Gaussian approximation of interferometric PDF for MLE derivation

The interferometry principle is usually implemented in sonar and radar systems to estimate the arrival angles of backscattered signals at time-sampling rate. This direction-finding method is based on phase-difference measurements between two close receivers. To quantify the associated bathymetric measurement quality, it is necessary to model the statistical properties of the interferometric-phase estimator. The classical interferometric estimator, i.e. arg{s₁s₂}, has a well known Gaussian-shaped probability density function (PDF). It is important to notice that the interferometric PDF is 2π periodic and is driven by two parameters: mean and correlation coefficient. Thanks to the relationship between these two parameters, the idea is to find a conversion technique between variance and correlation coefficient. A third-order polynomial function performs this easily while a simple affine function provides a rough but efficient approximation. The second step is to fit the PDF with finite support based on the normal distribution. This operation impacts the value of the PDF. From this it becomes possible to derive a convenient approximation of the maximum likelihood estimator with the Gaussian approximation which is well adapted for these calculations.