# A Software-Defined Radio Platform for Teaching Beamforming Principles
## 3 System Parametrization  
In beamforming systems, or any other system that requires synchronized transmission across multiple channels, accurate phase and amplitude calibration for all channels is crucial for ensuring optimal performance. Unlike traditional RF equipment such as spectrum analyzers (SAs) or oscilloscopes, SDRs are not factory-calibrated in terms of phase and amplitude response. Their versatility and flexibility come at the cost of requiring careful calibration to ensure that the system performs as intended. Without calibration, SDR devices can introduce errors that impact the precision and coherence of the signals. Parametrization ensures that the phase and amplitude corrections are not static but adaptive, meaning they can change depending on the deployed hardware or software architecture (cable type, frequency range, and GNU Radio parameter settings). This section details the parametrization carried out for deploying the beamforming system on a 1 \(\times\) 4 array of patch antennas in the 3.53 GHz frequency range. Both phase and amplitude parametrizations were carried out for the four TX channels and the RX channel. One must note that any change in hardware or software implementation could require a new parametrization. The phase variation was validated using an Rhode&Schwartz(R&S) RTO2064 oscilloscope featuring four channels (CH), 6 GHz bandwidth, and 20 GSa/s maximum sample rate. The four TX channels of the USRP were connected by RF cables (same type and length) to each of the four oscilloscope channels. Figure 6 presents a picture of the experimental setup used for phase alignment of the four USRP TX channels, with one group of channels (CH1 and 3) being set out of phase compared to the other group of channels (CH2 and 4). Figure 5: GNU Radio flowchart (**top**) and spectrum (**bottom**) for TX channel. Figure 6: Experimental setup for phase alignment. Figure 7 presents the result of the phase calibration for 3.53 GHz frequency after the values of the complex constants were set to correspond to 0\({}^{\circ}\) phase difference between the four TX channels. The signal waveforms were displayed as time domain diagrams and as an XY type diagram. In the XY type diagram the signal on the X axis was set to channel 1 and the signal on the Y axis to channels 2, 3, and 4, respectively. The system maintains the phase calibration each time it is powered on if the operating frequency and hardware (splitter and cables) remains unchanged. Calibration is required for each operating frequency and was observed to persist over multiple experiments and device functioning hours. To ensure proper system functioning, a new calibration needs to be performed each time the frequency of the system is changed. The time-amplitude view in Figure 6 (top) reveals that the four TX USRP channels perfectly overlap, while the XY diagrams show very good alignment between CH1 and CH2/CH3. In the second XY diagram (bottom middle), one can observe that the phase alignment between CH1 and CH3 is not optimal, with a maximum measured phase offset of 2.4\({}^{\circ}\). This deviation was attributed to the instrument limitations given by factors such as the frequency stability of the LO, noise, component tolerances, and even small physical damage of hardware. However, as will be demonstrated in the following subsection, the slight phase misalignment did not significantly impact the overall performance of the system that was intended for demonstration and learning purposes. The amplitude calibration of the TX channels was performed with an R&S FSV3013 SA. Figure 8a presents an example of the measured 3.53 GHz frequency signal for a gain value of 20 dB (TX1), with the visible SA settings and TX channel directly connected by RF cable. Figure 8b presents the amplitude linearity measured for one of the USRP TX channels. It was observed that all channels demonstrated similar variation in amplitude linearity, so in Figure 8b, we present the result of a single TX channel amplitude linearity measurement.