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> N_f = 2 \frac{L_D}{L_f}-1 </math> where <math>L_D</math> is the length of a dissipation rate estimate (diss-length) and <math>L_f</math> is the length of an fft-segment (fft-length).

The auto-spectrum is given by <math> E_i = \frac{2}{MN_ff_s} \sum^{N_f}_{k=1} |\mathscr{F}_{ik}^2| </math>