Woodward's ambiguity function: From foundations to applications

Woodward's ambiguity function, introduced in the literature in the mid-20th century, has been a staple topic in the study of radar performance. There exists an inherent trade-off in the ability of a signal to accurately measure both the range and velocity of a target. Woodward's ambiguity function measures this uncertainty for narrowband RF signals for monostatic radar. Despite its popularity and widespread description in both radar texts and the literature, there has been a wide variation in description of the ambiguity function. There are, for example, numerous similar albeit different definitions of the ambiguity function. There are also false statements made including claims that an ambiguity function is not invertable to its underlying signal and that best estimates of the range and Doppler of an object is found by analyzing the maximum of the magnitude of a correlation. In this paper, these and other questions are answered through derivation of the ambiguity function from first principles. Tables of ambiguity functions and their properties are presented, and 2-D Fourier analysis is shown to provide deeper insight into the ambiguity function structure.