# Frequency domain decomposition

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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.[1][2]

## 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 values $u_{ij}$, $S_i$ 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.[1]
1. Using the mode shapes, an input-output system realization can be written.

## References

1. ^ a b Brincker, R.; Zhang, L.; Andersen, P. (2001). "Modal identification of output-only systems using frequency domain decomposition". Smart Materials and Structures 10 (3): 441. doi:10.1088/0964-1726/10/3/303. edit
2. ^ Brincker, R.; Zhang, L.; Andersen, P. (February 7–10, 2000). "Modal Identification from Ambient Response Using Frequency Domain Decomposition". Proc. of the 18th International Modal Analysis Conference. San Antonio, TX. Retrieved March 11, 2012.