Frequency-domain Karhunen-Loeve method and its application to linear dynamic systems

For the first time, the Karhunen-Loeve (KL) procedure is derived in the frequency domain as a tool for calculating eigenmodes of linear systems. The new derivation is based on the discrete Fourier transform representation of a time average of profile energy including a proper system response and a profile function. Taking the variational problem posed as such with respect to the profile function leads to an eigenformulation in the frequency domain. Choice of a system response for efficient KL mode calculation and construction of reduced order systems using the KL eigenmodes are also discussed. To demonstrate the method, both mechanical and fluid dynamic models are considered. The method is equally useful in extracting eigenmodes of an experimentally generated database.