# Time-Delay beamforming delay computation

I'm trying to implement a beamforming algorithm in python and I'm stuck with the time-delay computation for a rectangular array.

I'm using the following equation to compute the delays based on the desired steering angle : $$d_{i,j} = \frac{1}{c}\sin(\theta)(\cos(\phi)x_{i,j} + \sin(\phi)y_{i,j})$$

Where $d_{i,j}$ is the delay for the element $(i,j)$, $(\theta,\phi)$ is the azimuth/elevation steering angle, and $x_{i,j},y_{i,j}$ are the coordinates of the element $(i,j)$.

I am using Matlab's PhasedArray toolbox as a ground truth to verify my work, but I'm getting different results.

Here is the Matlab code I use to get the time delays :

array = phased.URA('Size', [4,4],...
'ElementSpacing', 0.042);
delay = phased.ElementDelay('SensorArray',array, 'PropagationSpeed', 340.0);
tau = delay([-30;-20])*fs


And here the Python equivalent :

ang_dft = np.array([-30, -20], float)

myArray = pa.URA(4, 4, 0.042, 0.042)
TDBeamformer = bf.TimeDelayBeamformer(myArray, ang_dft, 340.0, 8000.0)

myArray.plot_array()
TDBeamformer.compute_delays()

print(TDBeamformer.sec2spl(TDBeamformer.delay))

plt.show()


Where the compute_delays method is the following :

def compute_delays(self):
for x in range(0, self.array.sizeX):
for y in range(0, self.array.sizeY):
self.delay[y, x] = (1/self.c *
np.sin(np.pi / 180. * self.steeringAngle) *
(np.cos(np.pi / 180. * self.steeringAngle) * self.array.coordinatesX[y, x] +
np.sin(np.pi / 180. * self.steeringAngle) * self.array.coordinatesY[y, x]))


Could you help me pinpoint where the problem lies? Is my equation wrong (I found it from this website)? Is it a geometry problem in the code?

• to debug these kinds of problems, it's usually the best to start with the simplest geometry and delays and check if both codes produce the same (and the same to what you expect). Then go step by step to more complex parameters. – Maximilian Matthé Jun 27 '18 at 11:50
• Yes, I actually have tried that and linear arrrays actually give coherent results. Things start to go bad when there's more than one row of elements – Florent Jun 28 '18 at 13:46