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Sensors & detection
Radar Doppler shift
Frequency shift of an echo from a target closing at speed v.
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- Result
- fd [kHz]
v is the closing speed of the target along the line of sight, f₀ the radar frequency in GHz and c the speed of light. The shift is doubled because the target both receives and re-transmits the signal at a shifted frequency. For a target moving at an angle, v is the part of its velocity along the line of sight, v cos θ.
Frequency shift of an echo from a target closing at speed v. It lets a radar measure radial speed and pick moving targets out of clutter.
Calculator
- Wavelength c/f₀2.998 cm
- Doppler shift per 1 m/s of closing speed66.71 Hz
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- At 10 GHz each 1 m/s of closing speed gives 66.7 Hz of shift, so a 300 m/s aircraft produces 20 kHz and a walking person (1.5 m/s) about 100 Hz.
- A target moving across the line of sight (v cos θ = 0) gives no Doppler shift, so a radar can miss a target flying a circle around it.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (kHz) | Local step | Change in result |
|---|---|---|---|---|---|---|
| v Closing speed | 1 to 3,000 m/s | 0.06671 | 200.1 | 200.1 | +10% | +10 % |
| f0 Radar frequency | 0.1 to 100 GHz | 0.2001 | 200.1 | 199.9 | +10% | +10 % |
Sweep of Closing speed
| Closing speed (m/s) | Doppler shift (kHz) |
|---|---|
| 1 | 0.06671 |
| 2.72 | 0.1815 |
| 7.401 | 0.4937 |
| 20.13 | 1.343 |
| 54.77 | 3.654 |
| 149 | 9.941 |
| 405.4 | 27.04 |
| 1,103 | 73.57 |
| 3,000 | 200.1 |
Sweep one variable
Variables and how each one contributes
Doppler shift
What the equation gives you. Everything else feeds it.
Closing speed
The closing speed along the line from the radar to the target. Directly proportional: double the closing speed gives double the Doppler shift.
Radar frequency
The transmitted radar frequency, in GHz. Directly proportional: a higher frequency gives a larger shift for the same speed, so shorter wavelengths measure speed more finely.
Worked example with the values above
- Convert frequency: f₀ = 10 GHz = 1×1010 Hz.
- Numerator: 2 v f₀ = 2 × 300 × 1×1010 = 6×1012.
- fd = 6×1012 / 2.998×108 = 20,010 Hz = 20.01 kHz.
Assumptions and limits
- Closing speed much less than the speed of light, so the non-relativistic form applies.
- Constant radial speed during the measurement, and a stationary radar.
Common mistakes
- Forgetting the factor of 2 (this gives the one-way shift).
- Using the target’s total speed rather than its component along the line of sight.
Statistics and uncertainty
f_d ∝ v f₀, so a 1 % error in the measured shift gives a 1 % error in speed. The accuracy of the shift itself is limited to about 1/T for a dwell time T, so measuring speed to ±1 m/s at 10 GHz needs a dwell of 15 ms or longer.
Where it is used
- Measuring target speed and separating movers from ground clutter.
- Speed guns and weather radar.
Maths you need
- Doppler effect: frequency shift from relative motion
- Wavelength and frequency: c = fλ