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

The auto-spectrum is given by $E_i=\frac{2}{M{N}_f{f}_s}\sum _{k=1}^{N_f}|{\mathfrak{𝔉}}_{ik}^2|$