## 4 Concluding Remarks  
Accelerometers are usually employed to measure vibration signals in engineering practice. To numerically derive random velocity and displacement from recorded acceleration, a Walsh Transform and Empirical Mode Decomposition (EMD)-based integral algorithm, WATEBI in short, is proposed in this study. It is actually the linear superposition of the integral kernels of Walsh series and can be handled by an ordinary computer with ease. The residual noise in the reconstructed vibration signals is eliminated by EMD. Unlike the popularly used low-frequency cutoff and low-frequency attenuation methods, this algorithm does not need to specify the critical parameters like cutoff frequency, target frequency and target accuracy. As shown in the flowchart (Figure 3), the trend term is removed by linear least-squares fit. To test the performance of the WATEBI algorithm, three distinct case studies are used. They are respectively the vibration of a SDOF system with given parameters, ground motions of El Centro earthquake and Kaikoura earthquake. In each case study, all reference velocity and displacement are available. Then for the comparison among five methods, trapezoidal integral, Matlab Ode45, low-frequency cutoff, low-frequency attenuation and WATEBI, the reconstructed signals of velocity and displacement are investigated. The average peak error, the maximum peak error and the root-mean-square error, are three criterions used to differentiate the level of quality of these methods. The graphical comparisons of both full duration and enlarged details in Figures 5, 6, 8, 9, 11 and 12 reveal that not only the time history of reconstructed velocity by WATEBI perfectly overlaps the reference signal, but also the reconstructed displacement by WATEBI tallies with the reference data with very little deviations, especially at peaks and troughs. The errors in Tables 1-6 also demonstrate that for both velocity and displacement, the overall performance of WATEBI is the best among all compared methods. Particularly for displacement, WATEBI behaves very stably, providing lowest errors for all three cases. The Ode45 algorithm works very well in Case 1 and Case 3 for velocity reconstruction, while the trapezoidal integral method is suitable for all cases. The low-frequency cutoff method gives accurate velocity only in Case 1, while the low-frequency attenuation method renders accurate displacement in Case 3. Nevertheless, none of these four methods surpass the WATEBI algorithm in nearly all cases. Additionally, the WATEBI algorithm is more adaptive, since, as depicted in the flowchart (Figure 3), the signal reconstruction does not require subjective parameters such as cutoff and target frequencies to be specified. Signals adopted in these cases are equally spaced. However, for the irregularly sampled data series, the ALLSSA method might be more robust for the preprocessing than higher-order spline methods. The merit of Walsh transform with constant kernels has inspired this research work. For the sake of limited length of this paper, many other case studies to validate the robustness of the proposed algorithm are not included. Different from the long and popularly used Fourier transform and Hilbert transform in signal processing, the Walsh transform still looks like a stranger to many researchers. Hopefully more virtues of the Walsh transform could be rediscovered to further invoke its wider application in engineering problems. Conceptualization, Q.Z. and X.Y.Z. ; methodology, Q.Z. ; software, Q.Z. ; validation, Q.Z. and X.Y.Z. ; resources, Q.Z. and X.Y.Z. ; data curation, Q.Z. ; writing--Original draft preparation, Q.Z. ; writing--Review and editing, X.Y.Z. ; project administration, X.Y.Z. ; funding acquisition, X.Y.Z. All authors have read and agreed to the published version of the manuscript. This research was funded by China National Key Research Scheme 2016 YFC0303706 and the Offshore Engineering Development Projects (Grant Nos. [2015]-75 and 20141120164551165) of Shenzhen Government.