Asymptotic evaluation of the ambiguity functions of high-gain FM matched filter sonar systems

The evaluation of radar and active sonar matched filter ambiguity functions is critical to waveform choice, the proper selection of reference signals, and the ability to estimate system performance. In the radar case the evaluation is facilitated by use of the narrow-band approximation wherein Doppler distortion of the transmitted modulation function is neglected. However, modern-day sonar signals are usually broad-band with respect to the carrier frequency and long in duration so that the narrow-band approximation cannot be employed. This creates difficulties in both analytic and computer ambiguity function evaluations. Analytic difficulties arise from the inability to solve the appropriate integrals; computer difficulties, based on direct cross correlation, arise from the long computation time required. This paper shows how evaluation of the ambiguity functions of FM sonar signals having high time-bandwidth products can be greatly facilitated by asymptotic approximations derived on the basis of the principle of stationary phase. Application of the theory allows many seemingly difficult ambiguity evaluation problems involving highly complicated integrals to be solved rapidly using algebraic techniques. As examples of the theory, the cases of linear FM under constant target velocity conditions and hyperbolic (Doppler invariant) FM under constant target acceleration conditions are worked out. In addition, some interesting adjustable Doppler tolerant characteristics of a special modulation technique (parabolic FM) are described.