Component evaluation testing and analysis algorithms. Page: 22 of 136
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As would be expected, for the case of non-overlapping, rectangular windows, this updated
confidence interval equation degenerates back to the original equation.
As an example, the plot below of theoretical confidence intervals was generated for an arbitrary
time series with 100,000 data points, a 1024 point window length, and over a range of step sizes
for various window functions (see 0 PSD Confidence for the Matlab script to generate this
plot).
1.2 .............................. .......... ...........,...........................
Rectangular
Hann
Hamming
Kaiser
1 - B a rt le tt ---- ---- ------ ------ ------ --- -. -
CJ
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0.6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Step Size as a fraction of window length
Figure 8 PSD Confidence Interval versus Window Length
From this confidence interval plot, it is observed that considerable gains may be made in
reducing the uncertainty of the PSD estimate by allowing for an overlap in the data segments.
However, beyond an overlap of approximately one-half of the window length, there are
diminishing returns, due to the decreased independence of the data in successive time segments.
Also, for window functions that have greater amounts of taper at the extremes (such as Hann or
Kaiser), there are greater potential gains in confidence interval that can be achieved by reducing
the step size. For these window functions, the step size can be reduced further without any
decrease in the independence between the data in successive time segments.22
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Hart, Darren M. & Merchant, Bion John. Component evaluation testing and analysis algorithms., report, October 1, 2011; United States. (https://digital.library.unt.edu/ark:/67531/metadc847174/m1/22/: accessed April 17, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.