## 4 Experimental Methodology and Measurement Results
### RRU Synchronism
#### 4.2.3 Phase Difference Measurement  
Since distributed antenna systems are one of the targeted use cases for BISDoF-based networks, plotting the phase difference between two RRUs versus time gives an intuitive view of the achievable performance.  
As illustrated in Figure 11, one RRU1 transceiver functioned as a transmitter, while another RRU1 transceiver and one RRU2 transceiver received the transmitted signal wirelessly. The two receivers were perfectly synchronized in sampling timing, carrier frequency, and sampling clock frequency. Instead of PRBSs, an OFDM signal was sent to RRU1 via the DL fiber.  
The overall channel frequency responses, \(\mathbf{H}_{1}\) and \(\mathbf{H}_{2}\), were estimated using the least square channel estimation in [25]. The estimation results, \(\mathbf{\hat{H}}_{1}\) and \(\mathbf{\hat{H}}_{2}\), both contain 128 complex values corresponding to 128 subcarriers.  
Figure 11: Measurement setup.  
Figure 10: Spectra of the received sine wave and the asynchronous phase noise. Measurement time duration: 25 ms.  
During the measurement period, the wireless channels were assumed to remain static. The phase difference between the two RRUs, caused by the asynchronous jitter/phase noise, can be calculated by  
\[\phi=\frac{1}{N_{\mathbf{D}}}\sum_{k\in\mathbf{D}}\text{angle}\bigg{(}\frac{ \hat{\text{H}}_{1,k}}{\hat{\text{H}}_{2,k}}\bigg{)} \tag{2}\]  
where \(k\) denotes the subcarrier index, \(\mathbf{D}\) is the index set of all non-zero subcarriers and \(N_{\mathbf{D}}\) is the number of indices in \(\mathbf{D}\). \(\hat{\text{H}}_{1,k}\) is the \(k\)-th element of \(\hat{\text{H}}_{1}\).  
Figure 12 plots \(\phi\) versus time. The box plots of \(\phi\) are included to better visualize the distribution. As in Section 4.2.1, the PLLs had to re-lock each time we changed the fibers, hence the change in the average phase difference. The average was observed to be stable once the PLLs were locked and would be contained in the overall channel frequency responses.  
For the back-to-back case, the standard deviation of \(\phi\) is \(6.28^{\circ}\); \(50\%\) of the values fall within the range of the box: \(-7.4^{\circ}\) and \(1.1^{\circ}\) (a range of \(8.5^{\circ}\)). The data range excluding outliers is \(31.5^{\circ}\). For the \(100\,\text{m}\) case, the standard deviation is \(6.44^{\circ}\); \(50\%\) of the values are between \(21.8^{\circ}\) and \(31.1^{\circ}\) (a range of \(9.3^{\circ}\)). The data range excluding outliers is \(31.2^{\circ}\).  
The back-to-back case might seem to have a slightly better result. However, it is important to mention that the performance difference could come from both the DL signal quality and the retrieved clock quality. Overall, it can be concluded that the fiber length does not cause significant performance degradation.  
Compared to co-located antenna systems in which little asynchronous phase noise is expected between transceivers, the fluctuation in phase difference can definitely degrade the digital beamforming accuracy. However, the rich spatial diversity brought by distributed schemes is expected to compensate for this drawback.