Univariate 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 Univariate spectral data.
Highlight column B in worksheet. Click the Time Series Analysis App icon in the Apps Gallery window.
Choose Spectral Analysis tab. Click Univariate Spectral Density with Lag Window icon to open the dialog.
In the Setting branch, choose Mean correction. Enter 0.2 in Tapering Proportion. Choose Parzen window type. Enter 50 in Cut-off of Lag Window.
Click Preview button to display smoothed spectrum.
Click OK button to output the report.

Algorithm

Tapering factors

$\left\{\begin{array}{ll}\frac{1}{2}(1-cos(\frac{\pi(t-\frac{1}{2})}{T}))&1\leqslant t\leqslant T \cr\frac{1}{2}(1-cos(\frac{\pi(n-t+\frac{1}{2})}{T}))&n+1-T\leqslant t\leqslant n \cr1&Otherwise\end{array}\right.$

where $T=[\frac{np}{2}]$ and $p$ is the tapering proportion.

Smoothed sample spectrum

$\hat{f}(\omega) = \frac{1}{2\pi}(C_0+2\sum_{k=1}^{M-1}\omega_kC_kcos(\omega_k))$

where $M$ is the window width, and is calculated for frequency values

$\omega_i = \frac{2\pi i}{L},i=0,1...,[L/2]$

where [ ] denotes the integer part.

Smoothing window

$\omega_k = W\frac{k}{M}, k\leqslant M-1$

which for the various windows is defined over $0< \alpha < 1$ by

rectangular: $W(\alpha)=1$

Bartlett: $W(\alpha)=1-\alpha$

Tukey: $W(\alpha)=\frac{1}{2}(1+cos(\pi \alpha))$

Parzen: $W(\alpha)=\left\{\begin{array}{ll}1-6\alpha^2+6\alpha^3&0\leqslant \alpha\leqslant \frac{1}{2} \cr2(1-\alpha)^3&\frac{1}{2}< \alpha <1\end{array}\right.$

Reference

nag_tsa_spectrum_univar_cov (g13cac)