content_image/408407.jpg content_image/408417.jpg content_image/408418.jpg content_image/408414.jpg
contains the measured GW-induced strain added to in- strumental and environmental noise n . Thus, x = n when no GW signal is present, and x = s (Θ) + n when there is. In our simulations, n is a realization of a sta- tionary, Gaussian noise process with a PSD, S$_{n}$ [ k ], that matches the initial LIGO [13] design sensitivity curve in shape [14]. 

We use the signal family associated with a non- spinning inspiraling binary system, computed up to the second post-Newtonian (2PN) order [15]. This sys- tem consists of two non-spinning compact stars (Neu- tron Stars or Black Holes) losing orbital binding energy through GW emission. Members of this signal family have chirp waveforms with monotonically increasing in- stantaneous amplitude and frequency. 

For the case of a single detector, the parameters speci- fying the 2PN signal waveforms can be grouped into two sets. The first set is that of the chirp-time [16] param- eters, { τ$_{a}$ } , a = 0 , 1 , 1 . 5 , 2, that are constructed out of 

## B. Signal waveforms 

the masses of the two components of the binary. Expres- sions for the chirp-time parameters are provided in Ap- pendix A. The second set consists of the time-of-arrival , t$_{a}$ , the initial phase , Φ$_{a}$ and the amplitude, A . Interfer- ometric ground-based detectors have a sharp rise in seis- mic noise below some frequency f$_{a}$ ( = 40 Hz for inital LIGO). The chirp signal from a binary inspiral is essen- tially unobservable when its instantaneous frequency is below f$_{a}$ . The time at which the signal becomes visible is t$_{a}$ and the corresponding instantaneous phase of the signal is Φ$_{a}$. 

Since all the four chirp-times depend on the masses of the two compact stars, only two of them are independent. We choose τ$_{0}$ and τ$_{1}$$_{.}$$_{5}$ as the two independent chirp-time parameters. Thus, the set of signal parameters is Θ = {A , Φ$_{a}$ , t$_{a}$, θ = { τ$_{0}$, τ$_{1}$$_{.}$$_{5}$ }} . 

As discussed in Sec. I, the computational cost of gen- erating waveforms on the fly is important for stochastic methods like PSO. The 2PN signal family is amenable to a fast implementation because a sufficiently accurate analytical form exists for the Fourier transform of these waveforms [17], 

<img src='content_image/408407.jpg'>

where, the lower cutoff frequency f$_{a}$ was explained above and the upper cutoff frequency f$_{c}$ follows from the termi- nation of the inspiral waveform when the binary reaches its last stable orbit. For our testbed, we set f$_{c}$ = 700 Hz although in a real search it depends on the mass of the binary system. The expression for ψ ( f ; θ ) is given in Ap- pendix A. The normalization constant N is defined such that, ‖ s ( {A = 1 , Φ$_{a}$ , t$_{a}$, θ } ) ‖ 2 = 1. It follows that A is the SNR of the signal. 

Later on, we use the fact that although not all values of θ correspond to valid binary mass components, Eq. 3 can still be used to generate perfectly normal waveforms. These waveforms are also chirps but their phase evolution does not correspond to any physical binary system. 

The function to be maximized is, 

<img src='content_image/408417.jpg'>

<img src='content_image/408418.jpg'>

## C. Fitness function 

This function is obtained by maximizing the log- likelihood, 〈 x , s (Θ) 〉 − (1 / 2) ‖ s (Θ) ‖ 2 analytically over A and Φ$_{a}$. 

For given θ , the evaluation of 〈 q $_{φ}$( t$_{a}$, θ ) , x 〉 over t$_{a}$ = mδ$_{s}$ , m = 0 , 1 , . . . , N − 1, is a cross-correlation operation that can be computed efficiently using the Fast Fourier Transform (FFT). Thus, the function that is maximized using PSO is, 

In the remainder of the paper, λ ( θ | x ) will be called the fitness function in keeping with the standard terminology used in much of the literature on stochastic methods. 

The presence of noise in x makes the fitness function highly multi-modal as shown in Fig 1. The large number of local maxima with random locations and sizes poses a strong challenge to stochastic methods. When the noise is stationary and Gaussian, and the signal present in the data is from the waveform family that one is searching for, certain characteristic features are present in the fit- ness function. For example, the shape of the peak in Fig. 1 is elongated on the average along a predictable di- rection. MCMC methods in the GW literature use this type of prior information about the fitness function in tuning the design variables [9]. 

<img src='content_image/408414.jpg'>

