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

• Published in: IEEE transactions on signal processing Vol. 72; p. 686
• Main Authors: Fortunati, Stefano, Gini, Fulvio, Greco, Maria S., Zoubir, Abdelhak M., Rangaswamy, Muralidhar
• Format: Journal Article
• Language: English
• Published: New York The Institute of Electrical and Electronics Engineers, Inc. (IEEE) 2024
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2024.3353889

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_{0}}{\partial\theta_{i}},\tag{38} \end{align*} \begin{align*} [\mathbf{s}_{\boldsymbol{\theta}_{0}}]_{i}& =- \mathrm{tr}\left(\mathbf{P}_{i}^{0}\right)-\psi_{0}(Q_{0})\\ &\quad\times\left(2\mathrm{Re} \left[(\mathbf{z}-\boldsymbol{\mu}_{0})^{H}\boldsymbol{\Sigma}_{0}^{-1}\boldsymbol{\mu}_{i}^{0}\right]+ \right. \\ &\quad \left.+(\mathbf{z}-\boldsymbol{\mu}_{0})^{H}\mathbf{S}_{i}^{0}(\mathbf{z}-\boldsymbol{\mu}_{0})\right),\;i=1,\ldots,d. \tag{40} \end{align*} \begin{align*} [\mathbf{s}_{\boldsymbol{\theta}_{0}}]_{i} & =_{d }-\psi_{0}(\mathcal{Q})\left(2\sqrt{\mathcal{Q}}\mathrm{Re}\left[\mathbf{u}^{H }\boldsymbol{\Sigma}_{0}^{H/2}\boldsymbol{\Sigma}_{0}^{-1}\boldsymbol{\mu}_{i}^{0}\right]+\right. \\ & \quad\left.+\mathcal{Q}\mathbf{u}^{H}\boldsymbol{\Sigma}_{0}^{H/2} \mathbf{S}_{i}^{0}\boldsymbol{\Sigma}_{0}^{1/2}\mathbf{u}\right)-\mathrm{tr}\left (\mathbf{P}_{i}^{0}\right) \\ & =-\psi_{0}(\mathcal{Q})\left(\!2\sqrt{\mathcal{Q}}\mathrm{Re}\!\left[\!\mathbf{u}^{H}\boldsymbol{\Sigma}_{0}^{-1/2}\boldsymbol{\mu}_{i}^{0}\right]\right.\\ &\quad + \left.\vphantom{2\sqrt{\mathcal{Q}}\mathrm{Re}\!\left[\!\mathbf{u}^{H}\boldsymbol{\Sigma}_{0}^{-1/2}\boldsymbol{\mu}_{i}^{0}\right]}\mathcal{Q} \mathbf{u}^{H}\mathbf{P}_{i}^{0}\mathbf{u}\!\right)+ \\ & \quad-{\mathrm{tr}}\left(\mathbf{P}_{i}^{0}\right),\;i=1,\ldots,d. \tag{41} \end{align*} \begin{align*} &[\Pi (\mathbf{s}_{\boldsymbol{\theta}_{0}}| \mathcal{T}_{h_{0}})]_{i}=E_{0|\sqrt{\mathcal{Q}}}\{[\mathbf{s}_{\boldsymbol{ \theta}_{0}}]_{i}|\sqrt{\mathcal{Q}}\} \\ &\quad =_{d}-\mathrm{tr}(\mathbf{P}_{i}^{0})-2\sqrt{\mathcal{Q}}\psi_{0 }(\mathcal{Q})\mathrm{Re}\left[E\{\mathbf{u}\}^{H}\boldsymbol{\Sigma}_ {0}^{-1/2}\boldsymbol{\mu}_{i}^{0}\right] \\ &\quad\quad -\mathcal{Q}\psi_{0}(\mathcal{Q})\mathrm{tr}\left(\mathbf{P}_{i} ^{0}E\{\mathbf{u}\mathbf{u}^{H}\}\right) \\ &\quad =-{\mathrm{tr}}(\mathbf{P}_{i}^{0})-N^{-1}\mathcal{Q}\psi_{0} (\mathcal{Q})\mathrm{tr}\left(\mathbf{P}_{i}^{0}\right),\;i=1,\ldots,d. \tag{42} \end{align*}