5.1 Matched filtering and optimal signal-to-noise ratio
Matched filtering is a data analysis technique that efficiently searches for a signal of known shape buried
in noisy data [188
]. The technique consists in correlating the output of a detector with a waveform,
variously known as a template or a filter. Given a signal
buried in noise
, the task
is to find an ‘optimal’ template
that would produce, on the average, the best possible
SNR. In this review, we shall treat the problem of matched filtering as an operational exercise.
However, this intuitive picture has a solid basis in the theory of hypothesis testing. The interested
reader may consult any standard text book on signal analysis, for example Helstrom [188
], for
details.
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
5.1.1 Optimal filter
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.
5.1.2 Optimal signal-to-noise ratio
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 104 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.
5.1.3 Practical applications of matched filtering
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.
5.1.3.1 Coalescing binaries.
In the case of chirping binaries, post-Newtonian theory (a perturbative
approximation to Einstein’s equations in which the relevant quantities are expanded as a power-series in
, where
is the speed of light) has been used to model the dynamics of these systems to a
very high order in
, where
is the relative speeds of the objects in the binary (see also
Section 6.5, in which binaries are discussed in more detail). This is an approximation that
can be effectively used to match filter the signal from binaries whose component bodies are of
equal, or nearly equal, masses and the system is still “far” from coalescence. In reality, one takes
the waveform to be valid until the last stable circular orbit (LSCO). In the case of binaries
consisting of two neutron stars, or a neutron star and a black hole, tidal effects might affect the
evolution significantly before reaching the LSCO. However, this is likely to occur at frequencies
well-above the sensitivity band of the current ground-based detectors, so that for all practical
purposes post-Newtonian waveforms are a good approximation to low-mass (
)
binaries.
As elucidated in Section 6.5.2, progress in analytical and numerical relativity has made it possible to
have a set of waveforms for the merger phase of compact binaries too. The computational cost in matched
filtering the merger phase, however, will not be high, as there will only be on the order of a few 100 cycles in
this phase. But it is important to have the correct waveforms to enhance signal visibility and, more
importantly, to enable strong-field tests of general relativity.
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].
5.1.3.2 Searching for Continuous Wave Signals.
In the case of continuous waves (CWs), the signal
shape is pretty trivial: a sinusoidal oscillation with small corrections to take account of the slow spin-down
of the neutron star/pulsar to account for the loss of angular momentum to gravitational waves
and other radiation/particles. However, what leads to an enormous computational cost here
is the Doppler modulation of the signal caused by Earth’s rotation, the motion of the Earth
around the solar system barycenter and the moon. The number of independent patches that
we have to observe so as not to lose appreciable amounts of SNR can be worked out in the
following manner. The baseline of a gravitational wave detector for CW sources is essentially
. For a source that emits gravitational waves at 100 Hz, the wavelength of the
radiation is
, and the angular resolution
of the antenna at an SNR of 1 is
, or a solid angle of
. In other words, the number of patches one
should search for is
. Moreover, for an observation that lasts for about a year
the frequency resolution is
. Searching over a frequency band
of 300 Hz, around the best sensitivity of the detector, gives the number of frequency bins to
be about 1010. Thus, it is necessary to search over roughly 1011 patches in the sky for each
of the 1010 frequency bins. This is a formidable task and one can only perform a matched
filter search over a short period (days/weeks) of the data or over a restricted region in the
sky, or just perform targeted searches for known objects such as pulsars, the galactic center,
etc. [92].
The severe computational burden faced in the case of CW searches has led to the development of
specialized searches that look for signals from known pulsars [5, 10, 12] using an efficient search algorithm
that makes use of the known parameters [116
, 151] and hierarchical algorithms that add power
incoherently with the minimum possible loss in signal visibility [226, 4, 342, 11
]. The most ambitious
project in this regard is the Einstein@Home project [153]. The goal here is to carry out coherent
searches for CW signals using wasted CPUs on idle computers at homes, offices and university
departments around the world. The project has been successful in attracting a large number of
subscriptions and provides the largest computational infrastructure to the LSC for the specific
search of CW signals and the first scientific results from such are now being published by the
LSC [13].
5.1.3.3
veto.
Towards the end of Section 4.8 we discuss a powerful way of rejecting triggers, whose
root cause is not gravitational wave signals but false alarms due to instrumental and environmental
artifacts. In this section we will further quantify the
veto [31
] by using the scalar product introduced
in the context of matched filtering.
The main problem with real data is that it can be glitchy in the form of high amplitude transients that
might look like damped sinusoids. An inspiral signal and a template employed to detect it are both
broadband signals. Therefore, the matched-filter SNR for such signals has contributions from a
wide range of frequencies. However, the statistic of matched filtering, namely the SNR, is an
integral over frequency and it is not sensitive to contributions from different frequency regions.
Imagine dividing the frequency range of integration into a finite number of bins
,
, such that their union spans the entire frequency band,
and
,
and further that the contribution to the SNR from each frequency bin is the same, namely,
Now, define the contribution to the matched filtering statistic coming from the
-th bin by [31
]
where, as before,
and
are the Fourier transforms of the detector output and the
template, respectively. Note that the sum
gives the full matched filtering statistic [31]:
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]).