Asymptotic and geometric procedures for estimating correlation functions for FM signals

Frequency Modulated (FM) signals with large time-bandwidth products are used in many areas, such as Sonar, Radar and Spread Spectrum Communications. In order to take advantage of the pulse compression properties of these signals, many of these applications require coherent or matched filter processing as the appropriate receiver. The receiver response is usually expressed in terms of the ambiguity functions for the signals involved. Although there are many types of FM signals which have pulse compression properties, their ambiguity functions can have quite different properties. For example, linear FM signals (chirp) have a single ridge with no pedestal, whereas V-chirp FM signals have two small ridges near the time-frequency origin and an additional pedestal. Thus, it is desirable to have simple and intuitive methods for evaluating these ambiguity functions. The first section of this paper uses the method of stationary phase in order to derive an asymptotic expression for the cross-correlation function for two FM (frequency modulation) signals whose instantaneous frequency curves intersect at one point in time-frequency space. Assuming slow variation in the amplitude and frequency modulation functions allows a simple geometric interpretation for the asymptotic result which is the square-root of an area measured in time-frequency space. This is shown in section two. The case for multiple intersections is discussed and shown, through an example, to depend on the relative phases of the individual contributions. This intuitive method of evaluating correlation functions explains why V-chirp and SQFM signals have pedestals in their ambiguity functions.