Let us first fix our notation. We shall use to denote the detector output, which is assumed to consist of a background noise and a useful gravitational wave signal . The Fourier transform of a quantity will be denoted and is defined as

The detector output is just a realization of noise , i.e., , when no signal is present. In the presence of a signal with an arrival time , takes the form,

The correlation of a template with the detector output is defined as In the above equation, is called the lag; it denotes the duration by which the filter function lags behind the detector output. The purpose of the above correlation integral is to concentrate all the signal energy at one place. The following analysis reveals how this is achieved; we shall work out the optimal filter that maximizes the correlation when a signal is present in the detector output. To do this let us first write the correlation integral in the Fourier domain by substituting for and , in the above integral, their Fourier transforms and , i.e., and , respectively. After some straightforward algebra, one obtains where denotes the complex conjugate of .Since is a random process, is also a random process. Moreover, correlation is a linear operation and hence obeys the same probability distribution function as . In particular, if is described by a Gaussian random process with zero mean, then is also described by a Gaussian distribution function, although its mean and variance will, in general, differ from those of . The mean value of , denoted by , is, clearly, the correlation of the template with the signal , since the mean value of is zero:

The variance of , denoted , turns out to be, Now the SNR is defined by .The form of integrals in Equations (77) and (78) leads naturally to the definition of the scalar product of waveforms. Given two functions, and , we define their scalar product to be [159, 161, 115, 129]

Note that [cf. Equation (67)], consequently, the scalar product is real and positive definite.Noting that the Fourier transform of a real function obeys , we can write down the SNR in terms of the above scalar product:

From this it is clear that the template that obtains the maximum value of is simply where is an arbitrary constant. From the above expression for an optimal filter we note two important things. First, the SNR is maximized when the lag parameter is equal to the time of arrival of the signal . Second, the optimal filter is not just a copy of the signal, but rather it is weighted down by the noise PSD. We will see below why this should be so.

We can now work out the optimal SNR by substituting Equation (81) for the optimal template in Equation (80),

We note that the optimal SNR is not just the total energy of the signal (which would be ), but rather the integrated signal power weighted down by the noise PSD. This is in accordance with what we would guess intuitively: the contribution to the SNR from a frequency bin where the noise PSD is high is smaller than from a bin where the noise PSD is low. Thus, an optimal filter automatically takes into account the nature of the noise PSD.

The expression for the optimal SNR Equation (82) suggests how one may compare signal strengths with the noise performance of a detector. Note that one cannot directly compare with , as they have different physical dimensions. In gravitational wave literature one writes the optimal SNR in one of the following equivalent ways

which facilitates the comparison of signal strengths with noise performance. One can compare the dimensionless quantities, and , or dimensionful quantities, and . Signals of interest to us are characterized by several (a priori unknown) parameters, such as the masses
of component stars in a binary, their intrinsic spins, etc., and an optimal filter must agree with both the
signal shape and its parameters. A filter whose parameters are slightly mismatched with that of a signal can
greatly degrade the SNR. For example, even a mismatch of one cycle in 10^{4} cycles can degrade the SNR by
a factor two.

When the parameters of a filter and its shape are precisely matched with that of a signal, what is the improvement brought about, as opposed to the case when no knowledge of the signal is available? Matched filtering helps in enhancing the SNR in proportion to the square root of the number of signal cycles in the detector band, as opposed to the case in which the signal shape is unknown and all that can be done is to Fourier transform the detector output and compare the signal energy in a frequency bin to noise energy in that bin. We shall see below that, in initial interferometers, matched filtering leads to an enhancement of order 30 – 100 for compact binary inspiral signals.

Matched filtering is currently being applied to mainly two sources: detection of (1) chirping signals from compact binaries consisting of black holes and/or neutron stars and (2) continuous waves from rapidly-spinning neutron stars.

In the general case of black-hole–binary inspiral the search space is characterized by 17 different parameters. These are the two masses of the bodies, their spins, eccentricity of the orbit, its orientation at some fiducial time, the position of the binary in the sky and its distance from the Earth, the epoch of coalescence and phase of the signal at that epoch, and the polarization angle. However, not all these parameters are important in a matched filter search. Only those parameters that change the shape of the signal, such as the masses, orbital eccentricity and spins, or cause a modulation in the signal due to the motion of the detector relative to the source, such as the direction to the source, are to be searched for and others, such as the epoch of coalescence and the phase at that epoch, are simply reference points in the signal that can be determined without any significant burden on computational power.

For binaries consisting of nonspinning objects that are either observed for a short enough period that the detector motion can be neglected, or last for only a short time in the sensitive part of a detector’s sensitivity band, there are essentially two search parameters – the component masses of the binary. It turns out that the signal manifold in this case is nearly flat, but the masses are curvilinear coordinates and are not good parameters for choosing templates. Chirp times, which are nonlinear functions of the masses, are very close to being Cartesian coordinates and template spacing is more or less uniform in terms of these parameters. Chirp times are post-Newtonian contributions at different orders to the duration of a signal starting from a time when the instantaneous gravitational-wave frequency has a fiducial value to a time when the gravitational wave frequency formally diverges and system coalesces. For instance, the chirp times and at Newtonian and 1.5 PN orders, respectively, are

where is the total mass and is the symmetric mass ratio. The above relations can be inverted to obtain and in terms of the chirp times:There is a significant amount of literature on the computational requirements to search for compact binaries [324, 146, 278, 280]. The estimates for initial detectors are not alarming and it is possible to search for these systems online. Searches for these systems by the LSC (see, for example, [8]) employs a hexagonal lattice of templates [119] in the two-dimensional space of chirp times. For the best LIGO detectors we need several thousand templates to search for component masses in the range [280]. Decreasing the lower-end of the mass range leads to an increase in the number of templates that goes roughly as and most current searches [2, 6] only begin at , with the exception of one that looked for black hole binaries of primordial origin [7], in which the lower end of the search was .

Inclusion of spins is only important when one or both of the components is rapidly spinning [41, 97]. Spins effects are unimportant for neutron star binaries, for which the dimensionless spin parameter , that is the ratio of its spin magnitude to the square of its mass, is tiny: . For ground-based detectors, even after including spins, the computational costs, while high, are not formidable and it should be possible to carry out the search on large computational clusters in real time [97]. Recently, the LSC has successfully carried out such a search [15].

Having chosen the bins and quantities as above, one can construct a statistic based on the measured SNR in each bin as compared to the expected value, namely

When the background noise is stationary and Gaussian, the quantity obeys the well-known chi-square distribution with degrees of freedom. Therefore, the statistical properties of the statistic are known. Imagine two triggers with identical SNRs, but one caused by a true signal and the other caused by a glitch that has power only in a small frequency range. It is easy to see that the two triggers will have very different values; in the first case the statistic will be far smaller than in the second case. This statistic has served as a very powerful veto in the search for signals from coalescing compact binaries and it has been instrumental in cleaning up the data (see, e.g., [2, 6]).http://www.livingreviews.org/lrr-2009-2 |
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