A Textbook Derivation — The Stochastic Crb For Array Processing

(from Slepian–Bangs formula): The log-likelihood (ignoring constants) is: [ L = -N \log \det \mathbfR - \sum_t=1^N \mathbfx^H(t) \mathbfR^-1 \mathbfx(t) ] Taking derivatives and expectations yields the above trace formula. 3. Partitioning the Unknown Parameters Let: [ \boldsymbol\eta = [\boldsymbol\theta^T, \ \mathbfp^T, \ \sigma^2]^T ] We want the CRB for ( \boldsymbol\theta ), i.e., the top-left ( d \times d ) block of ( \mathbfF^-1 ).

Let ( \mathbfB = \mathbfA \mathbfP^1/2 ). Then ( \mathbfR = \mathbfB \mathbfB^H + \sigma^2 \mathbfI ). The projection matrix onto the column space of ( \mathbfB ): [ \mathbfP_B = \mathbfB(\mathbfB^H \mathbfB)^-1 \mathbfB^H ] but ( \mathbfB^H \mathbfB = \mathbfP^1/2 \mathbfA^H \mathbfA \mathbfP^1/2 ). Let ( \mathbfB = \mathbfA \mathbfP^1/2 )

[ [\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) ] Let ( \mathbfB = \mathbfA \mathbfP^1/2 )

where ( \boldsymbol\eta ) is the real parameter vector. Let ( \mathbfB = \mathbfA \mathbfP^1/2 )