## 3 Digital Beamforming General Considerations
### Calibration Errors that Can Be Handled by the PAU-RAD Array  
Although DBF presents several clear advantages, it requires a previous hardware calibration. Signals to be combined by the DBF have to be previously calibrated in phase and amplitude. Thereby, since it is not possible to ensure this by system design, the whole receiver set has to be equalized in phase and amplitude. This point is a key issue to compose the digital array.  
In PAU-RAD, analog receivers are driven from an ovenized 10 MHz master clock that is distributed to all receivers to generate their own local oscillator. Hence, it can be assumed that each receiver's PLL has a fixed, but possibly slowly varying, random phase variable uniformly distributed within [0-2\(\pi\)). Furthermore, each receiver has a nominal 110 dB gain chain. Component tolerances can strongly affect this nominal gain. For this reason, each receiver has a potentiometer that controls the last amplifier stage at IF to manually equalize all receivers' gain within 110 dB \(\pm\) 1 dB. Even with this functionality, the gain has to be taken into consideration as a first approximation, a random variable with a normal probability density function with a standard deviation of 1 dB. The residual calibration errors will ultimately determine the quality of the synthesized array factor.  
There are some figures of merit to measure the quality of an array. In this study the main beam efficiency (MBE) has been selected because it unifies different parameters (main beamwidth, side lobe levels (SLL), backside lobe, _etc._) in a single one. In general, the MBE is defined as the ratio between the collected power by the main beam of the array and the collected total power. Theoretically, the main beam ends in the nulls between the main beam and the first side lobes, but in practice, these nulls are difficult to determine due to noise phase error and the angular resolution of the measurement system. In practice, to compute the edge of the main beam, it is considered that the main beam ends when it reaches the maximum side lobe amplitude level. In any case, this approximation gives a slightly lower MBE value than the real one.  
For radiometric applications a high MBE is required (\(>\)90%) to ensure that the main contribution of the collected power comes from the main beam. In this case, the PAU-RAD MBE target is \(>\)94% to ensure, at least, the same performance as in previous developed radiometers [23].  
Figure 3 shows the MBE degradation dependence with the residual calibration error for amplitude and phase, respectively. The determining parameter is the residual phase error, which corrupts the MBE rapidly. This is well known for radiometrics.  
Figure 3: Degradation of the MBE _vs._ the amplitude **(left)** and phase **(right)** residual calibration error, obtained using theoretical development and simulations, pointing to the boresight. The considered array has 4 \(\times\)4 elements with triangular illumination.  
To fulfill the radiometric requirements, a MBE equal or higher to 0.94 has to be obtained. As shown in Figure 3, this can be fulfilled with a residual standard deviation error of 10 \({}^{\circ}\) for the phase and 0.25 dB for the amplitude. Despite these residual values, more exigent residual standard deviation errors were selected as a calibration residual error target in order to ensure a high MBE. Actually, a residual standard deviation error of 2 \({}^{\circ}\) for the phase and 0.1 dB for the amplitude were chosen as targets. These target values (2 \({}^{\circ}\)and 0.1 dB) were already achieved in a preliminary work in which only one receiver was considered [11]. These are the specifications adopted for the whole hardware calibration performance. It is important to note that the phase and amplitude errors drift with temperature and time, so that hardware calibration needs to be periodically performed, each time the residual errors cross over these specified thresholds.