The Stochastic Crb For Array Processing A Textbook Derivation Access

After algebraic manipulation (see Stoica & Nehorai, 1989, or Van Trees, "Optimum Array Processing"), the stochastic CRB for ( \boldsymbol\theta ) simplifies to:

where ( \mu, \nu ) denote any real-valued scalar parameter in ( \boldsymbol\Theta ). This formula is a workhorse in array processing derivations. It stems from the general result for zero-mean complex Gaussian vectors: After algebraic manipulation (see Stoica & Nehorai, 1989,

: The signal vector, typically modeled as a zero-mean complex Gaussian process with covariance Pbold cap P : Additive White Gaussian Noise (AWGN) with variance σ2sigma squared The data covariance matrix Rbold cap R is defined as: Therefore, $\mathbfy(t)$ is also zero-mean complex Gaussian

Under these assumptions, the received data vector $\mathbfy(t)$ is a linear combination of independent Gaussian vectors. Therefore, $\mathbfy(t)$ is also zero-mean complex Gaussian. or Van Trees

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

bold cap R equals cap E open bracket bold x open paren t close paren bold x to the cap H-th power open paren t close paren close bracket equals bold cap A bold cap P bold cap A to the cap H-th power plus sigma squared bold cap I

[ \frac\partial \mathbfR\partial \sigma^2 = \mathbfI_M. ]