The Stochastic Crb For Array Processing A — Textbook Derivation ((install))

[ \frac\partial \mathbfR\partial \sigma^2 = \mathbfI ]

CRB sub bold theta equals the fraction with numerator sigma squared and denominator 2 cap T end-fraction the set Re open bracket bold cap H circled dot open paren bold cap P bold cap A to the cap H-th power bold cap R to the negative 1 power bold cap A bold cap P close paren to the cap T-th power close bracket end-set to the negative 1 power is the number of snapshots, bold cap R is the covariance matrix, and bold cap A is the array steering matrix. Mianzhi Wang 1. Define the Signal and Data Model [ \frac\partial \mathbfR\partial \sigma^2 = \mathbfI ] CRB

[ \mathrmCRB_\textstoch(\theta_k) = \frac\sigma^22N \left[ \operatornameRe\left( \mathbfD^H \mathbf\Pi_A^\perp \mathbfD \odot (\mathbfR_s \mathbfA^H \mathbfR^-1 \mathbfA \mathbfR s)^T \right) \right]^-1 kk ] For array processing, the stochastic CRB is often lower (i

The Cramér–Rao Bound (CRB) provides a lower bound on the variance of any unbiased estimator. For array processing, the stochastic CRB is often lower (i.e., better) than the deterministic CRB because it exploits knowledge of the signal distribution. This article derives the stochastic CRB step-by-step, as one would find in a graduate-level textbook (e.g., Stoica & Moses, Spectral Analysis of Signals , or Kay, Fundamentals of Statistical Signal Processing ). This formula is a workhorse in array processing derivations

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: