Matched filtering of gravitational waves from inspiraling compact binaries: Computational cost
and template placement
Benjamin J. Owen
Theoretical Astrophysics, California Institute of Technology, Pasadena, California 91125
and Max Planck Institut fu
¨
r Gravitationsphysik (Albert Einstein Institut), Am Mu
¨
hlenberg 5, 14476 Golm, Germany
B. S. Sathyaprakash
Theoretical Astrophysics, California Institute of Technology, Pasadena, California 91125
and Department of Physics and Astronomy, Cardiff University, Cardiff, CF2 3YB, United Kingdom
~Received 27 August 1998; published 22 June 1999!
We estimate the number of templates, computational power, and storage required for a one-step matched
filtering search for gravitational waves from inspiraling compact binaries. Our estimates for the one-step search
strategy should serve as benchmarks for the evaluation of more sophisticated strategies such as hierarchical
searches. We use a discrete family of two-parameter wave form templates based on the second post-Newtonian
approximation for binaries composed of nonspinning compact bodies in circular orbits. We present estimates
for all of the large- and mid-scale interferometers now under construction: LIGO ~three configurations!,
VIRGO, GEO600, and TAMA. To search for binaries with components more massive than m
min
50.2M
(
while losing no more than 10% of events due to coarseness of template spacing, the initial LIGO interferom-
eters will require about 1.03 10
11
flops ~floating point operations per second! for data analysis to keep up with
data acquisition. This is several times higher than estimated in previous work by Owen, in part because of the
improved family of templates and in part because we use more realistic ~higher! sampling rates. Enhanced
LIGO, GEO600, and TAMA will require computational power similar to initial LIGO. Advanced LIGO will
require 7.83 10
11
flops, and VIRGO will require 4.83 10
12
flops to take full advantage of its broad target noise
spectrum. If the templates are stored rather than generated as needed, storage requirements range from 1.5
3 10
11
real numbers for TAMA to 6.23 10
14
for VIRGO. The computational power required scales roughly as
m
min
28/3
and the storage as m
min
213/3
. Since these scalings are perturbed by the curvature of the parameter space at
second post-Newtonian order, we also provide estimates for a search with m
min
51M
(
. Finally, we sketch and
discuss an algorithm for placing the templates in the parameter space. @S0556-2821~99!05214-5#
PACS number~s!: 04.80.Nn, 07.05.Kf, 97.80.2d
I. INTRODUCTION
Close binary systems composed of compact objects ~such
as black holes and neutron stars! are expected to be an im-
portant source of gravitational waves for broadband laser in-
terferometers such as the Laser Interferometric Gravitational
Wave Observatory ~LIGO!, VIRGO, GEO600, and TAMA
@1–4#. The orbit of a compact binary decays under the influ-
ence of gravitational radiation reaction, emitting a gravita-
tional wave signal that increases in amplitude and ‘‘chirps’’
upward in frequency as the objects spiral in toward each
other just before their final coalescence. According to current
astronomical lore @5–7#, the rate of coalescences should be
about three per year within 200 to 300 Mpc @5# of the Earth
for neutron-star–neutron-star binaries, and within 400 Mpc
to 1 Gpc for black-hole–black-hole binaries. Signals from
inspiraling compact binaries at these distances are strong
enough to be detected by ‘‘enhanced’’ LIGO @8# and VIRGO
interferometers, but only if aided by a nearly optimal signal-
processing technique. Fortunately, the distinctive frequency
chirp has been calculated to a remarkable degree of precision
using a variety of approximations to the general relativistic
two-body problem ~e.g. @9,10#!. Because the functional form
of the chirp is quite well known, a search for inspiral signals
in noisy data is ideally suited to matched filtering.
Matched filtering @11# has long been known to be the
optimal linear signal-processing technique and is well dis-
cussed in the literature ~e.g. @12#!; therefore we will only
briefly summarize it here. In the frequency domain, a
matched filter is a best-guess template of the expected signal
waveform divided by the interferometer’s spectral noise den-
sity. The interferometer output is cross-correlated with the
matched filter at different time delays to produce a filtered
output. The signal-to-noise ratio, defined as the ratio of the
actual value of the filtered output to its rms value in the
presence of pure noise, is compared to a predetermined
threshold to decide if a signal is present in the noise. If the
signal from which the matched filter was constructed is
present, it contributes coherently to the cross-correlation,
while the noise contributes incoherently and thus is reduced
relative to the signal. Also, the weighting of the cross-
correlation by the inverse of the spectral noise density em-
phasizes those frequencies to which the interferometer is
most sensitive. Consequently, signals thousands of cycles
long whose unfiltered amplitude is only a few percent of the
rms noise can be detected.
A matched filtering search for inspiraling compact bina-
ries can be computationally intensive due to the variety of
possible waveforms. Although the inspiral signals are all ex-
pected to have the same functional form, this form depends
on several parameters—the masses of the two objects, their
spins, the eccentricity of their orbit, etc.—some of them
PHYSICAL REVIEW D, VOLUME 60, 022002
0556-2821/99/60~2!/022002~12!/$15.00 ©1999 The American Physical Society60 022002-1
quite strongly. A filter constructed from a waveform tem-
plate with any given parameter vector may do a very poor
job of detecting a signal with another parameter vector. That
is, the difference in parameter vectors can lead to a greatly
reduced cross-correlation between the two wave forms; and
in general, the greater the difference, the more the cross-
correlation is reduced. Because the parameter vector of a
signal is not known in advance, it is necessary to filter the
data with a family of templates located at various points in
parameter space—e.g., placed on a lattice—such that any
signal will lie close enough to at least one of the templates to
have a good cross-correlation with that template @13#.
There are several questions that must be answered in or-
der to determine the feasibility of a matched filtering search
strategy and, if feasible, to implement it. Which parameters
significantly affect the wave form? How should the spacing
of the template parameters ~lattice points! be chosen? Is there
a parametrization that is in some sense ‘‘preferred’’ by the
template wave forms? How many templates are needed to
cover a given region of interest in the parameter space, and
how much computing power and memory will it cost to pro-
cess the data through them? In the case of inspiraling com-
pact binaries, the full general-relativistic wave forms are not
exactly known, but are instead approximated ~e.g., using the
post-Newtonian scheme!; and we must also ask, what ap-
proximation to the true wave forms is good enough?
All of these questions have been addressed in recent
years, at least at some level. The current state of affairs is
summarized by the following brief review of the literature:
The standard measure for deciding what class of wave
forms is good enough is the fitting factor (FF) introduced by
Apostolatos @14#. The fitting factor is effectively the fraction
of optimal signal-to-noise-ratio obtained when filtering the
data with an approximate family of templates. Because bina-
ries are ~on large scales! uniformly distributed in space and
because the signal strength scales inversely with distance, the
fraction of event rate retained is approximately FF
3
. There-
fore it has become conventional to regard FF597%—i.e.,
10% loss of event rate—as a reasonable goal. Using the stan-
dard post-Newtonian expansion in the test-mass case ~i.e.,
when one body is much less massive than the other so that
the wave forms can be computed with arbitrarily high preci-
sion using the Teukolsky formalism!, Droz and Poisson @15#
found that second post-Newtonian ~2PN! signals had fitting
factors of 90% or higher. Damour, Iyer, and Sathyaprakash
@16# have devised a new way of rearranging the usual post-
Newtonian expansion ~similar to the way Pade
´
approximants
rearrange the coefficients of a Taylor expansion! to take ad-
vantage of physical intuition in constructing templates. They
find fitting factors of 95% or higher for the 2PN templates.
Research underway by the authors of @9# will lead to 3PN
templates that should easily achieve FF.97%.
Several people @17,14,18# have shown that it is insuffi-
cient to use templates that depend on just one shape param-
eter ~the ‘‘chirp mass,’’ which governs the rate of frequency
sweep at Newtonian-quadrupole order!. To achieve FF
.90% one must include the masses of both objects as tem-
plate parameters, as was done in the above 2PN analyses
@15,16#, and as is being done in the forthcoming 3PN tem-
plates.
Apostolatos @14,19# showed that, for binaries whose com-
ponents spin rapidly about their own axes which are orthogo-
nal to the orbital plane so that there is no precession, neglect-
ing the spin parameters ~i.e., using two-mass-parameter wave
forms based on the theory of spinless binaries! degraded the
fitting factors by less than 2%. With precession the situation
is much more complicated, and data analysis algorithms are
as yet poorly developed: It is clear that there are interesting
corners of parameter space ~most especially a neutron star in
a substantially nonequatorial, precessing orbit around a much
more massive, rapidly spinning black hole! in which the two-
mass-parameter spinless wave forms give FF!90%; to
search for such binaries will require wave forms with three
or more parameters @19#. However, the 2PN ~or 3PN! two-
mass-parameter wave forms do appear to cover adequately a
significant portion of the parameter space for precessing bi-
naries @14#.
Sathyaprakash @20# showed that in computations with the
two-mass-parameter wave forms, the best coordinates to use
on the parameter space are not the two masses, but rather the
inspiral times from some fiducial frequency to final merger,
as computed at Newtonian and first post-Newtonian order.
Working with the restricted first post-Newtonian wave forms
~see below! he found that the effective dimension of the pa-
rameter space is nearly one.
Sathyaprakash and Dhurandhar @21–23# developed a cri-
terion for putting templates at discrete points on a grid in
parameter space and numerically implemented their criterion
for a one-parameter ~Newtonian! family of templates and for
simple noise models. They introduced the concept of what
Owen @24# later called the minimal match ~analogous to the
fitting factor! as a measure of how well a discrete set of
templates covers the parameter space and estimated the com-
putational costs for an on-line search.
Owen @24#, building on the work of Sathyaprakash and
Dhurandhar, defined a metric on the parameter space from
which one can semi-analytically calculate ~i! the template
spacing needed to achieve a desired minimal match, ~ii! the
total number of templates needed, and ~iii! the computational
requirements to keep up with the data—for any family of
wave forms and any interferometer noise spectrum. Owen
combined this metric-based formalism with computational
counting procedures from Schutz @25# to estimate the com-
putational requirements for LIGO searches based on two-
parameter 1PN templates. These estimates were confirmed
by Apostolatos @19# using a numerical method in the vein of
~but more sophisticated than! the previous work of
Sathyaprakash and Durandhar @21–23#. Apostolatos also
showed that a search for precessing binaries that fully covers
all the nooks and crannies of the precessional parameter
space, using currently available templates and techniques, is
prohibitively costly.
Mohanty and Dhurandhar @26,27# have studied hierarchi-
cal search strategies. Such strategies reduce computational
costs by making a first pass of the data through a coarsely-
spaced template grid and a low signal-to-noise threshold to
identify candidate signals. Each candidate flagged by the first
BENJAMIN J. OWEN AND B. S. SATHYAPRAKASH PHYSICAL REVIEW D 60 022002
022002-2
pass is examined more closely with a second, finely-spaced
grid of templates and a higher threshold to weed out false
alarms. Such strategies can reduce the total computational
requirements by roughly a factor 25.
The purpose of this paper is to refine and update the
analyses by Owen @24# for the two-parameter, spinless tem-
plates that are likely to be used for binary-inspiral searches in
ground-based interferometers. This refinement is needed be-
cause the kilometer-scale interferometers will begin taking
data in about 2 years ~preliminary, engineering run!; people
are even now designing software to implement the simplest
matched filtering search algorithm; and in the context of
these implementations, the factor of 3 accuracy attempted in
Ref. @24# is no longer adequate. The numbers that are de-
rived in this paper should establish a reliable baseline cost to
which more sophisticated search strategies ~e.g., hierarchical
searches! can be compared.
The substantial differences between this paper and Ref.
@24# are that we now ~i! approximate the phase evolution of
the inspiral wave form to 2PN rather than 1PN order; ~ii!
give results for the noise spectra of several more interferom-
eters; and ~iii! use a better estimate of the sampling fre-
quency needed for each interferometer. We assume the fol-
lowing fiducial search: a minimal match of 97%
~corresponding to 10% loss of event rate due to coarse pa-
rameter space coverage!, second post-Newtonian wave
forms, and templates made for objects of minimum mass
m
min
50.2M
(
and up.
Our results for the computational requirements are given
in Tables II–IV. These tables show that the initial LIGO
interferometers need about twice as many templates and
triple the computational power estimated in Ref. @24#. These
increases result mainly from using 2PN wave forms rather
than the ~clearly inadequate! 1PN, and from using a higher
sampling rate ~as, it turns out, is required to keep time-step
discretization error from compromising the 97% minimal
match!. GEO600 requires slightly more templates and power
than LIGO because of its flatter noise spectrum, while
TAMA requires slightly less because its sensitivity is limited
to higher frequencies where there are fewer cycles. Initial
VIRGO, with its extremely broad and flat spectrum, requires
about the same as advanced LIGO.
The rest of this paper is organized as follows. In Sec. II
we analyze the application of matched filtering to a search
for inspiraling binaries and summarize the method of Ref.
@24# which uses differential geometry to answer important
questions about such a search. We use this method in Sec. III
to estimate the computational costs and other requirements
of a matched-filtering binary search for LIGO, VIRGO,
GEO600, and TAMA. In Sec. IV we illustrate a detailed
example of a template placement algorithm, and in Sec. V
we discuss our results.
II. FORMALISM
This section summarizes material previously presented in
@24# with several incremental improvements. We begin by
introducing some notation.
The Fourier transform of a function h(t) is denoted by
h
˜
(f), where
h
˜
~
f
!
[
E
2 `
`
dt e
i2
p
ft
h
~
t
!
. ~2.1!
We write the interferometer output h(t) as the sum of noise
n(t) and a signal As(t), where we have separated the signal
into a dimensionless, time-independent amplitude A and a
‘‘shape’’ function s(t) which is defined to have unit norm
@see Eq. ~2.4! below#.
The strain power spectral noise density of an interferom-
eter is denoted by S
h
(f). We use the one-sided spectral den-
sity, defined by
E
@
n
˜
~
f
1
!
n
˜
*
~
f
2
!
#
5
1
2
d
~
f
1
2 f
2
!
S
h
~
u
f
1
u
!
, ~2.2!
where E
@#
denotes the expectation value over an ensemble
of realizations of the noise and an asterisk denotes complex
conjugation.
We use geometrized units, i.e., Newton’s gravitational
constant G and the speed of light c have values of unity.
A. Matched filtering
First we flesh out the Introduction’s brief description of
matched filtering. In the simplest idealization of matched fil-
tering, the filtered signal-to-noise ratio is defined by @12#
S
N
[
^
h,u
&
rms
^
n,u
&
. ~2.3!
Here u is the template wave form used to filter the data
stream h, and the inner product
^
a,b
&
[4Re
F
E
0
`
df
a
˜
*
~
f
!
b
˜
~
f
!
S
h
~
f
!
G
~2.4!
is the noise-weighted cross-correlation between a and b ~cf.
@28#!. The denominator of Eq. ~2.3! is equal to
A
^
u,u
&
, the
norm of u ~see Sec. II B of Ref. @28# for a proof!. Because
the norm of u cancels out of Eq. ~2.3!, we can simplify our
calculations without loss of generality by considering all
templates to have unit norm.
When searching for a parametrized family of signals the
situation is somewhat more complicated. The parameter val-
ues of the signals are not known in advance; therefore one
must filter the data through many templates constructed at
different points in the parameter space. To develop a strategy
for searching the parameter space, one must know how much
the S/N is reduced by using a template whose parameter
values differ from those of the signal. Neglecting fluctuations
due to the noise, the fraction of the optimal S/N obtained by
using the wrong parameter values is given by the ambiguity
function
A
~
l,L
!
[
^
u
~
l
!
,u
~
L
!
&
~2.5!
MATCHED FILTERING OF GRAVITATIONAL WAVES . . . PHYSICAL REVIEW D 60 022002
022002-3
~see, e.g., Chaps. XIII and X of Ref. @12#!. Here l and L are
the parameter vectors of the signal and template ~it does not
matter which is which!. The ambiguity function, as its name
implies, is a measure of how distinguishable two wave forms
are with respect to the matched filtering process. It can be
regarded as an inner product on the wave form parameter
space and is fundamental to the theory of parameter estima-
tion @12,29#.
For the purposes of a search for inspiraling compact bi-
naries, the ambiguity function isn’t quite what is needed.
This is because the test statistic ~for a given set of parameter
values
u
) is not given by Eq. ~2.3!, but rather by
max
f
c
,t
c
^
h,u
~
u
!
e
i(2
p
ft
c
2
f
c
)
&
rms
^
n,u
~
u
!
&
. ~2.6!
Here
f
c
and t
c
are respectively the coalescence time and
coalescence phase. We separate these parameters out from
the rest: l5(
f
c
,t
c
,
u
), where
u
is the vector of intrinsic
parameters that determine the shape of the wave form and
f
c
and t
c
are extrinsic parameters @24#~also referred to as
kinematical and dynamical parameters @20#, respectively!.
The practical difference is that maximization over the extrin-
sic parameters is performed automatically by Fourier trans-
forming, taking the absolute value, and looking for peaks.
The use of Eq. ~2.6! as a detection statistic suggests the defi-
nition of a modified ambiguity function known as the match
@24#
M
~
u
1
,
u
2
!
[max
f
c
,t
c
^
u
~
u
1
!
,u
~
u
2
!
e
i(2
p
ft
c
2
f
c
)
&
, ~2.7!
where the templates u are assumed to have unit norm. The
use of this match function rather than the ambiguity function
takes into account the fact that a search can benefit from
systematic errors in the extrinsic parameters.
B. Applications of differential geometry
The match ~2.7! can be regarded as an inner product on
the space of template shapes and intrinsic template param-
eters, and correspondingly one can define a metric on this
space @24#:
g
ij
~
u
!
[2
1
2
]
2
M
~
u
,Q
!
]
Q
i
]
Q
j
U
Q
k
5
u
k
. ~2.8!
The metric ~2.8! is derived from the match ~2.7! in the same
way the information matrix G
ij
is derived from the ambiguity
function @29#, and plays a role in signal detection similar to
that played by the information matrix in parameter estima-
tion @30#. The g
ij
can be derived by expanding M(
u
,Q)
about Q5
u
, or equivalently by projecting G
ij
on the sub-
space orthogonal to
f
c
and t
c
.
The g
ij
can be used to approximate the match in the re-
gime 12 M!1by
M
~
u
,
u
1D
u
!
.12g
ij
D
u
i
D
u
j
, ~2.9!
which is simply another way of writing the Taylor expansion
of M(
u
,
u
1 D
u
) about D
u
5 0. ~The first derivative term van-
ishes because M takes its maximum value of unity at D
u
5 0.! We find that the quadratic approximation ~2.9! is good
typically for M.0.95 or greater, though this depends on the
wave form and noise spectrum used. Experience suggests
that the quadratic approximation generally underestimates
the true match; and thus the spacings and numbers of tem-
plates we calculate using Eq. ~2.9! err on the safe side. See
Fig. 1 for an example.
FIG. 1. Comparison of the full match to the quadratic approxi-
mation in the case of ~a! first post-Newtonian and ~b! second post-
Newtonian wave forms. In both cases the noise spectrum is LIGO I
~see Table I!. The elliptical solid line is the 97% contour of the
match with a reference wave form ~in the center of the ellipse!
where the match is given analytically by the metric in the quadratic
approximation @Eq. ~2.9!#. The dots are locations of the same con-
tour given by constructing stationary phase wave forms and numeri-
cally computing the full match from them. The reference wave form
is from two 1.4M
(
objects @the mass parameters
t
0
and
t
1
are
defined in Eq. ~3.2!#. The quadratic approximation is safe, in the
sense that its 97% contour always lies inside the numerical contour.
The quadratic approximation also works well for high mass bina-
ries, provided the numerical and analytical contours use the same
coalescence frequency.
BENJAMIN J. OWEN AND B. S. SATHYAPRAKASH PHYSICAL REVIEW D 60 022002
022002-4
In the limit of close template spacing, Eq. ~2.9! leads to a
simple, analytical way of placing templates on a lattice. We
discuss this in some detail in Sec. IV, but for now turn our
attention to the use of the quadratic approximation in calcu-
lating the number of templates needed for a lattice.
C. Computational costs
If the number N of templates needed to cover a region of
interest is large, it is well approximated by the ratio of the
proper volume of the region of interest to the proper volume
per template DV,
N5
~
DV
!
2 1
E
d
D
u
A
det
i
g
ij
i
, ~2.10!
where D is the dimension of the parameter space @24#. Equa-
tion ~2.10! underestimates N when not in the limit DV→0
(N→`). The reason is template spill over, i.e., the fact that
in any real algorithm for laying out templates, those on the
boundaries of the region of interest will to some extent cover
regions just outside. This effect is small in the limit of many
templates because it goes as the surface-to-volume ratio of
the region of interest.
The proper volume per template, DV, depends on the
packing algorithm used, which in turn depends on the num-
ber D of dimensions ~see Sec. IV!. For D5 2, the optimal
packing is a hexagonal lattice, and thus
DV5
3
A
3
2
~
12 MM
!
, ~2.11!
where MM is the minimal match parameter defined in Ref.
@24# as the match between signal and template in the case
when the signal lies equidistant between all the nearest tem-
plates ~i.e., the worst-case scenario!. There is no packing
scheme which is optimal for all D, but it is always possible
~though inefficient! to use a hypercubic lattice, for which
DV5 „2
A
~
12 MM
!
/D…
D
. ~2.12!
For inspiraling compact binaries, Ref. @24# has spelled out
a detailed prescription for obtaining the g
ij
needed to evalu-
ate the proper volume integral in Eq. ~2.10!. In summary,
first one obtains a metric including the t
c
parameter,
g
ab
5
1
2
~
J
@
c
a
c
b
#
2J
@
c
a
#
J
@
c
b
#
!
, ~2.13!
where
c
a
is the gradient of the wave form phase C in the
parameter space of intrinsic parameters plus t
c
and the mo-
ment functionals
J
@
a
#
[
^
f
2 7/3
,a
~
f
!
&
^
f
2 7/3
,1
&
~2.14!
can be expanded ~for binary chirp wave forms! in terms of
the noise moments @31#
J
~
p
!
[
^
~
f/f
0
!
2 p/3
,1
&
^
~
f/f
0
!
2 7/3
,1
&
~2.15!
where f
0
is the frequency of the minimum of S
h
(f) @32#.
Then one projects out the coalescence time t
c
to obtain
g
ij
5
g
ij
2
g
0i
g
0j
/
g
00
. ~2.16!
Once N has been found it is a relatively straightforward
matter to calculate the CPU power and storage required to
process all the templates in an on-line search. The interfer-
ometer data stream is broken up for processing into segments
of D samples ~real numbers!, such that D@ F where F is the
length ~in real numbers! of the longest filter. ~See Schutz
@25# for a discussion of the optimization of D/F, taking into
account the fact that successive data segments must overlap
by at least F to avoid circular correlations in the Fourier
transform.! Using the operations count for a real Fourier
transform @25#, filtering the data segment through N tem-
plates of length F requires
ND
~
161 3log
2
F
!
~2.17!
floating point operations. If we take the sampling frequency
to be 2f
u
~see Sec. III and Table I!, the computational power
required to keep pace with data acquisition is
P.Nf
u
~
321 6log
2
F
!
~2.18!
flops ~floating point operations per second!.
TABLE I. Analytical fits to noise power spectral densities S
h
(f) of the interferometers treated in this
paper. Here S
0
is the minimum value of S
h
(f), and f
0
is the frequency at which the minimum value occurs.
For our purposes S
h
(f) can be treated as infinite below the seismic frequency f
s
. The high-frequency cutoff
f
u
is chosen so that the loss of signal-to-noise ratio due to finite sampling rate 2f
u
is 0.75% ~see text!.
Detector Fit to noise power spectral density S
0
(Hz
21
) f
0
~Hz! f
s
~Hz! f
u
~Hz!
LIGO I S
0
/3
@
(f
0
/f)
4
1 2(f/f
0
)
2
#
4.43 10
2 46
175 40 1300
LIGO II S
0
/11
$
2(f
0
/f)
9/2
1 9/2
@
11 (f/f
0
)
2
#
%
7.93 10
2 48
110 25 900
LIGO III S
0
/5
$
(f
0
/f)
4
1 2
@
11 (f/f
0
)
2
#
%
2.33 10
2 48
75 12 625
VIRGO S
0
/4
@
290(f
s
/f)
5
1 2(f
0
/f)111(f/f
0
)
2
#
1.13 10
2 45
475 16 2750
GEO600 S
0
/5
@
4(f
0
/f)
3/2
2 21 3(f/f
0
)
2
#
6.63 10
2 45
210 40 1450
TAMA S
0
/32
$
(f
0
/f)
5
1 13(f
0
/f)1 9
@
11 (f/f
0
)
2
#
%
2.43 10
2 44
400 75 3400
MATCHED FILTERING OF GRAVITATIONAL WAVES . . . PHYSICAL REVIEW D 60 022002
022002-5
