## 4 Validation and Discussion  
As shown in Figure 7, this paper presents a high-precision angle measurement system for pointing mechanisms, utilizing the Point Ahead Angle Mechanism (PAAM) in conjunction with Differential Wavefront Sensing (DWS) technology [39]. The system includes key components such as the ultra-stable laser source SLS-INT-1064-200-1 (manufactured by Menlo Systems GmbH, Martinsried, Germany), the GD4542-20M QPD (produced by First Sensor AG, Berlin, Germany, now part of TE Connectivity), and a pointing device developed by our team. All tests were performed at an absolute temperature of around 30 degrees Celsius with a temperature control precision of 0.001 degrees Celsius, under a vacuum of 7.2 Pa.  
Figure 6: Evaluation of noise spectral density estimation under noise metric simulation. (**a**) Squared Bias, Variance, and MSE for LPSD, LISA-LPSD, and Adaptive Sine Multi-Taper Algorithms before and after improvement; (**b**) Comparison of computation time using LPSD, Adaptive Sine Multi-Taper, Original Adaptive Sine Multi-Taper, and LISA-LPSD Algorithms. The system analyzes pointing noise within the frequency range of 1 mHz to 1 Hz, validating the effectiveness of the algorithm. To simulate laser jitter-induced pointing, the experiment utilized PAAM to induce sinusoidal jitter in the incident laser at frequencies of 1 mHz, 10 mHz, and 100 mHz, each with an amplitude of 50 \(\upmu\)rad in the yaw direction. After a sufficient warm-up period, the experiment was conducted under these conditions, with the PAAM deflected to specific angles for durations ranging from 1 to 5 h.  
As shown in Figure 7, the DWS angular measurement system utilizes single-frequency light from an ultra-stable laser source, modulated into dual-frequency light by two AOM. The measurement beam, combined with the reference beam by the pointing mechanism, is directed onto a QPD1. The signal is then transmitted to a phase meter via a coherent radio frequency transmission line. The acquired data are processed through DWS phase conversion to yield yaw angle information, which is analyzed for amplitude spectral density (ASD) estimation using the adaptive sine multi-taper algorithm, its original version, the LPSD algorithm, and the LISA-LPSD algorithm. As illustrated in Figure 8(a1,b1,c1), the angular variations measured under PAAM vibrations at 100 mHz, 10 mHz, and 1 mHz are presented, reflecting the system's recorded noise. Figure 8(a2,b2,c2) indicate that the improved adaptive sine multi-taper algorithm shows fewer taper effects at these frequencies compared to the original algorithm. It also features fewer sidelobes and higher spectral resolution than both the original and other algorithms. Furthermore, the reduction in peak values appears less pronounced, suggesting that the proposed enhanced algorithm significantly mitigates pointing noise. These results provide robust support for subsequent research. Figure 7: Schematic of the angular measurement device (BS: Beam Splitter, PBS: Polarizing Beam Splitter, QPD: Quadrant Photodetector, AOM: Acousto-Optic Modulator). Figure 8: Comparison of taper numbers before and after improvements for LPSD, LISA-LPSD, and Adaptive Sine Multi-Taper Algorithm in sinusoidal excitation experiment. (**a**) 100 mHz, where a sinusoidal signal was applied to evaluate the algorithm’s performance at higher frequencies; (**b**) 10 mHz, where a sinusoidal stimulus was introduced to assess mid-frequency response and noise suppression; (**c**) 1 mHz, where a low-frequency sinusoidal excitation was used to test the algorithm’s effectiveness in resolving signals near the lower frequency limit. To evaluate the spectral density analysis algorithm's effectiveness at 1 mHz, the 5 h dataset was divided into ten groups. The mean PSD derived from the PAAM's motion serves as the true value, as shown in Figure 9a,b. Near 1 mHz, the improved adaptive sine multi-taper algorithm demonstrates reduced variance and bias compared to the original algorithm, resulting in a lower MSE.