## 4 System Overview
### ADFT Cores and Digital ADFT Spectral Estimation  
The discussed 32-point ADFT was implemented by utilizing Xilinx tools. The sparse matrices from the factorization shown in (5) were structured as parallel input-output blocks to optimize performance [97; 98]. The verification of these blocks was conducted through MATLAB Simulink 8.5, ensuring the reliability of the design before hardware deployment. Subsequently, the verified ADFT design was mapped onto the Xilinx Virtex-6 sx475T FPGA.  
To integrate the ADFT core into the ROACH-2 FPGA platform, we utilized two libraries from the Collaboration for Astronomy Signal Processing and Electronics Research (CASPER) community [100; 101]. This integration enabled the system to generate 32 simultaneous RF beams in parallel. Each beam corresponds to a discrete direction, calculated using the function \(\theta_{k}=\arcsin(k/32)\), where \(k\) ranges from \(-16\) to \(15\). The successful implementation of the ADFT on the ROACH-2 FPGA not only validated the algorithm effectiveness but also demonstrated the practical application of advanced digital hardware design techniques. This work highlights the potential of ADFT in various applications requiring efficient RF beamforming.  
We implemented a spatial windowing technique to address the issue of high side lobes in the ADFT, where the worst-case side lobe was observed to be 1.5 dB higher than that of the (exact) DFT. Four configurations were tested: three windowing functions (Butterworth, Hamming, and Hanning) and a baseline with no windowing [85]. The selection among these windowing functions was managed using a multiplexer equipped with a control selector pin.  
Figure 4 shows the spatial windowing unit coupled with the ADFT sparse matrices. In mathematical terms, the windowing is introduced as a diagonal matrix. The ADFT-based spectral estimation can be described by combining the windowing diagonal matrix (\(\mathbf{D}_{\text{windowing}}\)), the low-complexity ADFT core (\(\mathbf{\hat{F}}_{32}\)), and a power-normalizing diagonal matrix (\(\mathbf{D}_{\text{normalized}}\)). The mathematical representation of the proposed approximate spectrum estimation is given by  
\[\mathbf{ADFT}_{\text{estimator}}=\mathbf{D}_{\text{normalized}}\cdot\mathbf{ \hat{F}}_{32}\cdot\mathbf{D}_{\text{windowing}}. \tag{6}\]Figure 5 illustrates the ADFT response and its modifications under the windowing techniques. It is noteworthy that the introduction of windowing functions resulted in wider main lobes, which is a compromise made to suppress the side lobes. This trade-off between main lobe width and side lobe suppression is a consideration in the application of spatial windowing techniques. However, implementing spatial windowing increases the hardware complexity due to the multiplexers introduced by the windowing unit. Therefore, the user must select the appropriate configuration based on their specific requirements and constraints.  
Figure 4: ADFT architecture with spatial windowing and power normalizing units.  
Figure 5: Simulated magnitude response of 32-point ADFT and its modifications under three different windowing techniques. Subplot (**a**) shows the baseline response without any windowing. Subplots (**b–d**) demonstrate the ADFT responses after applying the Butterworth, Hamming, and Hanning windows, respectively.