The Stochastic Crb For Array Processing A Textbook Derivation -

(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 ).

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 ] Define the FIM as: [ \mathbfF = \beginbmatrix

This guide focuses on the derivation — showing the logical steps, assumptions, and mathematical manipulations required to arrive at the closed-form expression for the CRB when signals are modeled as stochastic (Gaussian) processes. We consider an array of ( M ) sensors receiving ( d ) narrowband signals from far-field sources. 1.1 Data Model (Stochastic Assumption) The ( M \times 1 ) snapshot vector at time ( t ) is: Define the FIM as: [ \mathbfF = \beginbmatrix

where ( \boldsymbol\eta ) is the real parameter vector. Define the FIM as: [ \mathbfF = \beginbmatrix