The Stochastic Crb For Array Processing A Textbook Derivation Jun 2026

: For ( \mathbfX = \mathbfA_i' \mathbfP \mathbfA^H ), one finds that only the projection onto the noise subspace survives due to: [ \mathbfR^-1 \mathbfA = \mathbfA (\mathbfA^H \mathbfA)^-1 (\mathbfP^-1 + \sigma^-2 \mathbfA^H \mathbfA)^-1 \sigma^-2 ] After substituting and taking trace, cross terms vanish because ( \mathbfR^-1 \mathbfA ) lies in column space of ( \mathbfA ), while ( \mathbf\Pi_A^\perp \mathbfA = 0 ).

The closed-form expression for the stochastic CRB of the DOA parameters bold theta

This is the Schur complement of the nuisance parameter block.

bold x open paren t close paren equals bold cap A open paren bold theta close paren bold s open paren t close paren plus bold n open paren t close paren steering matrix dependent on the DOAs bold theta : For ( \mathbfX = \mathbfA_i' \mathbfP \mathbfA^H

The genius of the Stoica et al. derivation is that it bypasses the need for large-sample ML proofs. Instead, it uses the to compute the Fisher Information Matrix (FIM) directly from the data covariance matrix.

[ \frac\partial \mathbfR\partial [\mathbfR s] ij = \mathbfA \mathbfE ij \mathbfA^H + \mathbfA \mathbfE ji \mathbfA^H ]

where the array covariance matrix is:

[ \mathbfx(t) = \mathbfA(\boldsymbol\theta) \mathbfs(t) + \mathbfn(t), \quad t = 1,\dots, N ]

Using the Slepian-Bangs structure, the final simplified expression (after substantial algebra) is:

where ( \boldsymbol\eta ) is the real parameter vector. derivation is that it bypasses the need for

For the deterministic model, ( \mathbfs(t) ) are unknown deterministic, and the CRB for DOAs is:

The unknown parameter vector ( \boldsymbol\Theta ) contains: