Desirable bandwidths for optimal frequency-modulated pulses

In a paper on radar frequency modulations, Schweppe (IEEE Trans. Military Electronics, vol. MIL-9, pp. 25-32, Jan. 1965) developed a simple design principle: "The target's position is obtained from the modulation while the target's motion (velocity and acceleration) is obtained from the RF phase." It was also pointed out that the modulation should be selected to be orthogonal to the target motion so the modulation will not interfere with the motion estimates. Having satisfied the orthogonality constraint, that modulation is selected that provides the best range estimate. The "best" estimate sought was that which minimized the variance of the range estimate. Designing a radar pulse so as to minimize the variance of an estimate corresponds to the designing of an ambiguity function with a sharp central peak at the true values of the parameters with no regard to the rest of its shape. This analysis was carried out (the solutions in each case are nonunique), the function is plotted for five and seven switching points, as well as 3 and 3. It is shown from these results that an optimal frequency-modulated pulse for an accelerating target with three switching points yields the most desirable ambiguity function.