Radar Range Equation

What is modelled

A pulse radar sends a short pulse, a small part of it returns from the target, and the receiver must tell the echo from its own thermal noise. The radar equation links the power, the antenna, the wavelength, the target size and the distance to the echo strength; the detection theory turns the signal to noise ratio into a probability.

Equations

• Single-pulse SNR = Pt G2 λ2 σ / ((4π)3 R4 k T0 B F L).
• Noise floor k T0 B F, about −174 dBm/Hz plus 10 log B plus the noise figure.
• Required SNR for N integrated pulses (Albersheim): −5 log10 N + [6.2 + 4.54/√(N+0.44)] log10(A + 0.12AB + 1.7B), A = ln(0.62/P_fa), B = ln(P_d/(1−P_d)).
• Exact single-pulse detection Pd = Q1(√(2 SNR), √(−2 ln Pfa)) (Marcum Q).

Checks run in the page

• Monte Carlo of noisy echoes against the Marcum Q integral.
• Albersheim against the exact Marcum result.
• The fourth-power range law, the solved range and the noise floor.
• The gain from pulse integration.

Honest simplifications

The target does not fluctuate, the antenna is pointed at it all the time, propagation is free space with no atmospheric loss, clutter and interference are absent, and the Albersheim equation is a fit that is only accurate for false alarm probabilities between about 1e-7 and 1e-3.

• Press Small drone and see how little range a small cross section gives.
• Double the pulses integrated and read the range gain.
• Raise the noise figure and the losses (Poor receiver) and watch the dashed 90 % ring shrink.