A 1-D search solution for localization from frequency measurements

A source that emits a constant frequency tone and moves at a constant course and speed can be localized through measurements of the Doppler shifted frequencies (DSF). With five unknowns, namely, the rest frequency and the positions and speeds in the x-y directions, five separate sensors would normally be necessary to give five DSP measurements for instantaneous localization. The equations are nonlinear, and the standard solution is by grid search or iteration. The high dimensionality leads to a large computational requirement. By incorporating DSF rates, a quantity available from frequency line trackers, a one-dimensional grid search solution is possible which requires only three sensors and reduces the computational load. The derivation of the grid search technique is given, together with simulation results. The conclusion is that at high signal-to-noise ratios (SNR), the scheme reaches the three-sensor Cramer-Rao lower bound; at lower SNR's or with increased sensors, the grid search answer is a good initializer for a nonlinear optimization algorithm that gives a maximum likelihood estimate