Characterizing the radar ambiguity functions

Ambiguity functions are expanded relative to cross-ambiguity functions associated with a special orthonormal basis of signal space related to a rectangular pulse. The cross-ambiguity functions are simply related to the well-known ambiguity function of a rectangular pulse. The description of ambiguity functions in terms of these well-known and interrelated cross-ambiguity functions facilitates the ease with which desired calculations can be made. Indeed, one can compute the ambiguity function of a signal as easily as taking the Fourier series of a periodic function. The characterization of ambiguity functions resulting from this expansion is applied to prove two general results about the set of all ambiguity functions. We prove that the set of ambiguity functions is closed on the square-integrable topology and that, except in a trivial case, the sum of two ambiguity functions is never an ambiguity function.