## 2 Input-Output Data  
The hammer test is performed, and the out-of-plane displacement of the structure under test is measured with a pair of high-speed cameras. The present work develops a frequency domain approach for estimating the FRFs utilizing DIC data. The time domain observations of excitation and displacement are transformed into the frequency domain ones. For a sampled temporal sequence \(x(nT_{s})\), where \(T_{s}\) is the sampling period, \(n=0,\cdots,N-1\), and \(N\) is the length of the sampled sequence, the discrete Fourier transform (DFT) is defined as  
\[X(k)=\frac{1}{\sqrt{N}}\sum_{n=0}^{N-1}x(n)\exp(-j2\pi\frac{n}{N}k) \tag{1}\]  
where \(k\)\(=0\), \(1\), \(\cdots\), \(N/2\), \(j^{2}=-1\). Note that \(k\) is the index of the discretized angular frequency \(\omega_{k}=2\pi k/NTs_{s}\), and the Fourier transforms of data at negative frequencies carry the same information for parameter estimation as those at positive frequencies, so \(k\) ranges from \(0\) to \(N/2\).  
Due to the limitations of the onboard memory capacity and the lightly damped property of a flexible structure, an exponential window should be applied to both force- and response-time histories to minimize the energy leakage caused by the Fourier transform. When the exponential window is \(w(t)=e^{-bt}\), the decay of the windowed signal \(x(t)\) is given by  
\[20\log_{10}\left|\frac{x(0)e^{0}}{x(T)e^{-bt}}\right|=20\log_{10}\left|\frac{ x(0)}{x(T)}\right|+\left(20T\log_{10}e\right)b \tag{2}\]  
where \(T=(N-1)T_{s}\). By considering the floor noise level, the left side of Equation (2) is user-determined, and then the decay rate \(b\) can be derived.  
By applying Equation (1), the DFT \(Y(k)\) of the windowed response at a spatial location and the DFT \(U(k)\) of the windowed input are related as  
\[Y(k)=G(\omega_{k})U_{0}(k)+N_{Y}(k) \tag{3}\]  
\[U(k)=U_{0}(k)+N_{U}(k) \tag{4}\]  
where \(G(\omega_{k})\) is the (windowed) FRF of the structure under test, \(U_{0}(k)\) is the DFT of the noise-free impact, and \(N_{U}(k)\) and \(N_{Y}(k)\) are the DFTs of input and output measurement noises, respectively. By conducting a modal impact hammer test, a high signal-to-noise ratio (SNR) can be achieved once the measurement noise that occurs after the impact is minimized using a window function. As the windowed \(U(k)\) approaches \(U_{0}(k)\), the windowed input is thereby used in this work for FRF identification.  
As mentioned in the introduction section, DIC data are disturbed by optical sensor noises and external vibrations when digital images are acquired, and the potential sources of unwanted camera system vibrations usually include high-speed camera cooling fans and ambient excitations. Under mild conditions, the DFT of output measurement noise (\(N_{Y}\)) can be considered as zero-mean asymptotically (\(N\)\(\rightarrow\)\(\infty\)) circular complex Gaussian distributed and independent over frequency due to the central limit theorem [19]. The diagonal covariance matrix of \(N_{Y}\) can be obtained by conducting a preprocessing experiment where the structure under test is in static equilibrium.  
In addition to the Gaussian property of DIC measurement noises, the frequency domain formulation allows one to easily choose a frequency band of interest for FRF identification. Let \(\mathcal{F}\) denote the set of frequency indices at which the FRF is estimated, which includes \(F\) spectrum lines. The DC component (\(k=0\)) and Nyquist frequency are excluded in this work.