\subsection{Short-Time Fourier Transform (STFT)}

The Short-Time Fourier Transform (STFT) is a fundamental technique used to analyze non-stationary signals, such as those generated by structures under dynamic load or white noise excitation. While the traditional Fourier Transform provides frequency-domain information, it lacks time resolution. STFT overcomes this limitation by applying the Fourier Transform over short overlapping segments of the signal, thereby producing a time-frequency representation.

Mathematically, the STFT of a signal $x(t)$ is given by:
\begin{equation}
X(t, \omega) = \int_{-\infty}^{\infty} x(\tau) w(\tau - t) e^{-j \omega \tau} d\tau
\end{equation}
where $w(\tau - t)$ is a window function centered at time $t$, and $\omega$ is the angular frequency.

In this study, the STFT is employed to extract the time-frequency features of the vibration signals collected from the structure. These features are then used as inputs to machine learning classifiers. This process captures localized frequency content over time, which is crucial in identifying structural changes due to damage.