# Field Research on the Wind-Induced Response of a Super High-Rise Building under Typhoon
## 3 Analytical Research Methods
### Improved NExT-ERA Method  
The basic idea of the Natural Excitation Technique (NExT) method is that the cross-correlation and impulse response functions between two points of the structure under white noise excitation have similar expressions, and a modal identification method in the time domain is used to identify the modal parameters after obtaining the cross-correlation function of the response between the two points. For a linear system with a degree of freedom of \(N\), when the \(k\) point of the system receives an excitation of force \(f_{k}(t)\), the response \(x_{ik}(t)\) of the system's \(i\) point can be expressed as:  
\[x_{ik}(t)=\sum_{r=1}^{2N}\phi_{ir}a_{kr}\int_{-\infty}^{t}e^{\lambda_{r}(t-p)}f _{k}(p)dp \tag{6}\]  
where \(\phi_{ir}\) is the \(r\)-th-order modal model of the \(i\)-th measuring point, and \(a_{kr}\) is a constant term that is related only to the excitation point \(k\) and the modal order r.  
An impulse response \(h_{ik}(t)\) of the system's \(i\) point is obtained when the system's \(k\) point is excited by a unit pulse and can be expressed as:  
\[h_{ik}(t)=\sum_{r=1}^{2N}\phi_{ir}a_{kr}e^{\lambda_{r}t} \tag{7}\]  
Figure 4: Sketch map of wind speeds and directions. The system's \(k\)-point input force is excited, and the responses obtained by the system's \(i\)- and \(j\)-point tests are \(x_{ik}(t)\) and \(x_{jk}(t)\), respectively. The expression of the cross-correlation functions of the two responses can be written as:  
\[ \begin{array}{l}R_{ijk}(\tau)=E[x_{ik}(t+\tau)x_{jk}(t)]\\ =\sum\limits_{r=1}^{2N}\sum\limits_{s=1}^{2N}\phi_{ir}\phi_{js}a_{kr}a_{bs} \int_{-\infty}^{t}\int_{-\infty}^{t+r}e^{\lambda_{r}(t+\tau-p)}e^{\lambda_{s}(t -q)}E[f_{k}(p)f_{k}(q)]dpdq\end{array}  \tag{8}\]  
Assuming that the stimulus \(f(t)\) is an ideal white noise, the definition of the correlation functions implies that:  
\[E[f_{k}(p)f_{k}(q)]=a_{k}\delta(p-q) \tag{9}\]  
where \(\delta(t)\) is an impulse function, and \(a_{k}\) is a constant term only related to the excitation point \(k\). Equation (5) substituted Equation (4) by integrating and simplifying it to obtain:  
\[R_{ijk}(\tau)=\sum\limits_{r=0}^{2N}b_{ir}\phi_{ir}e^{\lambda_{r}\tau} \tag{10}\]  
\[b_{ir}=\sum\limits_{s=1}^{2N}\phi_{js}a_{kr}a_{bs}a_{k}(-\frac{1}{\lambda_{r}+ \lambda_{s}}) \tag{11}\]  
where \(b_{ir}\) is a constant term that is related only to the reference point \(j\) and the modal order \(r\), \(\phi_{ir}\) is the \(r\)-th-order modal model of the \(i\)-th measuring point. \(\phi_{js}\) is the \(s\)-th-order modal model of the \(j\)-th measuring point. \(a_{kr}\) and \(a_{bs}\) are constant terms that are related only to the excitation point \(k\) and the modal order r and s, respectively, and \(a_{k}\) is a constant term only related to the excitation point \(k\). \(\lambda_{r}\) and \(\lambda_{s}\) are the modal participation factors of the \(r\)-th and \(s\)-th modes, respectively. The comparison of Equations (5) and (10) shows that the mathematical expressions of the cross-correlation function of the two-point response and the impulse response of the pulse excitation of the linear system under white noise excitation are completely identical in form. The cross-correlation function can be characterised as a superposition of a series of complex exponential functions. The NExT method assumes the excitation as white noise, selects the measurement point with less response as a reference point and calculates the cross-correlation function between other measurement points and the reference measurement point. The function is used as input data in combination with traditional modal identification methods for the parameter identification under environmental excitation. This study initially calculates the cross-power spectrum of the measured signal and then obtains the measured cross-correlation function by using an inverse Fourier transform based on the NExT method. This method adopts statistical averaging processing, and the obtained signal-to-noise ratio of the cross-correlation function is greatly increased.