Example: transferF program

0 This example uses the function GRnormalize() to calculate the response of the interferometer to a specified gravitational-wave strain $h(t)$. [Note: for clarity, in this example, we have NOT worried about getting the overall normalization correct.] The code includes two possible $h(t)$'s. The first of these is a binary-inspiral chirp (see Section ). Or, if you un-comment one line of code, you can see the response of the detector to a unit-impulse gravitational wave strain, in other words, the impulse response of the detector.

Note that to run this program, you must specify a path to the 40-meter data, for example by typing:
setenv GRASP_FRAMEPATH /usr/local/data/19nov94.3.frame
so that the code can find a frame containing a swept-sine calibration file to use.

The response of the detector to a pair of inspiraling stars is shown in Figure . You will notice that although the chirp starts at a (gravitational-wave) frequency of 140 Hz on the left-hand side of the figure, the low-frequency response of the detector is so poor that the chirp does not really become visible until about half-a-second later, at somewhat higher frequency. In the language of the audiophile, the IFO has crummy bass response! Of course this is entirely deliberate; the whitening filters of the instrument are designed to attenuate the low-frequency seismic contamination, and consequently also attenuate any possible low-frequency gravitational waves.

Figure: Output produced by the transfer example program. The top graph shows the gravitational-wave strain produced by an inspiraling binary pair. The lower graph shows the calculated interferometer output [channel.0 or IFO_DMRO] produced by this signal. Notice that because of the poor low-frequency response of the instrument, the IFO output does not show significant response before the input frequency has increased. The sample rate is slightly under 10 kHz.

The response of the detector to a unit gravitational strain impulse is shown as a function of time-offset in Figure . Here the predominant effect is the ringing of the anti-aliasing filter. The impulse response of the detector lasts about 30 samples, or 3 msec. For negative offset times the impulse response is quite close to zero; its failure to vanish is partly a wrap-around effect, and partly due to errors in the actual measurement of the transfer function.

Figure: Output produced by the transfer example program. This shows the calculated interferometer output [channel.0 or IFO_DMRO] produced by an impulse in the gravitational-wave strain at sample number zero. This (almost) causal impulse response lasts about 3 msec.

This is a good place to insert a cautionary note. Now that we have determined the transfer function $R(f)$ of the instrument, you might be tempted to ask: ``Why should I do any of my analysis in terms of the instrument output? After all, my real interest is in gravitational waves. So the first thing that I will do in my analysis is convert the instrument output into a gravitational wave strain $h(t)$ at the detector, by convolving the instrument's output with (the time-domain version of) $R(f)$." Please do not make this mistake! A few moment's reflection will show why this is a remarkably bad idea. The problem is that the response function $R(f)$ is extremely large at low frequencies. This is just a reflection of the poor low frequency response of the instrument: any low-frequency energy in the IFO output corresponds to an extremely large amplitude low frequency gravitational wave. So, if you calculate $h(t)$ in the way described: take a stretch of (perhaps zero-padded) data, FFT it into the frequency domain, multiply it by $R(f)$ and invert the FFT to take it back into the frequency domain, you will discover the following:

• Your $h(t)$ is dominated by a single low-frequency noisy sinusoid (whose frequency is determined by the low frequency cutoff imposed by the length of your data segment or the low-frequency cutoff of the response function).
• Your $h(t)$ has lost all the interesting information present at frequencies where the detector is quiet (say, around 600 Hz). Because the noise power spectrum (see Figure ) covers such a large dynamic range, you can not even represent $h(t)$ in a floating point variable (though it will fit, though barely, into a double). This is why the instrument uses a whitening filter in the first place.
• It is possible to construct ``$h(t)$" if you filter out the low-frequency garbage by setting $R(f)$ to zero below (say) 100 Hz.

If you are unconvinced by this, do the following exercise: calculate the power spectrum in the frequency domain as was done with Figure , then construct $h(t)$ in time time domain, then take $h(t)$ back into the frequency domain, and graph the power spectrum again. You will discover that it has completely changed above 100 Hz and is entirely domainted by numerical quantization noise (round-off errors).

Includes/transferF.tex

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

Comments: None.