The Stochastic Crb For Array Processing A Textbook Derivation -

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 \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 ] This guide focuses on the derivation — showing

where ( \boldsymbol\eta ) is the real parameter vector. The CRB for ( \boldsymbol\theta ) (with nuisance

The CRB for ( \boldsymbol\theta ) (with nuisance parameters ( \mathbfp, \sigma^2 )) is: [ \textCRB(\boldsymbol\theta) = \left( \mathbfF \theta\theta - [\mathbfF \theta p \ \mathbfF \theta \sigma^2] \beginbmatrix \mathbfF pp & \mathbfF p\sigma^2 \ \mathbfF \sigma^2 p & \mathbfF \sigma^2\sigma^2 \endbmatrix^-1 \beginbmatrix \mathbfF p\theta \ \mathbfF_\sigma^2\theta \endbmatrix \right)^-1 ] This guide focuses on the derivation — showing