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5.6.18 Bivariate Spectral Density with Lag Window
Tutorial
- Open the sample project file in Origin, go to Folder Spectral Analysis using the Project Explorer. Activate the workbook Bivariate spectral data.
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- Highlight column A and B in worksheet. Click the Time Series Analysis App icon
in the Apps Gallery window. - Choose Spectral Analysis tab. Click Bivariate Spectral Density with Lag Window icon to open the dialog.
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- In the Setting branch, choose No correction. Enter 0.2 in Tapering Proportion. Choose Tukey window type. Enter 50 and 0 in Cut-off of Lag Window and Aligment Shift between Two Time Series.
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- Click Preview button to display smoothed spectrum.
- Click OK button to output the report.
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Algorithm
The smoothed sample cross spectrum is a complex valued function of frequency \(\omega\), \(f_{xy}(\omega)=cf(\omega)+iqf(\omega)\), defined by its real part or co-spectrum
- \[cf(\omega)=\frac{1}{2\pi}\sum_{k=-M+1}^{M-1}\omega_kC_{xy}(k+S)cos(\omega k)\]
and imaginary part or quadrature spectrum:
- \[qf(\omega)=\frac{1}{2\pi}\sum_{k=-M+1}^{M-1}\omega_kC_{xy}(k+S)sin(\omega k)\]
where \(\omega_k =\omega_{-k}\), for \(k=0,1,...,M-1\), is the smoothing lag window as described in Univariate Spectral Density with Lag Window.
The results are calculated for frequency values
- \[\omega_j = \frac{2\pi j}{L},j=0,1...,[L/2]\]
where [ ] denotes the integer part.
Reference