Bivariate Spectral Density with Daniell Method

Tutorial

  1. Open the sample project file in Origin, go to Folder Spectral Analysis using the Project Explorer. Activate the workbook Bivariate spectral data.
    Bi dan pe data.png
  2. Highlight column A and B in worksheet. Click the Time Series Analysis App icon Time Series Analysis icon.png in the Apps Gallery window.
  3. Choose Spectral Analysis tab. Click Bivariate Spectral Density with Daniell Method icon to open the dialog.
    Bi dan toolbar.png
  4. In the Setting branch, choose Mean correction. Enter 0.2,20, 0 and 0.5 in Tapering Proportion, Smoothing Window Width, Aligment Shift between Two Time Series, and Shape Ratio of Trapezium Window respectively.
    Bi dan dialog.png
  5. Click Preview button to display smoothed spectrum.
  6. Click OK button to output the report.
    Bi dan report.png

Algorithm

The unsmoothed sample cross-spectrum
f^*_{xy}(\omega) = \frac{1}{2\pi n}(\sum_{t=1}^{n}y_texp(i\omega t))\times (\sum_{t=1}^{n}x_texp(-i\omega t))
for frequency values \omega_j = \frac{2\pi j}{K},0\le \omega_j \le \pi.
The smoothed spectrum is returned at a subset of these frequencies
v_l=\frac{2\pi l}{L},l=0,1,...,[L/2]
where [ ] denotes the integer part.
Its real part or co-spectrum cf(v_l), and imaginary part or quadrature spectrum qf(v_l) are defined by
f_{xy}(v_l)=cf(v_l)+iqf(v_l)=\sum_{|\omega_k|<\pi/M}^{}\hat{\omega}_kf^*_{xy}(v_l+\omega_k)
where \hat{\omega}_k=\omega_kexp(-2\pi iSk/L), and S is alignment shift.

Reference

  1. nag_tsa_spectrum_bivar (g13cdc)