Compensating for the Reference Transmitter Channel

In this tutorial, we will try to visualize the received phases of the receive antennas and apply a classical AoA estimation method to a DICHASUS dataset. We will see that before we can do any of this, we need to compensate for the constant receiver-specific carrier phase offsets (CPOs) and sampling time offsets (STOs).

But wait: Isn't it strange that there are any phase and time offsets in the first place? Looking at an overview of the system architecture, you can see that the antenna array receivers (AARX) should be able to receive both the reference transmitter (REFTX) for synchronization and mobile "main" transmitter (MOBTX) for channel measurements. Isn't the whole purpose of the over-the-air synchronization in DICHASUS to make all receivers coherent in frequency, time and also in phase?

Well, it's true that all receivers (AARX) are synchronous to the reference signal, also called synchronization broadcast. However, they can only be synchronous to their respective received synchronization signal, not to the one that was actually transmitted. In other words, the channel between the reference transmitter (REFTX) and each receiver (AARX) has an impact on the synchronization!

While almost all receivers should have a good (line-of-sight or, at least, low-attenuation) channel to the REFTX, each REFTX-receiver path may experience a different phase shift and path delay. This is because the REFTX signal still has to pass through a real-world physical environment channel, with different path lengths and possibly even multipath propagation.

As usual, let's fire up our Python / TensorFlow / NumPy environment and import the dataset. Since we are not doing any deep learning in this tutorial, it makes sense to already convert the CSI from a real / imaginary part format to a complex-valued tensor. Moreover, to simplify our mental model of the subcarriers, we can already apply the tf.signal.fftshift function so that subcarriers are order from lowest frequency (at index 0) to highest frequency (at index 1023).

In the following, we will create three datasets containing datapoints which are located in proxmity of the three afforementioned locations (frontal, left and right). We can do this by filtering the dataset according to a certain predicate that determines whether any given datapoint is within a small radius of the respective location.

Now that we have channel coefficient vectors for each of the three locations, we only need to plot the phases in a way that effectively generates a frontal view of the antenna array.

To do that, though, we first have to figure out where to draw the measured phases, i.e., how to assign each of the phase values to an antenna in the array. All the required information for this is given at the dataset's description page, namely in the antenna configuration section.

Most importantly, this is where you will find the "channel assignments" table. This table contains the mapping from the channel of the CSI vector (i.e., its antenna index in the CSI tensor, here in the range 0-31) to the position of the antenna in the array, as seen from a viewpoint in front of the array. For dichasus-015x, these assignments are also illustrated in the picture. Please note that the assignments may be different from one dataset to another!

plot_array_phases(points_frontal, 512, 8, "Frontal")
plot_array_phases(points_left, 512, 8, "Left")
plot_array_phases(points_right, 512, 8, "Right")

The results we obtain are rather... disappointing. To put it bluntly, the phases we measure appear to be completely random, there is no visible pattern whatsoever. Even worse, just looking at the phase plots, there is no way on earth we would be able to distinguish the three different transmitter positions.

Admittedly, this is not really unexpected: We already established that each antenna has a constant phase and time offset that we have not compensated for up until this point. The seemingly random phase plots only illustrate this point once again.

So, how exactly do the phase and time offsets manifest themselves in the measured channel coefficients? To understand this, we must first recap some basics about channel coefficients, modulation and fourier transforms.

points_frontal_calibrated = points_frontal.map(apply_calibration)
points_left_calibrated = points_left.map(apply_calibration)
points_right_calibrated = points_right.map(apply_calibration)

plot_array_phases(points_frontal_calibrated, 512, 8, "Frontal")
plot_array_phases(points_left_calibrated, 512, 8, "Left")
plot_array_phases(points_right_calibrated, 512, 8, "Right")

• If the transmitter is in front of the array, the received phase decreases from the bottom to the top. This is because the antenna array is mounted at a hight of approximately 1.2m above the transmit antenna, so that the electromagnetic wave arrives earliest at the lowest row.
• If the transmitter is to the left or to the right of the array, in addition to the phase decrease from bottom to top, we can also see a decrease from the left to the right or inversely, respectively.

But wait... doesn't this mean that, instead of applying neural networks to the task of transmitter position estimation, we could now just apply classical techniques? Well yes, it does! In the following, we are going to look at a very simple approach to this topic - note that much better, more advanced methods are available, such as the MUSIC algorithm. Within the scope of this tutorial, we only care about showing that classical AoA estimation is at all possible with DICHASUS data.

The following code snippets estimate the phase difference between antennas that are horizontal neighbors (for a small subcarrier range). Subsequently, this this measured phase difference is transformed to an AoA estimate using the well-known ULA AoA equation, i.e., $$\mathrm{\Delta }\phi =\frac{2\pi f_{\mathrm{c}}}{c}d\sin (\vartheta ),$$ where $\mathrm{\Delta }\phi$ is the estimated phase difference between neighboring antennas, $f_{\mathrm{c}}$ is the carrier frequency, $c$ is the speed of light and $\vartheta$ is the angle of arrival of the electromagnetic wave, measured relative to the perpendicular (see sketch).

Let's generate and plot some phase estimates! This may take a short while to compute.

positions, angle_estimates = estimate_angles(dataset_calibrated, 512, 520, 1.272e9, 50e6)

plt.figure(figsize=(10,10))
plt.hexbin(positions[:,0], positions[:,1], C = np.rad2deg(angle_estimates), gridsize = 25)
plt.xlabel("Transmitter Position X")
plt.ylabel("Transmitter Position Y")
plt.colorbar(label = "AoA Estimate [deg]")
plt.show()

Remember that, in our dataset, the antenna array is located at $(x,y)=(0,0)$ and points in the negative $x$ direction. The diagram that is generated is pretty convincing: We can clearly see the color gradient from the steep incident angles to the right of the antenna array (dark blue) to the angles on the left of the array (yellow).

Based on the position label of every datapoint, we can also compute an "ideal" AoA value for every datapoint and compare estimate and ideal value. Let's plot this estimation error in a histogram.

true_angles = np.arctan2(-positions[:,1], -positions[:,0] + 0.2)
est_error = np.rad2deg(np.angle(np.exp(1.0j * angle_estimates) * np.conj(np.exp(1.0j * true_angles))))
plt.xlabel("AoA Estimate Error")
plt.ylabel("Number of Occurences")
plt.hist(est_error, bins = 100)
plt.show()
print("MAE:", np.mean(np.abs(est_error)), "degrees")

Looking at a histogram of the AoA estimation error, we see that for most datapoints, the error is somewhere below ${10}^{\circ }$. In fact, the mean absolute error (MAE) we obtain is $9^{\circ }$. While this is worse than what we could have achieved using a neural network-based approach (typically $3^{\circ }-5^{\circ }$ MAE), it clearly shows that classical AoA estimation is possible.