In a range-Doppler map (RDM), you actually have two sample rates that define the two dimensions of the matrix:
- Fast-time dimension. This is usually established by the ADC to sample the pulse return. This tends to be the fastest of the two sample rates, hence the name. This is what is typically given by the symbol $f_s$ and establishes the range dimension of the RDM.
- Slow-time dimension. The second dimension is made up of the number of pulses collected. The fastest you can hope to gather these is at the $PRI$, and thus the sample rate across this dimension is the $PRF$.
So now the RDM captures two Nyquist bands:
- $[-f_s/2, fs/2)$ for the fast-time dimension
- $[-PRF/2, PRF/2)$ for the slow-time dimension
You can see directly from (2) that $\pm PRF/2$ is the maximum Doppler frequency you can measure unambiguously. You can apply the frequency-Doppler conversion formula to get the velocity limits that you can measure.
$$v_{max} = \pm\frac{f_{d_{max}}\lambda}{2} = \pm\frac{PRF\lambda}{4}$$
Example
We're going to start with some already-simulated target returns from a moving target. The pertinent parameters are
- Target range of 800 m and velocity of 250 m/s
- Wavelength $\lambda$ of 0.03 m
- PRF of 50 kHz
- Collected 500 fast-time samples and 256 slow-time samples (which is the number of pulses)
We then form the 500x256 RDM:

The mapping still has to be done in order to determine the target's range and velocity. Let's assume we've already done the range mapping, but now we need to do velocity.
Let's check that this system can unambiguously measure this velocity at the given PRF:
$$v_{max} = \pm\frac{PRF \lambda}{4} = \pm\frac{(50 \space kHz)0.03}{4}$$
$$v_{max} = \pm 375 \space m/s$$
It can, so now lets map the Doppler axis to velocity. Assuming that we did a N-point DFT in the slow-time dimension, the frequency bin size is now:
$$\Delta f = \frac{PRF}{N}$$
And using the Doppler-velocity mapping we get the velocity bin size:
$$\Delta v = \Delta f\frac{\lambda}{4}$$
We now have the bin size itself, and with a little work you can use it to transform the slow-time dimension from $[-PRF/2, PRF/2)$ to $[-v_{max}/2, v_{max}/2)$:

Update
You should be careful in defining the range axis with a negative frequency. This type of range mapping applies only to a very specific type of system that uses stretch-LFM processing, which I doubt you're doing.
For more traditional radars, it makes sense that you can't have a negative range so introducing $-f_s$ is not appropriate. Now, this was my fault since I did state that the fast-time dimension captures the band $[-f_s/2, fs/2)$. This is technically true in the mathematical sense, but is not used directly as shown to define the range axis. The velocity axis defined by $[-PRF/2, PRF/2)$ is still valid.
The calculations that you have won't work. Instead try something like this:
rangeAxis = (1:numRangeBins)*(c/(2*fs)) % Range bin size is c/(2*fs)
dopplerAxis = (-numDopplerBins/2:numDopplerBins/2 - 1).*PRF/numDopplerBins;
velocityAxis = dopplerAxis*lambda/2;
With the right values this should yield the correct range and velocity axes you're looking for.
tau
whentau
changes with time, so perhaps the plot is correct, and I've misunderstood it? I appreciate your help, thank you $\endgroup$