# Drive-by Bridge Damage Detection Using Continuous Wavelet Transform
## 3 Continuous Wavelet Transform and Development of the Proposed Method  
Traditionally, Fourier transform is extensively used for analyzing the frequency content of a signal. However, it has two primary limitations. First, it examines the entire signal without offering insights into potential variations in its frequency content over time. Additionally, the Fourier transform is constrained to using sinusoidal waves as the basis for signal decomposition. The Wavelet Transform (WT) addresses both shortcomings by offering detailed information about the temporal variation of frequency content and permitting the use of diverse basis functions, known as mother wavelets, for signal decomposition [57]. A mother wavelet, is a waveform of limited duration and has an average value of zero:  
\[\int_{-\infty}^{+\infty}\Psi(t)dt=0 \tag{4}\]  
where \(\Psi(t)\) is the mother wavelet function. The Continuous Wavelet Transform (CWT) of a signal \(f(t)\) is then given by:  
\[W(a,b)=\frac{1}{\sqrt{a}}\int_{-\infty}^{+\infty}f(t)\Psi(\frac{t-b}{a})dt=0 \tag{5}\]  
The function \(W(a,b)\) denotes the degree of correlation between the signal at a particular time interval and the mother wavelet. The scaling parameter, \(a\), is employed to stretch and dilate the mother wavelet and is associated with the frequency of the wavelet, while the translation parameter, \(b\), is utilized to shift the finite-duration wavelet in time. The function \(W(a,b)\) can be plotted as a wavelet coefficient map (WCM), which presents the variation of wavelet coefficients with time at different scales, which are correlated with the frequency. As such, a wavelet coefficient map provides information about the variation of the frequency content of the signal with time. Equation (6) shows the relationship between the scales used in CWT and frequency:  
\[f=\frac{F_{c}}{s\Delta t} \tag{6}\]  
where \(f\) is the pseudo-frequency corresponding to the scale \(s\), \(F_{c}\) is the central frequency of the mother wavelet, and \(\Delta t\) is the time step of the analysis. As indicated by Equation (6), scale and frequency are inversely proportional to each other; i.e., the frequency decreases as the scale increases and vice versa. The methodology developed here is inspired by the work by [PERSON] and [PERSON] [50], which shows that, when a vehicle crosses over a damaged bridge, the wavelet coefficient map of the accelerations recorded on the bridge reveals the damage and its location at scales that are significantly higher than the scale that corresponds to the lowest bridge  
Figure 2: (**a**) The road roughness profile generated (**b**) Power Spectral Density of the road profile. frequency. Later, [PERSON] and [PERSON] [51] further developed the method to detect damage on the bridges from the accelerations recorded on the vehicle. For this, they conducted a numerical study where the bridge and the vehicle were modeled in a similar way to this study while only one sensor was used to record the accelerations. In these studies, [PERSON] and [PERSON] [50; 51] showed that, damage can be observed as deviations at the damage locations in the horizontal sections of the WCM of the accelerations recorded both on the bridge [50] and on the vehicle [51], when the vehicle travels on a smooth bridge. However, when road roughness was included, detection and location of damage became very difficult when the accelerations were recorded on the bridge (i.e., direct monitoring) and impossible when the accelerations were recorded on the vehicle (drive-by monitoring). Further, in both studies, damage could only be detected and located only when compared with the corresponding information obtained from the undamaged bridge. In this study, we aim to negate the adverse effects of road roughness on damage detection using continuous wavelet analysis of accelerations recorded on the vehicle while proposing a standardized framework that will allow us to detect and locate damage in the absence of corresponding information from the undamaged bridge.