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The frequency domain decomposition (FDD) is an output-only
system identification The field of system identification uses statistical methods to build mathematical models of dynamical systems from measured data. System identification also includes the optimal design of experiments for efficiently generating informative data f ...
technique popular in
civil engineering Civil engineering is a professional engineering discipline that deals with the design, construction, and maintenance of the physical and naturally built environment, including public works such as roads, bridges, canals, dams, airports, sewa ...
, in particular in
structural health monitoring Structural health monitoring (SHM) involves the observation and analysis of a system over time using periodically sampled response measurements to monitor changes to the material and geometric properties of engineering structures such as bridges a ...
. As an output-only algorithm, it is useful when the input data is unknown. FDD is a
modal analysis Modal analysis is the study of the dynamic properties of systems in the frequency domain. Examples would include measuring the vibration of a car's body when it is attached to a shaker, or the noise pattern in a room when excited by a loudspeake ...
technique which generates a system realization using the frequency response given (multi-)output data.


Algorithm

# Estimate the
power spectral density The power spectrum S_(f) of a time series x(t) describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, ...
matrix \hat_(j\omega) at discrete frequencies \omega = \omega_i. # Do a
singular value decomposition In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any \ m \times n\ matrix. It is r ...
of the power spectral density, i.e. \hat_(j \omega_i) = U_i S_i U_i^H where U_i = _,u_,...,u_/math> is a
unitary matrix In linear algebra, a Complex number, complex Matrix (mathematics), square matrix is unitary if its conjugate transpose is also its Invertible matrix, inverse, that is, if U^* U = UU^* = UU^ = I, where is the identity matrix. In physics, esp ...
holding the
singular value In mathematics, in particular functional analysis, the singular values, or ''s''-numbers of a compact operator T: X \rightarrow Y acting between Hilbert spaces X and Y, are the square roots of the (necessarily non-negative) eigenvalues of the se ...
s u_, S_i is the diagonal matrix holding the singular values s_. # 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. ## Using the mode shapes, an input-output system realization can be written.


See also

* Eigensystem realization algorithm - an input/output identification technique


References

Systems theory {{engineering-stub