Radar Frequency Modulations for Accelerating Targets under a Bandwidth Constraint

An optimum frequency modulation for estimating the range, range rate, and range acceleration of a moving target is derived. The criterion of optimality is based on the estimate variances which are evaluated under the following assumptions: 1) additive white Gaussian observation noise, 2) high signal-to-noise ratio, 3) maximum-likelihood processing (matched filters), 4) RF phase used only for the range rate and range acceleration, and 5) carrier frequency much larger than the signal bandwidth. The choice of frequency modulation is constrained by the bandwidth of the transmitted signal. A large time-bandwidth product is assumed. The optimum frequency modulation consists of three appropriately placed frequency jumps between the limits imposed by the bandwidth constraint. This optimum modulation is compared with a third degree, power law modulation. The derivation of the optimum, originally done using Pontryagin´s Maximum Principle, leads to the following very simple design principle; the optimum modulation is orthogonal to the target´s motion.