Polarimetric detection for vector-sensor processing in quaternion proper Gaussian noises

Abstract

The polarimetric detection problem for 2-D vector sensor is reformulated based on quaternion technique. The quaternion-formed detectors are deduced and analyzed in different quaternion proper Gaussian noises, mainly for Q-proper and C-proper cases. The entropy method is applied to verify the equivalence of quaternion and complex second order statistics (SOS). The similar behaviors of quaternion and complex detectors are thus obtained when SOS is known a priori. Specially, in the case of unknown Q-proper SOS, quaternion and complex detection statistics follow the same F-distribution except for the distinctive second degrees of freedom. This means the improved behavior of quaternion adaptive detector can be achieved. Further, the estimate accuracy of SOS is evaluated fairly by Kullback–Leibler divergence. Consequently, the improved performance derives from the prior structure involved in the quaternion estimate. Numerical simulations also verify the analysis.

Appendices

Appendix 1: The proof of the distribution of QAED detection statistics

In this section, we will give the derivation of the distribution of QAED detection statistics and this derivation is an extension of real multivariate statistics (Anderson 2003). Firstly, we make the transformation

$$\begin{aligned} {\tilde{\mathbf{y}}^H}{{\hat{\tilde{\mathbf{R}}}^{ - 1}}}\tilde{\mathbf{y}} = {\tilde{\mathbf{z}}^H}{\left( \frac{1}{K}\sum _{i = 1}^K {{{\tilde{\mathbf{z}}}_i}\tilde{\mathbf{z}}_i^H} \right) ^{ - 1}}\tilde{\mathbf{z}} \end{aligned}$$

(58)

where $\tilde{\mathbf{z}} = {\tilde{\mathbf{R}}^{ - \frac{1}{2}}}\tilde{\mathbf{y}},{\tilde{\mathbf{z}}_i} = {\tilde{\mathbf{R}}^{ - \frac{1}{2}}}{\tilde{\mathbf{y}}_i},i = 1,2, \ldots K$. Under the ${H_0}$ hypothesis $\tilde{\mathbf{z}},{\tilde{\mathbf{z}}_i}$ follow ${\mathcal {QN}}({\mathbf{0}},{\mathbf{I}_{LN}})$, under the ${H_1}$ hypothesis, $\tilde{\mathbf{z}}$ follows ${\mathcal {QN}}({\tilde{\mathbf{R}}^{ - \frac{1}{2}}}\tilde{\mathbf{x}},{\mathbf{I}_{LN}})$, ${\tilde{\mathbf{z}}_i}$ follows ${\mathcal {QN}}({\mathbf{0}},{\mathbf{I}_{LN}})$. $\tilde{\mathbf{z}}$ and ${\tilde{\mathbf{z}}_i}$ are independent, and ${\tilde{\mathbf{z}}_i}$ is independent to ${\tilde{\mathbf{z}}_j}$ for $i \ne j$. In order to prove (29), the following lemmas and definitions are put forward.

Definition 1

If a quaternion random matrix $\tilde{\mathbf{X}} = {[{\tilde{\mathbf{x}}_1},{\tilde{\mathbf{x}}_2}, \ldots ,{\tilde{\mathbf{x}}_m}]_{p \times m}} \in {{\mathbb {Q}}^{p \times m}}$ and the column vectors are independent mutually, the matrix $\tilde{\mathbf{X}}$ is defined as the sample matrix from quaternion normal distribution ${\mathcal {QN}}({\tilde{\mathbf{u}}_{p \times 1}},{\tilde{{\varvec{\Sigma } }}_{p \times p}})$.

Definition 2

If a quaternion random matrix $\tilde{\mathbf{M}} = \tilde{\mathbf{X}}{\tilde{\mathbf{X}}^H} = \sum _{i = 1}^m {{{\tilde{\mathbf{x}}}_i}} \tilde{\mathbf{x}}_i^H \sim {\mathcal {Q}}{\mathcal {W}}_p(m,{{\varvec{\Sigma } }},{{\Lambda }})$, where $\tilde{\mathbf{X}}$ is the sample matrix from ${\mathcal {QN}}({\tilde{\mathbf{u}}_{p \times 1}},{\tilde{{\varvec{\Sigma }}}_{p \times p}})$, the matrix $\tilde{\mathbf{M}}$ is defined as quaternion Wishart matrix, ${{\tilde{\varvec{\Lambda }} }} = 4m{{\tilde{\mathbf {u}}}}{{{\tilde{\mathbf {u}}}}^H}$ is the non-centrality parameter (${\tilde{\varvec{\Lambda }}}=0$ means $\tilde{\mathbf{M}}$ is central Wishart matrix and it is ignored in the term ${\mathcal {Q}}{\mathcal {W}}_p(m,{{\varvec{\Sigma } }})$ ). Specially, if $p = 1$, $\tilde{\mathbf{M}} = \sum _{i = 1}^m {{{\left| {{{\tilde{x}}_i}} \right| }^2}} \sim \frac{1}{4}{\sigma ^2}\chi _{4m}^2(\lambda )$, where ${\sigma ^2}$ is the sample variance, $\lambda = 4m{\left| u \right| ^2}$ is the non-centrality parameter, and $u$ is the sample mean.

Lemma 1

For a quaternion invertible matrix $\tilde{\mathbf{A}} \in {{{\mathbb {Q}}}^{p \times p}}$ and a sample matrix $\tilde{\mathbf{X}}$ from ${\mathcal {QN}}({\tilde{\mathbf{u}}_{p \times 1}},{\tilde{{\varvec{\Sigma } }}_{p \times p}})$, we have ${\tilde{\mathbf{A}}\tilde{\mathbf{X}}}{({\tilde{\mathbf{A}}{\tilde{\mathbf{X}}}})^H} \sim {\mathcal {Q}}{\mathcal {W}}_p(m,\tilde{\mathbf{A}}\tilde{\varvec{\Sigma }} {\tilde{\mathbf{A}}^H}, 4m{\tilde{\mathbf {A}}\tilde{u}}{\tilde{\mathbf {u}}^{H}} {\tilde{\mathbf {A}}^{H}})$.

Proof

It can be easily deduced by $\tilde{\mathbf{A}}{\tilde{\mathbf{x}}_i} \sim {\mathcal {QN}}({\tilde{\mathbf{A}}\tilde{u}},{\tilde{\mathbf{A}}\tilde{\varvec{\Sigma }}}{\tilde{\mathbf{A}}^{H}})$. $\square$

Lemma 2

For a quaternion matrix $\tilde{\mathbf{U}} \in {{{\mathbb {Q}}}^{m \times a}},{\tilde{\mathbf{U}}^H}\tilde{\mathbf{U}} = {\mathbf{I}_a}$, and a sample matrix $\tilde{\mathbf{X}}\in {{\mathbb {Q}}^{p \times m}}$ from ${\mathcal {QN}}({\mathbf{0}},{\mathbf{I}_p})$ (Note that the following proof require covariance matrix is real), we have ${\tilde{\mathbf{X}}\tilde{U}}{\tilde{\mathbf{U}}^{H}}{\tilde{\mathbf{X}}^{H}} \sim {\mathcal {Q}}{\mathcal {W}}_p(a,{\mathbf{I}_{p}})$.

Proof

It is equivalent to prove ${\tilde{\mathbf{X}}\tilde{U}}$ is sample matrix from $QN({\mathbf{0}},{\mathbf{I}_{p}})$. Define $\tilde{\mathbf{Z}} = {\tilde{\mathbf{X}}^{H}}$, and the entries in $\tilde{\mathbf{Z}}$ are independent because covariance matrix is ${\mathbf{I}_{p}}$. Then we prove the row vectors of $\tilde{\mathbf{Z}}$ are i.i.d random vectors.

Consider $\tilde{\mathbf{Z}}$, we have ${\text {vec}}(\tilde{\mathbf{Z}}) \sim {\mathcal {QN}}({\mathbf{0}},{\mathbf{I}_p} \otimes {\mathbf{I}_m})$, where ${\text {vec}}( \cdot )$ is the vector operator. We have $vec({\tilde{\mathbf{U}}^H}\tilde{\mathbf{Z}}) = ({\mathbf{I}_p} \otimes {\tilde{\mathbf{U}}^H})vec(\mathbf{Z})$, therefore

$$\begin{aligned} {\text {vec}}({\tilde{\mathbf{U}}^H}\tilde{\mathbf{Z}})\sim & {} {\mathcal {QN}}({\mathbf{0}},({\mathbf{I}_p} \otimes {\tilde{\mathbf{U}}^H})({\mathbf{I}_p} \otimes {\mathbf{I}_m}){({\mathbf{I}_p} \otimes {\tilde{\mathbf{U}}^H})^H})\nonumber \\= & {} {\mathcal {QN}}({\mathbf{0}},{\mathbf{I}_p} \otimes {\tilde{\mathbf{U}}^H}\tilde{\mathbf{U}}) \end{aligned}$$

(59)

Note that the kronecker operator for quaternion requires the second term ${\mathbf{I}_p} \otimes {\mathbf{I}_m}$ in (59) is real. Then ${\text {vec}}({\tilde{\mathbf{U}}^H}\tilde{\mathbf{Z}}) \sim QN({\mathbf{0}},{\mathbf{I}_p} \otimes {\mathbf{I}_a})$, so the entries in ${\tilde{\mathbf{U}}^H}\tilde{\mathbf{Z}}$ are uncorrelated, then ${\tilde{\mathbf{X}}\tilde{\mathbf{U}}}$ is sample matrix from ${\mathcal {QN}}({\mathbf{0}},{\mathbf{I}_{p}})$. $\square$

Lemma 3

For random matrix $\tilde{\mathbf{M}} = \sum _{k = 1}^K {{{\tilde{\mathbf{z}}}_k}\tilde{\mathbf{z}}_k^H} = \tilde{\mathbf{Z}}{\tilde{\mathbf{Z}}^H} \sim {\mathcal {Q}}{\mathcal {W}}_p(m,{\mathbf{I}_p})$, we partition the matrices $\tilde{\mathbf{M}}$, ${\mathbf{I}_p}$ and $\tilde{\mathbf{Z}}$ into $\tilde{\mathbf{M}} = \left( {\begin{matrix}{{{\tilde{\mathbf{M}}}_{11}}} &{} {{{\tilde{\mathbf{M}}}_{12}}} \\ {{{\tilde{\mathbf{M}}}_{21}}} &{} {{{\tilde{\mathbf{M}}}_{22}}} \\ \end{matrix} } \right)$, ${\mathbf{I}_p} = \left( {\begin{matrix}{{\mathbf{I}_a}} &{} {\mathbf{0}} \\ {\mathbf{0}} &{} {{\mathbf{I}_b}} \\ \end{matrix} } \right)$, $\tilde{\mathbf{Z}} = \left( {\begin{matrix}{{{\tilde{\mathbf{Z}}}_1}} \\ {{{\tilde{\mathbf{Z}}}_2}} \\ \end{matrix} } \right)$, where ${\tilde{\mathbf{Z}}_1} \in {{\mathbb {Q}}^{a \times m}},{\tilde{\mathbf{Z}}_2} \in {{\mathbb {Q}}^{b \times m}}, a + b = p$, and ${\tilde{\mathbf{M}}_{ij}} = {\tilde{\mathbf{Z}}_i}\tilde{\mathbf{Z}}_j^H$. Define the invertible matrix ${\tilde{\mathbf{M}}^{ - 1}} = \left( {\begin{matrix}{{{\tilde{\mathbf{M}}}^{11}}} &{} {{{\tilde{\mathbf{M}}}^{12}}} \\ {{{\tilde{\mathbf{M}}}^{21}}} &{} {{{\tilde{\mathbf{M}}}^{22}}} \\ \end{matrix}} \right)$,$\mathbf{I}_p^{ - 1} = \left( {\begin{matrix}{{\mathbf{I}_a}} &{} {\mathbf{0}} \\ {\mathbf{0}} &{} {{\mathbf{I}_b}} \\ \end{matrix} } \right)$. Through principle of the inversion of partitioned matrix, we have ${({\tilde{\mathbf{M}}^{22}})^{ - 1}} = {\tilde{\mathbf{M}}_{2.1}} = {\tilde{\mathbf{M}}_{22}} - {\tilde{\mathbf{M}}_{21}}\tilde{\mathbf{M}}_{11}^{ - 1}{\tilde{\mathbf{M}}_{12}}$,

$$\begin{aligned} {\tilde{\mathbf{M}}_{2.1}} \sim {\mathcal {Q}}{\mathcal {W}}_b(m - a,{\mathbf{I}_b}) \end{aligned}$$

(60)

and ${\tilde{\mathbf{M}}_{2.1}}$ is independent to $\tilde{\mathbf{Z}}_1$.

Proof

Define the matrix $\tilde{\mathbf{P}} = (\mathbf{I} - \tilde{\mathbf{Z}}_1^H{({\tilde{\mathbf{Z}}_1}\tilde{\mathbf{Z}}_1^H)^{ - 1}}{\tilde{\mathbf{Z}}_1})$ which is an idempotent matrix, so the characteristic values are $1$ or $0$. The quaternion matrix $\tilde{\mathbf{P}}$ can be decomposed by

$$\begin{aligned} \tilde{\mathbf{P}} = \left( { \begin{matrix}{{{\tilde{\mathbf{U}}}_1}} &{} {{{\tilde{\mathbf{U}}}_2}} \\ \end{matrix} } \right) \left( {\begin{matrix}{{\mathbf{I}_{m - a}}} &{} {\mathbf{0}} \\ {\mathbf{0}} &{} {\mathbf{0}} \\ \end{matrix} } \right) \left( { \begin{matrix} {\tilde{\mathbf{U}}_1^H} \\ {\tilde{\mathbf{U}}_2^H} \\ \end{matrix} } \right) = {\tilde{\mathbf{U}}_1}\tilde{\mathbf{U}}_1^H \end{aligned}$$

(61)

where $\tilde{\mathbf{U}}_1^H{\tilde{\mathbf{U}}_1} = {\mathbf{I}_{m - a}}$.

According to lemma 2, we have

$$\begin{aligned} {\tilde{\mathbf{M}}_{2.1}} \sim {\mathcal {Q}}{\mathcal {W}}_b(m - a,{\mathbf{I}_b}) \end{aligned}$$

(62)

Especially, if $a = p - 1,b = 1$, we have

$$\begin{aligned} {({\tilde{\mathbf{M}}^{22}})^{ - 1}} = {\tilde{\mathbf{M}}_{2.1}} \sim {\mathcal {Q}}{\mathcal {W}}_1(n - p + 1,1) = \frac{1}{4}\chi _{4n - 4p + 4}^2 \end{aligned}$$

(63)

and due to the independence of $\tilde{\mathbf{Z}}_1$ and $\tilde{\mathbf{Z}}_2$, ${\tilde{\mathbf{M}}_{2.1}}$ is independent to $\tilde{\mathbf{Z}}_1$. $\square$

Lemma 4

For a quaternion vector $\tilde{{\alpha }}$, quaternion random matrix $\tilde{\mathbf{M}} = \sum _{k = 1}^K {{{\tilde{\mathbf{z}}}_k}\tilde{\mathbf{z}}_k^H} = \tilde{\mathbf{Z}}{\tilde{\mathbf{Z}}^H} \sim {\mathcal {Q}}{\mathcal {W}}_p(m,{\mathbf{I}_p})$ from ${\mathcal {QN}}({\mathbf{0}},{\mathbf{I}_p})$, we have

$$\begin{aligned} {\left\| {{\tilde{ {\alpha }}}} \right\| ^2}{({\tilde{{\alpha }}^H}{\tilde{\mathbf{M}}^{ - 1}}\tilde{{\alpha }})^{ - 1}} \sim \frac{1}{4}\chi _{4n - 4p + 4}^2 \end{aligned}$$

(64)

where ${\left\| {{{\tilde{\alpha }}}} \right\| ^2}={{\tilde{\alpha }}}^H{{\tilde{\alpha }}}$.

Proof

For a quaternion unitary matrix $\tilde{\mathbf{B}}$ where $\tilde{\mathbf{B}}^H\tilde{\mathbf{B}}=\mathbf{I}_p$, $\frac{{\tilde{\alpha }}}{{\left\| {\tilde{{\alpha }}} \right\| }}$ is the last column of $\tilde{\mathbf{B}}$. Define the matrix $\tilde{\mathbf{N}} = {\tilde{\mathbf{B}}^{ - 1}}\tilde{\mathbf{M}}{\tilde{\mathbf{B}}^{ - H}}$, and ${\tilde{\mathbf{N}}^{ - 1}} = {\tilde{\mathbf{B}}^H}{\tilde{\mathbf{M}}^{ - 1}}\tilde{\mathbf{B}}$, where $\tilde{\mathbf{M}} = \sum _{k = 1}^K {{{\tilde{\mathbf{z}}}_k}\tilde{\mathbf{z}}_k^H} = \tilde{\mathbf{Z}}{\tilde{\mathbf{Z}}^H} \sim {\mathcal {Q}}{\mathcal {W}}_p(m,{\mathbf{I}_p})$.

According to Lemma 1, we have $\tilde{\mathbf{N}} \sim {\mathcal {Q}}{\mathcal {W}}_p(m,{\mathbf{I}_p})$

According to Lemma 3, we have

$$\begin{aligned} {({\tilde{\mathbf{N}}^{22}})^{ - 1}} = {\left\| {{{\tilde{\alpha }}}} \right\| ^2} {({\tilde{{\alpha }}^H}{\tilde{\mathbf{M}}^{ - 1}}\tilde{{\alpha }})^{ - 1}} \sim \frac{1}{4}\chi _{4n - 4p + 4}^2 \end{aligned}$$

(65)

For detection statistic ${\tilde{\mathbf{z}}^H}{(\frac{1}{K}\sum _{i = 1}^K {{{\tilde{\mathbf{z}}}_i}\tilde{\mathbf{z}}_i^H} )^{ - 1}}\tilde{\mathbf{z}}$, we have

$$\begin{aligned} K{\tilde{\mathbf{z}}^H}{\left( \sum _{i = 1}^K {{{\tilde{\mathbf{z}}}_i}} \tilde{\mathbf{z}}_i^H\right) ^{ - 1}}\tilde{\mathbf{z}} = K\left( \frac{{{{\tilde{\mathbf{z}}}^H}{{\left( \sum _{i = 1}^K {{{\tilde{\mathbf{z}}}_i}} \tilde{\mathbf{z}}_i^H\right) ^{ - 1}}}\tilde{\mathbf{z}}}}{{{{\tilde{\mathbf{z}}}^H}\tilde{\mathbf{z}}}}\right) {\tilde{\mathbf{z}}^H}\tilde{\mathbf{z}} \end{aligned}$$

(66)

According to lemma 3 and lemma 4, we have

$$\begin{aligned} \frac{{{{\tilde{\mathbf{z}}}^H}\tilde{\mathbf{z}}}}{{{{\tilde{\mathbf{z}}}^H}{{\left( \sum _{i = 1}^K {{{\tilde{\mathbf{z}}}_i}} \tilde{\mathbf{z}}_i^H\right) ^{ - 1}}}\tilde{\mathbf{z}}}} \sim \frac{1}{4}\chi _{4K - 4LN + 4}^2 \end{aligned}$$

(67)

and

$$\begin{aligned} {\tilde{\mathbf{z}}^H}\tilde{\mathbf{z}} \sim \frac{1}{4}\chi _{4LN}^2(\mu _i) \end{aligned}$$

(68)

where $\mu _0 = 0$ and $\mu _1 = 4{\tilde{\mathbf{x}}^H}{\tilde{\mathbf{R}}^{ - 1}}\tilde{\mathbf{x}}$.

Hence, Eq. (29) is proved. $\square$

Appendix 2: Equivalence of non-centrality parameters

In this subsection, we give the proof that $\mu _1''=\mu _1'$, where $\mu _1'' = 2{\tilde{\mathbf{x}}^{cH}}{\tilde{\mathbf{R}}^{ - c}}{\tilde{\mathbf{x}}^c}$ and $\mu _1' = 2{\mathbf{x}^H}{{\mathbf{R}}^{ - 1}}\mathbf{x}$.

Firstly, according to the matrix theory that ${(\mathbf{A} \otimes {\mathbf{B}})^{ - 1}} = {\mathbf{A}^{ - 1}} \otimes {{\mathbf{B}}^{ - 1}}$, where $\mathbf{A},{\mathbf{B}} \in {{\mathbb {C}}^{n \times n}}$ are invertible matrices (Zhang 2004). we have

$$\begin{aligned} {\tilde{\mathbf{R}}^{-c}} = \frac{1}{{4{\sigma _1}{\sigma _2}}}\left[ {\begin{matrix} {({\sigma _1} + {\sigma _2}){{\mathbf{R}}_{st}^{-1}}} &{} {({\sigma _2} - {\sigma _1}){{\mathbf{R}}_{st}^{-1}}} \\ {({\sigma _2} - {\sigma _1}){\mathbf{R}}_{st}^{-1}} &{} {({\sigma _1} + {\sigma _2})\mathbf{R}_{st}^{-1}} \\ \end{matrix} } \right] . \end{aligned}$$

(69)

Then according to (69), we have

$$\begin{aligned} {\tilde{\mathbf{x}}^{cH}}{\tilde{\mathbf{R}}^{ - c}}{\tilde{\mathbf{x}}^c} = 2a{\tilde{\mathbf{x}}^H}{\mathbf{R}}_{st}^{ - 1}\tilde{\mathbf{x}} + 2b{\text {Re}} \{ {\tilde{\mathbf{x}}^{iH}}{\mathbf{R}}_{st}^{ - 1}\tilde{\mathbf{x}}\} \end{aligned}$$

(70)

where $a = \frac{{{\sigma _1} + {\sigma _2}}}{{4{\sigma _1}{\sigma _2}}}$ and $b = \frac{{{\sigma _2} - {\sigma _1}}}{{4{\sigma _1}{\sigma _2}}}$.

The first term ${\tilde{\mathbf{x}}^H}{\mathbf{R}}_{st}^{ - 1}\mathbf{x}$ in (70) is given by

$$\begin{aligned} {\tilde{\mathbf{x}}^H}{\mathbf{R}}_{st}^{ - 1}\tilde{\mathbf{x}}= & {} (\mathbf{x}_h^H - \mathbf{x}_v^Tj){\mathbf{R}}_{st}^{ - 1}({\mathbf{x}_h} + {\mathbf{x}_v}j)\nonumber \\= & {} \mathbf{x}_h^H{\mathbf{R}}_{st}^{ - 1}{\mathbf{x}_h} + \mathbf{x}_v^H{\mathbf{R}}_{st}^{ - 1}{\mathbf{x}_v} + ( - \mathbf{x}_v^T{\mathbf{R}}_{st}^{ - T}{\mathbf{x}_h^*} + \mathbf{x}_h^H{\mathbf{R}}_{st}^{ - 1}{\mathbf{x}_v})j \nonumber \\= & {} \mathbf{x}_h^H{\mathbf{R}}_{st}^{ - 1}{\mathbf{x}_h} + \mathbf{x}_v^H{\mathbf{R}}_{st}^{ - 1}{\mathbf{x}_v} \end{aligned}$$

(71)

where we employ that $jc=c^*j$ for complex number $c$ whose imaginary operator is $i$.

The second term ${\tilde{\mathbf{x}}^{iH}}\tilde{\mathbf{R}}_{st}^{ - 1}\tilde{\mathbf{x}}$ in (70) is given by

$$\begin{aligned} {\tilde{\mathbf{x}}^{iH}}\tilde{\mathbf{R}}_{st}^{ - 1}\tilde{\mathbf{x}}= & {} (\mathbf{x}_h^H + \mathbf{x}_v^Tj){\mathbf{R}}_{st}^{ - 1}({\mathbf{x}_h} + {\mathbf{x}_v}j)\nonumber \\= & {} \mathbf{x}_h^H{\mathbf{R}}_{st}^{ - 1}{\mathbf{x}_h} - \mathbf{x}_v^H{\mathbf{R}}_{st}^{ - 1}{\mathbf{x}_v} + ( \mathbf{x}_v^T{\mathbf{R}}_{st}^{ - T}{\mathbf{x}_h^* } + \mathbf{x}_h^H{\mathbf{R}}_{st}^{ - 1}{\mathbf{x}_v})j. \end{aligned}$$

(72)

According to (71) and (72), The non-centrality parameter $\mu$ is given by

$$\begin{aligned} u= & {} 2{\tilde{\mathbf{x}}^{cH}}{\tilde{\mathbf{R}}^{ - c}}{\tilde{\mathbf{x}}^c}\nonumber \\= & {} 4a(\mathbf{x}_h^H{\mathbf{R}}_{st}^{ - 1}{\mathbf{x}_h} + \mathbf{x}_v^H{\mathbf{R}}_{st}^{ - 1}{\mathbf{x}_v})+ 4b(\mathbf{x}_h^H{\mathbf{R}}_{st}^{ - 1}{\mathbf{x}_h} - \mathbf{x}_v^H{\mathbf{R}}_{st}^{ - 1}{\mathbf{x}_v})\nonumber \\= & {} 4(a+b)\mathbf{x}_h^H{\mathbf{R}}_{st}^{ - 1}{\mathbf{x}_h}+4(a-b)\mathbf{x}_v^H{\mathbf{R}}_{st}^{ - 1}{\mathbf{x}_v} \nonumber \\= & {} \frac{2}{{{\sigma _1}}}\mathbf{x}_h^H{\mathbf{R}}_{st}^{ - 1}{\mathbf{x}_h}+\frac{2}{{{\sigma _2}}}\mathbf{x}_v^H{\mathbf{R}}_{st}^{ - 1}{\mathbf{x}_v}. \end{aligned}$$

(73)

The non-centrality parameter $u'$ is given by

$$\begin{aligned} u'= & {} 2{\mathbf{x}^H}{\mathbf{R}}^{-1}\mathbf{x}\nonumber \\= & {} 2\left[ { \begin{array}{ll} {\mathbf{x}_h^H} &{}\quad {\mathbf{x}_v^H} \\ \end{array} } \right] {\left[ { \begin{array}{ll} {{\sigma _1}{{\mathbf{R}}_{st}}} &{}\quad {\mathbf{0}} \\ {\mathbf{0}} &{}\quad {{\sigma _2}{{\mathbf{R}}_{st}}} \\ \end{array} } \right] ^{ - 1}}\left[ { \begin{array}{ll} {{\mathbf{x}_h}} \\ {{\mathbf{x}_v}} \\ \end{array} } \right] \nonumber \\= & {} \frac{2}{{{\sigma _1}}}\mathbf{x}_h^H{\mathbf{R}}_{st}^{ - 1}{\mathbf{x}_h}+\frac{2}{{{\sigma _2}}}\mathbf{x}_v^H{\mathbf{R}}_{st}^{ - 1}{\mathbf{x}_v}. \end{aligned}$$

(74)

Comparing the (73) and (74), we have $\mu _1''=\mu _1'$.