A tetrahedral array of isotropic sensors, each suffering a random complex gain — the resulting hybrid Cramér-Rao bound for direction finding

Consider a tetrahedral sensor array consisting of isotropic sensors that are nominally identical. In the real world, this nominal identity is not perfectly achievable. What if these sensors were each to suffer an uncertainty in its complex-value gain, which is modeled here stochastically as complex-value Gaussian random variables? What then would happen to the entire tetrahedral array's overall polar-azimuth direction-finding accuracy? This problem is quantified here via the hybrid Cramér-Rao bound (HCRB) of the tetrahedral sensor's polar-azimuth direction-of-arrival estimates. This hybrid Cramér-Rao bound is found analytically to be same as the Cramér-Rao bound (if the complex-gain uncertainty does not exist) except by a multiplicative factor that equals one plus the variance of the complex-gain uncertainty.