The Accuracy of Maximum Likelihood Angle Estimates in Radar and Sonar

By extending the results of Kelly, Reed, and Root, formulas are derived for the variances of maximum likelihood estimates of azimuth, and azimuth and elevation, jointly, by dense, discrete and discrete-continuous apertures for the strong signal case. The accuracy of angle measurements depends upon 1) total signal energy captured by the aperture, 2) the mean-square aperture size, 3) carrier frequency, 4) the mean-square signal bandwidth. Mean-square quantities are the second moments about the centroids. The actual signal form and aperture form do not matter, except as they affect the mean-square quantities. When joint estimates of azimuth and elevation are made, the errors are generally coupled. Minimum variances are obtained when the errors are uncoupled. This condition is obtained, in the narrowband case, when the two-dimensional illumination function is factorable into the product of one-dimensional functions. The formulas for the dense and discrete apertures are identical in form, the various factors being discrete or continuous analogs of one another in which integrations are replaced by summations. The formulas for the discrete-continuous array differs in form by the presence of terms which reflect the anisotropy of the beam patterns.