Frequency domain decomposition

The frequency domain decomposition (FDD) is an output-only system identification technique popular in civil engineering, in particular in structural health monitoring. As an output-only algorithm, it is useful when the input data is unknown. FDD is a modal analysis technique which generates a system realization using the frequency response given (multi-)output data.

Algorithm

 * 1) Estimate the power spectral density matrix $$\hat{G}_{yy}(j\omega)$$ at discrete frequencies $$\omega = \omega_i$$.
 * 2) Do a singular value decomposition of the power spectral density, i.e. $$\hat{G}_{yy}(j \omega_i) = U_i S_i U_i^H$$ where $$U_i = [u_{i1},u_{i2},...,u_{im}]$$ is a unitary matrix holding the singular vectors $$u_{ij}$$, $$S_i$$ is the diagonal matrix holding the singular values $$s_{ij}$$.
 * 3) For an $$n$$ degree of freedom system, then pick the $$n$$ dominating peaks in the power spectral density using whichever technique you wish (or manually).  These peaks correspond to the mode shapes.
 * 4) Using the mode shapes, an input-output system realization can be written.