# Design of a Digital Array Signal Processing System with Full Array Element
## 2 System Architectural Design
### System Software Architecture Design
#### 2.2.2 Broadband Calibration Design  
The frequency coverage of the signal used in this design is 1 GHz. Due to the relatively large bandwidth, the impact of frequency characteristic mismatch between multiple channels in the entire frequency band must be considered. The mismatch in frequency characteristics can lead to problems such as directional graphic changes and pulse compression into pairs of the side lobe. To correct the frequency mismatch, a broadband equalization filter is added to each channel. Channel equalization is a prerequisite for achieving broadband beamforming, and the following equalization filter is designed using the frequency method for broadband calibration [21; 22; 23]. The frequency response of the equalization filter of each channel to be calibrated can be calculated from the frequency response of each channel to be calibrated and the frequency response of the reference channel and that of the equalization filter of the reference channel, as shown in Equation (5). \[H_{i}(m)=\frac{C_{ref}(m)}{C_{i}(m)}H_{ref}(m),i=1,2,\ldots,N,m=1,2,\ldots,M. \tag{5}\]  
where \(C_{ref}(m)\) is the frequency response of the reference channel, \(C_{i}(m)\) is the frequency response of each channel to be calibrated, and \(H_{ref}(m)\) is the frequency response of the reference channel equalization filter. There are a total of \(N\) channels to be calibrated and the broadband signal used for calibration consists of \(M\) sample points. Based on the algorithm in the literature [21; 22; 23], the plural weight \(h_{i}\) of the \(i\)-channel equalization filter is obtained using Equation (6). \[h_{i}=\mathbf{A}^{+}\mathbf{H} \tag{6}\]  
where \(\mathbf{A}^{+}\) is the pseudo-inverse of the frequency factor array. \(\mathbf{H}=\left[H_{i}(0),H_{i}(1),\ldots,H_{i}(M-1)\right]^{T}\). The frequency factor array \(\mathbf{A}\) is an M \(\times\)L matrix, and the elements of its m + 1 row and l + 1 column are shown in Equation (7). \[\mathbf{A}_{m+1J+1}=\exp(-j\frac{2\pi ml}{M}). \tag{7}\]  
The hardware implementation process of the broadband calibration is shown in Figure 13 and includes the FFT IP core, the IP core for the equalization filter weight coefficient calculation, the ROM for the pseudo-inverse memory, and the IP core for the equalization filter. A linear frequency modulation (LFM) signal with 1 GHz bandwidth and 11.5 GHz IF will be used as the channel calibration signal in the calibration process. Figure 12: Inter-channel delay distribution. The signal is acquired through each transceiver channel, and the acquired signal is first used to calculate the frequency response of the channel through the FFT IP core, and then the frequency response of the channel is input to the IP core of the equalization filter weight factor calculation. The IP core is designed and implemented by the XILINX High-Level Synthesis (HLS) Tool, which mainly performs the pseudo-inverse matrix multiplication of the equalization filter frequency response matrix and the frequency factor array for each channel. At the same time, the pseudo-inverse of the frequency factor array is read from the ROM and input to the weight coefficient calculation IP to finally calculate the weight coefficient of the equalization filter for the channel. The equalization filter IP core constructs a complex finite impulse response (FIR) filter based on the weight coefficients, and finally completes the construction of the equalization filter for the channel. After the broadband baseband signal passes through the filter during beamforming, the frequency mismatch can be corrected, and the channel equalization completed. After broadband channel calibration, the calibration is tested. Figure 14 shows time-domain waveforms between channels acquired by the high-speed oscilloscope after the broadband amplitude-phase calibration. The magnitude-phase response before and after calibration is shown in Figure 15. Figure 15a shows that the amplitude error between the different channels to be calibrated is large, and the maximum difference reaches 10 dB.