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Vetoing techniques ( time/frequency test)

The second technique vetoing or discrimination test operates in the frequency domain, and is described here. It is a very stringent test, which determines if the hypothetical chirp which has been found in the data stream is consistent with a true binary inspiral chirp summed with Gaussian interferometer noise. If this is true, it should be possible to subtract the (best fit) chirp from the signal, and be left with a signal stream that is consistent with Gaussian IFO noise. One of the nice features of this technique is that it can be statistically characterized in a rigorous way. We follow the same general course as in Section on Wiener filtering, first considering the case of a single phase" phase signal, then considering the case of an unknown phase" signal.

In the single-phase case, suppose that one of our optimal chirp filters is triggered with a large SNR at time . We suppose that the signal which was responsible for this trigger may be written in either the time or the frequency domain as

 (6.24.105)

We assume that we have identified what is believed to be the correct" template , by the procedure already described of maximizing the SNR over arrival time and template, and have used this to estimate . We assume that has been determined exactly (a good approximation since it can be estimated to high accuracy). Our goal is to construct a statistic which will indicate if our estimate of and identification of are credible.

We will denote the signal value at time offset by the real number :

 (6.24.106)

(Here, denotes the Nyqist frequency, one-half of the sampling rate.) The expected value of is , although of course since we are discussing a single instance, it's actual value will in general be different. The chirp template is normalized so that the expected value :
 (6.24.107)

We are going to investigate if this signal is really" due to a chirp by investigating the way in which gets its contribution from different ranges of frequencies. To do this, break up the integration region in this integral into a set of disjoint subintervals whose union is the entire range of frequencies from DC to Nyquist. Here is a small integer (for example, ). This splitup can be performed using the GRASP function splitup(). The frequency intervals:
 (6.24.108)

are defined by the condition that the expected signal contributions in each frequency band from a chirp are equal:
 (6.24.109)

A typical set of frequency intervals in shown in Figure .

Because the filter is optimal, this also means that the expected noise contributions in each band from the chirp is the same. The frequency subintervals are narrow in regions of frequency space where the interferometer is quiet, and are broad in regions where the IFO is noisy.

Now, define a set of signal values, one for each frequency interval:

 (6.24.110)

We have included both the positive and negative frequency subintervals to ensure that the are real. If the detector output is Gaussian noise plus a true chirp, the are normal random variables, with a mean value of and a variance determined by the expected value of the noise-squared:
 (6.24.111)

>From these signal values we can construct a useful time/frequency statistic.

To characterize the statistic, we will need the probability distribution of the . Because each of these values is a sum over different (non-overlapping) frequency bins, they are independent random Gaussian variables with unknown mean values. Their a-priori probability distribution is

 (6.24.112)

The statistic that we will construct addresses the question, are the measured values of consistent with the assumption that the measured signal is Gaussian detector noise plus ?" One small difficulty is that the value of that appears in () is not known to us: we only have an estimate of its value. To overcome this, we first construct a set of values denoted
 (6.24.113)

These are not independent normal random variables: they are correlated since vanishes. To proceed, we need to calculate the probability distribution of the , which we denote by . This quantity is defined by the relation that the integral of any function of variables with respect to the measure defined by this probability distribution satisfies
 (6.24.114)

[Note that in this expression and the following ones, all integrals are from to .] This may be used to find a closed form for : let . This gives
 (6.24.115)

To evaluate the integral, change variables from to defined by
 (6.24.116)

which can be inverted to yield
 (6.24.117)

The Jacobian of this coordinate transformation is:
 (6.24.118)

Using the linearity in rows of the determinant, it is straightforward to show that .

The integral may now be written as

 (6.24.119)

A few moments of algebra shows that the exponent may be expressed in terms of the new integration variables as
 (6.24.120)

and thus the integral yields
 (6.24.121)

This probability distribution arises because we do not know the true mean value of which is but can only estimate it using the actual measured value of . Similar problems arise whenever the mean of a distribution is not know but must be estimated (problem 14-7 of [24]). This probability distribution is as close as you can get to Gaussian" subject to the constraint that the sum of the must vanish. It is significant that this probability density function is completely independent of , which means that the properties of the do not depend upon whether a signal is present or not.

The individual have identical mean and variance, which may be easily calculated from the probability distribution function (). For example the mean is zero: . To calculate the variance, let in (). One finds

 (6.24.122)

Now that we have calculated the probability distribution of the it is straightforward to construct and characterize a -like statistic which we will call .

Define the statistic

 (6.24.123)

>From () it is obvious that for Gaussian noise plus a chirp the statistical properties of are independent of : it has the same statistical properties if a chirp signal is present or not. For this reason, the value of provides a powerful method of testing the hypothesis that the detector's output is Gaussian noise plus a chirp. If the detector's output is of this form, then the value of is unlikely to be much larger than its expected value (this statement is quantified below). On the other hand, if the filter was triggered by a spurious noise event that does not have the correct time/frequency distribution, then will typically have a value that is very different than the value that it has for Gaussian noise alone (or equivalently, for Gaussian noise plus a chirp).

The expected value of is trivial to calculate

 (6.24.124)

One can also easily compute the probability distribution of using (). The probability that in the presence of a true chirp signal is the integral of () over the region . In the -dimensional space, the integrand vanishes except on a -plane, where it is spherically-symmetric. To evaluate the integral, introduce a new set of orthonormal coordinates obtained from any orthogonal transformation on for which the new 'th coordinate is orthogonal to the hyperplane . Hence . Our statistic is also the squared radius in these coordinates. Hence
 (6.24.125)

It's now easy to do the integral over the coordinate , and having done this, we are left with a spherically-symmetric integral over :
 (6.24.126)

where is the volume of a unit-radius sphere . The incomplete gamma function is the same function that describes the likelihood function in the traditional test [the Numerical Recipes function gammq(a,x)]. Figure show a graph of for some different values of the parameter . The appearance of in these expressions reflects the fact that although is a sum of the squares of Gaussian variables, these variables are subject to a single constraint (their sum vanishes) and thus the number of degrees of freedom is .

In practice (based on CIT 40-meter data) breaking up the frequency range into intervals provides a very reliable veto for rejecting events that trigger an optimal filter, but which are not themselves chirps. The value of so if then one can conclude that the likelihood that a given trigger is actually due to a chirp is less than ; rejecting or vetoing such events will only reduce the true event" rate by . However in practice it eliminates almost all other events that trigger an optimal filter; a noisy event that stimulates a binary chirp filter typically has or larger!

The previous analysis for the single-phase" case assumes that we have found the correct template describing the signal. In searching for a binary inspiral chirp however, the signal is a linear combination of the two different possible phases:

 (6.24.127)

and the amplitudes and are unknown. The reader might well wonder why we can't simply construct a single properly normalized template as
 (6.24.128)

and then use the previously-described single phase" method. In principle, this would work properly. The problem is that we do not know the correct values of and . Since and , we can estimate the values of and from the real and imaginary parts of the measured signal, however these estimates will not give the true values. For this reason, an statistic and test can be constructed for the two-phase" case, but it has twice the number of degrees of freedom as the single-phase" case.

The description and characterization of the test for the two phase case is similar to the single-phase case. For the two phase case, the signal is a complex number

 (6.24.129)

The templates for the individual phases are normalized as before:
 (6.24.130)

This assume the same adiabatic limit discussed earlier: . In this limit, the frequency intervals are identical for either template. We define signal values in each frequency band in the same way as before, except now these are complex:
 (6.24.131)

The mean value of the signal in each frequency band is
 (6.24.132)

and the variance of either the real or imaginary part is as before, so that the total variance is twice as large as in the single phase case:
 (6.24.133)

The signal values are now characterized by the probability distribution
 (6.24.134)

Note that the arguments of this function are complex; for this reason the overall normalization factors have changed from the single-phase case. We now construct complex quantities which are the difference between the actual signal measured in a frequency band and the expected value for our templates and phases:
 (6.24.135)

The probability distribution of these differences is still defined by () but in that expression, the variables of integration and are integrated over the complex plane (real and imaginary parts from to ), and is any function of complex variables. As before, we can calculate by choosing correctly, in this case as , where . The same procedure as before then yields the probability distribution function
 (6.24.136)

It is now easy to see that the expectation of the signal differences is still zero but the variances are twice as large as in the single-phase case:
 (6.24.137)

The statistic is now defined by
 (6.24.138)

and has an expectation value which as twice as large as in the single-phase case:
 (6.24.139)

The calculation of the distribution function of is similar to the single phase case (but with twice the number of degrees of freedom) and gives the incomplete -function
 (6.24.140)

This is precisely the distribution of a statistic with degrees of freedom: each of the variables has 2 degrees of freedom, and there are two constraints since the sum of both the real and imaginary parts vanishes. In fact since the expectation value of the statistic is just the number of degrees of freedom:
 (6.24.141)

the relationship between the and statistic may be obtained by comparing equations () and (), giving
 (6.24.142)

Next: How does the test Up: GRASP Routines: Gravitational Radiation Previous: Vetoing techniques (time domain   Contents
Bruce Allen 2000-11-19