Passive bearing estimation of a broad-band source

The variance on the bearing estimates of a wide-band source has been given wide attention in the literature under the assumption of unbiased estimation process. This paper relaxes such a condition and derives the resulting statistical measures. Those lower bound measures depend on the tangent of the true bearing Pt to the source, and the bearing variance σ²βuas derived with an unbiased condition. In the far field, the resulting bias isgiven simply by A (tan βt)σ²βuwhere A is a constant. The bias size can be in the order of the unbiased bearing variance σ²βu. The variance is the sum of the unbiased variance, σ²βu, and a combination of the bias and σ²βu. The variance is contrasted against the Cramer Rao bound developed for the biased estimate. Such statistical results enable the sizing of the bias and its deteriorating effect on the estimation process for the specific case of interest. When the bias is unacceptable, averaging should be applied to the time delays rather than the bearings. The bias impact is also of interest when noisy bearings are applied to estimate other functions such as passive source motion at long ranges.