fly at right angles to that radar: the notch or beaming manoeuvre. The radial velocity then drops to almost zero and the radar throws the echo away together with the stationary terrain (Pulse-Doppler radar and Moving Target Indicator).
The radial velocity (vr) is the component of a target's velocity along the sensor's line of sight — the speed at which the target approaches or recedes from the sensor. This is not the same as the target's true (ground) speed.
A target flying straight at the radar has a radial velocity equal to its true speed. A target flying exactly across the line of sight (tangentially) has a radial velocity of zero — however fast it is going.
Why does this matter? Because Doppler radars (Doppler effect and Pulse-Doppler radar and Moving Target Indicator) measure radial velocity only. An aircraft flying tangentially produces no Doppler shift and can therefore "disappear" from the velocity filters of a pulse-Doppler radar — a principle exploited in practice with the so-called notch or beaming manoeuvre.