Example: power_spectrumF program

0 This example uses the function GRnormalize() to produce a normalized, properly calibrated power spectrum of the interferometer noise, using the gravity-wave signal and the swept-sine calibration information from the frames.

The output of this program is a 2-column file; the first column is frequency and the second column is the noise in units of $\rm meters/\sqrt{\rm Hz}$. To run this program, and display a graph, type
setenv GRASP_FRAMEPATH /usr/local/GRASP/18nov94.1frame
power_spectrumF > outputfile
xmgr -log xy outputfile

A couple of comments are in order here:

1. Even though we only need the modulus, for pedagogic reasons, we explicitly calculate both the real and imaginary parts of $\widetilde{\Delta l}(f) = R(f) \widetilde{C_{\rm IFO}}(f)$.

2. The fast Fourier transform of $\Delta l$, which we denote ${\rm FFT}[\Delta l]$, has the same units (meters!) as $\Delta l$. As can be immediately seen from Numerical Recipes equation (12.1.6) the Fourier transform $\widetilde{\Delta l}$ has units of meters-sec and is given by $\widetilde{\Delta l} =\Delta t \; {\rm FFT}[\Delta l]$, where $\Delta t$ is the sample interval. The (one-sided) power spectrum of $\Delta l$ in $\rm meters/\sqrt{\rm Hz}$ is $P=\sqrt{2 \over T} \vert \widetilde{\Delta l} \vert$ where $T=N \Delta t$ is the total length of the observation interval, in seconds. Hence one has

$$P=\sqrt{2 \over N \Delta t} \; \Delta t \; \vert {\rm FFT}[\... ... \sqrt{2 \Delta t \over N} \; \vert {\rm FFT}[\Delta l] \vert.$$ (4.11.26)

This is the reason for the factor which appears in this example.

3. To get a spectrum with decent frequency resolution, the time-domain data must be windowed (see the example program calibrate and the function avg_spec() to see how this works).

A sample of the output from this program is shown in Figure .

Figure: An example of a power spectrum curve produced with power_spectrumF. The spectrum produced off a data tape (with 100 point smoothing) is compared to that produced by the HP spectrum analyzer in the lab.

Includes/power_spectrumF.tex

Author: Bruce Allen, ballen@dirac.phys.uwm.edu

Comments: The IFO output typically consists of a number of strong line sources (harmonics of the 60 Hz line and the 180 Hz laser power supply, violin modes of the suspension, etc) superposed on a continuum background (electronics noise, laser shot noise, etc) In such situations, there are better ways of finding the noise power spectrum (for example, see the multi-taper methods of David J. Thompson [39], or the textbook by Percival and Walden [40]). Using methods such as the F-test to remove line features from the time-domain data stream might reduce the sidelobe contamination (bias) from nearby frequency bins, and thus permit an effective reduction of instrument noise near these spectral line features. Further details of these methods, and some routines that implemen them, may be found in Section .