Least-square synthesis of radar ambiguity functions

The synthesis of radar ambiguity functions is approached by minimizing the integrated square error between an arbitrary desired function and a realizable ambiguity function. The approximation is carried out via an orthonormal signal basis which generates an "induced basis" over the time-frequency plane consisting of all pair-wise cross-ambiguity functions of the signal basis. The minimum mean-square error and the corresponding signal are determined through an eigenvalue problem for 1) approximation by complex autoambiguity function and 2) approximation by complex cross-ambiguity function. A new form of the realizability theorem shows that the conditions for cross-ambiguity functions differ from those for autoambiguity functions only in the absence of the symmetry condition:
F(\tau, \omega) = F^{\ast}(-\tau, -\omega)
. Moreover, the approximations by cross- and autoambiguity functions coincide whenever the desired function has the above symmetry. When an ambiguity function realizable by a known signal is to be approximated on a finite basis, least-square approximation in signal space or in ambiguity-function space leads to equivalent results. The relation between the mean-square errors in the two spaces is obtained. For the phase incoherent radar case an iteration procedure is presented for successively modifying the arbitrary phase assigned to the desired function. Convergence is proved in the sense that the error is nonincreasing at each stage of the iteration, but arrival at the best approximation to the desired magnitude is not guaranteed. As an aid in numerical applications, a formulation based on a discrete sample grid in the time-frequency plane is derived. With the appropriate grid dimensions, the analytic procedures carry over directly into sampled-data representation.