Matched filtering a gravitational wave pulsar signal involving reciprocal gamma functions

The direct detection of Gravitational Waves (GW) is a challenging problem that involves elaborate experimental and data analysis techniques. The verification of a detected signal demands an effective way to distinguish the source signal from the background noise. One possibility is to perform matched filtering analysis using different templates. Matched filtering, a form of pattern recognition, is ubiquitous and finds innumerable and diverse applications. In the present work, we develop the matched filter analysis for a Fourier transformed, monochromatic, Doppler shifted, continuous GW pulsar signal, which incorporates the effects of the rotational and orbital motion of the Earth. The GW pulsar signal involves a product of the reciprocals of two Eulerian gamma functions containing the Fourier transformed bandwidth frequency in their arguments. We derive an exact analytic solution for the case of constant spectral noise density for the inner product of the template with a received signal, thereby obtaining a closed form expression for the fitting factor, a measure of how well the template matches the received signal. This result can in turn be used to determine the location of the GW source. Simpler cases of the spectral noise density for the French-Italian VIRGO GW detector and its special case for Gaussian white noise are also amenable to an analytic formulation. Our analysis shows that the fitting factor may exhibit simple symmetries with respect to the polar direction angle to the source. Approximate symmetries will also be useful in reducing the numerical computation times. Our current study confirms that the whole analysis lends itself well to parallel computation.