The Stochastic Crb For Array Processing A Textbook Derivation [upd] [ BEST ]

[ [\mathbfF(\boldsymbol\eta)]_ij = N \cdot \textTr\left( \mathbfR^-1 \frac\partial \mathbfR\partial \eta_i \mathbfR^-1 \frac\partial \mathbfR\partial \eta_j \right) ]

For ( N ) i.i.d. complex Gaussian vectors ( \mathbfx(t) \sim \mathcalCN(\mathbf0, \mathbfR(\boldsymbol\eta)) ), the Fisher Information Matrix (FIM) is:

Define the FIM as: [ \mathbfF = \beginbmatrix \mathbfF \theta\theta & \mathbfF \theta p & \mathbfF \theta \sigma^2 \ \mathbfF p\theta & \mathbfF pp & \mathbfF p\sigma^2 \ \mathbfF \sigma^2\theta & \mathbfF \sigma^2 p & \mathbfF_\sigma^2\sigma^2 \endbmatrix ]

[ \mathbfR(\boldsymbol\Theta) = \mathbfA(\boldsymbol\theta)\mathbfR_s\mathbfA(\boldsymbol\theta)^H + \sigma^2 \mathbfI_M. ]