PHYSICAL REVIEW D 73, 122004 (2006)

Adaptive filters for detection of gravitational waves from coalescing binaries

Antonio Eleuteri, Leopoldo Milano, Rosario De Rosa, and Fabio GarufiDip. di Scienze Fisiche, Universita di Napoli ‘‘Federico II’’, via Cintia, I-80126 Napoli, Italia

and INFN sez. Napoli, via Cintia, I-80126 Napoli, Italia

Fausto Acernese and Fabrizio BaroneDip. di Scienze Farmaceutiche, Universita di Salerno, via Ponte Don Melillo, 84084 Fisciano (SA), Italia

and INFN sez. Napoli, via Cintia, I-80126 Napoli, Italia

Lara Giordano and Silvio PardiDip. di Matematica ed Applicazioni, Universita di Napoli ‘‘Federico II’’, via Cintia, I-80126 Napoli, Italia

and INFN sez. Napoli, via Cintia, I-80126 Napoli, Italia(Received 22 September 2005; published 20 June 2006)

1550-7998=20

In this work we propose use of infinite impulse response adaptive line enhancer (IIR ALE) filters fordetection of gravitational waves from coalescing binaries. We extend our previous work and define anadaptive matched filter structure. Filter performance is analyzed in terms of the tracking capability anddetermination of filter parameters. Furthermore, following the Neyman-Pearson strategy, receiver oper-ating characteristics are derived, with closedform expressions for detection threshold, false alarm, anddetection probability. Extensive tests demonstrate the effectiveness of adaptive filters both in terms ofsmall computational cost and robustness.

DOI: 10.1103/PhysRevD.73.122004 PACS numbers: 04.80.Nn, 05.40.Ca, 07.05.Kf, 07.60.Ly

I. INTRODUCTION

Gravitational wave (hereafter GW) detection is certainlyone of the most challenging goals for today’s physics: itrepresents a very strong proof in favor of the Einsteingeneral relativity description of phenomena related to thedynamics of gravitation and the opening of a completelynew channel of information on astrophysical objects [1–3].Ground and space-based detectors are now operational orwill be operational in the next years, with different band-widths and sensitivities. The VIRGO/LIGO/GEO/TAMA([4–7]) network of ground-based kilometer-scale laserinterferometer gravitational wave detectors will probablybe the key to open up a new astronomical channel ofinformation in the frequency band 10 Hz to 10 kHz. Inaddition, when the proposed 5� 106 kilometers longspace-based interferometer LISA [8] will fly, another win-dow will be opened in the frequency band 10�5 � 1 Hz.The sensitivities of the operational interferometers arebecoming very interesting, producing reasonable hopes inthe researchers that the first detection of gravitationalwaves is becoming very close.

Starting from the estimate of the huge computationalcost of the data analysis for VIRGO/LIGO/GEO/TAMA(correlation with 105 � 107 templates), we decided to ex-plore approaches able to reduce the computational cost ofthe overall search procedure. We devised an adaptive filterto be used as a first level trigger, to select interesting datablocks to be further analyzed with more refined optimaltechniques based on matched filters. Our efforts wereaimed at the implementation of a rough analysis with thefollowing characteristics: small signal losses with respect

06=73(12)=122004(12) 122004

to the use of matched filters, robustness against falsedetection, and low computational cost (for real-timeexecution).

Finally, considering that the algorithm does not requirethe knowledge of templates, it can be effectively used notonly for the inspiral phase, but for the whole coalescenceprocess. On the other hand if we want to implement atrigger running in real time, algorithms like IIR ALE, aswe shall see, could be very useful. Despite the need of datawhitening the algorithm can be used to set up a veto onfalse detection in the output of detection algorithms (suchas a matched filter bank).

The main idea is to use the adaptive IIR ALE filter to‘‘reconstruct’’ at its output the coherent component (if any)at its input. If the filter is correctly tracking the signal, thereconstruction consists of a noisy replica of the inputcoherent components. If we view the reconstruction as anoisy template, then a correlator detector can be built,which matches the raw filter input with its output. It shouldbe noted that adaptivity of the filter is fundamental, since itallows the reconstruction of signals with a time varyingfrequency component. Therefore, the correlator ‘‘integra-tes’’ the local information obtained by the adaptive filterstep-by-step.

The outline of the paper is the following. In Sec. II wedefine the signal model to which our analysis applies. InSec. III we introduce the class of IIR ALE filters anddescribe their statistical properties in terms of impulseresponse and correlation functions; here we lay the groundfor the developments in Sec. IV, where we describe theproposed adaptive correlator detector structure and for-mally derive the detection statistic distribution. In Sec. V

-1 © 2006 The American Physical Society

ANTONIO ELEUTERI et al. PHYSICAL REVIEW D 73, 122004 (2006)

we show the performance of the filter in white, colored, andnonstationary noise environments. In the appendices weshow the mathematical details of the detector statistics.

It should be noted that this work is an extension ofpreviously published works [9,10].

II. SIGNAL MODEL

To establish the signal models, to use for our simula-tions, we will recall the up-to-date results of the theoreticalstudies on the last fate of a binary system constituted eitherof two neutron stars or two black holes. The theoreticalinvestigations are performed dividing the coalescenceevent in three main distinct phases [11]:

(i) A

diabatic inspiral phase; (ii) M erger phase;

(iii) R

ing-down phase. FIG. 1 (color). Examples of inspiralling gravitational wave-forms (chirps) post-Newtonian order 2 at sampling frequencyfs � 4096 Hz from top to bottom. On the left we see thewaveforms, on the right the spectrogram. Top: classical neutronstar-neutron star binary 1:4M� � 1:4M�; normalized units; start-ing frequency 60 Hz. Middle: neutron neutron star� black holebinary 1:4M� � 10M� with spinning BH; spin � 0:99; normal-ized units; starting frequency 30 Hz. Bottom: effective one bodyapproach (EOB) of two inspiralling BH 5M� � 10M�; normal-ized units; starting frequency 30 Hz.

During the adiabatic inspiral phase, the time scale of thegravitational radiation reaction is much longer than theorbital period. The distance between the two compactobjects composing the binary is large enough to considertheir dimensions negligible (point masses hypothesis).

Near the end of the inspiral phase, the system evolvestowards a dynamic instability, passing from a radiationreaction regime to a free-falling plunge, up to the finalmerging (the merger phase). This phase constitutes one ofthe most challenging problems for numerical relativity andis of concern for the experimentalist due to the lack ofcredited templates for matched filtering. Suitable templatesfor the binary black hole merger phase are the objective ofcurrent research (see e.g. [12,13] and references therein).The nonlinear phase of merging gradually goes towards astationary regime that can be described as oscillations ofthis final newborn compact object (BH/NS) quasinormalmodes. The emitted gravitational waves are described bythe superposition of exponential damped sinusoids andcarry information about the mass and the spin (see e.g.[14] for nonspinning binaries or [15,16] for spinning bi-naries) of the final newborn compact object (the ring-downphase).

From this evolutionary picture it is clear that the phasesfor which credited templates exist are the inspiral and ring-down phases, while for the merger phase, interesting re-sults on suitable classes of templates are coming fromrecent numerical relativity studies. In Fig. 1 some ex-amples of foreseen waveforms (called chirp) are shownduring the inspiral phase of either neutron star binaries orblack bole (hereinafter BH) of their process of coalescence.

In our analysis we consider the inspiral phase of coales-cence, and so, for the sake of simplicity, we can assume alocally sinusoidal, amplitude and frequency modulatedsignal model. The chirp waveform is an example of suchsignals:

s�t� � a�t� cos �t�; (1)

where a�t� and �t� are monotone, slowly growing power

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laws. Defining a slowly varying pseudoperiod as T�t� �2�= _ �t�, if a�t� has small relative changes over the pseu-doperiod, then an adaptive scheme can be devised to tracksuch a signal [17].

In the following, we shall consider a (discrete time)signal with additive white Gaussian noise:

xn � sn � un; (2)

where in a realistic setting the white Gaussian noise un canbe thought of as obtained through a whitening operation ofan interferometer data stream, and the signal sn is a whit-ened sinusoidal signal having the form of Eq. (1).

III. AN INFINITE IMPULSE RESPONSE ADAPTIVELINE ENHANCER FILTER

The adaptive line enhancer filter model [18] is designedto approximate the signal-to-noise ratio (SNR) gain ob-tained by the matched filter solution for a locally sinusoidalsignal.

The advantage of using ALE filters is that no a prioriknowledge of the signal parameters (sinusoidal frequen-cies, amplitudes, phases, or even the number of narrow-band components) is required. Therefore, center frequencyband-pass filters can be used when the signal to detect isnarrow band and buried in a wide-band noise. The disad-vantage of such filters is that they are only able to track alocally sinusoidal signal; therefore, if this assumption is

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ADAPTIVE FILTERS FOR DETECTION OF . . . PHYSICAL REVIEW D 73, 122004 (2006)

not verified, the tracking performance is greatly degraded.We must also consider the problem of pull-in time, i.e. thetime taken by the filter to effectively track a signal and‘‘tune’’ to its instantaneous frequency. In low signal-to-noise ratio environments, this prevents the tracking of veryshort signals. Such a problem can usually be addressed byrunning a bank of filters spanning a range of frequencies.

The most popular form of ALE is based on a finiteimpulse response (FIR) structure, which has the advantageof being stable and easy to adapt. However it requires alarge number of adaptive weights to provide an adequateenhancement of narrow-band signals, with a consequenthigh computational cost [17]. Adaptive infinite impulseresponse (IIR) filters, instead, provide a typically narrowerband with a much smaller number of weights, thus having alow computational cost; however, they are possibly subjectto instabilities, and their performance surfaces are multi-modal. These problems can be overcome by imposingsome constraints on the filter. For line enhancement, thepoles should be constrained to lay close (but inside) to theunit circle in the z plane.

In this paper, we implement the ALE as a constrainedsecond order IIR filter with a single adaptive parameter.The general structure of the ALE is shown in the left part ofFig. 2. The purpose of the delay element is to decorrelatethe input noise at the input of the filter, while leaving theresponse of the filter to sinusoidal excitation as unaltered aspossible. The enhanced version of the sinusoid can berecovered at the output of the filter. The filter we considerhas the following form, which is a reparametrization of thestructure in [19] (for convenience we use the same symbolfor the new parameter, i.e.w is the parameter to adapt and ris a fixed parameter determining the bandwidth of thefilter):

H1�z� ��1� r��w� z�1�

1� �1� r�wz�1 � rz�2 (3)

with the constraints:

0 r < 1; �1<w< 1: (4)

z

1H (z)

1

<t

>tT(x)N−1

n=0Σ

x(n)

−1

0

FIG. 2. Adaptive detector scheme. On the left part of thefigure, schematics of the ALE filter are shown. On the rightpart, the detection section is shown, composed of a correlatorwhose output is compared with a threshold t to choose whether asignal is present (H 1) or absent (H 0).

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With this filter structure, the processing of input sequencesis performed by the filter:

H�z� � z�1H1�z� ��1� r��wz�1 � z�2�

1� �1� r�wz�1 � rz�2 : (5)

The purpose of the delay element is to create the adaptivecorrelator structure itself. Since the filter H1�z� makes aone-step-ahead prediction of its input, we must delay thesource signal x�n�, so that we can compare the input andoutput of H1�z� at the same time step to evaluate theirproduct.

The above filter shares some analytic features withsecond order adaptive noise cancelers (in particular con-cerning poles, bandwidth, and peak location) of the generalform:

D�z� �G�z�

1� 2Az�1 cos!0 � Bz�2 (6)

which have the following characteristics [17]:

(i) t

-3

he poles form a complex-conjugate pair at angularfrequency !0, lying on a circle of approximateradius A 1 inside the unit circle;

(ii) t

he approximate 3 dB bandwidth is (1� B); (iii) t he magnitude response has a peak at angular fre-

quency !0.

These characteristics in the case of the H�z� filter can be

read as (by comparing the denominators in Eqs. (5) and (6):

(i) T he poles of the filter form a complex-conjugate

pair lying on a circle of approximate radius �1�r�=2 inside the unit circle.

(ii) T

he approximate 3 dB bandwidth is (1� r). (iii) T he magnitude response has a peak at angular

frequency:

!0 � arccosw: (7)

(iv) F

urthermore, at the above angular frequency thefilter has unity gain and zero phase shift, whichguarantee that the local amplitude and phase of aninput sinusoid at angular frequency !0 are leftunaltered by the filter.

In Fig. 3 we show examples of amplitude and phaseresponses of the H�z� filter.

The introduction of an adaptation mechanism on the wparameter has the effect of moving the peak response ofH�z� so as to track the instantaneous frequency of the inputsinusoid. The r parameter is usually chosen close to unityto give a narrow bandwidth.

We briefly note here that this filter structure is differentfrom the one reported in our previous works [9,10] thatexhibited a magnitude response larger than 1 around thecentral frequency, which worsened its noise reductioncapability.

0 0.2 0.4 0.6 0.8 1−200

−100

0

100

200

Normalized Frequency (×π rad/sample)

Pha

se (

degr

ees)

0 0.2 0.4 0.6 0.8 1−40

−30

−20

−10

0

10

Normalized Frequency (×π rad/sample)

Mag

nitu

de (

dB)

Amplitude and phase response of H(z), ω0=0.2, r=0.98

FIG. 3 (color online). Amplitude and phase response of asample ALE filter. The phase delay at !0 is zero, and theamplitude response has a peak at !0.

ANTONIO ELEUTERI et al. PHYSICAL REVIEW D 73, 122004 (2006)

A. Parameter adaptation

The output error formulation is quite natural in filteringapplications [18]; however, in the case of IIR filters, ingeneral the mean square error is not a unimodal function ofthe filter coefficients, thus it may have multiple minima;furthermore, filter stability during adaptation is notguaranteed.

The performance function we used for filter adaptationis the expected value of the squared filter output sequenceyn [19]:

E �y2n� � jH�e

i!0�j2S�1� r1� r

�2; (8)

where S is the (local) input signal power, �2 is the inputnoise power, and !0 is the instantaneous signal frequency.It can be shown that the value of w maximizing thisfunction is given by:

w � cos!0: (9)

The comparison of the above equation with Eq. (7) showsthat the maximization of the performance function corre-sponds to tracking the local frequency of the inputsinusoid.

An important feature of this performance function is thatit is unimodal for all admissible values ofw and r. This factsuggests the use of a gradient search technique to find itsminimum. Following [18], we derive a normalized IIRleast mean squares-type algorithm. The filter output isgiven by:

yn � �1� r�wnyn�1 � ryn�2 � �1� r��wnxn�1 � xn�2�

(10)

so that the instantaneous estimate of the gradient is given

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by:

J �@y2

n

@wn� 2yn

@yn@wn

: (11)

The gradient of yn is:

�n�@yn@wn

��1�r�wn�n�1�r�n�2��1�r�yn�1��1�r�xn�1:

(12)

With these elements, we can build the parameter adapta-tion equation:

wn�1 � wn ��yn�n=Rn�1; (13)

where Rn�1 is an adaptive estimate of the error functionHessian [17]:

Rn�1 � �Rn � �2n: (14)

The parameters � and � are adaptation coefficients. In thefollowing, to ease the notation, we shall usew to denote theinstantaneous value of the sequence wn.

B. Computational cost of the adaptive correlatordetector

The computational cost of the adaptive detector in Fig. 2can be quantified by counting the number of floating pointoperations per sample. The relevant costs are due to theoperations of the H1�z� filter (output evaluation, filterparameters adaptation), plus the product of filter inputand output samples. By inspection of Eqs. (10)–(14), wehave:

(i) y

-4

n: 5 flops;

(ii) � n: 5 flops;

(iii) w

n�1: 3 flops; (iv) R n�1: 3 flops. The total cost to process a single sample (including the

product of filter input and output) is therefore 17 flops. Fora stream of N samples, the cost is 17N, which can bewritten, using the notation for algorithmic complexity,O�N�, meaning that the cost of the filter operation is linearin the size of the data it processes. For comparison, the costof a full matched filter analysis of a data stream is reportedin [20]. As an example, for a binary coalescence of 1M� �1M� with a SNR recovery of 90%, the computational costis estimated to be 1:1� 1010 flops.

C. Filter response characterization

By applying the inverse z transform to the rationalfunction expressing the filter impulse response in the zdomain, we get the time-domain impulse response:

hn �2�n�1� r�A�r; w�

�w�C�r; w�n � B�r; w�n� � 2�C�r; w�n�1

� B�r; w�n�1��n�1�; (15)

where ���� is the unit-step function and:

ADAPTIVE FILTERS FOR DETECTION OF . . . PHYSICAL REVIEW D 73, 122004 (2006)

A�r; w� ����������������������������������������4r� �1� r�2w2

q;

B�r; w� � w� rw� A�r; w�;

C�r; w� � w� rw� A�r; w�:

(16)

0 200 400 600 800 1000−0.01

−0.008

−0.006

−0.004

−0.002

0

0.002

0.004

0.006

0.008

0.01Impulse response (w=cos(0.1), r=0.99)

n

h(n)

0 200 400 600 800 1000−6

−4

−2

0

2

4

6x 10−3

k

g(k)

Sample autocorrelation (w=cos(0.1), r=0.99)

FIG. 4 (color online). Right: impulse response; Left: sampleautocorrelation function.

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We note that the system is strictly stable, since h0 � 0.The knowledge of the filter impulse response is neces-

sary to characterize the output statistics of the filter. Thefirst step to this end is the evaluation of the sample auto-correlation function gn. It can be written as [21]:

gk �X�1n�0

hnhn�k �

8<:

2�k�1�1�r�A�r;w��1�r� �w�1� r��B�r; w�

k � C�r; w�k� � �A�r; w��B�r; w�k � C�r; w�k�� k � 11�r1�r k � 0

: (17)

Given an input white noise sequence un, the autocorrela-tion sequence of the filter output yn can be written as [21]:

�yyk � gk�2; (18)

where �2 is the variance of the white noise. The evaluationof the autocorrelation at lag zero gives us the expectedpower at the filter output:

�2y � �yy0 �

1� r1� r

�2; (19)

where the factor �1� r�=�1� r� is the noise reductionfactor.

The cross-correlation function between the noise se-quences at filter input un and filter output vn can beevaluated as:

�uvnX�1k�0

hk�uun�k �X�1k�0

hk�2�n�k � �2hn; (20)

where we have used the fact that �uun�k � �n�k since u is awhite noise sequence. In Fig. 4 are shown the impulseresponse and the sample autocorrelation function in whiteGaussian noise of the IIR ALE filter.

IV. ADAPTIVE CORRELATOR DETECTOR

We have seen that, due to the filter structure, for a whitenoise input the cross correlation between input and outputat zero lag (which is a power) is exactly zero (ruv0 ��2h0 � 0). In the presence of a coherent componenttracked by the filter the cross correlation is going to bedifferent from zero.

It seems therefore natural to use an estimate of the cross-correlation function at zero lag as a detection statistic:

Tn � E�xy� 1

N

Xn�N�1

k�n

xkyk �1

Nhxn; yni; (21)

where we have expressed the statistic as a stochastic pro-cess. However, for the sake of simplicity, and since we areonly interested in deciding after observing nonoverlappingsequences N samples, we ignore the dependence of T fromn.

It can be shown [22] that in the case of Gaussian input, Tcorresponds to a maximum likelihood estimate of thecross-correlation function, and that this estimate is asymp-totically unbiased and consistent.

The detector would then work by comparing the value ofthe statistics T to a threshold t (previously set in corre-spondence to a desired false alarm rate) and decide that asignal is present (H 1) if T > t or absent (H 0) if T < t(also see Fig. 2).

Since a correctly tracking ALE produces estimates ofthe coherent components at its input, operation of theabove correlator can be thought of as matched filtering ofthe input sequence with a noisy template signal given bythe output of the ALE filter. Since ALE estimates are noisy,we can expect degraded performance with respect to aconventional matched filter operating in the hypothesis ofa perfectly known signal. The main advantage is that we donot need the knowledge of a signal template.

In the next sections we shall evaluate the test statisticdistributions both for the case of noise only input, and inthe case of presence of signal plus noise.

A. Detection statistic distribution: Noise only

Let us consider the case where no signal is present, sothat the input of the filter is a white Gaussian noise se-quence, xn � un, and the output of the filter is the corre-lation of the filter impulse response hn with un:

yn � hn xn � hn un � vn; (22)

where is the correlation operator. The sequence vn isGaussian correlated noise. The test statistic can in this casebe written:

-5

−0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05 0.060

0.2

0.4

0.6

0.8

1

T

F(T

)

Test statistic CDFs (N=512, r=0.9233)

−0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.05 0.06−4

−2

0

2

4x 10

−3

T

∆ F

(T)

Difference between model and empirical test statistic CDFs

ModelEmpirical

FIG. 5. Example of the adaptive detector output statisticsdistribution (model and empirical). Their difference is alsoshown.

ANTONIO ELEUTERI et al. PHYSICAL REVIEW D 73, 122004 (2006)

T �1

N

XN�1

n�0

xnyn �1

N

XN�1

n�0

unvn �1

Nhu; vi: (23)

To evaluate the distribution of T, we start by consideringthe extended random vector � �u; v�0 (the prime symbol 0

denotes the transpose operation) whose joint distributioncan be derived from the marginal distributions of its com-ponents. We have u�N �0; �2I�, v�N �0; G�, so:

�N

�0;

1

N��; � �

�2I KK0 G

� �; (24)

where G is the autocorrelation matrix over a window of Nsamples [21], K is the cross-correlation matrix over awindow of N samples [21], I is the N � N identity matrix,and ‘‘�’’ denotes the distribution of a random variable(Gaussian in this case).

We can express T as a quadratic form in the variable byusing a permutation matrix �:

T �1

20�; � �

0 II 0

� �: (25)

The above equation is in the form of a delta-gamma-normal model, which in terms of independent variatescan be written, after a change of variables (see Eq. (A1)):

T �X2N�1

i�0

1

2i�

2i ; (26)

where the i are the eigenvalues of the �� matrix (whichcan be calculated, for example, by reduction to a triangularform via Schur’s decomposition, and then taking the ele-ments on the diagonal).

It should be noted that since the filter used in theanalytical derivation of the T distribution is not adaptive,we must assess if (and how much) filter adaptivity altersthe theoretical derivation. We shall show by a statisticaltest that the analytical derivation is not affected in a signi-ficative way.

In Fig. 5 we show the empirical and model distributionsof T. The former is evaluated through the direct operationof the adaptive detector on white noise sequences, the latteris obtained by sampling the model distribution given byEq. (26) (which assumes a static filter). The difference �Fbetween the statistics is also shown, to give a better idea ofthe goodness of the model.

For the case in figure, a two-sided Kolmogorov-Smirnovtest [23] shows that the null hypothesis that the two dis-tributions are equal cannot be rejected at significance level0.05. The Kolmogorov distance between the two distribu-tions is 2:8� 10�3.

While for evaluation of the detector performance nu-merical Fourier inversion can be used, in a practical op-eration of the detector we only need to calculate thethreshold corresponding to a prescribed false alarm proba-bility �, which is the (1� �)-quantile of the distribution of

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T [24]. Since this is typically small, it can be accuratelyevaluated from an asymptotic expansion of the right tail ofthe distribution of T. In Appendix B, by following theapproach in [25], we show a closedform expression toevaluate the detection threshold for a prescribed falsealarm probability �.

An essential element for the approximation is the evalu-ation of the eigenvalues spectrum of the �� matrix; thisproblem, however, for large values of N can be computa-tionally very intensive. Although the eigenvalues must beevaluated only once for a given filter structure (defined bybandwidth and starting frequency N), it may be necessaryto define different filter structures, for example to create anarray of filters operating in parallel at different startingfrequencies.

In general, it is difficult to say how the filter parametersaffect the eigenvalue spectrum (and, as a consequence, thethreshold). However, simulations can be used to have anempirical insight into the problem. As an example, in Fig. 6we show how the choice of the bandwidth affects thethreshold, given false alarm probability (hereinafter Pfa),sampling frequency, and N. For a given Pfa, the thresholdincreases with the bandwidth B. A higher threshold, how-ever, means that the detection probability decreases, so oneshould use as low a threshold as possible, which implies alow bandwidth. But a very low bandwidth reduces thetracking capabilities of the filter. It is therefore clear thatthere is a trade off between bandwidth and detectioncapability.

These kind of analyses could be used to fit a regressionmodel, which, given as inputs the filter bandwidth, the falsealarm probability, andN, gives an approximate value of thethreshold. We are currently working on a neural network-based regression model.

-6

0 500 1000 1500 2000 2500 3000 35000.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

N

Thr

esho

ld

Pfa

=1e−4; fs=4096; B=400 (r=0.9023)

Pfa

=1e−4; fs=4096; B=200 (r=0.9512)

FIG. 6. Relation between bandwidth (B) and threshold, givenfalse alarm probability (Pfa), sampling frequency (fs), and N.

ADAPTIVE FILTERS FOR DETECTION OF . . . PHYSICAL REVIEW D 73, 122004 (2006)

B. Detection statistic distribution: Signal plus noise

In the case in which a signal is present, xn � sn � unand the output of the filter is given by Gaussian correlatednoise with an added signal component:

yn � hn xn � hn �sn� un� � hn sn� vn � sn� vn:

(27)

The test statistic can in this case be written:

T �1

N

XN�1

n�0

xnyn �1

N

XN�1

n�0

�sn � un��sn � vn�: (28)

If the ALE filter is correctly tracking the signal, so that itdoes not introduce a phase delay and its magnitude isapproximately one, we can assume that sn sn, so wecan write:

T �1

N

XN�1

n�0

xnyn �1

N

XN�1

n�0

�sn � un��sn � vn�

�1

N

XN�1

n�0

s2n �

1

N

XN�1

n�0

sn�un � vn� �1

N

XN�1

n�0

unvn

� E �1

Nhs;u� vi �

1

Nhu; vi; (29)

where E � s0s=N is a quadratic measure of the signalpower. If we introduce the signal vector � � 1���

Np �s; s�0,

so that 1N hs;u� vi � �0, we have:

T � E � �0� 120� (30)

which is a quadratic form (see Appendix A). So, thepresence of a signal is reflected in the formal expressionfor the case of noise only, by additional constant and linearterms in .

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C. Comparison with matched filter detector

It is instructive to briefly compare the detection statisticof the ALE correlator detector, with the detection statisticof the matched filter. It is well known [26] that the lattercan be written:

TM �XN�1

n�0

xnsn � hx; si; (31)

where, as before, x is the observed data, and s is the(known) signal vector (template). This expression showsthat the detection statistic is a linear form in the data, whilethe detection statistic for the ALE detector in Eq. (30) is aquadratic form in the data. Therefore the detectors areessentially different, notwithstanding the fact that at thehighest level the two detectors work based on the sameprinciple.

V. SIMULATION RESULTS

In all large baseline laser interferometric GW antennasoutput data, starting from the 40 m Caltech interferometerand going to VIRGO, LIGO, GEO, and TAMA, manyrelevant effects are present. Violin resonances in the sus-pensions, main power harmonics, ring-down noise fromservo control systems, electronic noises, transients, non-Gaussianity, nonstationarity, and nonlinearity of the darkfringe data are common features of the measured noise.Our goal is to assess the performance of IIR ALE, thus wemust evaluate, taking into account the experimental noise,what will be the behavior of IIR ALE as GW receiver bothin white noise and in not perfectly whitened noise. To thisaim we performed simulations using realistic Virgo-likedata segments, whitened by using a linear autoregressivefilter with 16 384 coefficients estimated with Burg’s algo-rithm [17]. The complexity of the model was estimated bystatistical cross validation [27]. The number of parametersin a model is a measure of its complexity. As the number ofparameters increases, a model will be able to better ap-proximate the data it is fitted on. However, this does notguarantee that the same model will perform equally wellon data it has never seen before; the model will act as aninterpolator, rather than a regressor. Therefore, it is neces-sary to control the model complexity, and cross validationis a statistical technique that allows such a control. Inpractice, a range of models with different number of pa-rameters is fitted on a block of data, and its performance istested on a different block of data. The prediction perfor-mances of all models are evaluated on this new block, andthe model with the best performances is chosen as thereference model.

Finally, to assess the influence of nonstationarities andincorrect whitening, we evaluated the decision statistics fordifferent noise sources.

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FIG. 8 (color). Examples of inspiralling gravitational wave-forms (chirps) at sampling frequency fs � 4096 Hz. Top:neutron star� black hole binary 5M� � 10M�, in normalizedunits, waveform computed by effective one body (EOB) ap-proach; starting frequency 30 Hz, embedded in white noise(SNR � 20). Bottom left: theoretical waveform (red) and de-tected waveform (blue). Bottom right: time-frequency behaviorof the detected waveform by IIR ALE.

ANTONIO ELEUTERI et al. PHYSICAL REVIEW D 73, 122004 (2006)

A. Simulation in white noise

In this section we will show the simulated trackingperformances of IIR ALE for varying frequency of inputsinusoid in white noise, such that the corresponding fre-quency variation is a frequency step. The simulations wereperformed with the following specifications of the inputand the filter: for 0 � t � 0:5 sec , f � 180 Hz and for0:5< t � 1 sec , f � 220 Hz; amplitude A �

���2p

; sam-pling frequency fs � 4096 Hz; SNR � 26:5; filter pa-rameters � � 0:1, � � 0:97, r � 0:97, R0 � 10 000 000.Before the occurrence of the step, care is taken to ensurethat the adaptive filter is in steady state, by consideringnear steady state initial conditions and allowing sufficienttime i.e. at least 1000 samples. In Fig. 7 there are shown theresults of the simulation. In Fig. 7 (top left) there is shownthe theoretical frequency response (dotted curve) vs timeand the simulated mean response (blue curve) embedded inthe standard error curves; these are computed on 2000realizations of the simulation. The signal is corrupted bywhite noise of zero mean and variance �2 � 6:25. Fromthe frequency step response it is clear that there is a delayof about 0.1 sec for the filter to adapt as it is also possible tosee in Fig. 7 (bottom left) where it is shown the amplituderesiduals (vs time), between the tracked signal and thetheoretical one. We must note that from the output of thefilter we can extract two kinds of information in real timethat is either the output amplitude or the frequency of thetracked signal (if any) at the input. In Fig. 7 (top right)there is shown the noisy input and in red the output of IIRALE. In Fig. 7 (bottom right) there are shown the linearSNR for each of the 2000 noise realizations. The red linesrepresent the mean SNR (26.5) embedded in the standard

0 0.2 0.4 0.6 0.8 1170

180

190

200

210

220

230

Time (sec)

Fre

quen

cy (

Hz)

0 0.2 0.4 0.6 0.8 1−15

−10

−5

0

5

10

15

Time (sec)

Am

plitu

de

0 0.2 0.4 0.6 0.8 1−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Time (sec)

Am

plitu

de D

iffer

ence

(O

bs−

The

or)

0 500 1000 1500 200024.8

25

25.2

25.4

25.6

25.8

26

26.2

26.4

26.6

26.8

Number of realizationsSig

nal t

o N

oise

Rat

io x

rea

lizat

ion

(SN

R)

FIG. 7 (color). Frequency step from 180 Hz to 220 Hz appliedin a sinusoidal signal of amplitude

���2p

and duration 1 sec atsampling frequency fs � 4096 Hz embedded in white noise(SNR � 26:5). The parameter of IIR ALE were � � 0:1, � �0:97, r � 0:97, R0 � 10 000 000.

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SNR error. In the following, we also show three examplesof simulated chirp detections in white noise:

(i) 1

FIG. 9forms dclassicanormaliwhite no

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:4M� � 1:4M� Binary inspiralling towards themerger phase, post-Newtonian order 2.5, normal-ized units, embedded in white noise with SNR 24.5,sampling frequency of 4096 Hz and starting fre-quency of 60 Hz.

(color). Examples of inspiralling gravitational wave-etection at sampling frequency fs � 4096 Hz. Top left:l neutron star-neutron star binary 1:4M� � 1:4M� inzed unit with starting frequency 60 Hz, embedded inise (SNR � 24:5).

0Power Spectral Density Estimate via Welch of ARMA(2,2) process

ADAPTIVE FILTERS FOR DETECTION OF . . . PHYSICAL REVIEW D 73, 122004 (2006)

(ii) 1

−5)

:4M� � 10M� Binary-spinning BH post-Newtonian order 2 embedded in white noise withSNR 20, sampling frequency of 4096 Hz, startingfrequency of 30 Hz, BH Spin � 0:99 (Fig. 8).

mpl

e

(iii) 5

−10

ency

(dB

/rad

/sa

M� � 10M� Black hole-black hole (BH) effectiveone body (EOB) approach post-Newtonian order 2embedded in white noise with SNR 20, samplingfrequency of 4096 Hz and starting frequency of30 Hz (Fig. 9).

0 0.2 0.4 0.6 0.8 1−25

−20

−15

Normalized Frequency (×π rad/sample)

Pow

er/fr

equ

FIG. 10 (color online). Power spectral density of coloredARMA (2,2) process used as noise source.

−0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

T

F(T

)

Test statistic CDFs for different stochastic processes (N=2048, r=0.939, ω0 = 0.077)

nonstationaryARMA gaussianwhite gaussian

FIG. 11 (color online). Empirical test statistic distributions fordifferent noise sources.

As shown in Figs. 8 and 9, the promising characteristicof this kind of trigger is also the possibility to have aninsight of the waveform amplitude and of the time-frequency behavior. The weak point is connected withthe loss in amplitude of the final part of the chirp, due tothe nonadiabatic behavior of the frequency evolution.Nonetheless the frequency variation is nearly tracked, asit is possible to see from the time-frequency diagrams.

B. Test statistics for colored and nonstationary noisesources

We evaluated how much possible nonstationarities andimperfect whitening of real interferometric noise impacton the performance of the adaptive detector. In particular,we assessed the influence of nonwhite noise sources on thedetection threshold. Since it is difficult to analyticallyevaluate such effects, we used extensive Monte Carlosimulation runs. For each experiment, we used 1.300.000samples; such a choice guarantees that the smallest proba-bility which can be evaluated is 0.05, and that the relativeerror e in the probability estimate satisfies:

P�jej> 0:01� � 0:01 (32)

which means that e is larger than 0.01 (in absolute value)with probability at most 0.01 [26].

The first test involves an autoregressive moving average(ARMA) process of second order, fed by white Gaussiannoise with unity variance. The power spectral density ofthe process is shown in Fig. 10.

The second test involves the same process as above, butits amplitude is modulated in time by the function n1=4,where n is the discrete time index. The process obtained isnonstationary, and its probability density is non-Gaussianand leptokurtotic (i.e. more peaked than a Gaussian proba-bility density).

Figure 11 shows the simulated test statistic for the abovenoise sources; comparing also the test statistic for whiteGaussian noise.

Since the ALE is designed to track slowly varyingsinusoids, it is not able to exactly track the spectral featuresof the above noise sources. However, this influences thefilter response, and consequently the test statistic distribu-tion. As it clearly appears from Fig. 11, for a given detec-tion threshold there is a large increment in the false alarmprobability. For example, for a threshold T � 0:05, thefalse alarm probability in the white Gaussian case is

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10�11, while for the ARMA and nonstationary cases theyare 0:42 and 0:91, respectively. So, there is a largeloss in efficiency, depending on the increasing nonstatio-narity of the noise sources and the residual colored noiseafter the whitening procedure. The main effect is an in-crease in the false alarm rate.

Figure 12 shows receiver operating characteristic (ROC)curves for the proposed adaptive matched filter. ROCcurves compactly describe the performances of a detector,in terms of the relationship between false alarm and detec-tion probabilities. There is a natural trade off between thesetwo quantities, since by incrementing the probability ofdetection, we also increment the probability of falsedetections.

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10 20 30 40 50 60 70 80 90 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

False alarms / hour

Det

ectio

n pr

obab

ility

ROC curves for ALE detector (N=8192, fs=4096)

SNR 5SNR 7SNR 10SNR 15SNR 20

FIG. 12 (color online). Receiver operating characteristics forALE detector with whitened input. The SNR ranges from 5 to 20.

ANTONIO ELEUTERI et al. PHYSICAL REVIEW D 73, 122004 (2006)

We embedded a chirp signal (post-Newtonian order 2.5,chirp mass 1.4, lower cutoff frequency 50 Hz, samplingfrequency fs � 4096 Hz) in simulated Virgo noise, thenwhitened the resulting sum. The whitened data were thenfed to an ALE matched filter with an integration time ofN � 8192 samples (corresponding to two seconds). TheSNR ranges from 5 to 30. It should be noted that theseSNRs refer to the whole chirp (whose length is 44 532samples). The ALE works only on 8192 samples, so theeffective SNR ‘‘seen’’ by the filter is just a fraction of thetotal SNR which could be obtained by using the full signal.Since the chirp amplitude is growing, tracking is easiertowards the final stages of signal evolution, so we canexpect a large contribution to the detection coming fromthese stages. However, as we have shown in the simula-tions, when the signal frequency becomes too steep, ALEis not able to track it anymore.

Since the adaptive filter operates starting from a fixedfrequency, in actual use of the filter it is conceivable to usean array of filters starting from different frequencies toprocess data. Work is under way to characterize the deci-sion statistics of this multifilter structure. We can concludethat the performances of the IIR ALE filters are encourag-ing by using them as a first level trigger to isolate interest-ing segments of data for further analysis with matchedfilters.

VI. CONCLUSION

In this work we have extended our previous results onthe use of ALE filters for GW detection. We have definedan adaptive matched filter structure, in which a noisytemplate probability (if a signal is present), is adaptivelybuilt by the ALE, whose output is then fed to a correlatordetector. The theoretical adaptive matched filter perform-

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ance characteristics have been evaluated and comparedwith the classic matched filter’s ones. The performanceloss in terms of probability of detection has been shown notto be excessive under the hypothesis of white Gaussiannoise; the performance loss is balanced by the low compu-tational cost of the filter. This adaptive matched filterstructure (IIR ALE� correlator) can be a useful first leveltrigger to select interesting data segments to be later ana-lyzed by use of classic matched filter-based techniques. IIRALE is not useful for very short signals, like GW bursts,due to the response time of the filter. It could be useful forGW signal from pulsars, but the intrinsic weakness of thosekind of signals rules out any practical use.

APPENDIX A: THE DELTA-GAMMA-NORMALFAMILY OF MODELS

To evaluate the detection performance of the proposeddetector, the distribution of the test statistic T must becomputed. As we shall see, this problem is not an easytask, since the distributions we get cannot be expressed inclosed form. In particular, we shall show that after suitablealgebraic manipulation the test statistic can be written as aquadratic form in normal variates :

T � ���0� 120�: (A1)

This model is termed delta-gamma-normal [25,28].To ease analytical treatment of the statistic T, we must

write it as a sum of independent random variates that arefunctions of standard normal random variables �i. This canbe done by solving the generalized eigenvalue problem, i.e.finding matrices C and � such that:

CC0 � �; C0�C � �: (A2)

By defining:

� C�; � � C0�; � � diag�0; . . . ; N�1�

(A3)

we can write:

T � ��X2N�1

i�0

��i�i �

1

2i�2

i

�: (A4)

APPENDIX B: ASYMPTOTIC TAILS ANDQUANTILES

Let the generalized eigenvalues of T in Eq. (26) besorted in increasing order. Suppose there are M � 2Ndistinct eigenvalues, and let ij be the highest index of thejth distinct eigenvalue, and �j be its multiplicity. We havethen i1 < i2 < . . .< iM . For j � 1; . . . ;M, define:

Tj �1

2ij

Xijl�ij�1�1

�2l (B1)

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ADAPTIVE FILTERS FOR DETECTION OF . . . PHYSICAL REVIEW D 73, 122004 (2006)

then the Tj are independent, and we have:

T �XMj�1

Tj: (B2)

The above equation essentially defines T as a sum ofindependent �2 random variables with different degreesof freedom and scales. Specifically, if g��;�j� is the �2

�j

density, then the density of Tj is:

fj�x� �2

jij jg�

2

ijx;�j

�: (B3)

Since we are interested in the upper quantiles of thedistribution of T, according to the results in [25] we shoulddistinguish three cases, based on the sign of the largesteigenvalue. However, in our case it is possible to show thatthe largest eigenvalue of the �� matrix is always positive.In fact the � matrix acts on � with a block permutation:

�� �1

NK0 G�2I K

� �: (B4)

It is possible to show that [21]:

Tr ���� � 2�2h0 � 0: (B5)

Since for every square matrix the sum of its eigenvaluesequals its trace, we have:

XNi�1

i � 0; (B6)

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where i are the eigenvalues of ��. Excluding the degen-erate case in which all eigenvalues are zeros (since in thiscase, by Eq. (B2), we would get a degenerate density), thenwe have both positive, negative, and zero eigenvalues, suchthat their sum is zero. Therefore, there must exist at leastone positive eigenvalue (possibly of multiplicity greaterthan one) which is largest. So, we are allowed to consideronly the case in which the largest eigenvalue is positive.

We can now apply the following result from [25]. Let�! 0, then the upper quantile of T satisfies the followingequation:

x1�� � iM lnbM �iM2�2

1��;�M �O�1=�������������������2

1��;�M

q�;

(B7)

where �21��;�M

is the (1� �)-quantile of the �2�M

distri-bution and:

bM �YM�1

j�1

�1�

ijiM

���j=2

: (B8)

It should be noted that in practice the false alarm proba-bility must be smaller than 10�2 for the approximation tohold.

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