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