## 5 Discussion  
This paper proposed a cross-channel dynamic weighting RPCA algorithm to improve the de-noising performance of a multi-channel pulse wave processing procedure. Wavelet transform, VMD, RPCA, WRPCA, and the proposed cross-channel DWRPCA were evaluated for 3 DPI noise removal performance with time-domain and frequency-domain analysis, run-time complexity analysis, and non-deviation error evaluation. The de-noising results show that the RPCA-based methods are better than the traditional wavelet transform and VMD methods, whereby smooth and consistent periodic signals were extracted from the original signal at every energy level (see Figures 6-9). Since the energy distribution of noise is mixed with that of the physiological signal, the results processed by the traditional methods are not much different from the original ones. While the noise is always random in tactile sensors, we strengthen the periodicity of the signal by segmenting and aligning it by heartbeat cycles, which enables RPCA-based method to better de-noise the original signal. Compared with the cross-channel DWRPCA and WRPCA, RPCA overprocessed the signals and reduced the amplitudes of extracted signals, even approaching a straight line in Figure 10d. In the cross-channel DWRPCA and WRPCA, the weighted nuclear norm (WNN) replaced the nuclear norm used in RPCA to apply different thresholds for different features. This WNN technique prevents the main features of the pulse signal from being removed. Furthermore, by applying the corresponding CSF to different channels, the weighting factors estimated in the cross-channel DWRPCA were smaller compared to the WRPCA in sensor channels with relatively low signal energy, thus performing better (see Figures 9 and 10). Notably, the CSF values were predicted from our custom nine-layer CNN. The CNN structure was basic but decent enough for this study, and we can naturally expect that different sensor types require different neural network designs for appropriate CSF estimation. In brief, the basic CNN model used in this paper is one of the solutions for CSF estimation and can be developed in further studies. Figure 13: Boxplot of time complexity comparison for five algorithms. The \(y\)-axis is in a log scale. The non-deviation errors of key physiological points revealed that, with different individuals and heartbeat cycles, the TAEs and AREs of the cross-channel DWRPCA and WRPCA algorithm were significantly smaller compared to other algorithms. This result indicates that the physiological information of these two methods was relatively less distorted. In the analysis of noise comparison, we found that the amplitude of the noise extracted by RPCA (Figure 11d) was not constant, indicating that some of the signal features were still mixed with noise. Moreover, a time complexity experiment was conducted to evaluate the total run time required by our algorithms. Our method was the least time-consuming in this experiment since the dynamic weighing technique could save compute time. Overall, the proposed cross-channel DWRPCA achieved better performance than other well-known algorithms through several evaluation studies, such as evaluation in the time and frequency domains. There exist hardware and software modules for denoising in a standard multi-channel sensor system; however, for the reasons mentioned earlier, it is difficult to eliminate noise that is mixed with the signal. Despite not being the focus of this paper, a redesigned hardware is expected to fundamentally improve de-noising performance and deserves further study. Now that the system has output the noise mixed with signal, this study makes a novel contribution by proposing and verifying a generalizable de-noising method considering inter-channel correlation. A limitation of the current research is the relatively small sample size, and some sharp noises (e.g., the jitter in Figure 7) were rarely collected.