Tracker-based selection of radar waveforms with range and range-rate measurements: Analytical results

The impact of waveform selection in tracking problems is effected through measurement noise covariances depending on the waveform parameters. For unit energy waveforms of constant frequency (CF) and linearly frequency modulated (LFM) types, the parameters are the pulse width (CF and LFM) and the sweep rate (LFM). In the presence of a single target, it makes sense to employ waveform optimization to minimize the size of the state error covariance. In the first part of the paper we derive closed form approximations for waveforms minimizing two metrics encountered in the literature: the trace of the state error covariance and the determinant of the innovation covariance. We first show that a fixed upsweep LFM waveform minimizes an approximation of the trace of the steady state error covariance. Second, we show that a fixed CF waveform minimizes an approximation of the determinant of the steady state innovation covariance. The actual minima in the two cases are determined through grid searches, and the validity of the approximations is confirmed. In the presence of multiple targets, waveforms tuned to a particular track are not necessarily the best, due to the need for disambiguating the measurements. In the second part of the paper, we derive a closed form expression for a time-varying waveform which maximize the Bhattacharyya distance between the gating distributions of two tracks. Computer simulations are used to show the effectiveness of this waveform in contrast with waveforms tuned to a single track. Waveform optimization and waveform adaptation are complex design problems. In addition to providing optimal solutions under idealized conditions, closed form expressions can provide a first, overall assessment of complex design problems.