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.¹²

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