*4.1. Instantaneous Frequency*

Traditionally, a Fourier spectrum is evaluated using sine and cosine basis functions with a fixed amplitude. However, signals vary with time limiting the applicability of the fast Fourier transform; moreover, obtaining the instantaneous frequency (IF) at any specific time is impractical. For a structure affected by an earthquake, understanding the frequency variation is imperative. The Hilbert transform is widely used for nonlinear and nonstationary cases, facilitating the analysis of a time-varying signal. A measured signal can be expressed in the form of a complex number to determine the instantaneous amplitude *a*(*t*) and instantaneous phase θ(*t*), and the IF ω(*t*) can then be determined [59,60]. The Hilbert transform can be defined as the convolution between *X*(*t*) and 1/*t*. For any time series *<sup>X</sup>*(τ), the Hilbert transform *Y*(*t*) can be expressed as

$$Y(t) = \frac{1}{\pi} P \int\_{-\infty}^{\infty} \frac{X(\tau)}{t - \tau} d\tau \tag{10}$$

where *P* represents the Cauchy principal value.

Combining *X*(*t*) and *Y*(*t*) into a conjugate complex number yields an analytic signal *Z(t)*

$$Z(t) = X(t) + iY(t) = a(t)e^{i\theta(t)}\tag{11}$$

For example,

$$a(t) = \sqrt{\mathcal{X}^2(t) + \mathcal{Y}^2(t)}\tag{12}$$

$$\theta(t) = \tan^{-1}(Y(t)/X(t))\tag{13}$$

$$
\rho \omega(t) = \left( d\theta(t) / dt \right) \tag{14}
$$

According to the analysis, the time–frequency–amplitude distribution of the time series can be obtained.
