3dFFT to compute angle of arrival in FCMW radars

Question

I have a simulated doppler-range map of a 1Tx - 4Tx (azimuthally displaced) FMCW radar and I am doing something wrong to calculate the angle of arrival (AoA).

The scenario is the following. The radar is looking at two small spheres (0.1 m each in diameter) positioned on the same plane of the radar. One sphere is at a distance of 10 meters away from the radar, and is located 2 meters on the left of the radar boresight. The second small sphere is located 7m away from the radar and 1 m to the right of the boresight of the radar. The 2DFFT range-doppler output is fine and makes sense with the scenario, as you might see from the picture, representing the first antenna (Rx0) 2D FFT output amplitude (10*np.log10(np.abs(FFToutput))), with Hann window applied.

The issue I have is when I want to apply a new FFT to compute the AoA of the signal. As far as I understood, I have to compute the FFT on the doppler-range FFT across the virtual antennas (in my case they are the 4 Rx antennas).

To compute this 3rd FFT, what I do is the following:

1. I start with an array of complex numbers representing the range-doppler output with Hann windowing (size [dim_va x dim_vel x dim_range], respectively dimensions of virtual antennas array -in my case only the 4 Rx's-, resolution of velocity, resolution of range, these last two based on the chosen 2D FFT size),
2. compute the probable objects position in the range-doppler map, with a 1D OS-CFAR. The CFAR is applied on the non-coherent sum of the the doppler-range complex vector (basically the sum of the doppler-range real parts across antennas), hence I have a 1D array dim_vel x dim_range long.
3. compute the max of the peak for every object candidate peak found in the previous step, so to have only a single point per "candidate object".
4. compute 256-points FFT on the matrix given by the doppler-range map across antennas, considering the candidate targets only: basically, for every peak (object candidate) found, I use its index to isolate the respective doppler-range cell for each antenna. I have now a 2D matrix of size dim_va x dim_candidates.
5. the maximum of the 3D FFTs (one per candidate target/object) should be the angle of arrival.

Here is some Python code that should do the trick:

n_fft = 256 #size of fft
AoA = np.fft.fft(rv_tg_va, n_fft, axis=0)/rv_tg_va.shape[1] #rv_tg_va is the range-doppler complex array for the target candidates of size numberOfCandidates * n_fft
AoA = np.fft.fftshift(AoA)
# AoA_magnitude = 10.*np.log10(np.abs(AoA))
wavelength = 0.0039189 # 76.5 GHz
d = wavelength/2. # distance btw receivers 
f = np.arcsin(np.linspace(-0.5*wavelength/d , 0.5*wavelength/d , AoA.shape[0]))
maxFFT_idx = np.argmax(AoA, axis=0) #where the angles are supposed to be

plt.plot(np.rad2deg(f), AoA)
plt.grid()
plt.show()

In this picture, the maxima should be the AoA of the various candidates. Unfortunately, the identified maximum is around 5deg, that is not correct and in any case it is only a single value for the two spheres in front of the Radar.

Do you know what I am doing wrong?

Answer 1

This is an issue of angular resolution. At boresight angular resolution is 2/num_antenna in radians. For a 4 antenna system that's ~28 degree angular resolution. Given the reflectors you synthesized your system will find a single angular peak at the "center of gravity" of the reflectors.