Methods of Computing the Power Spectrum for Equally Spaced Time Series of Finite Length

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Abstract
Two conventional methods of computing the power spectrum, via the autocovariance function or via the fast Fourier transform (referred to as the lagged product method and the FFT method respectively for simplicity), have been examined analytically and numerically for equally spaced time series of finite length. It is found that the two methods are equivalent to each other, and that the only difference between them lies in regard to the spectral window. Spectral windows for the FFT method are superior to those for the lagged product method in that they do not show any negative values and that their influence is band-limited in frequency domain. There is little difference in spectral estimates between the two methods. In many cases the FFT method is economical in computation time, but for the case of large data points and small maximum lag the lagged product method is the more economical. It is proved that in the strict sense the power spectrum for higher frequencies than the Nyquist frequency is not folded linearly over lower frequencies both in the FFT method and the lagged product method. Finally it is discussed whether or not the use of original data repeatedly is consistent with the analysis of random phenomena.

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