The Stochastic Crb For Array Processing A Textbook Derivation [PRO]

where:

where ( \hat\mathbfR s = \frac1N\sum t=1^N \mathbfs(t)\mathbfs(t)^H ). The difference is that in the stochastic case, ( \mathbfR_s \mathbfA^H \mathbfR^-1 \mathbfA \mathbfR_s ) appears instead of ( \hat\mathbfR_s ). Since ( \mathbfA^H \mathbfR^-1 \mathbfA = (\mathbfR_s^-1 + \frac1\sigma^2 \mathbfA^H \mathbfA)^-1 ) (by matrix inversion lemma), we have ( \mathbfR_s \mathbfA^H \mathbfR^-1 \mathbfA \mathbfR_s = \mathbfR_s - \mathbfR_s(\mathbfR_s + \sigma^2 (\mathbfA^H \mathbfA)^-1)^-1 \mathbfR_s ), which is generally larger than ( \mathbfR_s ) in the positive definite sense. Hence the stochastic CRB is smaller. where: where ( \hat\mathbfR s = \frac1N\sum t=1^N

The signal power $\sigma_k^2$ appears only in the diagonal matrix $\mathbfP$. $$ \frac{\partial Hence the stochastic CRB is smaller

For simplicity, we assume ( \mathbfR_s ) is unknown but unstructured (except Hermitian positive definite). Then the number of real parameters: ( K ) DOAs + ( K^2 ) real parameters from ( \mathbfR_s ) (since ( K^2 ) real + imaginary parts, but Hermitian symmetry reduces to ( K^2 ) real) + 1 for ( \sigma^2 ). Then the number of real parameters: ( K

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

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

[ E\left[ \frac\partial \mathcalL\partial \mu \frac\partial \mathcalL\partial \nu \right] = N \cdot \mathrmtr\left( \mathbfR^-1 \frac\partial \mathbfR\partial \mu \mathbfR^-1 \frac\partial \mathbfR\partial \nu \right). ]