Slepian-Bangs formula and Cramer Rao bound for circular and non-circular complex elliptical symmetric distributions

TL;DR

This paper extends the Slepian-Bangs formula and analyzes the Cramer-Rao bound for non-circular complex elliptical symmetric distributions, revealing new insights into their statistical properties and bounds.

Contribution

It introduces a new stochastic representation theorem for NC-CES distributions and extends the Slepian-Bangs formula to these distributions, including non-circular complex Gaussian cases.

Findings

Gaussian distribution does not always maximize the stochastic CRB

Derived closed-form SCRBs for noisy mixture models

Established relations between CES and Gaussian CRBs

Abstract

This paper is mainly dedicated to an extension of the Slepian-Bangs formula to non-circular complex elliptical symmetric (NC-CES) distributions, which is derived from a new stochastic representation theorem. This formula includes the non-circular complex Gaussian and the circular CES (CCES) distributions. Some general relations between the Cramer Rao bound (CRB) under CES and Gaussian distributions are deduced. It is proved in particular that the Gaussian distribution does not always lead to the largest stochastic CRB (SCRB) as many authors tend to believe it. Finally a particular attention is paid to the noisy mixture where closedform expressions for the SCRBs of the parameters of interest are derived.

Equations

$$\mathrm{vec}(\mathbf{A}\mathbf{B}\mathbf{C})=({\mathbf{C}}^T\otimes \mathbf{A})\mathrm{vec}(\mathbf{B}),$$

$$(\mathbf{A}\otimes \mathbf{B})(\mathbf{C}\otimes \mathbf{D})=\mathbf{A}\mathbf{C}\otimes \mathbf{B}\mathbf{D},$$

$$\mathrm{Tr}(\mathbf{A}\mathbf{B})={\mathrm{vec}}^H({\mathbf{A}}^H)\mathrm{vec}(\mathbf{B}),$$

$$\mathrm{Tr}(\mathbf{A}\mathbf{B}\mathbf{C}\mathbf{D})={\mathrm{vec}}^H({\mathbf{A}}^H)({\mathbf{D}}^T\otimes \mathbf{B})\mathrm{vec}(\mathbf{C}),$$

$$\mathrm{Tr}(\mathbf{A}\otimes \mathbf{B})=\mathrm{Tr}(\mathbf{A})\mathrm{Tr}(\mathbf{B}),$$

$$\mathrm{Tr}[\mathbf{K}(\mathbf{A}\otimes \mathbf{B})]=\mathrm{Tr}(\mathbf{A}\mathbf{B}),$$

$$(\mathbf{A}+\mathbf{B}\mathbf{C}\mathbf{D}{)}^{-1}={\mathbf{A}}^{-1}-{\mathbf{A}}^{-1}\mathbf{B}({\mathbf{C}}^{-1}+\mathbf{D}{\mathbf{A}}^{-1}\mathbf{B}{)}^{-1}\mathbf{D}{\mathbf{A}}^{-1},$$

$$\mathrm{E}[({\tilde{\mathbf{y}}}^H\tilde{\mathbf{A}}\tilde{\mathbf{y}})({\tilde{\mathbf{y}}}^H\tilde{\mathbf{B}}\tilde{\mathbf{y}})]=\mathrm{Tr}(\tilde{\mathbf{A}})\mathrm{Tr}(\tilde{\mathbf{B}})+2\mathrm{Tr}(\tilde{\mathbf{A}}\tilde{\mathbf{B}}),$$

$$\mathrm{E}[({\tilde{\mathbf{y}}}^H\tilde{\mathbf{A}}\tilde{\mathbf{y}})({\tilde{\mathbf{y}}}^H\tilde{\mathbf{B}}\tilde{\mathbf{y}})]=\mathrm{Tr}[({\tilde{\mathbf{A}}}^T\otimes \tilde{\mathbf{B}})\mathrm{E}({\tilde{\mathbf{y}}}^{\ast }{\tilde{\mathbf{y}}}^T\otimes \tilde{\mathbf{y}}{\tilde{\mathbf{y}}}^H)],$$

$$\mathrm{E}({\tilde{\mathbf{y}}}^{\ast }{\tilde{\mathbf{y}}}^T\otimes \tilde{\mathbf{y}}{\tilde{\mathbf{y}}}^H)=\mathbf{I}\otimes \mathbf{I}+\mathbf{K}({\mathbf{J}}^{\prime }\otimes {\mathbf{J}}^{\prime })(\mathbf{I}\otimes \mathbf{I})+\mathrm{vec}(\mathbf{I}){\mathrm{vec}}^T(\mathbf{I}),$$

$$\mathrm{E}[({\tilde{\mathbf{y}}}^H\tilde{\mathbf{A}}\tilde{\mathbf{y}})({\tilde{\mathbf{y}}}^H\tilde{\mathbf{B}}\tilde{\mathbf{y}})]$$

$$+$$

$$\mathrm{Tr}[({\tilde{\mathbf{A}}}^T\otimes \tilde{\mathbf{B}})(\mathbf{I}\otimes \mathbf{I})]=\mathrm{Tr}(\tilde{\mathbf{A}})\mathrm{Tr}(\tilde{\mathbf{B}})$$

$$\mathrm{Tr}[({\tilde{\mathbf{A}}}^T\otimes \tilde{\mathbf{B}})\mathbf{K}({\mathbf{J}}^{\prime }\otimes {\mathbf{J}}^{\prime })(\mathbf{I}\otimes \mathbf{I})]=\mathrm{Tr}(\tilde{\mathbf{A}}\tilde{\mathbf{B}})$$

$$\mathrm{Tr}[({\tilde{\mathbf{A}}}^T\otimes \tilde{\mathbf{B}})\mathrm{vec}(\mathbf{I}){\mathrm{vec}}^T(\mathbf{I})]=\mathrm{Tr}(\tilde{\mathbf{A}}\tilde{\mathbf{B}})$$

$$\mathbf{z}=\mathbf{\mu }+\mathbf{\Psi }{\mathbf{z}}_0+\mathbf{\Phi }{\mathbf{z}}_0^{\ast },$$

$$\mathbf{\Sigma }=\mathbf{A}{\mathbf{A}}^H=\frac{\mathrm{E}({\mathcal{R}}^2)}{N{\sigma }_c}\left(\mathbf{\Psi }{\mathbf{\Psi }}^H+\mathbf{\Phi }{\mathbf{\Phi }}^H\right)\text{mbox}and\mathbf{\Omega }=\mathbf{A}{\mathbf{\Delta }}_{\kappa }{\mathbf{A}}^T=\frac{\mathrm{E}({\mathcal{R}}^2)}{N{\sigma }_c}\left(\mathbf{\Psi }{\mathbf{\Phi }}^T+\mathbf{\Psi }{\mathbf{\Phi }}^T\right),$$

$$\mathbf{A}{\mathbf{A}}^H=\mathbf{\Psi }{\mathbf{\Psi }}^H+\mathbf{\Phi }{\mathbf{\Phi }}^H\text{mbox}and\mathbf{A}{\mathbf{\Delta }}_{\kappa }{\mathbf{A}}^T=\mathbf{\Psi }{\mathbf{\Phi }}^T+\mathbf{\Psi }{\mathbf{\Phi }}^T.$$

$$\mathbf{I}={\mathbf{\Psi }}^{\prime }{\mathbf{\Psi }}^{{}^{\prime }H}+\mathbf{\Phi }{\mathbf{\Phi }}^{{}^{\prime }H}\text{mbox}and{\mathbf{\Delta }}_{\kappa }={\mathbf{\Psi }}^{\prime }{\mathbf{\Phi }}^{{}^{\prime }T}+{\mathbf{\Psi }}^{\prime }{\mathbf{\Phi }}^{{}^{\prime }T}.$$

$$\mathbf{I}={\mathbf{\Delta }}_1^2+{\mathbf{\Delta }}_2^2\text{mbox}and{\mathbf{\Delta }}_{\kappa }=2{\mathbf{\Delta }}_1{\mathbf{\Delta }}_2,$$

$$\mathbf{z}{=}_d\mathbf{\mu }+\mathcal{R}[\mathbf{\Psi }\mathbf{u}+\mathbf{\Phi }{\mathbf{u}}^{\ast }]=\mathbf{\mu }+\mathcal{R}\mathbf{A}[{\mathbf{\Delta }}_1\mathbf{u}+{\mathbf{\Delta }}_2{\mathbf{u}}^{\ast }].$$

$$\tilde{\mathbf{z}}{=}_d\tilde{\mathbf{\mu }}+\mathcal{R}{\tilde{\mathbf{\Gamma }}}^{1/2}\tilde{\mathbf{u}}$$

$$\mathrm{E}\left(\frac{\partial \log p(\mathbf{z};\mathbf{\alpha })}{\partial {\alpha }_k}\right)=0.$$

$$\frac{\partial \log p(\mathbf{z};\mathbf{\alpha })}{\partial {\alpha }_k}=-\frac{1}{2}\mathrm{Tr}({\tilde{\mathbf{\Gamma }}}^{-1}{\tilde{\mathbf{\Gamma }}}_k)+\varphi (\tilde{\eta })\frac{\partial \tilde{\eta }}{\partial {\alpha }_k}.$$

$$\frac{\partial \tilde{\eta }}{\partial {\alpha }_k}=-\mathrm{Re}\left({\tilde{\mathbf{\mu }}}_k^H{\tilde{\mathbf{\Gamma }}}^{-1}(\tilde{\mathbf{z}}-\tilde{\mathbf{\mu }})\right)-\frac{1}{2}(\tilde{\mathbf{z}}-\tilde{\mathbf{\mu }}{)}^H{\tilde{\mathbf{\Gamma }}}^{-1}{\tilde{\mathbf{\Gamma }}}_k{\tilde{\mathbf{\Gamma }}}^{-1}(\tilde{\mathbf{z}}-\tilde{\mathbf{\mu }}),$$

$$\frac{1}{2}(\tilde{\mathbf{z}}-\tilde{\mathbf{\mu }}{)}^H{\tilde{\mathbf{\Gamma }}}^{-1}{\tilde{\mathbf{\Gamma }}}_k{\tilde{\mathbf{\Gamma }}}^{-1}(\tilde{\mathbf{z}}-\tilde{\mathbf{\mu }}){=}_d\frac{1}{2}\mathcal{Q}{\tilde{\mathbf{u}}}^H{\tilde{\mathbf{H}}}_k\tilde{\mathbf{u}}$$

$$\mathrm{E}\left(\varphi (\tilde{\eta })\frac{\partial \tilde{\eta }}{\partial {\alpha }_k}\right)=-\mathrm{E}\left({\mathcal{Q}}^{1/2}\varphi (\mathcal{Q})\mathrm{Re}({\tilde{\mathbf{\mu }}}_k^H{\tilde{\mathbf{\Gamma }}}^{-1/2}\tilde{\mathbf{u}})\right)-\frac{1}{2}\mathrm{E}[\mathcal{Q}\varphi (\mathcal{Q}){\tilde{\mathbf{u}}}^H{\tilde{\mathbf{H}}}_k\tilde{\mathbf{u}}].$$

$$\mathrm{E}\left({\mathcal{Q}}^{1/2}\varphi (\mathcal{Q})\mathrm{Re}({\tilde{\mathbf{\mu }}}_k^H{\tilde{\mathbf{\Gamma }}}^{-1/2}\tilde{\mathbf{u}})\right)=0$$

$$\mathrm{E}[\mathcal{Q}\varphi (\mathcal{Q}){\tilde{\mathbf{u}}}^H{\tilde{\mathbf{H}}}_k\tilde{\mathbf{u}}]=\mathrm{E}[\mathcal{Q}\varphi (\mathcal{Q})]\mathrm{Tr}[{\tilde{\mathbf{H}}}_k\mathrm{E}(\tilde{\mathbf{u}}{\tilde{\mathbf{u}}}^H)]=-\mathrm{Tr}({\tilde{\mathbf{H}}}_k)=-\mathrm{Tr}({\tilde{\mathbf{\Gamma }}}^{-1}{\tilde{\mathbf{\Gamma }}}_k).$$

$$\mathrm{E}\left(\varphi (\tilde{\eta })\frac{\partial \tilde{\eta }}{\partial {\alpha }_k}\right)=\frac{1}{2}\mathrm{Tr}({\tilde{\mathbf{\Gamma }}}^{-1}{\tilde{\mathbf{\Gamma }}}_k),$$

$$[{\mathbf{I}}_{\mathrm{CES}}^{\mathrm{NC}}{]}_{k,l}=\mathrm{E}\left(\frac{\partial \log p(\mathbf{z};\mathbf{\alpha })}{\partial {\alpha }_k}\frac{\partial \log p(\mathbf{z};\mathbf{\alpha })}{\partial {\alpha }_l}\right)=-\frac{1}{4}\mathrm{Tr}({\tilde{\mathbf{\Gamma }}}^{-1}{\tilde{\mathbf{\Gamma }}}_k)\mathrm{Tr}({\tilde{\mathbf{\Gamma }}}^{-1}{\tilde{\mathbf{\Gamma }}}_l)+\mathrm{E}\left({\varphi }^2(\tilde{\eta })\frac{\partial \tilde{\eta }}{\partial {\alpha }_k}\frac{\partial \tilde{\eta }}{\partial {\alpha }_l}\right).$$

Full text

Detailed proofs of paper [1]

Slepian-Bangs formula and Cramér Rao bound for circular and non-circular complex elliptical symmetric distributions

Habti Abeida and Jean-Pierre Delmas

I Useful relations and lemma

I-A Useful relations

We will make use of the following well known relations which hold for any conformable matrices ${\bf A}$, ${\bf B}$, ${\bf C}$ and ${\bf D}$.

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where ${\bf K}$ is the vec-permutation matrix which transforms ${\rm vec}({\bf C})$ to ${\rm vec}({\bf C}^{T})$ for any square matrix ${\bf C}$,

[TABLE]

where ${\bf A}$, ${\bf C}$ and ${\bf C}^{-1}+{\bf D}{\bf A}^{-1}{\bf B}$ are assumed invertible.

I-B Useful lemma for the proof of Result 2

Lemma 1

Let \widetilde{\bf A}=\left(\begin{array}[]{cc}{\bf A}_{1}&{\bf A}_{2}\\ {\bf A}_{2}^{*}&{\bf A}_{1}^{*}\\ \end{array}\right) and \widetilde{\bf B}=\left(\begin{array}[]{cc}{\bf B}_{1}&{\bf B}_{2}\\ {\bf B}_{2}^{*}&{\bf B}_{1}^{*}\\ \end{array}\right) be two $2M\times 2M$ partitioned matrices with ${\bf A}_{1}$ and ${\bf B}_{1}$ are $M\times M$ Hermitian matrices, ${\bf A}_{2}$ and ${\bf B}_{2}$ are $M\times M$ complex symmetric matrices, and suppose that ${\bf y}\sim\mathbb{C}\mathcal{N}_{M}({\bf 0},{\bf I})$. Then

[TABLE]

where $\widetilde{\bf y}\stackrel{{\scriptstyle{\rm def}}}{{=}}({\bf y}^{T},{\bf y}^{H})^{T}$.

Proof:

We get from (4) then (2)

[TABLE]

where from e.g. [2, Appendix B]

[TABLE]

where {\bf J}^{\prime}\stackrel{{\scriptstyle{\rm def}}}{{=}}\left(\begin{array}[]{cc}{\bf 0}&{\bf I}\\ {\bf I}&{\bf 0}\\ \end{array}\right). Plugging (10) in (9), we get:

[TABLE]

where we have successively

[TABLE]

from (2) and (5),

[TABLE]

from (2), (6) and ${\bf J}^{\prime}\widetilde{\bf A}^{T}{\bf J}^{\prime}=\widetilde{\bf A}$, and

[TABLE]

from (4). Plugging these three expressions in (11), (8) follows.

II Proof of Result 1 and Eq. (5) of [1]

Since a linear transform in $\mathbb{R}^{2M}$ is tantamount to $\mathbb{R}$-linear transform in $\mathbb{C}^{M}$, the definition of GCES given in [3] is equivalent to saying that111Note that if ${\bf\Phi}={\bf 0}$, ${\bf z}$ is C-CES distributed.

[TABLE]

where ${\bf\Psi}$ and ${\bf\Phi}$ are $M\times M$ fixed complex-valued matrices and ${\bf z}_{0}$ is a complex spherical distributed r.v. with stochastic representation ${\bf z}_{0}=_{d}\mathcal{R}{\bf u}$ [4, th. 3]. Since ${\rm E}({\bf u}{\bf u}^{H})=\frac{1}{M}{\bf I}$ and ${\rm E}({\bf u}{\bf u}^{T})={\bf 0}$ [4, lemma 1b], we get if ${\rm E}(\mathcal{R}^{2})<\infty$,

[TABLE]

where $\sigma_{c}$ is defined by ${\rm E}[({\bf z}\!-\!{\boldsymbol{\mu}})({\bf z}\!-\!{\boldsymbol{\mu}})^{H}]=\sigma_{c}{\bf\Sigma}$ and ${\rm E}[({\bf z}\!-\!{\boldsymbol{\mu}})({\bf z}\!-\!{\boldsymbol{\mu}})^{T}]=\sigma_{c}{\bf\Omega}$ whose value is ${\rm E}(\mathcal{R}^{2})/N$ [4, (14)]. Consequently (13) reduces to

[TABLE]

By the one to one change of variable (because ${\bf A}$ is nonsingular): ${\bf\Psi}^{\prime}={\bf A}{\bf\Psi}$ and ${\bf\Phi}^{\prime}={\bf A}{\bf\Phi}$, (14) is equivalent to:

[TABLE]

It is clear that the solution of (15) is not unique, but we can look for solutions in real-valued diagonal form $({\bf\Psi},{\bf\Phi})=({\bf\Delta}_{1},{\bf\Delta}_{2})$ with

[TABLE]

whose solutions are ${\bf\Delta}_{1}=\frac{{\bf\Delta}_{+}+{\bf\Delta}_{-}}{2}$ and ${\bf\Delta}_{2}=\frac{{\bf\Delta}_{+}-{\bf\Delta}_{-}}{2}$ where ${\bf\Delta}_{+}\stackrel{{\scriptstyle{\rm def}}}{{=}}\sqrt{{\bf I}+{\bf\Delta}_{\kappa}}$ and ${\bf\Delta}_{-}\stackrel{{\scriptstyle{\rm def}}}{{=}}\sqrt{{\bf I}-{\bf\Delta}_{\kappa}}$. Consequently

[TABLE]

If ${\rm E}(\mathcal{R}^{2})$ is not finite, the scatter and pseudo-scatter matrices of ${\bf z}$ given by (17) are also ${\bf\Sigma}={\bf A}{\bf A}^{H}$ and ${\bf\Omega}={\bf A}{\bf\Delta}_{\kappa}{\bf A}^{T}$, respectively.

From the eigenvalue decomposition \left(\begin{array}[]{cc}\!{\bf I}\!&\!{\bf\Delta}_{\kappa}\!\\ \!{\bf\Delta}_{\kappa}\!&\!{\bf I}\!\\ \end{array}\right)\!=\!\left[\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}\!{\bf I}\!&\!{\bf I}\!\\ \!{\bf I}\!&\!-{\bf I}\!\\ \end{array}\!\right)\!\right]\!\left(\begin{array}[]{cc}\!{\bf I}+{\bf\Delta}_{\kappa}\!&\!{\bf 0}\!\\ \!{\bf 0}\!&\!{\bf I}+{\bf\Delta}_{\kappa}\!\\ \end{array}\right)\!\left[\!\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}\!{\bf I}\!&\!{\bf I}\!\\ \!{\bf I}\!&\!-{\bf I}\!\\ \end{array}\!\right)\!\right], we deduce from \widetilde{\bf\Gamma}=\left(\begin{array}[]{cc}\!{\bf A}\!&\!{\bf 0}\!\\ \!{\bf 0}\!&\!{\bf A}^{*}\!\\ \end{array}\right)\!\!\left(\begin{array}[]{cc}\!{\bf I}\!&\!{\bf\Delta}_{\kappa}\!\\ \!{\bf\Delta}_{\kappa}\!&\!{\bf I}\!\\ \end{array}\right)\!\!\left(\begin{array}[]{cc}\!{\bf A}^{H}\!&\!{\bf 0}\!\\ \!{\bf 0}\!&\!{\bf A}^{T}\!\\ \end{array}\right) that \widetilde{\bf\Gamma}^{1/2}=\left(\begin{array}[]{cc}\!{\bf A}\!&\!{\bf 0}\!\\ \!{\bf 0}\!&\!{\bf A}^{*}\!\\ \end{array}\right)\!\left(\begin{array}[]{cc}\!\!{\bf\Delta}_{1}\!&\!{\bf\Delta}_{2}\!\!\\ \!\!{\bf\Delta}_{2}\!&\!{\bf\Delta}_{1}\!\!\\ \end{array}\right). Consequently, the stochastic representation ${\bf z}=_{d}{\boldsymbol{\mu}}+\mathcal{R}{\bf A}{\bf v}$ is equivalent to

[TABLE]

with $\widetilde{\bf u}\stackrel{{\scriptstyle{\rm def}}}{{=}}({\bf u}^{T},{\bf u}^{H})^{T}$. It follows directly $\frac{1}{2}(\tilde{\bf z}-\tilde{\bf\boldsymbol{\mu}})^{H}\tilde{\bf\Gamma}^{-1}(\tilde{\bf z}-\tilde{\bf\boldsymbol{\mu}})=_{d}\frac{1}{2}\mathcal{R}^{2}\|\widetilde{\bf u}\|^{2}=\mathcal{Q}$.

III Proof of Result 2

To prove this result, we follows the different steps of [5, sec. 3]. First, we cheek that the p.d.f. $p({\bf z};{\bf\boldsymbol{\alpha}})$ satisfies the ”regularity” condition

[TABLE]

Taking the derivative of the p.d.f. [1, (1)] w.r.t. $\alpha_{k}$, yields

[TABLE]

It follows from the definition of $\tilde{\eta}$ that

[TABLE]

where $\widetilde{\boldsymbol{\mu}}_{k}\stackrel{{\scriptstyle{\rm def}}}{{=}}\frac{\partial\widetilde{\boldsymbol{\mu}}}{\partial\alpha_{k}}$ and $\widetilde{\boldsymbol{\Gamma}}_{k}\stackrel{{\scriptstyle{\rm def}}}{{=}}\frac{\partial\widetilde{\boldsymbol{\Gamma}}}{\partial\alpha_{k}}$. Making use of the extended stochastic representation (18), the second term of (21) is given by

[TABLE]

where $\widetilde{\bf H}_{k}\stackrel{{\scriptstyle{\rm def}}}{{=}}{\widetilde{\bf\Gamma}}^{-1/2}{\widetilde{\bf\Gamma}}_{k}{\widetilde{\bf\Gamma}}^{-1/2}$. Thus using $\tilde{\eta}=_{d}\mathcal{Q}$ [1, (5)], we get:

[TABLE]

Since $\mathcal{Q}$ and ${\bf u}$ are independent, $\mathcal{Q}$ and $\tilde{\bf u}$ are also independent. It follows then from ${\rm E}(\tilde{\bf u})={\bf 0}$, ${\rm E}(\tilde{\bf u}\tilde{\bf u}^{H})=\frac{1}{M}{\bf I}$ and ${\rm E}(\mathcal{Q}\phi(\mathcal{Q}))=-M$ [5, (11)] that

[TABLE]

and

[TABLE]

Thus

[TABLE]

which proves (19).

Now, we evaluate the elements of the FIM. It follows from (20), using (24), that

[TABLE]

It follows from (18) that $\widetilde{\bf\Gamma}^{-1/2}(\widetilde{\bf z}-\widetilde{\boldsymbol{\mu}})=_{d}\sqrt{\mathcal{Q}}\ \widetilde{\bf u}$ and hence from (21) we get

[TABLE]

The first term of (26) can be further simplified as

[TABLE]

and thanks to the independence between $\mathcal{Q}$ and $\tilde{\bf u}$, the expected value of the first term of (26) is given by

[TABLE]

using ${\rm E}(\tilde{\bf u}\tilde{\bf u}^{H})=\frac{1}{M}{\bf I}$ and ${\rm E}(\tilde{\bf u}^{*}\tilde{\bf u}^{H})=\frac{1}{M}{\bf J}$, $\widetilde{\bf\Gamma}^{-*1/2}{\bf J}^{\prime}\widetilde{\bf\Gamma}^{-1/2}={\widetilde{\bf\Gamma}}^{-*}{\bf J}^{\prime}$ and ${\bf J}^{\prime}\tilde{\boldsymbol{\mu}}_{l}=\tilde{\boldsymbol{\mu}}_{l}^{*}$. The expected value of the second and third terms of (26) are zero because the third-order moments of ${\bf u}$ are zero. Because ${\bf y}=_{d}\|{\bf y}\|{\bf u}$, where $\|{\bf y}\|$ and ${\bf u}$ are independent when ${\bf y}\sim\mathbb{C}\mathcal{N}_{M}({\bf 0},{\bf I})$, we get

[TABLE]

Noting that $\widetilde{\bf H}_{k}$ and $\widetilde{\bf H}_{l}$ are structured as $\widetilde{\bf A}$ and $\widetilde{\bf B}$ of the Lemma 1, this lemma applies to the couples $(\widetilde{\bf H}_{k},\widetilde{\bf H}_{l})$ and $({\bf I},{\bf I}$) giving ${\rm E}[(\widetilde{\bf y}^{H}\widetilde{\bf H}_{k}\widetilde{\bf y})(\widetilde{\bf y}^{H}\widetilde{\bf H}_{l}\widetilde{\bf y})]={\rm Tr}(\widetilde{\bf H}_{k}){\rm Tr}(\widetilde{\bf H}_{l})+2{\rm Tr}(\widetilde{\bf H}_{k}\widetilde{\bf H}_{l})$ and ${\rm E}[\|\widetilde{\bf y}\|^{4}]=4M(M+1)$. Consequently the expected value of the last term of (26) is given by

[TABLE]

Gathering (27) (28) in (25) concludes the proof.

IV Proof of Eq. (9) of [1]

Using that [1, (4)] is a p.d.f. with $\int_{0}^{\infty}\delta^{-1}_{M,g}\mathcal{Q}^{M-1}_{t}g(\mathcal{Q}_{t})d\mathcal{Q}_{t}=1$ and that ${\rm E}(\mathcal{Q})={\rm E}(\mathcal{R}^{2})<\infty$, we get

[TABLE]

It follows from Cauchy-Schwarz inequality that

[TABLE]

Next, note that

[TABLE]

Plugging (31) in (30) proves Eq. (9) of [1].

V Proof of Result 4

Because $\xi_{2}=1$ for Gaussian distributions, we get for NC-CES distributions:

[TABLE]

where $({\widetilde{\bf\Gamma}}^{-T}\otimes\widetilde{\bf\Gamma}^{-1})+\frac{1}{2}{\rm vec}(\widetilde{\bf\Gamma}^{-1}){\rm vec}^{H}(\widetilde{\bf\Gamma}^{-1})$ is positive definite. Replacing ${\widetilde{\bf\Gamma}}$ by ${\bf\Gamma}$, the proof is identical for C-CES distributions.

VI Proof of Result 5

We note first that the general expressions of the SCRB proved here is valid for arbitrary parameterization of ${\bf A}_{\theta}$ if the real-valued parameter of interest $\boldsymbol{\theta}\in\mathbb{R}^{L}$ is characterized by the subspace generated by the columns of the full column rank $M\times K$ matrix ${\bf A}_{\theta}$ with $K<M$. It can be applied for example to near or far-field DOA modeling with scalar or vector-sensors for an arbitrary number of parameters per source $s_{t,k}$ (with ${\bf s}_{t}\stackrel{{\scriptstyle{\rm def}}}{{=}}(s_{t,1},..,s_{t,K})^{T}$ and many other modelings as the SIMO and MIMO modelings. Let us start with the circular case for which ${\bf\Omega}={\bf 0}$ and thus $\widetilde{\bf\Gamma}={\rm Diag}({\bf\Sigma},{\bf\Sigma}^{*})$ where ${\bf\Sigma}={\bf A}_{\theta}{\bf R}_{s}{\bf A}^{H}_{\theta}+\sigma^{2}_{n}{\bf I}$. The SCRB form for this case can be then written through the compact expression of the general FIM given in Result 2, using (1) and (2), as follows:

[TABLE]

The SCRB of ${\boldsymbol{\theta}}$ alone can be deduced from (33) as follows:

[TABLE]

with ${\bf G}\stackrel{{\scriptstyle{\rm def}}}{{=}}{\bf T}^{1/2}_{i}({{\bf\Sigma}}^{-T/2}\otimes{\bf\Sigma}^{-1/2})\frac{\partial{\rm vec}({\bf\Sigma})}{\partial{\bf\boldsymbol{\theta}}^{T}}$ and ${\bf\Delta}\stackrel{{\scriptstyle{\rm def}}}{{=}}{\bf T}^{1/2}_{i}({{\bf\Sigma}}^{-T/2}\otimes{\bf\Sigma}^{-1/2})\frac{\partial{\rm vec}({\bf\Sigma})}{\partial{\bf\boldsymbol{\alpha}}^{T}_{n}}$ where

[TABLE]

Let’s further partition the matrix ${\bf\Delta}$ as ${\bf\Delta}={\bf T}^{1/2}_{i}({\bf\Sigma}^{-T/2}\otimes{\bf\Sigma}^{-1/2})\left[\frac{\partial{\rm vec}({\bf\Sigma})}{\partial\boldsymbol{\rho}^{T}}\mid\frac{\partial{\rm vec}({\bf\Sigma})}{\partial\sigma_{n}^{2}}\right]\stackrel{{\scriptstyle{\rm def}}}{{=}}\left[{\bf V}\mid{\bf u}_{n}\right]$. In the sequel, the proofs presented here follow the lines of the proof presented in [6] for circular Gaussian distributed observations. It follows from [6, rel. (14)] that

[TABLE]

Using $\frac{\partial{\rm vec}({\bf\Sigma})}{\partial\sigma_{n}^{2}}={\rm vec}({\bf I})$, we obtain

[TABLE]

Consequently using (34) and (36), if ${\bf g}_{k}$ denotes the kth column of ${\bf G}$, the $(k,l)$ element of ${\rm SCRB}^{-1}_{\rm CES}({\boldsymbol{\alpha}})$ can be written elementwise as

[TABLE]

Let us proceed now to determine the expression of ${\bf g}_{k}$. Letting ${\bf A}^{\prime}_{\theta_{k}}\stackrel{{\scriptstyle{\rm def}}}{{=}}\frac{\partial{\bf A}_{\theta}}{\partial\theta_{k}}$, we get

[TABLE]

Hence, using (1), the kth column of ${\bf G}$ in (38) is given by

[TABLE]

Next, we determine ${\bf V}$ and then ${\bf\Pi}^{\bot}_{{\bf V}}$. Since ${\bf R}_{s}$ is a Hermitian matrix, it can be then factorized as

[TABLE]

where ${\bf J}$ is a $K^{2}\times K^{2}$ constant nonsingular matrix. It follows, using (1), that ${\bf V}$ can be be expressed as

[TABLE]

Note from (38) that the SCRB depends on ${\bf V}$ only via ${\bf\Pi}^{\bot}_{{\bf V}}$, that can be expressed as

[TABLE]

After some algebraic manducation, using (1) and (2), we obtain

[TABLE]

where ${\bf U}\stackrel{{\scriptstyle{\rm def}}}{{=}}{\bf A}^{H}_{\theta}{\bf\Sigma}^{-1}{\bf A}_{\theta}$ is a $K\times K$ Hermitian nonsingular matrix. It follows from matrix inverse lemma (given by (7)), that its inverse can be expressed as

[TABLE]

where $\eta\stackrel{{\scriptstyle{\rm def}}}{{=}}\frac{\xi_{2}-1}{\xi^{2}_{2}(1+\frac{\xi_{2}-1}{\xi_{2}}{\rm vec}^{H}(\tilde{\bf U})(\tilde{\bf U}^{-*}\otimes\tilde{\bf U}^{-1}){\rm vec}(\tilde{\bf U}))}$ can be simplified, using (4), as $\eta\stackrel{{\scriptstyle{\rm def}}}{{=}}\frac{\xi_{2}-1}{\xi^{2}_{2}(1+\frac{\xi_{2}-1}{\xi_{2}}K)}$. Thus, using (1) and (2), we obtain

[TABLE]

where ${\bf H}_{1}\stackrel{{\scriptstyle{\rm def}}}{{=}}{{\bf\Sigma}}^{-1/2}{\bf A}_{\theta}{\bf U}^{-1}{\bf A}^{H}_{\theta}{{\bf\Sigma}}^{-1/2}$. Therefore, (42) becomes

[TABLE]

Now let us show that ${\bf u}^{H}_{n}{\bf\Pi}^{\bot}_{{\bf V}}{\bf g}_{k}=0$. It follows from (37) and (40), using (44), that

[TABLE]

It follows, after some algebraic manipulation, using (1), (3) and (43) that

[TABLE]

using ${\bf H}^{2}_{1}={\bf H}_{1}$ and ${\rm Tr}({\bf H}_{1})=K$. Using the definition (35) for ${\bf T}_{i}$ and (3), the first term of (45) can be expressed as

[TABLE]

using ${\rm Tr}({\bf\Sigma}^{-1}({\bf Z}_{k}+{\bf Z}^{H}_{k}))=2{\rm Re}({\rm Tr}({\bf\Sigma}^{-2}{\bf A}_{\theta}{\bf R}_{s}{\bf A}^{{}^{\prime}H}_{\theta_{k}}))$ and ${\rm Tr}({\bf Z}_{k}+{\bf Z}^{H}_{k})=2{\rm Re}({\rm Tr}({\bf\Sigma}^{-1}{\bf A}_{\theta}{\bf R}_{s}{\bf A}^{{}^{\prime}H}_{\theta_{k}}))$. After simple algebraic manipulations, using (46), (1) and (3), and that ${\rm Tr}({\bf Z}_{k}+{\bf Z}^{H}_{k})={\rm Tr}(({\bf Z}_{k}+{\bf Z}^{H}_{k}){\bf H}_{1})={\rm Tr}({\bf H}_{1}({\bf Z}_{k}+{\bf Z}^{H}_{k}){\bf H}_{1})=2{\rm Re}({\rm Tr}({\bf\Sigma}^{-1}{\bf A}_{\theta}{\bf R}_{s}{\bf A}^{{}^{\prime}H}_{\theta_{k}}))$ and ${\rm Tr}({\bf\Sigma}^{-1}{\bf H}^{2}_{1})={\rm Tr}({\bf\Sigma}^{-1}{\bf H}_{1})$, the second term of (45) can be simplified as

[TABLE]

where the first term in the last line is obtained using ${\bf A}_{\theta}{\bf U}^{-1}{\bf A}^{H}_{\theta}{\bf\Sigma}^{-2}{\bf A}_{\theta}={\bf\Sigma}^{-1}{\bf A}_{\theta}$. It follows, therefore, from (45), (47) and (48) that

[TABLE]

This identity together with (40) and (44) allows us to rewrite the individual elements of (38) as

[TABLE]

After simple algebraic manipulations, using the definition (35) for ${\bf T}_{i}$, (1) and (3), the first term in (49) can be simplified as

[TABLE]

Similarly, after some algebraic manipulations, using (46), (1) and (4), the second term in (49) can be simplified as

[TABLE]

It follows then from (50) and (51) that (49) can be simplified as

[TABLE]

where the second equality is obtained using ${\bf\Sigma}^{-1}-{\bf\Sigma}^{-1}{\bf A}{\bf U}^{-1}{\bf A}^{H}{\bf\Sigma}^{-1}=\frac{1}{\sigma^{2}_{n}}{\bf\Pi}^{\bot}_{{\bf A}}$ thanks to ${\bf A}{\bf U}^{-1}{\bf A}^{H}{\bf\Sigma}^{-1}={\bf A}({\bf A}^{H}{\bf A})^{-1}{\bf A}^{H}$. Using (4), we can write (52) in matrix form as is shown in Result 5.

In the noncircular case, the proof follows the similar above steps by replacing ${\bf T}_{i}$ by $\widetilde{\bf T}_{i}\stackrel{{\scriptstyle{\rm def}}}{{=}}\frac{\xi_{2}}{2}{\bf I}+\frac{\xi_{2}-1}{4}{\rm vec}({\bf I}){\rm vec}^{T}({\bf I})$, and ${\bf\Sigma}$ by $\widetilde{\bf\Gamma}$ where (39) is replaced by $\frac{\partial\widetilde{\bf\Gamma}}{\partial\theta_{k}}=\tilde{\bf A}^{\prime}_{\theta_{k}}{\bf R}_{\tilde{s}}\tilde{\bf A}^{H}_{\theta}+\tilde{\bf A}_{\theta}{\bf R}_{\tilde{s}}\tilde{\bf A}^{{}^{\prime}H}_{\theta_{k}}$ with $\tilde{\bf A}_{\theta}\stackrel{{\scriptstyle{\rm def}}}{{=}}{\rm Diag}({\bf A}_{\theta},{\bf A}^{*}_{\theta})$ and $\tilde{\bf A}^{\prime}_{\theta_{k}}\stackrel{{\scriptstyle{\rm def}}}{{=}}\frac{\partial\tilde{\bf A}_{\theta}}{\partial\theta_{k}}$.

VII Proof of Result 6

The proof of this result follows similar steps as the proof of Result 5 based on [7, th. 1] by replacing ${\bf\Sigma}$ by $\widetilde{\bf\Gamma}=\tilde{\bf A}_{\omega}{\bf R}_{r}\tilde{\bf A}^{H}_{\omega}+\sigma^{2}_{n}{\bf I}$, ${\bf A}_{\theta}$ by \widetilde{\bf A}_{\omega}=\left(\begin{array}[]{cc}{\bf A}_{\theta}{\bf\Delta}_{\phi}\\ {\bf A}^{*}_{\theta}{\bf\Delta}^{*}_{\phi}\\ \end{array}\right) where $\boldsymbol{\omega}\stackrel{{\scriptstyle{\rm def}}}{{=}}(\boldsymbol{\theta}^{T},\boldsymbol{\phi}^{T})^{T}$ with $\boldsymbol{\phi}\stackrel{{\scriptstyle{\rm def}}}{{=}}(\phi_{1},...,\phi_{K})^{T}$, and also by pointing out that ${\bf R}_{r}\in\mathbb{R}^{K\times K}$ is symmetric which lead us to replace ${\bf J}$ in (41) by ${\bf D}_{\rho}$ defined in [7, th. 1] to get ${\rm vec}({\bf R}_{r})={\bf D}_{\rho}\boldsymbol{\rho}$. Thus, ${\bf V}$ becomes ${\bf V}=\widetilde{\bf T}^{1/2}_{i}{\bf W}{\bf D}_{\rho}$ with ${\bf W}=({\widetilde{\bf\Gamma}}^{-T/2}{\widetilde{\bf A}}^{*}_{\omega}\otimes{\widetilde{\bf\Gamma}}^{-1/2}\widetilde{\bf A}_{\omega})$. Hence ${\bf\Pi}^{\bot}_{{\bf V}}$ in [7, th. 1] takes here the following key form expression: ${\bf\Pi}^{\bot}_{{\bf V}}={\bf I}-\widetilde{\bf T}^{1/2}_{i}{\bf\mathcal{B}}\widetilde{\bf T}^{1/2}_{i}$ with ${\bf\mathcal{B}}=\frac{2}{\xi_{2}}{\bf W}({\bf U}^{-1}\otimes{\bf U}^{-1}){\bf N}_{K}{\bf W}^{H}-\tilde{\eta}{\rm vec}({\bf H}_{1}){\rm vec}^{H}({\bf H}_{1})$ where ${\bf U}\stackrel{{\scriptstyle{\rm def}}}{{=}}\widetilde{\bf A}^{H}_{\omega}\widetilde{\bf\Gamma}^{-1}\widetilde{\bf A}_{\omega}$, ${\bf N}_{K}$ is defined in [7, th. 1] and $\tilde{\eta}\stackrel{{\scriptstyle{\rm def}}}{{=}}\frac{\xi_{2}-1}{\xi^{2}_{2}(1+\frac{\xi_{2}-1}{2\xi_{2}}K)}$. The rest of the proof follows the same lines of arguments as that of the proof of Result 5.