Theory

The DFT is the only not strictly data-driven decomposition present in MODULO. In fact, the temporal basis \(\mathbf{\psi}_F\) is the Fourier basis, computed a priori. This reads:

\[\mathbf{\psi}_F = \frac{1}{\sqrt{n_t}} \exp{(2 \pi f_r t)}\]

in which \(f_r=r\Delta f\) is the frequency of the \(r\)-th Fourier mode, \(\Delta f=f_s/n_t\) is the frequency resolution, and \(n_t\) is the number of time samples. The term \(\frac{1}{\sqrt{n_t}}\) is a normalization factor, which ensures that the Fourier basis is orthonormal, e.g. \(\|\mathbf{\psi}_F\|^2=1\).

Considering the DFT of the signal at one specific location \(\mathbf{x}_i\), the Fourier coefficients are computed as:

\[\mathbf{C}_F(\mathbf{x}_i) = \frac{1}{\sqrt{n_t}} \sum_{t=0}^{n_t-1} \mathbf{u}(\mathbf{x}_i,t) \exp{(-2 \pi f_r t)}\]

that effectively projects the signal onto the Fourier basis. For each of these projections, one can compute the norm as:

\[\sigma_F = \|\mathbf{C}_F\|\]

that leads to the normalized projected fields:

\[\mathbf{\Phi}_F = \frac{\mathbf{C}_F}{\sigma_F}\;,\]

that are the spatial structure of the DFT. This completes the DFT decomposition, presented here as a matrix factorization:

\[\mathbf{D} = \mathbf{\Phi}_F \mathbf{\Sigma}_F \mathbf{\Psi}_F^T\;,\]

where \(\mathbf{\Sigma}_F\) is a diagonal matrix containing the modal amplitudes \(\sigma_F\).