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Forthee rsttime,theKarhunen ‐Loeve(KL)procedureisderivedinthefrequencydomainasatoolforcalculating eigenmodes of linear systems. The new derivation is based on the discrete Fourier transform representation of a time average of proe le energy including a proper system response and a proe le function. Taking the variational problem posed as such with respect to the proe le function leads to an eigenformulation in the frequency domain. Choice of a system response for efe cient KL mode calculation and construction of reduced-order systems using the KL eigenmodes are also discussed. To demonstrate themethod, both mechanical and e uid dynamic models are considered. The method is equally useful in extracting eigenmodes of an experimentally generated database. Nomenclature c = wing chord length F = snapshot matrix as dee ned in Eq. (13) F = Fourier operator GIk = impulse response for the kth input GSk = step response for the kth input i
Taehyoun Kim (Sun,) studied this question.