  The matched filter (MF) is widely used to detect signals hidden within the
noise. If the noise is Gaussian, its performances are well-known and
describable in an elegant analytical form. The treatment of non-Gaussian noises
is often cumbersome as in most cases there is no analytical framework. This is
true also for Poisson noise which, especially in the low-number count regime,
presents the additional difficulty to be discrete. For this reason in the past
methods have been proposed based on heuristic or semi-heuristic arguments.
Recently, an analytical form of the MF has been introduced but the computation
of the probability of false detection or false alarm (PFA) is based on
numerical simulations. To overcome this inefficient and time consuming approach
we propose here an effective method to compute the PFA based on the saddle
point approximation (SA). We provide the theoretical framework and support our
findings by means of numerical simulations. We discuss also the limitations of
the MF in practical applications.
