## 1 Introduction  
With the rapid development of modern science and technology, the research on mechanical and structural vibration has promoted great advances in numerous engineering fields and areas, such as aerospace engineering, bridge engineering, earthquake engineering and offshore engineering [1; 2]. In mechanical engineering, vibration tests have attracted considerable attention, as they are of significant value for the health monitoring of machinery [3], including system operational reliability and mechanical breakdown detection. In earthquake engineering, ground motion signals are the key to evaluating seismic magnitude and intensity [4]. Records of ground motions in a strong seismic event are an essential input to time-domain dynamic response calculation for structures located in earthquake-prone zones. Thus, it is important to obtain accurate vibration responses, either directly through in situ real-time monitoring, or indirectly through vibration tests and numerical reconstruction through data post-processing. Usually the responses of interest are acceleration, velocity, displacement, and stress signals of a structure. They can be measured by a wide variety of sensors. In most situations it is acceleration that is measured. Please note that the measured acceleration is an absolute quantity, while the measured displacement is relative to a reference object. In reality, it is not straightforward to install a fixed displacement sensor due to the geometrical complexity of a structure and its restricted ambient space [5]. By contrast, as a result of their small size, acceleration sensors need neither a large space nor a stationary reference [1]. Among concurrent motion sensors, acceleration sensors (forinstance accelerometers) are technically the most versatile. It is very convenient in both laboratory tests and field observations to carry out acceleration logging. To acquire data of unknown velocity and displacement, a traditional and popular approach is the integral of measured acceleration. There are two main integral schemes. One is by integrating the circuit and the other uses a numerical integral algorithm [6]. Circuit integration is rarely used as it is sensitive to the low frequency of acceleration, while the numerical integral scheme has been widely adopted for years. The numerical scheme can be divided into two categories, time-domain integral [5] and frequency-domain conversion [7]. The former includes known methods like trapezoid formula, Simpson's rule, Newton-Cotes formula and first to _Nth_-order integral operator based on the Runge-Kutta algorithm. However, these conventional methods have several disadvantages [1]. The result usually exhibits drift and errors in the trend term [8] because of the contained direct current component, and high-frequency or low-frequency noise. The frequency-domain conversion method by contrast is numerically much faster, by transforming the time-domain acceleration into a curve synthesizing complex coefficients through finite Fourier series. Various Fast Fourier transform (FFT) algorithms have been widely applied to build the corresponding relationship between frequencies and amplitudes [9]. To solve the problem of spectral leakage in FFT, the antileakage least-squares spectral analysis (ALLSSA) [10] and multichannel ALLSSA [11] have been proposed to regularize the irregularly sampled series and to attenuate the random noise. They were proven more robust than the antileakage Fourier transform (ALFT) [12] and the arbitrarily sampled Fourier transform (ASFT) [13]. It follows that by the principle of trigonometric function integrals, the velocity and displacement signals in frequency domain could be obtained by simply dividing the Fourier coefficients of acceleration with transfer functions (\(i\omega\)) and (\(i\omega\))\({}^{2}\). Then, the time series of velocity and displacement are recovered by applying the Inverse Fourier Transform (IFT) [7]. Many attempts have been proposed to improve the rationality and accuracy of the aforementioned methods.