On a definition of the ambiguity function for non-coherent radars

The ambiguity function is an important tool to study the performance of radar detectors. In this paper, motivated by Neyman-Pearson testing principles, we propose an alternative definition of the ambiguity function that directly associates with each pair of true and assumed target parameters the probability that the radar will declare a target present. We show that the original ambiguity function definition of Woodward and Davies for single antenna systems (and its extensions to multichannel systems that use coherent processing) are essentially equivalent to the proposed definition. Further, for radars that perform non-coherent processing, we show the extensions to Woodward's ambiguity function proposed in literature are not equivalent to our proposed definition - and therefore may not accurately reflect detection performance. Simulations results demonstrate the differences between these different ambiguity function definitions for non-coherent radars.