Effects of Parametric Constraints on the CRLB in Gain and Phase Estimation Problems

The problem of estimating the direction-independent gain and phase characteristics of sensor arrays requires a boundary condition to solve the phase ambiguity of the solution. It has become common practice to use the constraint that the phase of the first element of the array is zero. By Cramer-Rao lower bound (CRLB) analysis, we show analytically for calibration on a single point source of an array of identical elements that the variance of the phase estimates decreases by a factor 2p/(p-1) under the boundary condition that the sum of all phases equals zero, where p is the number of elements in the array. We also show that this constraint is the one that among all possible constraints minimizes the total variance on all estimated parameters. Our analysis suggests that this conclusion also holds for arbitrary source models and arrays of nonidentical elements. This statement is confirmed by repeating the CRLB analysis in simulation, showing that the CRLB under this constraint coincides within the numerical accuracy with the minimal CRLB