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Conventional resolution theory is based on the assumption of constant range rate, and it neglects distortions of the modulation function as well as effects of the antenna pattern. Among the generalizations of the theory reported in the literature, some include the antenna pattern and others retain higher-order range derivatives in both carrier term and modulation function. The first part of this paper is tutorial and discusses the significance of the various generalizations of the ambiguity function. It is shown that an important class of problems can be accommodated when the conventional theory is extended by permitting higher-order range derivatives in the carrier term. Target resolvability on the basis of motion then depends only on the relative phase shifts introduced over the signal duration. In the second part, the ambiguity function is formally extended to arbitrary target motion. The modified theory retains the simplicity of the conventional approach and allows utilization of well-known results.