The Stochastic Crb For Array Processing A Textbook Derivation -

We partition the parameters into:

Let ( \mathbf\Pi_A^\perp = \mathbfI - \mathbfA(\mathbfA^H\mathbfA)^-1\mathbfA^H ) (projector onto noise subspace).

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 ). We partition the parameters into: Let ( \mathbf\Pi_A^\perp

$$ \mathcalL(\boldsymbol\eta) = -N \ln \det(\mathbfR) - N \texttr(\mathbfR^-1\hat\mathbfR) + \textconstant $$

[ \frac\partial \mathbfR\partial \sigma^2 = \mathbfI_M. ] For sensors and narrowband sources, the received signal

Define ( \mathbf\Pi_A^\perp = \mathbfI - \mathbfA(\mathbfA^H\mathbfA)^-1\mathbfA^H ) (projector onto noise subspace). After some algebraic manipulations (using Woodbury and matrix identities), one can show that:

[ \frac\partial \mathbfR\partial \theta_k = \mathbfA_k' \mathbfP \mathbfA^H + \mathbfA \mathbfP (\mathbfA_k')^H ] where ( \mathbfA_k' = \frac\partial \mathbfA\partial \theta_k = [\mathbf0, \dots, \mathbfa'(\theta_k), \dots, \mathbf0] ) (derivative of the ( k )-th column). For sensors and narrowband sources

The first step is establishing the mathematical representation of the array output. For sensors and narrowband sources, the received signal vector is modeled as: