Optimal mismatched filter design for radar ranging, detection, and resolution

In a multiple target environment a radar signal processor often uses weighting filters which are not necessarily matched to the transmitted waveform. In this paper expressions for the mean-square range-estimation error, the detection signal-to-noise (SNR) and the effects of sidelobes are derived in terms of the impulse response of an arbitrary mismatched filter. It is desired to find that impulse response which results in the minimum range estimate variance subject to preassigned constraints on the sidelobes and the detection SNR. This optimization problem is first formulated in state-space in which the optimal control law is sought. Pontryagin's maximum principle is used to obtain necessary conditions for the optimum impulse response, from which it is possible to deduce the structure of the optimum filter. Certain mathematical details which detract from the rigor of the time domain formulation are resolved by formulating the problem in the frequency domain and applying Hilbert space techniques. It is shown that for the problem of detecting the radar target and estimating its range, the optimum filter is a modified transversal equalizer. If only the detection function is to be performed the optimum filter reduces to the transversal equalizer. This establishes the optimality of this important practical device as the solution to the radar detection problem in a multiple target environment. The tap weights and spaces of the delay line as well as certain other parameters upon which the solution depends can be found by solving a non-linear programming problem. Numerical results are given for an interesting class of transmitted waveforms which shows the tradeoffs of the various filter parameters. Accepted for the Air Force Franklin C. Hudson Chief, Lincoln Laboratory Office iii