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} |\mathscr{F}_{ik}|^2 }[/math] where [math]\displaystyle{ \mathscr{F}_{ik} }[/math] is the fast Fourier transform of the k-th fft-segment, i is the frequency index ranging from [math]\displaystyle{ i=1 }[/math] to [math]\displaystyle{ i=1+M/2 }[/math], [math]\displaystyle{ M }[/math] is the number of samples in an fft-segment, [math]\displaystyle{ N_f }[/math] is the number of fft-segments within a dissipation segment, and [math]\displaystyle{ f_s }[/math] is the sampling rate of the data. The frequency of each spectral value is [math]\displaystyle{ f_i = \frac{i-1}{M} f_s }[/math]. It is assumed that the cosine window applied to each fft-segment has mean-square of 1, i.e., it does not change the variance of the windowed data with respect to the original data. A suitable cosine window is [math]\displaystyle{ W = \sqrt{\frac{2}{3}}\, \left(1 + \cos\left(\pi [j-1] /M\right) \right) }[/math] where [math]\displaystyle{ j = 1, ..., M }[/math] is the index to the samples in an fft-segment.

This spectrum, [math]\displaystyle{ E_i }[/math] has the property that its integral (obtained by a suitable numerical approximation) equals the variance of the signal. The very first spectral value, [math]\displaystyle{ E_1 }[/math] represents the mean and should be set to zero. The very last spectral value represents the Nyquist frequency and should be very small if the data have been properly sampled using an anti-aliasing filter.

Cross-spectra are calculated similarly to the auto-spectra, except that the [math]\displaystyle{ |\mathscr{F}_{ik}|^2 }[/math] is replaced by [math]\displaystyle{ \mathscr{F}_{xik} \mathscr{F}^*_{yik} }[/math] where [math]\displaystyle{ \mathscr{F}_{xik} }[/math] and [math]\displaystyle{ \mathscr{F}^*_{yik} }[/math] are the fast Fourier transforms of the two signals x and y and the asterisk indicates a complex conjugate. Cross-spectra are complex and are used in the vibration-coherent noise removal algorithm of Goodman et al (2006).