Consistent real-valued and one-sided spectral density functions

Due to the properties of the Fourier transform, the spectral density (SD) functions are not only defined over the Nyquist band, but over the double interval. Normally that includes negative frequencies and a corresponding SD matrix that is known to contain the same information as the SD matrix in the positive frequency band. In this paper, we will use the Parseval's theorem that expresses the equality between the sum over all SD matrices and the response covariance function as a basis to define a real-valued SD matrix that is only defined over all non-negative frequency bins. This new real and one-sided SD matrix fulfil the Parseval equation and we will also illustrate how it can be used successfully to perform identification in the operational modal analysis where modal parameters are to be identified from the operating response without any pre-knowledge about the excitation forces.