  Gravitational Wave Inspiral search with a global network of interferometers
when carried in a phase coherent fashion would mimic an effective
multi-detector network with synthetic streams constructed by the linear
combination of the data from different detectors. For the first time, we
demonstrate that the two synthetic data streams pertaining to the two
polarizations of Gravitational Wave can be derived prior to the
maximum-likelihood analysis in a most natural way using the technique of
singular-value-decomposition applied to the network signal-to-noise ratio
vector. We construct the network matched filters in combined network plus
spectral space which capture both the synthetic streams. We further show that
the network LLR is then sum of the LLR of each synthetic stream. The four
extrinsic parameters are mapped to the two amplitudes and two phases. The
maximization over these is a straightforward approach closely linked to the
single detector approach. Towards the end, we connect all the previous works
related to the multi-detector Gravitational Wave inspiral search and express in
the same notation in order to bring under the same footing.
