Corrections to “Semiparametric CRB and Slepian-Bangs Formulas for Complex Elliptically Symmetric Distributions”

Errors in [1] are corrected below. 1. In Eq. (17), $\mathrm{vecs}(\boldsymbol{\Sigma}_{0})$ should be $\mathrm{vec}(\boldsymbol{\Sigma}_{0})$. Specifically, the correct version of Eq. (17) is: \begin{align*} \mathbf{s}_{\boldsymbol{\phi}_{0}}\triangleq\nabla_{\boldsymbol{\phi}}\ln p_{Z}(\mathbf{z};\boldsymbol{\phi}_{0},h_{0})=[\mathbf{s}^{T}_{\boldsymbol{\mu}_{0}},\mathbf{s}^{T}_{\boldsymbol{\mu}^{*}_ {0}},\mathbf{s}^{T}_{\mathrm{vec}(\boldsymbol{\Sigma}_{0})}]^{T}.\tag{17} \end{align*} 2. In the first line after Eq. (18), $\mathbf{s}^{T}_{\mathrm{vecs}(\boldsymbol{\Sigma}_{0})}$ should be $\mathbf{s}^{T}_{\mathrm{vec}(\boldsymbol{\Sigma}_{0})}$. 3. A minus “-” is missing in front of the right-hand side of Eq. (25). The correct equation is: \begin{align*} &\bar{\mathbf{s}}_{\mathrm{vec}(\boldsymbol{\Sigma}_{0})}= _{d}-\mathcal{Q}\psi_{0}(\mathcal{Q})\times \\ &\quad \times(\boldsymbol{\Sigma}_{0}^{-*/2}\otimes\boldsymbol{\Sigma}_{0}^{-1/2} \mathrm{vec}(\mathbf{u}\mathbf{u}^{H})-N^{-1}\mathrm{vec}(\boldsymbol{\Sigma}_{0}^{-1})) . \tag{25} \end{align*} 4. A minus “-” is missing in front of $\mathrm{tr}(\mathbf{P}_{i}^{0})$ in Eqs. (38), (40), (41), (42). The correct equations are: \begin{align*} [\mathbf{s}_{\boldsymbol{\theta}_{0}}]_{i} & \triangleq\left.\frac{\partial\ln p_{Z}\left(\mathbf{z};\boldsymbol{\theta},h_{0} \right)}{\partial\theta_{i}}\right|_{\boldsymbol{\theta}=\boldsymbol{\theta}_{0}}\\ &=-\mathrm{tr} (\mathbf{P}_{i}^{0})+\psi_{0}(Q_{0})\frac{\partial Q_...
Source: IEEE Transactions on Signal Processing - Category: Biomedical Engineering Source Type: research