Unambiguous Doppler measurement requires the pulse repetition frequency to be at least four times the target velocity divided by the wavelength.

\(v_{max}=\pm\dfrac{\lambda\,f_{PRF}}{4}\quad\Rightarrow\quad f_{PRF}=\dfrac{4v_{max}}{\lambda}\)

Step 1 — convert the units.

\(v=360\ \text{km/h}=\dfrac{360\times1000}{3600}=100\ \text{m/s}\)

\(\lambda=\dfrac{c}{f}=\dfrac{3\times10^{8}}{12\times10^{9}}=0.025\ \text{m}\)

Step 2 — substitute.

\(f_{PRF}=\dfrac{4\times100}{0.025}=16{,}000\ \text{Hz}=16\ \text{kHz}\)

— option 1.

Where the factor of four comes from — and this is the heart of the question. Two separate factors of two combine.

The first two is in the Doppler shift itself. The target's motion shortens the path on the way out and on the way back, so the shift is doubled compared with a one-way path:

\(f_{d}=\dfrac{2v}{\lambda}\)

The second two is Nyquist. A pulsed radar samples the returning phase once per pulse, so the Doppler frequency is a sampled signal and can only be measured unambiguously up to half the sampling rate:

\(|f_{d}|\le\dfrac{f_{PRF}}{2}\)

Combining the two gives \(2v/\lambda\le f_{PRF}/2\), hence \(f_{PRF}\ge4v/\lambda\).

What happens if the PRF is too low. The Doppler frequency aliases, and a fast target is reported at a wrong — often reversed — velocity. In weather radar this is the notorious velocity folding, where the strongest outbound winds of a tornado appear on the display as inbound.

The unavoidable trade-off. Raising the PRF to measure high velocities shortens the interval between pulses and so reduces the unambiguous range:

\(R_{max}=\dfrac{c}{2f_{PRF}}\)

At 16 kHz that is only 9.4 km. The product of the two limits is fixed at \(c\lambda/8\), which is why weather radars alternate between several PRFs and unfold the ambiguities afterwards.

Hence, the minimum PRF is 16 kHz.