The focus of this paper is on co-existence between a communication system and
a pulsed radar sharing the same bandwidth. Based on the fact that the
interference generated by the radar onto the communication receiver is
intermittent and depends on the density of scattering objects (such as, e.g.,
targets), we first show that the communication system is equivalent to a set of
independent parallel channels, whereby pre-coding on each channel can be
introduced as a new degree of freedom. We introduce a new figure of merit,
named the {\em compound rate}, which is a convex combination of rates with and
without interference, to be optimized under constraints concerning the
signal-to-interference-plus-noise ratio (including {\em signal-dependent}
interference due to clutter) experienced by the radar and obviously the powers
emitted by the two systems: the degrees of freedom are the radar waveform and
the afore-mentioned encoding matrix for the communication symbols. We provide
closed-form solutions for the optimum transmit policies for both systems under
two basic models for the scattering produced by the radar onto the
communication receiver, and account for possible correlation of the
signal-independent fraction of the interference impinging on the radar. We also
discuss the region of the achievable communication rates with and without
interference. A thorough performance assessment shows the potentials and the
limitations of the proposed co-existing architecture
