# unambiguous velocity $V_\max$ in radar doppler processing

In literature, $$V_\max = \lambda\cdot \operatorname{PRF} / 2$$; should it be $$V_\max = \lambda \cdot \operatorname{PRF} / 4$$, which would be consistent with FMCW $$V_\max = \lambda / (4 \cdot T_c)$$?

This is because $$\operatorname{PRF} = 2 f_d = 2 \cdot (2 V_\max / \lambda)$$.

• hi Linda, great having you back! I've edited your question to make the formulas easier to read. I think it's identical to what it was before, but it'd be nice if you checked. Mar 23 at 16:51
• Can somebody answer my question here? I am very confused the this factor of 2 difference. Mar 23 at 19:28
• @Linda I posted an answer below. I can post a real example, but it will take some time as I'm running out of it these days. Mar 25 at 3:41

Both are actually correct, but it depends on what Doppler $$f_D$$ frequencies we want to observe.

If we only want to observe positive Doppler frequencies, that is, targets with closing (approaching) radial velocities then the maximum unambiguous observable Doppler span given a pulse-repetition frequency (PRF) is

$$[0, PRF)$$

Thus the maximum observable unambiguous velocity (within one bin given the maximum frequency yielded by the DFT) when $$f_D = PRF$$ is

$$V_{max} = \frac{PRF\lambda}{2}$$

If we want to consider both positive and negative frequencies (we want to observe targets moving towards and away from the radar), the captured Doppler span given a PRF is now

$$[\frac{-PRF}{2}, \frac{PRF}{2})$$

In the same fashion the maximum observable unambiguous velocity when $$f_D = \pm PRF/2$$ is

$$V_{max} = \pm\frac{PRF\lambda}{4}$$

## Example

I think you would also benefit from viewing this answer, which goes into a little more detail. I'm also using the same parameters for the simulation.

To see the effect of our choice in defining the observed Doppler span, we assume the following:

• Target range of 800 m and velocity of 250 (closing) m/s
• Wavelength λ of 0.03 m
• PRF of 50 kHz

Choosing to observe $$[-\frac{PRF}{2}, \frac{PRF}{2})$$ we get the maximum unambiguous velocities we can observe:

$$v_{max} = \pm\frac{PRF \lambda}{4} = \pm\frac{(50 \space kHz)0.03}{4}$$ $$v_{max} = \pm 375 \space m/s$$

We're able to unambiguously observe targets either moving away or towards the radar as long as the target is no faster than 375 m/s as shown in the range-Doppler map above. If the target does go beyond this limit, the target will alias. Let's say the target speeds up to 450 m/s towards the radar:

Performing a measurement here would give you not only a wrong value of its absolute speed, but you would perceive it as going away from you. Not good!

However if we choose to observe the Doppler span as $$[0, PRF)$$ then we get

$$v_{max} = \frac{PRF \lambda}{2} = \frac{(50 \space kHz)0.03}{2}$$ $$v_{max} = 750 \space m/s$$

And forming the range-Doppler map:

Measuring the target now will give us the correct absolute speed and direction, but we're relegated to only being able to observe positive Doppler (closing) targets, albeit with a potentially higher velocity (double in this case).