Frequency spectra and cross-spectra of shear and vibrations

Auto- and cross-spectra should be calculated using the windowed and overlapped periodogram method.

The window should be a periodic cosine window. The overlap should be 50%. Although this is not the optimal overlap, it is a convenient one and it is one that provides 90% of the maximum degrees of freedom attainable with the cosine window (Nuttall, 1971). The number of fft-segments within a diss-length segment of data is [math]\displaystyle{ N_f = 2 \frac{L_D}{L_f}-1 }[/math] where [math]\displaystyle{ L_D }[/math] is the length of a dissipation rate estimate (diss-length) and [math]\displaystyle{ L_f }[/math] is the length of an fft-segment (fft-length).

The auto-spectrum is given by [math]\displaystyle{ E_i = \frac{2}{MN_ff_s} \sum^{N_f}_{k=1} |\mathfrak{F}_{ik}^2| }[/math]