the addition of corresponding time samples. For later convenience we allow each sample to take complex values. A natural basis for this vector space is the [*time basis*]{} ${\bf e}_r^a = 
\delta^a_r$ where $r$ and $a$ are the vector and component indices respectively. Another basis which we shall use extensively is the Fourier basis.

A gravitational wave signal from a coalescing binary system can be characterised by a set of parameters $\mu^a, a = 0,1, ...,m-1$ belonging to some open set of the $m$-dimensional real space $R^m$. The set of such signals ${\bf s}(t; \bbox{\mu})$ constitutes a $m$-dimensional manifold $\cal S$ which is embedded in the vector space $\cal V$. Note that Greek characters in boldface denote the full collection of parameters characterising the signal. The parameters of the binary can be regarded as coordinates on the manifold. The basic problem of signal analysis is thus to determine whether the detector output vector $\bf x$ is the sum of a signal vector and a noise vector, ${\bf x} = {\bf s} + {\bf n}$, or just the noise vector, ${\bf x} = {\bf n}$, and furthermore to identify which particular signal vector, among all possible. The latter is relevant to 