*3.1. Accelerometer-Derived Displacement Reconstruction*

The frequency-domain integration approach (FDIA) is based on Fourier transform, which derives from its integral property as follows:

$$\mathcal{F}\left[\int\_{-\infty}^{t} a(t)dt\right] = \frac{1}{j\omega} \cdot \mathcal{F}[a(t)]\tag{11}$$

where F represents the Fourier transform symbol, *a*(*t*) denotes the acceleration signal, *j* is the imaginary number, and *ω* means the frequency. The above formula illustrates that the Fourier transform of acceleration signal integration is equivalent to the Fourier transform of the signal divided by the factor *i<sup>ω</sup>*, simplifying the complex integration into division.

In FDIA, the signal is transformed into a frequency-domain signal by Fourier transform, and the operation is completed in the frequency domain. The velocity and displacement information in the time domain is obtained by inverse Fourier transform. The calculation procedure is represented in Figure 4.

**Figure 4.** Algorithm of frequency-domain integration.

The frequency spectrum of the acceleration signal after Fourier transform can be expressed by

$$A(k) = \sum\_{n=0}^{N-1} a(n)e^{-j(2\pi nk/N)}\tag{12}$$

where *N* is the number of acquisition points, *a*(*n*) means the discrete expression of *<sup>a</sup>*(*t*), and *n* and *k* stand for positive integers.

According to the formula, the single integration is obtained as follows:

$$V(n) = \frac{A(k)}{j\omega} = \sum\_{k=0}^{N-1} \frac{1}{j2\pi k\Delta f} H(k)a(n)e^{-j2\pi nk/N} \tag{13}$$

The results of double integration are given by

$$X(n) = -\frac{A(k)}{\omega^2} = \sum\_{k=0}^{N-1} \frac{1}{-\left(j2\pi k \Delta f\right)^2} H(k)a(n)e^{-j2\pi nk/N} \tag{14}$$

$$\text{As for } H(k) = \left\{ \begin{array}{c} 1 \ (f\_d \le k\Delta f \le f\_u) \\ 0 \text{ (other)} \end{array} , \Delta f \text{ is the frequency resolution while } f\_d \text{ and } f\_u.$$

are the upper and lower cutoff frequencies, respectively. After all Fourier components of different frequencies are calculated according to the frequency-domain relationship, time-domain signals can be obtained by inverse Fourier transform.
