#  Dual Power Spectrum Manifold and Toeplitz HPD Manifold: Enhancement and
Analysis for Matrix CFAR Detection

Hao Wu, Yongqiang Cheng, Xixi Chen, Zheng Yang, Xiang Li and Hongqiang Wang
This work was supported by the NSFC of China under grant No. 61871472.
(Corresponding author: Yongqiang Cheng.)Hao Wu, Yongqiang Cheng, Xixi Chen,
Zheng Yang, Xiang Li, Hongqiang Wang are with the College of Electronic
Science and Technology, National University of Defense Technology, Changsha
410000, China (Email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>oliverwhq@tom.com).

###### Abstract

Recently, an innovative matrix CFAR detection scheme based on information
geometry, also referred to as the geometric detector, has been developed
speedily and exhibits distinct advantages in several practical applications.
These advantages benefit from the geometry of the Toeplitz Hermitian positive
definite (HPD) manifold $\mathcal{M}_{\mathcal{T}H_{++}}$, but the
sophisticated geometry also results in some challenges for geometric
detectors, such as the implementation of the enhanced detector to improve the
SCR (signal-to-clutter ratio) and the analysis of the detection performance.
To meet these challenges, this paper develops the dual power spectrum manifold
$\mathcal{M}_{\text{P}}$ as the dual space of
$\mathcal{M}_{\mathcal{T}H_{++}}$. For each affine invariant geometric measure
on $\mathcal{M}_{\mathcal{T}H_{++}}$, we show that there exists an equivalent
function named induced potential function on $\mathcal{M}_{\text{P}}$. By the
induced potential function, the measurements of the dissimilarity between two
matrices can be implemented on $\mathcal{M}_{\text{P}}$, and the geometric
detectors can be reformulated as the form related to the power spectrum. The
induced potential function leads to two contributions: 1) The enhancement of
the geometric detector, which is formulated as an optimization problem
concerning $\mathcal{M}_{\mathcal{T}H_{++}}$, is transformed to an equivalent
and simpler optimization on $\mathcal{M}_{\text{P}}$. In the presented example
of the enhancement, the closed-form solution, instead of the gradient descent
method, is provided through the equivalent optimization. 2) The detection
performance is analyzed based on $\mathcal{M}_{\text{P}}$, and the
advantageous characteristics, which benefit the detection performance, can be
deduced by analyzing the corresponding power spectrum to the maximal point of
the induced potential function.

###### Index Terms:

geometric detector, Toeplitz HPD manifold, dual power spectrum manifold,
affine invariant geometric measure, information geometry.

##  I Introduction

Detection of targets, especially in the clutter background, is a common
challenge in radar, sonar, navigation and communication fields, and thus has
been studied and gained much interest during the last decades[1, 2, 3, 4, 5,
6, 7, 8]. Recently, an innovative matrix CFAR detection scheme based on
information geometry[9] is developed speedily and exhibits distinct advantages
in the low SCR (signal-to-clutter ratio) regime, short pulse sequences,
heterogeneous clutter backgrounds, etc[10, 11, 12, 13]. In this detection
scheme, the detection issues are transformed to the geometric problems, that
the observed data are modeled as covariance matrices and treated as points on
the matrix manifold. Then, the decision is made based on the geometric
measurement of the dissimilarity between the corresponding points of primary
data and secondary data on the matrix manifold. The Riemannian distance
(RD)[14, 15] is the initial geometric measure in the matrix CFAR detection
scheme, and it is applied in target detection with X-band and high frequency
surface wave radars[16, 10], burg estimation of the scatter matrix[17] and the
monitoring of wake vortex turbulences[18, 19]. To improve the detection
performance and circumvent the expensive computation, the matrix CFAR
detection scheme has been extended by replacing the RD with other different
geometric measures, such as Kullback-Leibler divergence (KLD)[20], Jensen-
Shanon divergence (JSD)[21], log-determinant divergence (LDD)[22] and their
modified extension[23, 24, 25], etc[11, 26]. By these pioneered works, the
geometry-based detection method has been widely investigated from both
theoretical and practical perspective, and is developing to a critical and
effective approach in detection issues because of the extraordinary detection
performance and little requirement of the statistical characteristics of
clutter environment[27, 13, 28, 29, 30]. Therefore, the matrix CFAR detection
scheme has a wide application prospect and is deserved to be further studied.

The geometry-based detection methods benefit from the high-dimensional and
non-flat geometry of the matrix manifold, which can more accurately describe
the intrinsic dissimilarity among the matrices. However, the sophisticated
geometry also results in the complex principle and expressions of these
detectors. Moreover, there are twofold challenges in these methods. The first
challenge concerns the low SCR regime and the enhancement of the geometry-
based detection method. Under the low SCR regime, the target echo and the
clutter are hard to discriminative. An effective way is to map the manifold
into a more discriminative space. But, the sophisticated geometry leads to the
difficulty of the determination of the effective enhanced mapping. Another is
about the analysis of the detection performance. The complex structure of
these detectors lead to the absence of the analytic and closed PDF of the test
statistic. Thus, the detection performance of the geometry-based detection
method is hard to deduce. Moreover, because of the unavailable analysis of the
detection performance, the relation between the performance and
characteristics of the target echo cannot be studied, which would impede the
further research about the geometry-based detection methods.

This paper mainly focuses on these two challenges and seeks for some new
theoretical tools to deal with them. In the traditional signal theory, the
duality of two signal spaces often play a pivotal role in the signal
processing issues. For example, by the Fourier transformation, the time domain
and frequency domain can be mutually transformed from each other, and the
analysis and filtering operations of the signal on the time domain can be
easily implemented on the frequency domain. Inspired by the duality between
the time domain and the frequency domain, the dual manifold of the matrix
manifold is desired to be investigated for the new theoretical tools.

This paper is mainly concentrated on the matrix CFAR detection scheme, and the
covariance matrix is considered as a Toeplitz Hermitian positive definite
(HPD) matrix and located on the Toeplitz HPD manifold
$\mathcal{M}_{\mathcal{T}H_{++}}$. As the dual manifold of
$\mathcal{M}_{\mathcal{T}H_{++}}$, the dual power spectrum manifold
$\mathcal{M}_{\text{P}}$ is established by using the Wiener-Khinchin theorem.
By the duality between these two manifolds, every geometric detectors with
affine invariant geometric measures can be equivalently transformed to a
relatively straightforward form with the induced potential function on the
dual power spectrum manifold. Moreover, the duality between these two
manifolds can transform the mentioned twofold challenges to relatively simple
problems on the dual power spectrum manifold. Specifically, the main
contributions of this paper are listed as follows:

  1. 1.

Toeplitz HPD manifold: For the geometry-based detection methods, the
covariance matrix of observed data is shown to be located on a Toeplitz HPD
matrix manifold. On the Toeplitz HPD manifold, the geometric measures are
established to quantify the intrinsic dissimilarity between two Toeplitz HPD
matrix. Moreover, the definition of the affine invariant measure is provided,
and we show that the commonly used RD, KLD, JSD and LDD belong to the affine
invariant measures.

  2. 2.

Dual power spectrum manifold: As the lower-dimensional dual manifold of the
Toeplitz HPD manifold, the dual power spectrum manifold is established. The
relation between the Toeplitz HPD manifold and dual power spectrum manifold is
studied, that the power spectrum of observed data equals to the Fourier
transform of the first row of the corresponding Toeplitz HPD matrix. Moreover,
another significant property between these two manifold is shown, that the
eigenvalues of the Toeplitz HPD matrix asymptotically tend to the
corresponding power spectrum for each observed data.

  3. 3.

Induced potential function: By the duality between the two manifolds, we have
shown: for every affine invariant geometric measure on
$\mathcal{M}_{\mathcal{T}H_{++}}$, there exists an equivalent function on
$\mathcal{M}_{\text{P}}$, which is defined as the induced potential function
for $\mathcal{M}_{\text{P}}$. Therefore, by the induced potential function,
the measurements of the dissimilarity between two matrices can be implemented
on $\mathcal{M}_{\text{P}}$, and the geometric detectors can be reformulated
as the form related to the power spectrum. The form with the induced potential
function is more straightforward than it with the geometric measure, so the
induced potential function is of great potentials to address the challenges
concerning the geometric detector.

  4. 4.

Equivalent enhancement on dual power spectrum manifold: To improve the
detection performance in the low SCR regime, the enhancement of the matrix
CFAR detection is proposed and formulated as an optimization problem with
respect to the Toeplitz HPD matrix. By the duality between the Toeplitz HPD
manifold and the dual power spectrum manifold, we transform this optimization
problem to an equivalent optimization on the dual power spectrum manifold.
This equivalent optimization problem is simpler and easier to solve than the
original optimization because the decision variable has lower dimension and
the constraint of each component is independent. In the presented example, the
closed-form solution of the equivalent optimization is figured out, that it
was used to be solved by the gradient descent method from the perspective of
the original optimization.

  5. 5.

Performance analysis based on dual power spectrum manifold: The analytic
method for detection performance is proposed based on the induced potential
function. Because the geometric detector using the affine invariant geometric
measure can be reformulated as the form based on the induced potential
function, the detection performance is related to the value of the induced
potential function. Therefore, the advantageous characteristics, which benefit
the detection performance, can be deduced by analyzing the corresponding power
spectrum to the maximal point of the induced potential function. As the
examples, the detection performance of RD, KLD and LDD is analyzed by this
method. Moreover, by the induced potential function, the optimal dimension of
the enhanced mapping in the enhanced geometric detector is deduced.

The remainder of this paper is organized as follows. Section II reviews the
covariance matrix model of the observed data for target detection issues, and
the matrix formed binary hypothesis model is introduced. In section III, the
Toeplitz HPD manifold, the dual power spectrum manifold, the affine invariant
measure and the induced potential function are introduced. Section IV reports
on the equivalent enhancement on the dual power spectrum manifold. Section V
presents the analytic method based on the dual power spectrum manifold for
detection performance. Finally, section VI provides the brief conclusions of
the paper.

The following notations are adopted in this paper: the math italic $x$,
lowercase bold italic $\bm{x}$ and uppercase bold $\mathbf{A}$ denote the
scalars, vectors and matrices, respectively. Symbol $(\cdot)^{H}$ indicates
the conjugate transpose operator. $\text{rk}(\mathbf{A})$,
$\text{tr}(\mathbf{A})$ and $|\mathbf{A}|$ mean the rank, trace and
determinant of matrix $\mathbf{A}$, respectively. And, constant matrix
$\mathbf{I}$ indicates the identity matrix.

##  II Toeplitz Covariance Matrix Model for Target Detections

In detection issues, the observed data consist of a series of data which can
be classified into two sorts. The first one is primary data that may contains
the target echo, and the second one is secondary data which provides the
reference of background clutters. By the primary data and the secondary data,
the detection problem can be formulated as following binary hypothesis test
model,

$\begin{split}&\mathcal{H}_{0}:\begin{cases}\bm{x}=\bm{w}\\\
\bm{x}_{k}=\bm{w}_{k}\quad k=1,\dots,K,\\\ \end{cases}\\\
&\mathcal{H}_{1}:\begin{cases}\bm{x}=\bm{s}+\bm{w}\\\
\bm{x}_{k}=\bm{w}_{k}\quad k=1,\dots,K,\\\ \end{cases}\end{split}$ (1)

where $\bm{x}$ is the primary data, $\bm{s}$ is the target echo, $\bm{w}$ is
the noise (clutter), $\bm{x}_{k}$ is the $k$-th secondary data and $K$ is the
number of secondary data. Suppose the noise $\bm{w}$ and
$\bm{w}_{k}\;(k=1,\dots,K)$ are the independently and identically distributed
stochastic variable, thus the power and other statistical characteristic of
$\bm{w}$ can be estimated by $\bm{w}_{k}\;(k=1,\dots,K)$.

To deal with the detection problem, the covariance matrix as the second order
statistics is often employed to represent the statistical characteristics of
observed data, which is shown as the following Toeplitz structure

$\mathbf{C}_{\bm{x}}=\begin{bmatrix}c_{0}&c_{1}&\dots&c_{m-1}\\\
c_{-1}&c_{0}&\dots&c_{m-2}\\\ \vdots&\vdots&\ddots&\vdots\\\
c_{1-m}&c_{2-m}&\dots&c_{0}\\\ \end{bmatrix},$ (2)

where

$c_{k}=c_{-k}^{*}=\sum_{i=k+1}^{m}x_{i}x_{i-k}^{*}\quad(k=0,\dots,m-1).$ (3)

Suppose that the noise $\bm{w}$ and the target echo $\bm{s}$ are independent,
then the correlation coefficient, under hypothesis $\mathcal{H}_{1}$, can be
decomposed as

$c_{k}|_{\mathcal{H}_{1}}=\sum_{i=k+1}^{m}x_{i}x_{i-k}^{*}=\sum_{i=k+1}^{m}s_{i}s_{i-k}^{*}+\sum_{i=k+1}^{m}w_{i}w_{i-k}^{*}.$
(4)

Therefore, the covariance matrix $\mathbf{C}_{\bm{x}}$ under $\mathcal{H}_{1}$
can be decomposed to

$\mathbf{C}_{\bm{x}}|_{\mathcal{H}_{1}}=\mathbf{C}_{\bm{s}}+\mathbf{C}_{\bm{w}}.$
(5)

By the covariance matrix $\mathbf{C}_{\bm{x}}$, the binary hypothesis test
model can be reformulated as

$\begin{split}&\mathcal{H}_{0}:\begin{cases}\mathbf{C}_{\bm{x}}=\mathbf{C}_{\bm{w}}\\\
\mathbf{C}_{\bm{x}_{k}}=\mathbf{C}_{\bm{w}_{k}}\quad k=1,\dots,K,\\\
\end{cases}\\\
&\mathcal{H}_{1}:\begin{cases}\mathbf{C}_{\bm{x}}=\mathbf{C}_{\bm{s}}+\mathbf{C}_{\bm{w}}\\\
\mathbf{C}_{\bm{x}_{k}}=\mathbf{C}_{\bm{w}_{k}}\quad k=1,\dots,K.\\\
\end{cases}\end{split}$ (6)

The covariance matrix is used to represent the statistical characteristic of
the observed data, but the operation performed to the matrix is expensive
because of the high dimension.

##  III Toeplitz HPD Manifold and Dual Power Spectrum Manifold

This section would introduce the Toeplitz HPD manifold and its dual manifold.
Briefly, the main contents of this section are presented as follows. Firstly,
the covariance matrix $\mathbf{C}_{\bm{x}}$ in (2) is shown to be Toeplitz HPD
matrix, and the geometry of the Toeplitz HPD manifold
$\mathcal{M}_{\mathcal{T}H_{++}}$ is studied. Then, the dual manifold of
$\mathcal{M}_{\mathcal{T}H_{++}}$, named dual power spectrum manifold, is
builded, and two ways of connecting these two manifold are deduced. Finally,
we show that there exists an induced potential function on dual power spectrum
manifold for every affine invariant geometric measure, which is equivalent to
the corresponding geometric measure.

###  III-A Toeplitz HPD Manifold

For the covariance matrix $\mathbf{C}_{\bm{x}}$, the following theorem state
that $\mathbf{C}_{\bm{x}}$ is a Toeplitz HPD matrix.

######  Theorem 1.

Given
$\bm{x}=(x_{1},x_{2},\dots,x_{m})\in\mathbb{C}^{m}\backslash\\{\bm{0}\\}$, the
corresponding matrix $\mathbf{C}_{\bm{x}}$ is a Toeplitz Hermitian positive
definite matrix.

###### Proof.

According to the (2) and (3), the matrix $\mathbf{C}_{\bm{x}}$ is obviously
Hermitian and Toeplitz. Then, the positive definiteness is shown below.

For any
$\bm{y}=(y_{1},y_{2},\dots,y_{m})\in\mathbb{C}^{m}\backslash\\{\bm{0}\\}$, the
quadratic form is

$\bm{y}^{H}\mathbf{C}_{\bm{x}}\bm{y}=\sum^{m}_{i=1}\sum^{m}_{j=1}c_{i-j}y_{i}y_{j}^{*}.$
(7)

By (3), it can be expressed as

$\begin{split}\bm{y}^{H}\mathbf{C}_{\bm{x}}\bm{y}=&\sum^{m}_{i=1}\sum^{m}_{j=1}\sum_{1\leq
i+k\leq m\atop 1\leq j+k\leq m}x_{i+k}x_{j+k}^{*}y_{i}y_{j}^{*}\\\
=&\sum^{m-1}_{k=1-m}\sum_{1\leq i+k\leq m\atop 1\leq i\leq m}\sum_{1\leq
j+k\leq m\atop 1\leq j\leq m}x_{i+k}x_{j+k}^{*}y_{i}y_{j}^{*}\\\
=&\sum^{m-1}_{k=1-m}\left(\sum_{1\leq i+k\leq m\atop 1\leq i\leq
m}x_{i+k}y_{i}\right)\left(\sum_{1\leq j+k\leq m\atop 1\leq j\leq
m}x_{j+k}y_{j}\right)^{*}\\\ =&\sum^{m-1}_{k=1-m}\left\Arrowvert\sum_{1\leq
i+k\leq m\atop 1\leq i\leq m}x_{i+k}y_{i}\right\Arrowvert^{2}\geq 0.\\\
\end{split}$ (8)

And, the quadratic form is zero, if and only if

$\sum_{1\leq i+k\leq m\atop 1\leq i\leq m}x_{i+k}y_{i}=0,\quad 1-m\leq k\leq
m-1,$ (9)

i.e.,

$\tilde{\mathbf{X}}\bm{y}=\begin{bmatrix}&&&x_{1}\\\ &&x_{1}&x_{2}\\\
&\iddots&\vdots&\vdots\\\ x_{1}&\dots&x_{m-1}&x_{m}\\\ x_{2}&\dots&x_{m}&\\\
\vdots&\iddots&&\\\ x_{m}&&&\\\ \end{bmatrix}\begin{bmatrix}y_{1}\\\ y_{2}\\\
\vdots\\\ y_{m}\end{bmatrix}=\bm{0}.$ (10)

Actually, as $\bm{y}\neq\bm{0}$, $\tilde{\mathbf{X}}\bm{y}=\bm{0}$ means
$\text{rk}(\tilde{\mathbf{X}})<m$. However, the $m$ columns of
$\tilde{\mathbf{X}}$ are linearly independent as $\bm{x}\neq\bm{0}$, i.e.,
$\text{rk}(\tilde{\mathbf{X}})=m$. Therefore, for any
$\bm{y}\in\mathbb{C}^{m}\backslash\\{\bm{0}\\}$ the quadratic form
$\bm{y}^{H}\mathbf{C}_{\bm{x}}\bm{y}>0$, i.e., the matrix
$\mathbf{C}_{\bm{x}}$ is positive definite.

Totally, $\mathbf{C}_{\bm{x}}$ is a Toeplitz Hermitian positive definite
matrix. ∎

Therefore, the matrices $\mathbf{C}_{\bm{x}}$ and
$\mathbf{C}_{\bm{w}_{k}}\;(k=1,\dots,K)$ are located on the Toeplitz HPD
manifold, which is defined as follows.

######  Definition 1 (Toeplitz HPD manifold).

The Toeplitz HPD manifold is

$\mathcal{M}_{\mathcal{T}H_{++}}=\\{\mathbf{C}|\mathbf{C}\in\mathcal{T}(m,\mathbb{C}),\mathbf{C}^{H}=\mathbf{C},\mathbf{C}\succ\mathbf{0}\\},$
(11)

where $\mathcal{T}(m,\mathbb{C})$ is the set of $m\times m$ dimensional
complex Toeplitz matrices.

On the Toeplitz HPD matrix manifold, the geometric measure, which is used for
quantifying the difference between two matrices
$\mathbf{C}_{1},\mathbf{C}_{2}$, is the function
$\mathcal{D}:\;\mathcal{M}_{\mathcal{T}H_{++}}\times\mathcal{M}_{\mathcal{T}H_{++}}\to\mathbb{R}$
satisfying[9]:

  * •

$\forall\mathbf{C}_{1},\mathbf{C}_{2}\in\mathcal{M}_{\mathcal{T}H_{++}},\;\mathcal{D}(\mathbf{C}_{1},\mathbf{C}_{2})\geq
0$;

  * •

$\mathcal{D}(\mathbf{C}_{1},\mathbf{C}_{2})=0$ if and only if
$\mathbf{C}_{1}=\mathbf{C}_{2}$.

The geometric measure $\mathcal{D}$ can be regarded as the extended distance
which is not limited by the symmetry and triangular inequality.

There are some typical geometric measures on the matrix manifold:

  * •

Riemannian Distance:

$\mathcal{D}_{\text{RD}}(\mathbf{C}_{1},\mathbf{C}_{2})=\left\Arrowvert\log\left(\mathbf{C}_{1}^{-\frac{1}{2}}\mathbf{C}_{2}\mathbf{C}_{1}^{-\frac{1}{2}}\right)\right\Arrowvert_{F}^{2};$
(12)

  * •

Kullback-Leibler Divergence:

$\mathcal{D}_{\text{KL}}(\mathbf{C}_{1},\mathbf{C}_{2})=\text{tr}(\mathbf{C}_{1}\mathbf{C}_{2}^{-1}-\mathbf{I})-\log\left|\mathbf{C}_{1}\mathbf{C}_{2}^{-1}\right|;$
(13)

  * •

Jensen-Shannon Divergence:

$\begin{split}&\mathcal{D}_{\text{JS}}(\mathbf{C}_{1},\mathbf{C}_{2})\\\
=&\frac{1}{2}\left[\mathcal{D}_{\text{KL}}\left(\mathbf{C}_{1},\frac{\mathbf{C}_{1}+\mathbf{C}_{2}}{2}\right)+\mathcal{D}_{\text{KL}}\left(\mathbf{C}_{2},\frac{\mathbf{C}_{1}+\mathbf{C}_{2}}{2}\right)\right];\end{split}$
(14)

  * •

log-determinant Divergence:

$\mathcal{D}_{\text{LD}}(\mathbf{C}_{1},\mathbf{C}_{2})=\log\left|\frac{\mathbf{C}_{1}+\mathbf{C}_{2}}{2}\right|-\log\sqrt{\left|\mathbf{C}_{1}\mathbf{C}_{2}\right|}.$
(15)

In fact, these geometric measures are affine invariant measures[31], which
remain invariant under affine transformations.

######  Definition 2 (Affine Invariant Measure).

The geometric measure $\mathcal{D}(\mathbf{C}_{1},\mathbf{C}_{2})$ is called
affine invariant measure, if and only if
$\forall\mathbf{W}\in\text{GL}(m,\mathbb{C})$ (general linear group) the
following equation is established,

$\mathcal{D}(\mathbf{W}^{H}\mathbf{C}_{1}\mathbf{W},\mathbf{W}^{H}\mathbf{C}_{2}\mathbf{W})=\mathcal{D}(\mathbf{C}_{1},\mathbf{C}_{2}).$
(16)

######  Lemma 1.

The geometric measure
$\mathcal{D}_{\text{RD}},\mathcal{D}_{\text{KL}},\mathcal{D}_{\text{JS}},\mathcal{D}_{\text{LD}}$
are affine invariant measures.

###### Proof.

See Appendix A. ∎

Figure 1: The relation between Toeplitz HPD matrix manifold and dual power
spectrum manifold. There are two ways which can connect the two manifolds: the
Fourier transforms of the elements of $\mathbf{C}_{\bm{x}}$ equals to the
power spectrum of $\bm{x}$, and the eigenvalues of $\mathbf{C}_{\bm{x}}$ tend
to the power spectrum of $\bm{x}$.

###  III-B Dual Power Spectrum Manifold

By the Wiener-Khinchin theorem, the power spectrum is the Fourier transform of
the autocorrelation. Moreover, the manifold composed by the power spectrum is
the dual manifold of $\mathcal{M}_{\mathcal{T}H_{++}}$. Specifically, the dual
power spectrum manifold $\mathcal{M}_{\text{P}}$ is defined as follows.

######  Definition 3 (Dual Power Spectrum Manifold).

The dual power spectrum manifold is

$\begin{split}\mathcal{M}_{\text{P}}=\left\\{P=\left(|y_{0}|^{2},\dots,|y_{m-1}|^{2}\right)\,\Big{|}\;\bm{y}=\mathcal{F}(\bm{x}),\bm{x}\in\mathbb{C}^{m}\backslash\bm{0}\right\\},\end{split}$
(17)

where $\mathcal{F}(\bm{x})=(y_{0},\dots,y_{m-1})$ that

$y_{k}=\sum^{m}_{i=1}x_{i}e^{-j\frac{2\pi ki}{m}}.$ (18)

In fact, the points on Toeplitz HPD manifold and dual power spectrum manifold
are both generated by the received signals $\bm{x}$, and they can be connected
by the following lemma.

######  Lemma 2.

Given $\bm{x}=(x_{1},x_{2},\dots,x_{m})\in\mathbb{C}^{m}\backslash\bm{0}$, the
power spectrum $P=\left(|y_{0}|^{2},|y_{1}|^{2},\dots,|y_{m-1}|^{2}\right)$
satisfies

$|y_{k}|^{2}=\sum^{m-1}_{i=1-m}c_{i}e^{-j\frac{2\pi ki}{m}},$ (19)

where $\bm{c}=(c_{0},c_{1},\dots,c_{m-1})$ is the first row of
$\mathbf{C}_{\bm{x}}$.

###### Proof.

$\begin{split}|y_{k}|^{2}=&\sum^{m}_{i=1}x_{i}e^{-j\frac{2\pi
ki}{m}}\left(\sum^{m}_{l=1}x_{l}e^{-j\frac{2\pi kl}{m}}\right)^{*}\\\
=&\sum^{m}_{i=1}\sum^{m}_{l=1}x_{i}x_{l}^{*}e^{-j\frac{2\pi k(i-l)}{m}}\\\
=&\sum^{m-1}_{i=1-m}\sum_{1\leq t\leq m\atop 1\leq t-i\leq
m}x_{t}x_{t-i}^{*}e^{-j\frac{2\pi ki}{m}}\\\
=&\sum^{m-1}_{i=1-m}c_{i}e^{-j\frac{2\pi ki}{m}}\\\ \end{split}$ (20)

∎

Actually, according to the properties of autocorrelation sequences, the
$c_{k}$ tends to zero, i.e., $\lim_{k\to\infty}c_{k}=0$. Then, the eigenvalue
of $\mathbf{C}_{\bm{x}}$ satisfies the following theorem.

######  Lemma 3.

While $m\to\infty$, if $\sum_{k}c_{k}$ is absolute convergence, the eigenvalue
$\lambda_{k}$ of $\mathbf{C}_{\bm{x}}$ satisfies

$\lambda_{k}=|y_{k}|^{2}=\sum^{m-1}_{i=1-m}c_{i}e^{-j\frac{2\pi
ki}{m}}\quad(k=0,\dots,m-1),$ (21)

i.e.,
$\Lambda=(\lambda_{0},\lambda_{1},\dots,\lambda_{m-1})=P=\left(|y_{0}|^{2},|y_{1}|^{2},\dots,|y_{m-1}|^{2}\right)$.

###### Proof.

Let the circulant matrix

$\mathbf{C}^{*}=\begin{bmatrix}c_{0}&\cdots&c_{l}&c_{1-m+l}&\cdots&c_{-1}\\\
c_{-1}&\cdots&c_{l-1}&c_{l}&\cdots&c_{-2}\\\ \vdots&&\vdots&\vdots&&\vdots\\\
c_{1}&\cdots&c_{1-m+l}&c_{2-m+l}&\cdots&c_{0}\end{bmatrix},$ (22)

where $l=[m/2]$ is the nearest integer of $m/2$.

As $\lim_{k\to\infty}c_{k}=0$ and the properties of banded Toeplitz matrix and
circulant matrix[32], while $m\to\infty$,
$\mathbf{C}_{\bm{x}}\to\mathbf{C}^{*}$ and the eigenvalues
$\lambda_{k}\;(k=0,\dots,m-1)$ of $\mathbf{C}_{\bm{x}}$ satisfies

$\lambda_{k}=\sum^{l}_{i=1-m+l}c_{i}e^{-j\frac{2\pi ki}{m}}.$ (23)

Because $\sum_{k}c_{k}$ is absolute convergence, we can get

$\begin{split}&\lim_{m\to\infty}\left|\sum^{m-1}_{i=l+1}c_{i}e^{-j\frac{2\pi
ki}{m}}\right|\leq\lim_{m\to\infty}\sum^{m-1}_{i=l+1}|c_{i}|=0,\\\
&\lim_{m\to\infty}\left|\sum^{-m+l}_{i=1-m}c_{i}e^{-j\frac{2\pi
ki}{m}}\right|\leq\lim_{m\to\infty}\sum^{-m+l}_{i=1-m}|c_{i}|=0.\end{split}$
(24)

Then, the following equation is established under $m\to\infty$,

$\lambda_{k}=\sum^{l}_{i=1-m+l}c_{i}e^{-j\frac{2\pi
ki}{m}}=\sum^{m-1}_{i=1-m}c_{i}e^{-j\frac{2\pi ki}{m}}.$ (25)

And, as Lemma 19, we can get $\lambda_{k}=|y_{k}|^{2}$ and
$\Lambda=(\lambda_{0},\lambda_{1},\dots,\lambda_{m-1})=P=\left(|y_{0}|^{2},|y_{1}|^{2},\dots,|y_{m-1}|^{2}\right)$.
∎

######  remark 1.

Lemma 3 reveals that the eigenvalues of $\mathbf{C}_{\bm{x}}$,
$\lambda_{0},\dots,\lambda_{m-1}$, asymptotically equal to the power spectrum
of $\bm{x}$. That means the point $\mathbf{C}_{\bm{x}}$ on
$\mathcal{M}_{\mathcal{T}H_{++}}$ can be related to the
$(\lambda_{0},\dots,\lambda_{m-1})$ on $\mathcal{M}_{\text{P}}$. Moreover, as
the properties of banded Toeplitz matrix and circulant matrix, while
$m\to\infty$[32],

$\mathbf{C}_{\bm{x}}\to\mathbf{F}\,\text{diag}(\lambda_{0},\lambda_{1},\dots,\lambda_{m-1})\,\mathbf{F}^{H},$
(26)
$\mathbf{C}_{\bm{x}}\to\mathbf{F}\,\text{diag}\left(|y_{0}|^{2},|y_{1}|^{2},\dots,|y_{m-1}|^{2}\right)\,\mathbf{F}^{H},$
(27)

where $\mathbf{F}$ is a constant matrix that does not depend on
$\mathbf{C}_{\bm{x}}$.

Totally, the duality between $\mathcal{M}_{\mathcal{T}H_{++}}$ and
$\mathcal{M}_{\text{P}}$ are illustrated as Fig.1. There are two crucial ways
which can connect the two manifolds, and the asymptotical relationship
($\bm{\lambda}(\mathbf{C}_{\bm{x}})\to|\mathcal{F}(\bm{x})|^{2}$) will play a
more important role in the next contents.

###  III-C Induced Potential Function on Dual Power Spectrum Manifold

In signal processing theory, the whitening processing is an effective method
to eliminate the statistical characteristics of the noise in the whitened
signal. Consider the matrices $\mathbf{C}_{1},\mathbf{C}_{2}$ are generated by
the signal $\bm{s}+\bm{w}$ and $\bm{w}$, respectively. The result of whitening
processing can be regarded as $\mathbf{C}_{1}\mathbf{C}_{2}^{-1}$. This is
because

$\begin{split}&\mathbf{C}_{1}\mathbf{C}_{2}^{-1}\to\mathbf{F}\,\text{diag}(\lambda_{0}/\lambda^{\prime}_{0},\dots,\lambda_{m-1}/\lambda^{\prime}_{m-1})\,\mathbf{F}^{H}\,(m\to\infty)\\\
&\begin{cases}\lambda_{k}=|y_{k}|^{2},\quad\bm{y}=\mathcal{F}(\bm{s}+\bm{w}),\\\
\lambda^{\prime}_{k}=|y^{\prime}_{k}|^{2},\quad\bm{y}^{\prime}=\mathcal{F}(\bm{w}).\\\
\end{cases}\end{split}$ (28)

We formulate the result of whitening processing to represent the difference
between $\mathbf{C}_{1},\mathbf{C}_{2}$, then the following mapping from
$\mathcal{M}_{\mathcal{T}H_{++}}$ to $\mathcal{M}_{\text{P}}$ can be defined.

######  Definition 4 (Whitening Spectrum Mapping).

Define the whitening spectrum mapping
$\mathcal{P}:\mathcal{M}_{\mathcal{T}H_{++}}\times\mathcal{M}_{\mathcal{T}H_{++}}\to\mathcal{M}_{\text{P}}$,
that

$\mathcal{P}(\mathbf{C}_{1},\mathbf{C}_{2})=\Lambda_{\mathbf{C}_{1}\mathbf{C}_{2}^{-1}}=(\lambda_{0},\lambda_{1},\dots,\lambda_{m-1}),$
(29)

where $\Lambda_{\mathbf{C}}=(\lambda_{0},\lambda_{1},\dots,\lambda_{m-1})$ are
the eigenvalues of $\mathbf{C}$.

######  remark 2.

As a well-known property, the eigenvalues of $\mathbf{A}\mathbf{B}$ equal to
the eigenvalues of $\mathbf{B}\mathbf{A}$ when
$\mathbf{A},\mathbf{B}\in\mathbb{C}^{m\times m}$. Therefore, the eigenvalues
of
$\mathbf{C}_{1}\mathbf{C}_{2}^{-1}=(\mathbf{C}_{1}\mathbf{C}_{2}^{-\frac{1}{2}})\mathbf{C}_{2}^{-\frac{1}{2}}$
equals to that of
$\mathbf{C}_{2}^{-\frac{1}{2}}\mathbf{C}_{1}\mathbf{C}_{2}^{-\frac{1}{2}}$.
That means
$\mathcal{P}(\mathbf{C}_{1},\mathbf{C}_{2})=\Lambda_{\mathbf{C}^{\prime}}$ is
also established, where
$\mathbf{C}^{\prime}=\mathbf{C}_{2}^{-\frac{1}{2}}\mathbf{C}_{1}\mathbf{C}_{2}^{-\frac{1}{2}}$.

Through such mapping, the difference between two matrices can also be
quantified by a function on $\mathcal{M}_{\text{P}}$.

######  Theorem 2.

For each affine invariant measure
$\mathcal{D}(\mathbf{C}_{1},\mathbf{C}_{2})$, there exists a function
$\varphi_{\mathcal{D}}\,:\,\mathcal{M}_{\text{P}}\to\mathbb{R}$ satisfying

$\varphi_{\mathcal{D}}(\mathcal{P}(\mathbf{C}_{1},\mathbf{C}_{2}))=\mathcal{D}(\mathbf{C}_{1},\mathbf{C}_{2}),\quad\forall\mathbf{C}_{1},\mathbf{C}_{2}\in\mathcal{M}_{\mathcal{T}H_{++}}.$
(30)

###### Proof.

As affine invariant property,
$\forall\mathbf{C}_{1},\mathbf{C}_{2}\in\mathcal{M}_{\mathcal{T}H_{++}}$,

$\mathcal{D}(\mathbf{C}_{1},\mathbf{C}_{2})=\mathcal{D}(\mathbf{C}_{2}^{-\frac{1}{2}}\mathbf{C}_{1}\mathbf{C}_{2}^{-\frac{1}{2}},\mathbf{I}).$
(31)

Because
$\mathbf{C}_{2}^{-\frac{1}{2}}\mathbf{C}_{1}\mathbf{C}_{2}^{-\frac{1}{2}}$ is
a symmetric matrix, it can be orthogonally decomposed to

$\mathbf{C}_{2}^{-\frac{1}{2}}\mathbf{C}_{1}\mathbf{C}_{2}^{-\frac{1}{2}}=\mathbf{Q}^{H}\,\text{diag}\,(\lambda_{0},\dots,\lambda_{m-1})\mathbf{Q},$
(32)

where $\mathbf{Q}^{H}\mathbf{Q}=\mathbf{Q}\mathbf{Q}^{H}=\mathbf{I}$ and
$\lambda_{0},\dots,\lambda_{m-1}$ are the eigenvalues of
$\mathbf{C}_{2}^{-\frac{1}{2}}\mathbf{C}_{1}\mathbf{C}_{2}^{-\frac{1}{2}}$.
Therefore,

$\begin{split}\mathcal{D}(\mathbf{C}_{1},\mathbf{C}_{2})=&\mathcal{D}(\mathbf{Q}\mathbf{C}_{2}^{-\frac{1}{2}}\mathbf{C}_{1}\mathbf{C}_{2}^{-\frac{1}{2}}\mathbf{Q}^{H},\mathbf{Q}\mathbf{Q}^{H})\\\
=&\mathcal{D}(\text{diag}\,(\lambda_{0},\dots,\lambda_{m-1}),\mathbf{I}).\end{split}$
(33)

That means $\mathcal{D}(\mathbf{C}_{1},\mathbf{C}_{2})$ only depending on the
eigenvalues of
$\mathbf{C}_{2}^{-\frac{1}{2}}\mathbf{C}_{1}\mathbf{C}_{2}^{-\frac{1}{2}}$,
i.e., $\Lambda_{\mathbf{C}_{1}\mathbf{C}_{2}^{-1}}$.

Then, let
$\varphi_{\mathcal{D}}(\lambda_{0},\dots,\lambda_{m-1})=\mathcal{D}(\text{diag}\,(\lambda_{0},\dots,\lambda_{m-1}),\mathbf{I})$,
we can get
$\varphi_{\mathcal{D}}(\mathcal{P}(\mathbf{C}_{1},\mathbf{C}_{2}))=\mathcal{D}(\mathbf{C}_{1},\mathbf{C}_{2})$.
∎

######  remark 3.

The function $\varphi_{\mathcal{D}}$ can be constructed by
$\varphi_{\mathcal{D}}(\mathcal{P}(\mathbf{C}_{1},\mathbf{C}_{2}))=\mathcal{D}(\text{diag}\,(\lambda_{0},\dots,\lambda_{m-1}),\mathbf{I})\;(\mathcal{P}(\mathbf{C}_{1},\mathbf{C}_{2})=\Lambda_{\mathbf{C}_{1}\mathbf{C}_{2}^{-1}}=(\lambda_{0},\dots,\lambda_{m-1}))$.

Figure 2: The relation between the geometric measure and induced potential
function.

######  Definition 5 (Induced Potential Function).

For the affine invariant measure $\mathcal{D}$, the function
$\varphi_{\mathcal{D}}$ is called the $\mathcal{D}$ induced potential
function.

Totally, the relation between the geometric measure and induced potential
function is shown in Fig.2.

In addition, the induced potential functions of the typical geometric measures
are as follows:

  * •

Riemannian Distance:

$\varphi_{\mathcal{D}_{\text{RD}}}(\lambda_{0},\dots,\lambda_{m-1})=\sum^{m-1}_{k=0}\log^{2}\lambda_{k};$
(34)

  * •

Kullback-Leibler Divergence:

$\varphi_{\mathcal{D}_{\text{KL}}}(\lambda_{0},\dots,\lambda_{m-1})=\sum^{m-1}_{k=0}\lambda_{k}-1-\log\lambda_{k};$
(35)

  * •

Jensen-Shannon Divergence:

$\begin{split}\varphi_{\mathcal{D}_{\text{JS}}}(\lambda_{0},\dots,\lambda_{m-1})=\sum^{m-1}_{k=0}\log\frac{\lambda_{k}+1}{2\sqrt{\lambda_{k}}};\end{split}$
(36)

  * •

log-determinant Divergence:

$\varphi_{\mathcal{D}_{\text{LD}}}(\lambda_{0},\dots,\lambda_{m-1})=\sum^{m-1}_{k=0}\log\frac{\lambda_{k}+1}{2\sqrt{\lambda_{k}}}.$
(37)

The derivation of the induced potential functions is presented in Appendix B.
According to the induced potential function and Theorem 30, JSD equals to LDD
for each pair of
$\mathbf{C}_{1},\mathbf{C}_{2}\in\mathcal{M}_{\mathcal{T}H_{++}}$. Therefore,
only the LDD is considered in the next discussion.

In summary, the induced potential function is equivalent to the corresponding
geometric measure on the Toeplitz HPD manifold. Moreover, compared with the
RD, KLD and LDD, the corresponding induced potential functions have simpler
forms, that means the operations and analysis of the induced potential
functions are easier to implement than geometric measures.

##  IV Enhanced Matrix CFAR Detection Based on Dual Power Spectrum Manifold

This section formulate the enhancement of the matrix CFAR detection as an
optimization problem and then transform it to an equivalent optimization
problem on the dual power spectrum manifold, which has a simpler form and is
easier to solve than the original optimization. Finally, some examples of the
enhancement, which are solved by the equivalent transformation, are provided.

###  IV-A Matrix CFAR Detection

In the geometric detector, the difference between the Toeplitz covariance
matrix of the primary data and the secondary data is quantified by the
selected geometric measure, then the geometric measure is compared with the
threshold to make the decision. Totally, the matrix CFAR detection scheme is
depicted in Fig.3.

Figure 3: The diagram of the matrix CFAR detection scheme, where
$\mathbf{C}_{\bm{x}}$ denotes the covariance matrix of the primary data, and
$\mathbf{C}_{\bm{x}_{k}}$ is the covariance matrix of the secondary data.

The detection is performed by a sliding window, which divides the data into
the primary data and secondary data. The detailed steps are summarized as
follows:

  1. 1.

The Toeplitz covariance matrices are calculated using the primary data and the
secondary data, respectively.

  2. 2.

The mean matrix $\hat{\mathbf{C}}$ of secondary data is estimated using
$\mathbf{C}_{\bm{x}_{1}},\dots,\mathbf{C}_{\bm{x}_{K}}$.

  3. 3.

The selected geometric measure is employed to quantify the difference between
$\mathbf{C}_{\bm{x}}$ and $\hat{\mathbf{C}}$.

  4. 4.

The decision is made by the comparison between the test statistics and the
threshold, i.e.,

$\mathcal{D}(\mathbf{C}_{\bm{x}},\hat{\mathbf{C}})\underset{\mathcal{H}_{0}}{\overset{\mathcal{H}_{1}}{\gtrless}}\eta.$
(38)

where $\eta$ is the threshold under false alarm probability $P_{f}$ and often
obtained by $100P_{f}^{-1}$ independent Monte Carlo runs[11].

TABLE I: Solution or iteration formula of estimate $\hat{\mathbf{C}}$ based on different geometric measures. Geometric measure | Formula
---|---
Riemannian Distance | $\hat{\mathbf{C}}_{t+1}=\hat{\mathbf{C}}_{t}^{\frac{1}{2}}\exp\left(-\epsilon_{t}\sum\limits^{K}_{k=1}\log\left(\hat{\mathbf{C}}_{t}^{-\frac{1}{2}}\mathbf{C}_{\bm{x}_{k}}\hat{\mathbf{C}}_{t}^{-\frac{1}{2}}\right)\right)\hat{\mathbf{C}}_{t}^{\frac{1}{2}}$
KL Divergence | $\hat{\mathbf{C}}=\left(\frac{1}{K}\sum\limits^{K}_{k=1}\mathbf{C}^{-1}_{\bm{x}_{k}}\right)^{-1}$
log-determinant Divergence | $\hat{\mathbf{C}}_{t+1}=\left(\frac{1}{K}\sum\limits^{K}_{k=1}\left(\frac{\mathbf{C}_{\bm{x}_{k}}+\hat{\mathbf{C}}_{t}}{2}\right)^{-1}\right)^{-1}$

The estimate $\hat{\mathbf{C}}$ in step 2 is often deduced from the following
optimization problem

$\hat{\mathbf{C}}=\arg\min_{\mathbf{C}}\sum^{K}_{k=1}\mathcal{D}(\mathbf{C}_{\bm{x}_{k}},\mathbf{C})$
(39)

the solutions or iteration formulas with different geometric measures are
listed in the Table I.

###  IV-B Enhancement of Matrix CFAR Detection

The dimensionality reduction on HPD matrix is an effective way to make the
$\mathcal{H}_{0}$ and $\mathcal{H}_{1}$ data more discriminative by the
enhanced mapping[33], as shown in Fig.4.

Figure 4: Dimensionality reduction on Toeplitz HPD manifold: The data on the
high-dimensional Toeplitz HPD manifold can be mapped to a lower-dimensional
manifold, where the $\mathcal{H}_{0}$ data and $\mathcal{H}_{1}$ data are more
discriminative.

Define the enhanced mapping as

$\Pi_{\mathbf{W}}(\mathbf{C})=\mathbf{W}^{H}\mathbf{C}\mathbf{W},$ (40)

where $\mathbf{W}\in\mathbb{C}^{m\times n}$ is a full column rank matrix,
i.e., $\text{rk}(\mathbf{W})=n$. In this paper, we only consider $n\leq m/2$.
By the enhanced mapping, the modified geometric measure between
$\mathbf{C}_{1}$ and $\mathbf{C}_{2}$ is[34]

$\mathcal{D}_{m}(\Pi_{\mathbf{W}}(\mathbf{C}_{1}),\Pi_{\mathbf{W}}(\mathbf{C}_{2}))=\mathcal{D}_{n}(\mathbf{W}^{H}\mathbf{C}_{1}\mathbf{W},\mathbf{W}^{H}\mathbf{C}_{2}\mathbf{W}),$
(41)

where $\mathcal{D}$ is an affine invariant geometric measure, and
$\mathcal{D}_{m},\mathcal{D}_{n}$ is the implementation of $\mathcal{D}$ in
$m\times m$ and $n\times n$ dimensional matrix, respectively.

Without loss of generality, for any fixed $n\leq m/2$, the crucial point of
the enhanced geometric measure is the full column rank mapping matrix
$\mathbf{W}$, which is determined by the following optimization problem

$\begin{split}\max_{\mathbf{W}}&\;f(\mathcal{D}_{n}(\mathbf{W}^{H}\mathbf{C}_{1}\mathbf{W},\mathbf{W}^{H}\mathbf{C}_{2}\mathbf{W}))\\\
\text{s.t.}&\quad\mathbf{W}\in\mathbb{C}^{m\times
n},\,\text{rk}(\mathbf{W})=n,\end{split}$ (42)

where $f$ is a real function in $\mathbb{R}$. In most cases, $f$ is the
trivial function $f(x)=x$.

Let $\mathbf{C}_{1}=\mathbf{C}_{\bm{x}}$ and
$\mathbf{C}_{2}=\hat{\mathbf{C}}$, the detection decision of the enhanced
detection method is given by

$\mathcal{D}_{n}(\Pi_{\mathbf{W}}(\mathbf{C}_{\bm{x}}),\Pi_{\mathbf{W}}(\hat{\mathbf{C}}))\underset{\mathcal{H}_{0}}{\overset{\mathcal{H}_{1}}{\gtrless}}\eta^{\prime},$
(43)

where $\eta^{\prime}$ is the threshold of the enhanced detector and still
obtained by independent Monte Carlo runs.

###  IV-C Equivalent Enhancement Based on Dual Power Spectrum Manifold

The optimization (42) is an optimization problem on the Grassmann manifold and
often solved by the Riemannian conjugate gradient method[33] which requires
many iterations and results in the expensive computational burden. This part
aims to obtain a simpler solution by equivalently transforming the
optimization (42) onto $\mathcal{M}_{\text{P}}$.

According to Theorem 30, there is a induced potential function
$\varphi_{\mathcal{D}_{m}}$ for the geometric measure $\mathcal{D}_{m}$
satisfying
$\varphi_{\mathcal{D}_{m}}(\mathcal{P}(\mathbf{C}_{1},\mathbf{C}_{2}))=\mathcal{D}_{m}(\mathbf{C}_{1},\mathbf{C}_{2})$.
Moreover, considering the enhanced mapping, the following equation is also
established

$\varphi_{\mathcal{D}_{n}}(\mathcal{P}(\Pi_{\mathbf{W}}(\mathbf{C}_{1}),\Pi_{\mathbf{W}}(\mathbf{C}_{2})))=\mathcal{D}_{n}(\Pi_{\mathbf{W}}(\mathbf{C}_{1}),\Pi_{\mathbf{W}}(\mathbf{C}_{2})).$
(44)

By the mapping $\mathbf{W}$, the power spectrum
$\mathcal{P}(\mathbf{C}_{1},\mathbf{C}_{2})$ is transformed to
$\mathcal{P}(\Pi_{\mathbf{W}}(\mathbf{C}_{1}),\Pi_{\mathbf{W}}(\mathbf{C}_{2}))$.
In oder to transform the optimization (42) onto $\mathcal{M}_{\text{P}}$, the
impact of the mapping $\mathbf{W}$ on $\mathcal{M}_{\text{P}}$ should be
studies firstly. In the next contents, the relationship between
$\mathcal{P}(\mathbf{C}_{1},\mathbf{C}_{2})$ and
$\mathcal{P}(\Pi_{\mathbf{W}}(\mathbf{C}_{1}),\Pi_{\mathbf{W}}(\mathbf{C}_{2}))$
would be discussed.

For the matrices
$\mathbf{C}_{1},\mathbf{C}_{2}\in\mathcal{M}_{\mathcal{T}H_{++}}$, let
$\mathbf{C}^{\prime}_{1}=\mathbf{C}_{2}^{-\frac{1}{2}}\mathbf{C}_{1}\mathbf{C}_{2}^{-\frac{1}{2}}$,
$\mathbf{C}^{\prime}_{2}=\mathbf{I}$,
$\mathbf{W}^{\prime}=\mathbf{C}_{2}^{\frac{1}{2}}\mathbf{W}$ and suppose the
QR decomposition of $\mathbf{W}^{\prime}$ is
$\mathbf{W}^{\prime}=\mathbf{Q}\mathbf{R}\;(\mathbf{Q}^{H}\mathbf{Q}=\mathbf{I},\,\mathbf{R}\in
GL(m,\mathbb{C}))$, the following equation is established,

$\begin{split}&\mathcal{D}_{n}(\mathbf{W}^{H}\mathbf{C}_{1}\mathbf{W},\mathbf{W}^{H}\mathbf{C}_{2}\mathbf{W})\\\
=&\mathcal{D}_{n}(\mathbf{W}^{\prime
H}\mathbf{C}_{2}^{-\frac{1}{2}}\mathbf{C}_{1}\mathbf{C}_{2}^{-\frac{1}{2}}\mathbf{W}^{\prime},\mathbf{W}^{\prime
H}\mathbf{W}^{\prime})\\\
=&\mathcal{D}_{n}(\mathbf{R}^{H}\mathbf{Q}^{H}\mathbf{C}_{2}^{-\frac{1}{2}}\mathbf{C}_{1}\mathbf{C}_{2}^{-\frac{1}{2}}\mathbf{Q}\mathbf{R},\mathbf{R}^{H}\mathbf{R})\\\
\overset{(a)}{=}&\mathcal{D}_{n}(\mathbf{Q}^{H}\mathbf{C}_{2}^{-\frac{1}{2}}\mathbf{C}_{1}\mathbf{C}_{2}^{-\frac{1}{2}}\mathbf{Q},\mathbf{I}),\end{split}$
(45)

where $(a)$ is established as the affine invariant property of
$\mathcal{D}_{n}$. Overall, equation (45) means that finding the enhanced
mapping $\mathbf{W}$ is equivalent to figuring out the orthogonal mapping
$\mathbf{Q}$, and the relationship between them is shown in Fig.5.

Figure 5: The relationship between the enhanced mapping $\mathbf{W}$ and the
orthogonal mapping $\mathbf{Q}$.

Combining the (44) and (45), the power spectrum
$\mathcal{P}(\Pi_{\mathbf{W}}(\mathbf{C}_{1}),\Pi_{\mathbf{W}}(\mathbf{C}_{2}))$
can be formulated as

$\mathcal{P}(\Pi_{\mathbf{W}}(\mathbf{C}_{1}),\Pi_{\mathbf{W}}(\mathbf{C}_{2}))=\mathcal{P}(\mathbf{Q}^{H}\mathbf{C}^{\prime}\mathbf{Q},\mathbf{I})=\Lambda_{\mathbf{Q}^{H}\mathbf{C}^{\prime}\mathbf{Q}},$
(46)

where
$\mathbf{C}^{\prime}=\mathbf{C}_{2}^{-\frac{1}{2}}\mathbf{C}_{1}\mathbf{C}_{2}^{-\frac{1}{2}}$.

Therefore, by the mapping $\mathbf{W}$, the power spectrum is changed from the
eigenvalues of $\mathbf{C}^{\prime}$ to that of
$\mathbf{Q}^{H}\mathbf{C}^{\prime}\mathbf{Q}$. In order to investigate the
more detailed impact of the mapping $\mathbf{W}$, the varying rule of the
eigenvalues is discussed in the following contents. Firstly, the Cauchy
interlace theorem is introduced.

######  Theorem 3 (Cauchy Interlace Theorem[35]).

Let $\mathbf{A}$ be a Hermitian matrix of order $m$, and let $\mathbf{B}$ be a
principle submatrix of $\mathbf{A}$ of order $m-1$. If
$\lambda_{m-1}\leq\lambda_{m-2}\leq\cdots\leq\lambda_{0}$ lists the
eigenvalues of $\mathbf{A}$ and $\mu_{m-2}\leq\mu_{m-3}\leq\cdots\leq\mu_{0}$
the eigenvalues of $\mathbf{B}$, then
$\lambda_{m-1}\leq\mu_{m-2}\leq\lambda_{m-2}\leq\cdots\leq\lambda_{1}\leq\mu_{0}\leq\lambda_{0}$.

######  Corollary 1.

When the order of the principle submatrix $\mathbf{B}$ is $n$, then
$\lambda_{i+m-n}\leq\mu_{i}\leq\lambda_{i}\,(0\leq i\leq n-1)$.

###### Proof.

By repeating the Cauchy interlace theorem $m-n$ times, this corollary can be
derived. ∎

By the Cauchy interlace theorem and its corollary, the following lemma is
established.

######  Lemma 4.

Given matrix $\mathbf{Q}\in\mathbb{C}^{m\times n}$ satisfying
$\mathbf{Q}^{H}\mathbf{Q}=\mathbf{I}$ and suppose the matrix
$\mathbf{C}\in\mathcal{M}_{\mathcal{T}H_{++}}$, if the eigenvalues of
$\mathbf{C}$ are
$\lambda_{0},\dots,\lambda_{m-1}\,(\lambda_{m-1}\leq\cdots\leq\lambda_{0})$,
then the eigenvalues of $\mathbf{Q}^{H}\mathbf{C}\mathbf{Q}$,
$\mu_{0},\dots,\mu_{n-1}\,(\mu_{n-1}\leq\cdots\leq\mu_{0})$, satisfy
$\lambda_{i+m-n}\leq\mu_{i}\leq\lambda_{i}\,(0\leq i\leq n-1)$.

###### Proof.

Let $\widetilde{\mathbf{Q}}$ with order $m$ is the orthonormal expansion of
$\mathbf{Q}$, i.e.,
$\widetilde{\mathbf{Q}}^{H}\widetilde{\mathbf{Q}}=\widetilde{\mathbf{Q}}\widetilde{\mathbf{Q}}^{H}=\mathbf{I}$
and

$\widetilde{\mathbf{Q}}=\begin{bmatrix}\mathbf{Q}&\mathbf{Q}^{\prime}\end{bmatrix}.$
(47)

Because the eigenvalues of
$\widetilde{\mathbf{Q}}^{H}\mathbf{C}\widetilde{\mathbf{Q}}$ equal to the
eigenvalues of
$\mathbf{C}\widetilde{\mathbf{Q}}\widetilde{\mathbf{Q}}^{H}=\mathbf{C}$, the
eigenvalues of $\widetilde{\mathbf{Q}}^{H}\mathbf{C}\widetilde{\mathbf{Q}}$
are $\lambda_{0},\dots,\lambda_{m-1}$.

And,

$\widetilde{\mathbf{Q}}^{H}\mathbf{C}\widetilde{\mathbf{Q}}=\begin{bmatrix}\mathbf{Q}^{H}\\\
\mathbf{Q}^{\prime
H}\end{bmatrix}\mathbf{C}\begin{bmatrix}\mathbf{Q}&\mathbf{Q}^{\prime}\end{bmatrix}=\begin{bmatrix}\mathbf{Q}^{H}\mathbf{C}\mathbf{Q}&\mathbf{Q}^{H}\mathbf{C}\mathbf{Q}^{\prime}\\\
\mathbf{Q}^{\prime H}\mathbf{C}\mathbf{Q}&\mathbf{Q}^{\prime
H}\mathbf{C}\mathbf{Q}^{\prime}\\\ \end{bmatrix},$ (48)

thus the matrix $\mathbf{Q}^{H}\mathbf{C}\mathbf{Q}$ is the principle
submatrix of $\widetilde{\mathbf{Q}}^{H}\mathbf{C}\widetilde{\mathbf{Q}}$.
According to the Cauchy interlace theorem, eigenvalue $\mu_{i}\,(0\leq i\leq
n-1)$ satisfies $\lambda_{i+m-n}\leq\mu_{i}\leq\lambda_{i}\,(0\leq i\leq
n-1)$. ∎

Lemma 4 reveals that the eigenvalues of all the matrices shaped like
$\mathbf{Q}^{H}\mathbf{C}\mathbf{Q}$ should satisfies the inequalities
$\lambda_{i+m-n}\leq\mu_{i}\leq\lambda_{i}\,(0\leq i\leq n-1)$. The remaining
question is whether there exists an orthogonal mapping $\mathbf{Q}$ for each
eigenvalues $\mu_{1},\mu_{2},\dots,\mu_{n}$ satisfying inequalities
$\lambda_{i+m-n}\leq\mu_{i}\leq\lambda_{i}\,(0\leq i\leq n-1)$, which makes
the eigenvalues of $\mathbf{Q}^{H}\mathbf{C}\mathbf{Q}$ actually equal to
$\mu_{1},\mu_{2},\dots,\mu_{n}$. The following lemma answers this problem.

######  Lemma 5.

Given the matrix $\mathbf{C}\in\mathcal{M}_{\mathcal{T}H_{++}}$ and suppose
its eigenvalues are
$\lambda_{0},\dots,\lambda_{m-1}\,(\lambda_{m-1}\leq\cdots\leq\lambda_{0})$.
Then, for each $\mu_{i}\,(0\leq i\leq n-1)$ satisfying
$\lambda_{i+m-n}\leq\mu_{i}\leq\lambda_{i}\,(0\leq i\leq n-1)$, there exists a
matrix $\mathbf{Q}\in\mathbb{C}^{m\times n}$ satisfying
$\mathbf{Q}^{H}\mathbf{Q}=\mathbf{I}$, which makes the eigenvalues of
$\mathbf{Q}^{H}\mathbf{C}\mathbf{Q}$ are $\mu_{0},\dots,\mu_{n-1}$.

###### Proof.

Suppose the eigenvalue decomposition of matrix $\mathbf{C}$ is

$\mathbf{C}=\mathbf{V}^{H}\text{diag}\,(\lambda_{0},\dots,\lambda_{m-1})\mathbf{V},$
(49)

where $\mathbf{V}^{H}=[\bm{v}_{0}\;\bm{v}_{1}\;\cdots\;\bm{v}_{m-1}]$ is an
orthogonal matrix.

For each $\mu_{i}\,(0\leq i\leq n-1)$ satisfying
$\lambda_{i+m-n}\leq\mu_{i}\leq\lambda_{i}\,(0\leq i\leq n-1)$, let

$t_{i}=\frac{\mu_{i}-\lambda_{i+m-n}}{\lambda_{i}-\lambda_{i+m-n}},$ (50)

then $0\leq t_{i}\leq 1$ and
$\mu_{i}=t_{i}\lambda_{i}+(1-t_{i})\lambda_{i+m-n}$.

Let

$\mathbf{Q}^{H}=\begin{bmatrix}\sqrt{t_{0}}\bm{v}^{H}_{0}+\sqrt{1-t_{0}}\bm{v}^{H}_{m-n}\\\
\sqrt{t_{1}}\bm{v}^{H}_{1}+\sqrt{1-t_{1}}\bm{v}^{H}_{1+m-n}\\\ \vdots\\\
\sqrt{t_{n-1}}\bm{v}^{H}_{n-1}+\sqrt{1-t_{n-1}}\bm{v}^{H}_{m-1}\end{bmatrix},$
(51)

then we can get $\mathbf{Q}^{H}\mathbf{Q}=\mathbf{I}$ because

$\begin{split}&(\sqrt{t_{i}}\bm{v}^{H}_{i}+\sqrt{1-t_{i}}\bm{v}^{H}_{i+m-n})(\sqrt{t_{j}}\bm{v}_{j}+\sqrt{1-t_{j}}\bm{v}_{j+m-n})\\\
=&\sqrt{t_{i}t_{j}}\bm{v}^{H}_{i}\bm{v}_{j}+\sqrt{t_{i}(1-t_{j})}\bm{v}^{H}_{i}\bm{v}_{j+m-n}\\\
&+\sqrt{(1-t_{i})t_{j}}\bm{v}^{H}_{i+m-n}\bm{v}_{j}+\sqrt{(1-t_{i})(1-t_{j})}\bm{v}^{H}_{i+m-n}\bm{v}_{j+m-n}\\\
\overset{\text{(a)}}{=}&\sqrt{t_{i}t_{j}}\bm{v}^{H}_{i}\bm{v}_{j}+\sqrt{(1-t_{i})(1-t_{j})}\bm{v}^{H}_{i+m-n}\bm{v}_{j+m-n}=\begin{cases}0&i\neq
j\\\ 1&i=j\end{cases}\end{split}$ (52)

where $(a)$ holds as $i\neq j+m-n$, $j\neq i+m-n$ ($n\leq m/2$, so $0\leq
i,j\leq n-1<m-n\leq i+m-n,j+m-n\leq m-1$).

And, we can get

$\mathbf{V}\mathbf{Q}=\begin{bmatrix}\sqrt{t_{0}}&&&\\\ &\sqrt{t_{1}}&&\\\
&&\ddots&\\\ &&&\sqrt{t_{n-1}}\\\ &&&\\\ &&&\\\ \sqrt{1-t_{0}}&&&\\\
&\sqrt{1-t_{1}}&&\\\ &&\ddots&\\\ &&&\sqrt{1-t_{n-1}}\\\ \end{bmatrix}$ (53)

and

$\begin{split}\mathbf{Q}^{H}\mathbf{C}\mathbf{Q}=&\mathbf{Q}^{H}\mathbf{V}^{H}\text{diag}\,(\lambda_{0},\dots,\lambda_{m-1})\mathbf{V}\mathbf{Q}\\\
=&\text{diag}\,(\mu_{0},\dots,\mu_{n-1}).\end{split}$ (54)

That means the eigenvalues of $\mathbf{Q}^{H}\mathbf{C}\mathbf{Q}$ are
$\mu_{0},\dots,\mu_{n-1}$. ∎

######  remark 4.

For each $\mu_{i}\,(0\leq i\leq n-1)$ satisfying
$\lambda_{i+m-n}\leq\mu_{i}\leq\lambda_{i}\,(0\leq i\leq n-1)$, the above
proof provides a construction approach for the orthogonal matrix $\mathbf{Q}$.

Therefore, according to Theorem 30, Lemma 5 and above discussion, the
optimization problem (42) can be transferred onto the dual power spectrum
manifold as shown in the following theorem.

######  Theorem 4.

Given two optimization problems

$\begin{split}\max_{\mathbf{W}}\;&f(\mathcal{D}_{n}(\mathbf{W}^{H}\mathbf{C}_{1}\mathbf{W},\mathbf{W}^{H}\mathbf{C}_{2}\mathbf{W}))\\\
\text{s.t.}&\quad\mathbf{W}\in\mathbb{C}^{m\times
n},\,\text{rk}(\mathbf{W})=n,\end{split}$ (55)

and

$\begin{split}&\max_{P=\\{\mu_{0},\dots,\mu_{n-1}\\}}f(\varphi_{\mathcal{D}_{n}}(P))\\\
s.t.\quad&\lambda_{i+m-n}\leq\mu_{i}\leq\lambda_{i}\quad(0\leq i\leq n-1),\\\
\end{split}$ (56)

where $\lambda_{m-1}\leq\cdots\leq\lambda_{0}$ and
$\mathcal{P}(\mathbf{C}_{1},\mathbf{C}_{2})=(\lambda_{0},\dots,\lambda_{m-1})$.
Suppose the solution of the optimization (56) is
$P^{*}=(\mu^{*}_{0},\dots,\mu^{*}_{n-1})$, then there exists a matrix
$\mathbf{W}_{*}\in\mathbb{C}^{m\times n}$ satisfying
$\mathcal{P}(\mathbf{W}_{*}^{H}\mathbf{C}_{1}\mathbf{W}_{*},\mathbf{W}_{*}^{H}\mathbf{C}_{2}\mathbf{W}_{*})=(\mu^{*}_{0},\dots,\mu^{*}_{n-1})$
and $\text{rk}(\mathbf{W}_{*})=n$. The matrix $\mathbf{W}_{*}$ is the solution
of the optimization (55).

###### Proof.

Suppose the solution of the optimization (56) is
$P^{*}=(\mu^{*}_{0},\dots,\mu^{*}_{n-1})$ and the solution of the optimization
(55) is $\mathbf{W}^{\prime}$.

According to Lemma 5, there exists a matrix $\mathbf{Q}_{*}$ satisfying
$\mathbf{Q}_{*}^{H}\mathbf{Q}_{*}=\mathbf{I}$ and that the eigenvalues of
$\mathbf{Q}_{*}^{H}\mathbf{C}_{2}^{-\frac{1}{2}}\mathbf{C}_{1}\mathbf{C}_{2}^{-\frac{1}{2}}\mathbf{Q}_{*}$
are $\mu_{0},\dots,\mu_{n-1}$. Let
$\mathbf{W}_{*}=\mathbf{C}_{2}^{-\frac{1}{2}}\mathbf{Q}_{*}$, then we can get
$\mathcal{D}_{n}(\mathbf{W}_{*}^{H}\mathbf{C}_{1}\mathbf{W}_{*},\mathbf{W}_{*}^{H}\mathbf{C}_{2}\mathbf{W}_{*})=\mathcal{D}_{n}(\mathbf{Q}_{*}^{H}\mathbf{C}_{2}^{-\frac{1}{2}}\mathbf{C}_{1}\mathbf{C}_{2}^{-\frac{1}{2}}\mathbf{Q}_{*},\mathbf{I})$.
As $\text{rk}(\mathbf{W}_{*})=n$, we can get
$f(\mathcal{D}_{n}(\mathbf{W}^{\prime
H}\mathbf{C}_{1}\mathbf{W}^{\prime},\mathbf{W}^{\prime
H}\mathbf{C}_{2}\mathbf{W}^{\prime}))\geq
f(\mathcal{D}_{n}(\mathbf{W}_{*}^{H}\mathbf{C}_{1}\mathbf{W}_{*},\mathbf{W}_{*}^{H}\mathbf{C}_{2}\mathbf{W}_{*}))=\mathcal{D}_{n}(\mathbf{Q}_{*}^{H}\mathbf{C}_{2}^{-\frac{1}{2}}\mathbf{C}_{1}\mathbf{C}_{2}^{-\frac{1}{2}}\mathbf{Q}_{*},\mathbf{I})=f(\varphi_{\mathcal{D}_{n}}(P^{*}))$.

Besides, according to (45), there exists a matrix $\mathbf{Q}$ satisfying
$\mathbf{Q}^{H}\mathbf{Q}=\mathbf{I}$ and $\mathcal{D}_{n}(\mathbf{W}^{\prime
H}\mathbf{C}_{1}\mathbf{W}^{\prime},\mathbf{W}^{\prime
H}\mathbf{C}_{2}\mathbf{W}^{\prime})=\mathcal{D}_{n}(\mathbf{Q}^{H}\mathbf{C}_{2}^{-\frac{1}{2}}\mathbf{C}_{1}\mathbf{C}_{2}^{-\frac{1}{2}}\mathbf{Q},\mathbf{I})$.
As Lemma 4, the eigenvalues $\mu^{\prime}_{0},\dots,\mu^{\prime}_{n-1}$ of
$\mathbf{Q}^{H}\mathbf{C}_{2}^{-\frac{1}{2}}\mathbf{C}_{1}\mathbf{C}_{2}^{-\frac{1}{2}}\mathbf{Q}$
should satisfy the inequality
$\lambda_{i+m-n}\leq\mu^{\prime}_{i}\leq\lambda_{i}\;(0\leq i\leq n-1)$.
Therefore, we can get $f(\mathcal{D}_{n}(\mathbf{W}^{\prime
H}\mathbf{C}_{1}\mathbf{W}^{\prime},\mathbf{W}^{\prime
H}\mathbf{C}_{2}\mathbf{W}^{\prime}))=f(\varphi_{\mathcal{D}_{n}}(\mu^{\prime}_{1},\dots,\mu^{\prime}_{n}))\leq
f(\varphi_{\mathcal{D}_{n}}(P^{*}))$.

Totally, we can get $f(\mathcal{D}_{n}(\mathbf{W}^{\prime
H}\mathbf{C}_{1}\mathbf{W}^{\prime},\mathbf{W}^{\prime
H}\mathbf{C}_{2}\mathbf{W}^{\prime}))=f(\mathcal{D}_{n}(\mathbf{W}_{*}^{H}\mathbf{C}_{1}\mathbf{W}_{*},\mathbf{W}_{*}^{H}\mathbf{C}_{2}\mathbf{W}_{*}))=f(\varphi_{\mathcal{D}_{n}}(P^{*}))$,
i.e., $\mathbf{W}_{*}$ is also the solution of optimization (55).

∎

Suppose the solution of (56) is $(\mu^{*}_{0},\dots,\mu^{*}_{n-1})$, then the
solution $\mathbf{W}_{*}$ of (55) can be obtained by the optimal solution
$(\mu^{*}_{0},\dots,\mu^{*}_{n-1})$ and matrices
$\mathbf{C}_{1},\mathbf{C}_{2}$, which is shown in algorithm 1.

Algorithm 1 Enhanced mapping calculation by $\mu^{*}_{0},\dots,\mu^{*}_{n-1}$

0: The optimal power spectrum $P=(\mu^{*}_{0},\dots,\mu^{*}_{n-1})$.The
matrices $\mathbf{C}_{1},\mathbf{C}_{2}$.

0: The enhanced mapping $\mathbf{W}_{*}$.

1: Calculating
$\mathbf{C}=\mathbf{C}_{2}^{-\frac{1}{2}}\mathbf{C}_{1}\mathbf{C}_{2}^{-\frac{1}{2}}$.

2: Performing the eigenvalue decomposition of $\mathbf{C}$ that
$\mathbf{C}=\mathbf{V}^{H}\text{diag}\,(\lambda_{0},\dots,\lambda_{m-1})\mathbf{V},\;\mathbf{V}^{H}=[\bm{v}_{0}\;\bm{v}_{1}\;\cdots\;\bm{v}_{m-1}]$.

3: Calculating $t_{0},\dots,t_{n-1}$ with $\lambda_{0},\dots,\lambda_{m-1}$
and $\mu^{*}_{0},\dots,\mu^{*}_{n-1}$ by (50).

4: Calculating $\mathbf{Q}$ with $t_{0},\dots,t_{n-1}$ and
$\bm{v}_{0},\dots,\bm{v}_{m-1}$ by (51).

5: Calculating enhanced mapping by
$\mathbf{W}_{*}=\mathbf{C}_{2}^{-\frac{1}{2}}\mathbf{Q}$.

Compared with the optimization (55) and (56), the decision variable of (55) is
a matrix $\mathbf{W}$ with dimension $m\times n$, and it is
$(\mu_{0},\dots,\mu_{n-1})$ with dimension $n$ for (56), that means the
dimension of gradient is $m\times n$ for (55) and $n$ for (56) in the gradient
descent method. Moreover, the constraint condition of (55) is
$\text{rk}(\mathbf{W})=n$ which is independent for each element of
$\mathbf{W}$, and that of (56) is just
$\lambda_{i+m-n}\leq\mu_{i}\leq\lambda_{i}$ that the feasible region is just a
$n$-dimensional cube.

###  IV-D Numerical Example

Compared with the optimization (55), the optimization (56) is easier to solve.
In the following example, the closed-form solution of the optimization (56) is
figured out.

(a) Riemannian distance (b) Kullback-Leibler divergence (c) log-determinant
divergence

Figure 6: The detection performance of enhanced detector with SCR from -5 dB
to 10 dB. In these figures, the detection probabilities are calculated based
on $10^{4}$ Monte Carlo runs.

######  Example 1.

Consider the radar target detection scenario, the details of the simulation
parameters are listed in the Table II. The primary data is from the cell under
test, and the secondary data are from the reference cells.

TABLE II: Settings of the Simulation. Parameters or Variables | Setting
---|---
Number of range cells | 17
Number of pulses | 15
False-alarm Probability | $P_{f}=10^{-3}$
Target cell | $9^{\text{th}}$
Pulse repetition frequency |  $f_{r}=1000$ Hz
K-distribution clutter | Shape parameter 1
Scalar parameter 0.5

The target echo steering vector is

$\bm{s}=A[1\;\text{exp}(j2\pi
f_{d}f_{r}^{-1})\;\cdots\;\text{exp}(j2\pi(M-1)f_{d}f_{r}^{-1})],$ (57)

where $A$ is the power of target echo, and $f_{d}$ is the Doppler frequency
which is 135 Hz in this example.

Let the objective function in (55) be $f(x)=x$, the enhancements of
$\mathcal{D}_{\text{RD}},\mathcal{D}_{\text{KL}},\mathcal{D}_{\text{LD}}$ are
discussed.

The enhancement of the geometric measures
$\mathcal{D}_{\text{RD}},\mathcal{D}_{\text{KL}},\mathcal{D}_{\text{LD}}$ are
formulated as follows,

$\begin{split}&\max_{\mathbf{W}}\mathcal{D}_{\text{RD}}(\mathbf{W}^{H}\mathbf{C}_{1}\mathbf{W},\mathbf{W}^{H}\mathbf{C}_{2}\mathbf{W})\\\
=&\max_{\mathbf{W}}\Arrowvert\log((\mathbf{W}^{H}\mathbf{C}_{1}\mathbf{W})^{-\frac{1}{2}}\mathbf{W}^{H}\mathbf{C}_{2}\mathbf{W}(\mathbf{W}^{H}\mathbf{C}_{1}\mathbf{W})^{-\frac{1}{2}})\Arrowvert_{F}^{2},\end{split}$
(58)
$\begin{split}\max_{\mathbf{W}}\mathcal{D}_{\text{KL}}(\mathbf{W}^{H}\mathbf{C}_{1}\mathbf{W},\mathbf{W}^{H}\mathbf{C}&{}_{2}\mathbf{W})\\\
=\max_{\mathbf{W}}\text{tr}(\mathbf{W}^{H}\mathbf{C}_{1}\mathbf{W}(\mathbf{W}\mathbf{C}_{2}\mathbf{W}^{H}&)^{-1}-\mathbf{I})\\\
-\log&\left|\mathbf{W}^{H}\mathbf{C}_{1}\mathbf{W}(\mathbf{W}\mathbf{C}_{2}\mathbf{W}^{H})^{-1}\right|,\end{split}$
(59)
$\begin{split}&\max_{\mathbf{W}}\mathcal{D}_{\text{RD}}(\mathbf{W}^{H}\mathbf{C}_{1}\mathbf{W},\mathbf{W}^{H}\mathbf{C}_{2}\mathbf{W})\\\
=&\max_{\mathbf{W}}\log\left|\frac{\mathbf{W}^{H}\mathbf{C}_{1}\mathbf{W}+\mathbf{W}^{H}\mathbf{C}_{2}\mathbf{W}}{2}\right|-\log\sqrt{\left|\mathbf{W}^{H}\mathbf{C}_{1}\mathbf{W}\mathbf{W}^{H}\mathbf{C}_{2}\mathbf{W}\right|}.\end{split}$
(60)

By Theorem 4, the equivalent enhancements on $\mathcal{M}_{\text{P}}$ are

$\begin{split}&\max_{P=\\{\mu_{0},\dots,\mu_{n-1}\\}}\varphi_{\mathcal{D}_{\text{RD}}}(P)=\sum^{n-1}_{k=0}\log^{2}\mu_{k}\\\
s.t.&\quad\lambda_{i+m-n}\leq\mu_{i}\leq\lambda_{i}\quad(0\leq i\leq n-1),\\\
\end{split}$ (61)
$\begin{split}&\max_{P=\\{\mu_{0},\dots,\mu_{n-1}\\}}\varphi_{\mathcal{D}_{\text{KL}}}(P)=\sum^{n-1}_{k=0}\mu_{k}-1-\log\mu_{k}\\\
&s.t.\quad\lambda_{i+m-n}\leq\mu_{i}\leq\lambda_{i}\quad(0\leq i\leq n-1),\\\
\end{split}$ (62)
$\begin{split}&\max_{P=\\{\mu_{0},\dots,\mu_{n-1}\\}}\varphi_{\mathcal{D}_{\text{LD}}}(P)=\sum^{n-1}_{k=0}\log\frac{\mu_{k}+1}{2\sqrt{\mu_{k}}}\\\
&s.t.\quad\lambda_{i+m-n}\leq\mu_{i}\leq\lambda_{i}\quad(0\leq i\leq n-1),\\\
\end{split}$ (63)

where $\lambda_{m-1}\leq\cdots\leq\lambda_{0}$ and
$\mathcal{P}(\mathbf{C}_{1},\mathbf{C}_{2})=\\{\lambda_{0},\dots,\lambda_{m-1}\\}$.

The optimization problems of (61-63) are much easier than problems of (58-60),
because the variables $\mu_{i}\,(i=0,\dots,n-1)$ are independent in the
objective function and constraints. Therefore, these optimization problems can
be divided into the summation of subproblem as

$\sum^{n-1}_{k=0}\max_{\lambda_{k+m-n}\leq\mu_{k}\leq\lambda_{k}}\log^{2}\mu_{k},$
(64)
$\sum^{n-1}_{k=0}\max_{\lambda_{k+m-n}\leq\mu_{k}\leq\lambda_{k}}\mu_{k}-1-\log\mu_{k},$
(65)
$\sum^{n-1}_{k=0}\max_{\lambda_{k+m-n}\leq\mu_{k}\leq\lambda_{k}}\log\frac{\mu_{k}+1}{2\sqrt{\mu_{k}}}.$
(66)

Thus, they can be solved by the derivate of the objective function

$\frac{\text{d}\log^{2}\mu_{k}}{\text{d}\mu_{k}}=2\frac{\log\mu_{k}}{\mu_{k}},$
(67)
$\frac{\text{d}\mu_{k}-1-\log\mu_{k}}{\text{d}\mu_{k}}=1-\frac{1}{\mu_{k}},$
(68)
$\frac{\text{d}\log\frac{\mu_{k}+1}{2\sqrt{\mu_{k}}}}{\text{d}\mu_{k}}=\frac{\mu_{k}-1}{2(1+\mu_{k})\mu_{k}}.$
(69)

According to their derivates, these three sorts of objective functions are
monotonically decreasing in $(0,1]$ and monotonically increasing in
$[1,\infty)$. Therefore, the optimal $\mu^{*}_{i}$ is chosen from the two ends
of the interval $[\lambda_{i+m-n},\lambda_{i}]$, i.e.,

$\mu^{*}_{i}|_{\varphi_{\mathcal{D}_{\text{RD}}}}=\arg\max_{\mu_{i}\in\\{\lambda_{i},\lambda_{i+m-n}\\}}\log^{2}\mu_{i},$
(70)
$\mu^{*}_{i}|_{\varphi_{\mathcal{D}_{\text{KL}}}}=\arg\max_{\mu_{i}\in\\{\lambda_{i},\lambda_{i+m-n}\\}}\mu_{i}-1-\log\mu_{i},$
(71)
$\mu^{*}_{i}|_{\varphi_{\mathcal{D}_{\text{LD}}}}=\arg\max_{\mu_{i}\in\\{\lambda_{i},\lambda_{i+m-n}\\}}\log\frac{\mu_{i}+1}{2\sqrt{\mu_{i}}}.$
(72)

Then the enhanced mapping can be obtained by the algorithm 1, which is

$\mathbf{W}_{*}=\mathbf{C}_{2}^{-\frac{1}{2}}\begin{bmatrix}\sqrt{\frac{\mu^{*}_{0}-\lambda_{m-n}}{\lambda_{0}-\lambda_{m-n}}}\bm{v}^{H}_{0}+\sqrt{\frac{\lambda_{0}-\mu^{*}_{0}}{\lambda_{0}-\lambda_{m-n}}}\bm{v}^{H}_{m-n}\\\
\sqrt{\frac{\mu^{*}_{1}-\lambda_{m-n+1}}{\lambda_{1}-\lambda_{m-n+1}}}\bm{v}^{H}_{1}+\sqrt{\frac{\lambda_{1}-\mu^{*}_{1}}{\lambda_{1}-\lambda_{m-n+1}}}\bm{v}^{H}_{m-n+1}\\\
\vdots\\\
\sqrt{\frac{\mu^{*}_{n-1}-\lambda_{m-1}}{\lambda_{n-1}-\lambda_{m-1}}}\bm{v}^{H}_{n-1}+\sqrt{\frac{\lambda_{n-1}-\mu^{*}_{n-1}}{\lambda_{n-1}-\lambda_{m-1}}}\bm{v}^{H}_{m-1}\end{bmatrix},$
(73)

where $\bm{v}_{0},\bm{v}_{1},\cdots,\bm{v}_{m-1}$ are the corresponding
eigenvectors of $\lambda_{0},\dots,\lambda_{m-1}$.

_Computational efficiency_ : The computation of solving the optimization
(61-63) is without iterations, and the main computational burden is from
working out each $\mu^{*}_{k}$ by a closed analytic expression. So, the
computational complexity is $\mathcal{O}(m)$ for solving (61-63). The previous
methods[34] need the gradient descent-based method to solve the optimization
problem, and the SVD (singular value decomposition) is required in each
iteration. That means the computational complexity is $\mathcal{O}(km^{3})$
($k$ indicates the number of iteration) for directly solving (58-60) on
Grassmannian manifold. In addition, the transformations from (58-60) to
(61-63) depends on the computation of eigenvalues, so the computation
complexity of the transformations is $\mathcal{O}(m^{3})$. Totally, the
computational complexity of solving the optimization (58-60) descends from
$\mathcal{O}(km^{3})$ to $\mathcal{O}(m^{3})$ by transforming them to (61-63).

The detection performances of the enhanced detectors are shown in Fig.6.
According to this experiment, the following conclusions can be drawn:

  * •

The closed-form solution of the enhanced mapping is figured out by
transforming (58-60) to (61-63).

  * •

The complexity of computing the enhanced mapping descends from
$\mathcal{O}(km^{3})$ to $\mathcal{O}(m^{3})$.

  * •

KL divergence is the best among these three sorts of geometric measures.

  * •

The improvement of the enhancement of KLD is less than others.

  * •

The smaller dimension $n$ results in the better performance of the enhanced
detector.

##  V Analytic Method Based on Dual Power Spectrum Manifold for Detection
Performance

As discussed in section III, the affine invariant geometric measures on
$\mathcal{M}_{\mathcal{T}H_{++}}$ is equivalent to a corresponding induced
potential function on $\mathcal{M}_{\text{P}}$. So, the geometric detector can
be expressed as a simpler form by the induced potential function than the
original form, and the simpler form can more obviously reveal the relation
between the test statistics and the characteristic of the signal, which would
benefit the analysis of the detection performance. Moreover, the remaining
problem in the enhanced detection, that the selection criterion of the
dimension of the enhanced mapping, is also possible to be solved from the view
of the induced potential function. This section would introduce the analytic
method of the detection performance and discuss the optimal dimension of the
enhanced mapping on $\mathcal{M}_{\text{P}}$.

###  V-A Performance Analysis Based on the Induced Potential Function

As the decision rule (38), the large geometric measure means that the signal
has a great chance to be detected, and the small one is the opposite. In
addition, according to Theorem 30, the geometric detector using the affine
invariant geometric measure can be reformulated as the form based on the
induced potential function, so the detection performance is related to the
value of the induced potential function. As illustrated by Fig.7, the maximal
point of the induced potential function means the best detection performance,
and the advantageous characteristics, which benefit the detection performance,
can be deduced by analyzing the corresponding power spectrum to the maximal
point. Similarly, the disadvantageous characteristics can be analyzed by the
corresponding power spectrum to the minimal point. Without loss of generality,
only the advantageous characteristics for the detection performance is
considered in this part. In the next contents, we use the partial derivative
to obtain the maximal point and deduce the advantageous characteristics.

Figure 7: The variable of the induced potential function corresponds to the
power spectrum of the observed data. The advantageous and disadvantageous
characteristics for detection performance are embedded in the corresponding
power spectrums to the minimal point and the maximal point.

Firstly, the background settings are introduced as follows. Suppose that the
estimate $\hat{\mathbf{C}}$ precisely equals to the Toeplitz covariance matrix
of the clutter in the primary data. That means the Toeplitz HPD matrix
$\mathbf{C}_{\bm{x}}$ and matrix $\hat{\mathbf{C}}$ can be expressed as

$\begin{cases}\mathbf{C}_{\bm{x}}=\mathbf{C}_{\bm{s}}+\mathbf{C}_{\bm{w}},\\\
\hat{\mathbf{C}}=\mathbf{C}_{\bm{w}},\\\ \end{cases}$ (74)

where $\mathbf{C}_{\bm{w}}$ is the Toeplitz covariance matrix of the clutter
in the primary data.

According to Lemma 3 and the attached remark, the matrix can be approximated
to the following form,

$\begin{cases}\mathbf{C}_{\bm{x}}\approx\mathbf{F}\text{diag}\,(\lambda_{0}^{s}+\lambda^{\prime}_{0},\dots,\lambda^{s}_{m-1}+\lambda^{\prime}_{m-1})\mathbf{F}^{H},\\\
\hat{\mathbf{C}}\approx\mathbf{F}\text{diag}\,(\lambda^{\prime}_{0},\dots,\lambda^{\prime}_{m-1})\mathbf{F}^{H},\\\
\end{cases}$ (75)

where $\lambda^{s}_{k}\,(k=0,\dots,m-1)$ are the eigenvalues of
$\mathbf{C}_{\bm{s}}$, and $\lambda^{\prime}_{k}\,(k=0,\dots,m-1)$ are the
eigenvalues of $\mathbf{C}_{\bm{w}}$. Let
$\lambda^{*}_{k}=\lambda^{s}_{k}/\lambda^{\prime}_{k}$, the $\lambda^{*}_{k}$
can be regarded as the power of the $k^{\text{th}}$ frequency component of the
target echo whitened by the clutter according to Lemma 3 and Definition 4.
Then
$\mathcal{P}(\mathbf{C}_{\bm{x}},\hat{\mathbf{C}})=(\lambda^{*}_{0}+1,\dots,\lambda^{*}_{m-1}+1)$
is established. Because the summation of eigenvalues equals to the trace, the
following equation is established,

$\sum^{m-1}_{k=0}\lambda^{*}_{k}=mc_{0}|_{\mathcal{H}_{0}}=m\sigma^{2}\;\Leftrightarrow\;\lambda^{*}_{0}=m\sigma^{2}-\sum^{m-1}_{k=1}\lambda^{*}_{k},$
(76)

where $\sigma^{2}$ indicates the SCR. Then, under fixed SCR $\sigma^{2}$, the
induced potential function can be reformulated as the following adjusted form,

$\begin{split}&\varphi^{\prime}_{\mathcal{D}}(\lambda^{*}_{1},\dots,\lambda^{*}_{m-1})\\\
=&\varphi_{\mathcal{D}}\left(1+m\sigma^{2}-\sum^{m-1}_{k=1}\lambda^{*}_{k},1+\lambda^{*}_{1},\dots,1+\lambda^{*}_{m-1}\right).\end{split}$
(77)

As instances, the typical geometric measures are analyzed by this approach.
The adjusted induced potential functions with respect to
$\lambda^{*}_{1},\dots,\lambda^{*}_{m-1}$ are as follows,

$\begin{split}\varphi^{\prime}_{\mathcal{D}_{\text{RD}}}&(\lambda^{*}_{1},\dots,\lambda^{*}_{m-1})\\\
&=\sum^{m-1}_{k=1}\log^{2}(1+\lambda^{*}_{k})+\log^{2}(1+m\sigma^{2}-\sum^{m-1}_{k=1}\lambda^{*}_{k}),\end{split}$
(78)
$\begin{split}\varphi^{\prime}_{\mathcal{D}_{\text{KL}}}&(\lambda^{*}_{1},\dots,\lambda^{*}_{m-1})\\\
&=m\sigma^{2}-\sum^{m-1}_{k=0}\log(1+\lambda^{*}_{k})-\log(1+m\sigma^{2}-\sum^{m-1}_{k=1}\lambda^{*}_{k}),\end{split}$
(79)
$\begin{split}\varphi^{\prime}_{\mathcal{D}_{\text{LD}}}&(\lambda^{*}_{1},\dots,\lambda^{*}_{m-1})\\\
&=\sum^{m-1}_{k=1}\log\frac{2+\lambda^{*}_{k}}{2\sqrt{\lambda^{*}_{k}+1}}+\log\frac{2+m\sigma^{2}-\sum^{m-1}_{k=1}\lambda^{*}_{k}}{2\sqrt{1+m\sigma^{2}-\sum^{m-1}_{k=1}\lambda^{*}_{k}}},\end{split}$
(80)

and the domain of definition is

$\left\\{(\lambda^{*}_{1},\dots,\lambda^{*}_{m-1})\;\bigg{|}\;\lambda^{*}_{k}\geq
0,\,\sum^{m-1}_{k=1}\lambda^{*}_{k}\leq m\sigma^{2}\right\\}.$ (81)

(a) Riemannian distance (b) Kullback-Leibler Divergence (c) log-determinant
Divergence

Figure 8: The curve of induced potential functions with respect to
$\lambda^{*}_{1}$, where other eigenvalues
$\lambda^{*}_{2},\dots,\lambda^{*}_{m}$ are fixed. The $r$ in the legends is
$r=m\sigma^{2}-\sum^{m-1}_{k=2}\lambda^{*}_{k}=\lambda^{*}_{0}+\lambda^{*}_{1}$.

The curves of induced potential functions (78-80) with respect to
$\lambda^{*}_{1}$ (suppose $\lambda^{*}_{2},\dots,\lambda^{*}_{m-1}$ is fixed)
are shown in Fig.8. For $\mathcal{D}_{\text{RD}}$ and
$\mathcal{D}_{\text{LD}}$, the ratio of the medium to the two ends is
increasing when $r$ ($r=\lambda^{*}_{0}+\lambda^{*}_{1}$) is ascending. The
maximal point is in the medium when $r$ is large, and the maximal point is in
the two ends when $r$ is small. Moreover, for $\mathcal{D}_{\text{KL}}$, the
maximal point is always in the two ends regardless of the value of $r$, and
the KLD reaches the minimum when $\lambda^{*}_{1}$ is in the medium.

Moreover, the partial derivate of these functions are as follows,

$\frac{\partial\varphi^{\prime}_{\mathcal{D}_{\text{RD}}}}{\partial\lambda^{*}_{i}}=2\left(\frac{\log(1+\lambda^{*}_{i})}{1+\lambda^{*}_{i}}-\frac{\log(1+m\sigma^{2}-\sum\limits^{m-1}_{k=1}\lambda^{*}_{k})}{1+m\sigma^{2}-\sum\limits^{m-1}_{k=1}\lambda^{*}_{k}}\right),$
(82)
$\frac{\partial\varphi^{\prime}_{\mathcal{D}_{\text{KL}}}}{\partial\lambda^{*}_{i}}=-\frac{1}{1+\lambda^{*}_{i}}+\frac{1}{1+m\sigma^{2}-\sum\limits^{m-1}_{k=1}\lambda^{*}_{k}},$
(83)
$\begin{split}\frac{\partial\varphi^{\prime}_{\mathcal{D}_{\text{LD}}}}{\partial\lambda^{*}_{i}}=&\frac{\lambda^{*}_{i}}{2(2+\lambda^{*}_{i})(1+\lambda^{*}_{i})}\\\
&-\frac{m\sigma^{2}-\sum\limits^{m-1}_{k=1}\lambda^{*}_{k}}{2(2+m\sigma^{2}-\sum\limits^{m-1}_{k=1}\lambda^{*}_{k})(1+m\sigma^{2}-\sum\limits^{m-1}_{k=1}\lambda^{*}_{k})}.\end{split}$
(84)

By the partial derivative of the induced potential function, the maximal
points and the minimal points can be calculated. In the following contents, we
discuss the maximal points of these induced potential function to conclude the
factors which benefit the detection performance.

Analyzing the monotonic intervals of (78-80) by their partial derivates
(82-84), the maximal value is reached over the following situations.

  * •

Riemannian Distance and log-determinant Divergence:

$\lambda^{*}_{i}\in\left\\{0,m\sigma^{2}-\sum\limits_{1\leq k\leq m-1\atop
k\neq i}\lambda^{*}_{k},\frac{m\sigma^{2}}{2}-\sum\limits_{1\leq k\leq
m-1\atop k\neq i}\frac{\lambda^{*}_{k}}{2}\right\\};$ (85)

  * •

Kullback-Leibler Divergence:

$\lambda^{*}_{i}\in\left\\{0,m\sigma^{2}-\sum\limits_{1\leq k\leq m-1\atop
k\neq i}\lambda^{*}_{k}\right\\}.$ (86)

######  remark 5.

The equation (85) corresponds to the Fig.8, each component of the maximal
point is chosen from two ends or the medium, and which situation is maximal
depends on the value of $m\sigma^{2}$, i.e., SCR. The equation (86) also
reveals same conclusions as Fig.8, that the maximal points are in the two
ends.

By using the adjustment method to the (85) and (86), the maximal values of
these considered induced potential functions are from the following power
spectrum, respectively.

  * •

Riemannian Distance and log-determinant Divergence:

$P^{*}_{k}=\bigg{(}0,\dots,0,\underbrace{\frac{m\sigma^{2}}{k},\dots,\frac{m\sigma^{2}}{k}}_{k}\bigg{)}\quad(k=1,\dots,m),$
(87)

$P^{*}_{k}$ is the candidate of the maximal power spectrum, i.e.,
$P^{*}_{\max}\in\\{P^{*}_{1},\dots,P^{*}_{m}\\}$, and which one is maximal
depends on the value of $m\sigma^{2}$.

  * •

Kullback-Leibler Divergence:

$P^{*}_{\max}=\left(0,\dots,0,m\sigma^{2}\right).$ (88)

Equation (88) represents the power spectrum with only one nonzero component.
It means that the narrower spectrum encourages larger induced potential
function when the geometric measure is KL divergence. Moreover, equation (87)
represents the flat spectrums with different bandwidths, and which bandwidth
is maximal depends on the SCR ($m\sigma^{2}$). So, the maximal test statistic
is selected from the flat spectrum when the geometric measure is one of other
twos. Besides, according to the curves in Fig.8, the optimal $k$ is
incremental by the ascending SCR.

(a) Riemannian distance (b) Kullback-Leibler divergence (c) log-determinant
divergence

Figure 9: The normalized geometric measures versus SCR with
$\bm{s}_{1},\dots,\bm{s}_{5}$. For each SCR, the geometric measure of
$\bm{s}_{k}$ is normalized by the mean of $\bm{s}_{1},\dots,\bm{s}_{5}$ to
clearly show the relative value of them. Moreover, each normalized geometric
measure is the average of $10^{5}$ Monte Carlo runs.

(a) Riemannian distance (b) Kullback-Leibler divergence (c) log-determinant
divergence

Figure 10: The detection performance of $\bm{s}_{k}\,(k=1,\dots,5)$, under SCR
from -20 dB to 10 dB. In this figure, the detection probability is calculated
based on $5\times 10^{5}$ Monte Carlo runs. And, the curves of the detection
probability under low SCR (-20 dB to 0 dB) are attached in the center of
figures.

#### Numerical Example

The above conclusions are verified by the following sample.

######  Example 2.

The majority of settings are as same as example II, which are shown in Table
II. Differently, there are five sorts of target echo steering vectors,

$\bm{s}_{k}=A\mathcal{F}^{-1}\bigg{(}0,\dots,0,\underbrace{\frac{1}{\sqrt{k}},\dots,\frac{1}{\sqrt{k}}}_{k}\bigg{)},\quad
k=1,\dots,5,$ (89)

where $A$ is the amplitude of the target echo and $\mathcal{F}^{-1}$ denotes
the inverse discrete Fourier transform. For convenience, we define the
discrete bandwidth $B$ of $\bm{s}_{k}$ is $k$, then the discrete bandwidths of
these selected target echo are $1,\dots,5$, respectively.

To verify the proposed analytic method, the test statistics of geometric
detectors with $\bm{s}_{1},\dots,\bm{s}_{5}$ would be discussed firstly. To
coincide with the above analysis, we suppose the reference data is actually
equivalent to the component of the clutter in the primary data. For showing
the order changing, the test statistics are normalized by the mean of
$\bm{s}_{1},\dots,\bm{s}_{5}$ for each SCR. The normalized test statistics of
geometric detectors with $\bm{s}_{1},\dots,\bm{s}_{5}$ are presented in Fig.9.
The results correspond to the conclusion that the discrete bandwidth $k$ of
the maximal test statistic depends on the SCR for RD and LDD, and the maximal
$k$ equals to 1 for KLD. Moreover, the results prove the discussion that the
optimal $k$ of the RD and the LDD ascends when SCR is increasing.

Furthermore, the detection performance of this example with the geometry-based
detection scheme (as Fig.3 depicted) is shown in Fig.10. For KLD, the
detection probabilities of the five signals $\bm{s}_{1},\dots,\bm{s}_{5}$ are
similar. In the magnified figure, we can see that $\bm{s}_{1}$ has the largest
detection probability and is followed by
$\bm{s}_{2},\bm{s}_{3},\bm{s}_{4},\bm{s}_{5}$, successively. For the RD and
LDD, the order of the detection probabilities in descending is
$\bm{s}_{5},\bm{s}_{4},\bm{s}_{3},\bm{s}_{2},\bm{s}_{1}$ under high SCR, and
it is $\bm{s}_{1},\bm{s}_{2},\bm{s}_{3},\bm{s}_{4},\bm{s}_{5}$ under low SCR.

(a) Enhanced Riemannian distance of $\bm{s}_{1}$ (b) Enhanced Kullback-Leibler
divergence of $\bm{s}_{1}$ (c) Enhanced log-determinant divergence of
$\bm{s}_{1}$ (d) Enhanced Riemannian distance of $\bm{s}_{1}$ (e) Enhanced
Kullback-Leibler divergence of $\bm{s}_{2}$ (f) Enhanced log-determinant
divergence of $\bm{s}_{2}$ (g) Enhanced Riemannian distance of $\bm{s}_{3}$
(h) Enhanced Kullback-Leibler divergence of $\bm{s}_{3}$ (i) Enhanced log-
determinant divergence of $\bm{s}_{3}$

Figure 11: The detection performance of enhanced detector with SCR from -5 dB
to 10 dB. In these figures, the detection probabilities are calculated based
on $10^{4}$ Monte Carlo runs.

In addition, two problems about the experimental results should be discussed:
1) The cross points (the order changed points) of detection performance curves
are inconsistent with the normalized geometric measures in the SCR axis for
the RD and LDD; 2) The detection probability descends when SCR increases from
-20 dB to -8 dB (KLD is from -18 dB to -16 dB). This two problems both result
from the accuracy of the mean matrix estimates. For the first problem, the
normalized geometric measures in Fig.9 are calculated without the estimate of
the mean matrix, but this estimate is considered in the detection. Therefore,
under same SCR, the test statistics with the estimate constitute more clutter
components than another. So that, the critical points of detection probability
are larger than the normalized geometric measures in terms of SCR. As for the
second problem, the difference between the estimate and the ground truth of
the clutter covariance matrix results in that the eigenvalues in
$\mathcal{P}(\mathbf{C}_{\bm{x}},\hat{\mathbf{C}})$ might be lower than 1,
i.e., $\lambda^{*}_{k}<0\,(1+\lambda^{*}_{k}<1)$ might be encountered. But,
$\varphi_{\mathcal{D}_{\text{RD}}}$, $\varphi_{\mathcal{D}_{\text{KL}}}$ and
$\varphi_{\mathcal{D}_{\text{LDD}}}$ is monotonously decreasing when
$\lambda^{*}_{k}<0$. So, the test statistics descend when SCR increases from
-10 dB to -5 dB, then the detection probability also descends.

Totally, the proposed analytic method is effective that the detection
performance is consistent with the analysis. By the proposed analytic method,
we deduce the maximal points of the induced potential functions, and then the
advantageous characteristics in terms of detection performance are concluded
by the maximal points.

###  V-B Optimal Dimension for Enhanced Matrix CFAR Detection

As depicted in Fig.6, the different dimension $n$ of the enhanced mapping
results in the distinct detection performance. This part would discuss the
relation between the optimal $n$ and the signal from the view of dual power
spectrum manifold.

Consider the geometric measures $\mathcal{D}_{\text{RD}}$,
$\mathcal{D}_{\text{KL}}$ and $\mathcal{D}_{\text{LD}}$. Actually, the induced
potential functions of them can be commonly expressed as the following form

$\varphi_{\mathcal{D}}(\lambda_{1},\dots,\lambda_{m})=\sum^{m}_{k=0}\varphi_{\mathcal{D}}^{*}(\lambda_{k}),$
(90)

where $\varphi_{\mathcal{D}}^{*}(\lambda)=\log^{2}\lambda$,
$\varphi_{\mathcal{D}}^{*}(\lambda)=\lambda-1-\log\lambda$ and
$\varphi_{\mathcal{D}}^{*}(\lambda)=\log\frac{\lambda+1}{2\sqrt{\lambda}}$
correspond to the $\varphi_{\mathcal{D}_{\text{RD}}}$,
$\varphi_{\mathcal{D}_{\text{KLD}}}$ and $\varphi_{\mathcal{D}_{\text{LD}}}$,
respectively. According to the relation between the eigenvalues and the power
spectrum, the eigenvalue $\lambda_{k}$ corresponds to the $k$-th component of
the power spectrum. Suppose the discrete bandwidth of the target echo $\bm{s}$
is $B$, which is defined as the bandwidth in the DFT. In the summation (90),
only the $B$ of the $m$ addends
($\varphi_{\mathcal{D}}^{*}(\lambda_{k}),\;k=1,\dots,m$) contain the
components of $\bm{s}$, and other $m-B$ are pure clutter. So, the dimension of
$\mathcal{M}_{\text{P}}$ is suggested to be reduced to $B$. That means the
optimal $n$ equals to the discrete bandwidth of the target echo.

This conclusion would be verified by the following sample.

######  Example 3.

The majority of settings are same as example II, which are shown in Table II.
Differently, there are three target echo steering vectors,

$\bm{s}_{k}=A\mathcal{F}^{-1}\bigg{(}0,\dots,0,\underbrace{\frac{1}{\sqrt{k}},\dots,\frac{1}{\sqrt{k}}}_{k}\bigg{)},\quad
k=1,\dots,3,$ (91)

where $A$ is the power of target echo and $\mathcal{F}^{-1}$ denotes the
inverse discrete Fourier transform. The discrete bandwidth $B$ of
$\bm{s}_{1}$, $\bm{s}_{2}$ and $\bm{s}_{3}$ are 1, 2 and 3, respectively.

The experimental results illustrated in Fig.11 correspond to the conclusions
discussed above, that the optimal $n$ equals to the discrete bandwidth $B$. In
addition, it is worth pointing that the detection performance of enhanced
detector with $n=1$ is worser than the original detector without enhancement
when the target echo is $\bm{s}_{3}$. Moreover, the following conclusions can
be drawn:

  * •

The optimal dimension of the enhanced mapping equals to the discrete bandwidth
of target echo for enhanced detector.

  * •

The closer dimension of the enhanced mapping to the discrete bandwidth
encourages the better detection performance.

  * •

The improvement of detection performance is obvious, while the discrete
bandwidth of the target echo is small, i.e., the spectrum is narrow.

  * •

When the bandwidth is large, the enhanced mapping with low dimension would
degrade the detection performance of the geometric detector.

##  VI Conclusions

In this paper, the dual power spectrum manifold $\mathcal{M}_{\text{P}}$, as
the dual manifold of Toeplitz Hermitian positive definite (HPD) manifold
$\mathcal{M}_{\mathcal{T}H_{++}}$, have been studied. By the duality of these
two manifolds, we have shown that there exists an equivalent induced potential
function on $\mathcal{M}_{\text{P}}$ for every affine invariant geometric
measure on $\mathcal{M}_{\mathcal{T}H_{++}}$. Based on that, the geometric
detectors can be reformulated as the form related to the power spectrum, and
two problems have been circumvented: 1) The enhancement of the geometric
detector is reformulated as a simpler problem and is easier to solve than it
used to be. In the presented examples, the analytic expression of the enhanced
mapping is provide by the proposed method, that the enhanced mapping was
obtained by the gradient descent-based methods. Moreover, the effectiveness of
the enhanced detector has been verified by the experimental results. 2) The
performance of geometric detectors have been analyzed by the induced potential
functions. Besides, the optimal dimension of the enhance mapping also have
been deduced by the induced potential function. Experimentally, these deduced
conclusions have been validated.

##  Appendix A The Affine Invariant Properties of the Geometric Measures
$\mathcal{D}_{\text{RD}},\mathcal{D}_{\text{KL}},\mathcal{D}_{\text{JS}},\mathcal{D}_{\text{LD}}$

###  A-A Riemannian Distance $\mathcal{D}_{\text{RD}}$

The Riemannian distance can be transformed to

$\begin{split}\mathcal{D}_{\text{RD}}(\mathbf{C}_{1},\mathbf{C}_{2})=&\Arrowvert\log(\mathbf{C}_{1}^{-\frac{1}{2}}\mathbf{C}_{2}\mathbf{C}_{1}^{-\frac{1}{2}})\Arrowvert_{F}^{2}\\\
=&\text{tr}\left(\log^{2}(\mathbf{C}_{1}^{-\frac{1}{2}}\mathbf{C}_{2}\mathbf{C}_{1}^{-\frac{1}{2}})\right)\\\
=&\sum_{k=1}^{m}\lambda_{k}\left(\log^{2}(\mathbf{C}_{1}^{-\frac{1}{2}}\mathbf{C}_{2}\mathbf{C}_{1}^{-\frac{1}{2}})\right)\\\
\overset{\text{(a)}}{=}&\sum_{k=1}^{m}\log^{2}\left(\lambda_{k}(\mathbf{C}_{1}^{-\frac{1}{2}}\mathbf{C}_{2}\mathbf{C}_{1}^{-\frac{1}{2}})\right)\\\
\overset{\text{(b)}}{=}&\sum_{k=1}^{m}\log^{2}\left(\lambda_{k}(\mathbf{C}_{2}\mathbf{C}_{1}^{-1})\right),\\\
\end{split}$ (92)

where $\lambda_{k}(\mathbf{A})$ indicates the $k$-th eigenvalue of matrix
$\mathbf{A}$, (a) holds by the property of matrix logarithm function
$\lambda(\log(\mathbf{C}))=\log(\lambda(\mathbf{C}))$ and (b) holds by the
well-known property of matrix eigenvalue that matrices
$\mathbf{C}_{1}\mathbf{C}_{2}$ and $\mathbf{C}_{2}\mathbf{C}_{1}$ have the
same eigenvalues. Therefore,

$\begin{split}&\mathcal{D}_{\text{RD}}(\mathbf{W}^{H}\mathbf{C}_{1}\mathbf{W},\mathbf{W}^{H}\mathbf{C}_{2}\mathbf{W})\\\
=&\sum_{k=1}^{m}\log^{2}\left(\lambda_{k}(\mathbf{W}^{H}\mathbf{C}_{2}\mathbf{W}\mathbf{W}^{-1}\mathbf{C}_{1}^{-1}(\mathbf{W}^{H})^{-1})\right)\\\
=&\sum_{k=1}^{m}\log^{2}\left(\lambda_{k}(\mathbf{W}^{H}\mathbf{C}_{2}\mathbf{C}_{1}^{-1}(\mathbf{W}^{H})^{-1})\right)\\\
=&\sum_{k=1}^{m}\log^{2}\left(\lambda_{k}(\mathbf{C}_{2}\mathbf{C}_{1}^{-1})\right)=\mathcal{D}_{\text{RD}}(\mathbf{C}_{1},\mathbf{C}_{2}).\end{split}$
(93)

###  A-B Kullback-Leibler Divergence $\mathcal{D}_{\text{KL}}$

$\begin{split}&\mathcal{D}_{\text{KL}}(\mathbf{W}^{H}\mathbf{C}_{1}\mathbf{W},\mathbf{W}^{H}\mathbf{C}_{2}\mathbf{W})\\\
=&\text{tr}(\mathbf{W}^{H}\mathbf{C}_{1}\mathbf{W}\mathbf{W}^{-1}\mathbf{C}_{2}^{-1}(\mathbf{W}^{H})^{-1}-\mathbf{I})\\\
&-\log\left|\mathbf{W}^{H}\mathbf{C}_{1}\mathbf{W}\mathbf{W}^{-1}\mathbf{C}_{2}^{-1}(\mathbf{W}^{H})^{-1}\right|\\\
=&\text{tr}(\mathbf{C}_{1}\mathbf{C}_{2}^{-1}(\mathbf{W}^{H})^{-1}\mathbf{W}^{H}-\mathbf{I})\\\
&-\log\left(\left|\mathbf{W}^{H}\right|\left|\mathbf{C}_{1}\mathbf{C}_{2}^{-1}\right|\left|(\mathbf{W}^{H})^{-1}\right|\right)\\\
=&\text{tr}(\mathbf{C}_{1}\mathbf{C}_{2}^{-1}-\mathbf{I})-\log\left|\mathbf{C}_{1}\mathbf{C}_{2}^{-1}\right|=\mathcal{D}_{\text{KL}}(\mathbf{C}_{1},\mathbf{C}_{2}).\\\
\end{split}$ (94)

###  A-C Jensen-Shannon Divergence $\mathcal{D}_{\text{JS}}$

$\begin{split}&\mathcal{D}_{\text{JS}}(\mathbf{W}^{H}\mathbf{C}_{1}\mathbf{W},\mathbf{W}^{H}\mathbf{C}_{2}\mathbf{W})\\\
=&\frac{1}{2}\left[\mathcal{D}_{\text{KL}}\left(\mathbf{W}^{H}\mathbf{C}_{1}\mathbf{W},\frac{\mathbf{W}^{H}\mathbf{C}_{1}\mathbf{W}+\mathbf{W}^{H}\mathbf{C}_{2}\mathbf{W}}{2}\right)\right.\\\
&+\left.\mathcal{D}_{\text{KL}}\left(\mathbf{W}^{H}\mathbf{C}_{2}\mathbf{W},\frac{\mathbf{W}^{H}\mathbf{C}_{1}\mathbf{W}+\mathbf{W}^{H}\mathbf{C}_{2}\mathbf{W}}{2}\right)\right]\\\
=&\frac{1}{2}\left[\mathcal{D}_{\text{KL}}\left(\mathbf{C}_{1},\frac{\mathbf{C}_{1}+\mathbf{C}_{2}}{2}\right)+\mathcal{D}_{\text{KL}}\left(\mathbf{C}_{2},\frac{\mathbf{C}_{1}+\mathbf{C}_{2}}{2}\right)\right]\\\
=&\mathcal{D}_{\text{JS}}(\mathbf{C}_{1},\mathbf{C}_{2}).\end{split}$ (95)

###  A-D log-determinant Divergence $\mathcal{D}_{\text{LD}}$

$\begin{split}&\mathcal{D}_{\text{LD}}(\mathbf{W}^{H}\mathbf{C}_{1}\mathbf{W},\mathbf{W}^{H}\mathbf{C}_{2}\mathbf{W})\\\
=&\log\left|\frac{\mathbf{W}^{H}\mathbf{C}_{1}\mathbf{W}+\mathbf{W}^{H}\mathbf{C}_{2}\mathbf{W}}{2}\right|-\log\sqrt{\left|\mathbf{W}^{H}\mathbf{C}_{1}\mathbf{W}\mathbf{W}^{H}\mathbf{C}_{2}\mathbf{W}\right|}\\\
=&\log\frac{\left|\mathbf{W}^{H}\right|\left|\frac{1}{2}\left(\mathbf{C}_{1}+\mathbf{C}_{2}\right)\right|\left|\mathbf{W}\right|}{\sqrt{\left|\mathbf{C}_{1}\mathbf{C}_{2}\right|}|\mathbf{W}|\left|\mathbf{W}^{H}\right|}\\\
=&\log\left|\frac{\mathbf{C}_{1}+\mathbf{C}_{2}}{2}\right|-\log\sqrt{\left|\mathbf{C}_{1}\mathbf{C}_{2}\right|}=\mathcal{D}_{\text{LD}}(\mathbf{C}_{1},\mathbf{C}_{2}).\end{split}$
(96)

##  Appendix B Induced Potential Function of the Geometric Measures
$\mathcal{D}_{\text{RD}},\mathcal{D}_{\text{KL}},\mathcal{D}_{\text{JS}},\mathcal{D}_{\text{LD}}$

Let
$\mathbf{C}=\mathbf{C}_{2}^{-\frac{1}{2}}\mathbf{C}_{1}\mathbf{C}_{2}^{-\frac{1}{2}}$,
the induced potential functions of
$\mathcal{D}_{\text{RD}},\mathcal{D}_{\text{KL}},\mathcal{D}_{\text{JS}},\mathcal{D}_{\text{LD}}$
are shown as follows.

###  B-A Riemannian Distance $\mathcal{D}_{\text{RD}}$

$\begin{split}&\varphi_{\mathcal{D}_{\text{RD}}}(\mathcal{P}(\mathbf{C}_{1},\mathbf{C}_{2}))=\mathcal{D}_{\text{RD}}(\mathbf{C}_{1},\mathbf{C}_{2})=\mathcal{D}_{\text{RD}}(\mathbf{C},\mathbf{I})\\\
=&\parallel\log\mathbf{C}^{-1}\parallel^{2}_{F}=\text{tr}(\log^{2}\mathbf{C}^{-1})=\sum^{m-1}_{k=0}\log^{2}\lambda_{k}.\end{split}$
(97)

###  B-B Kullback-Leibler Divergence $\mathcal{D}_{\text{KL}}$

$\begin{split}&\varphi_{\mathcal{D}_{\text{KL}}}(\mathcal{P}(\mathbf{C}_{1},\mathbf{C}_{2}))=\mathcal{D}_{\text{KL}}(\mathbf{C}_{1},\mathbf{C}_{2})=\mathcal{D}_{\text{KL}}(\mathbf{C},\mathbf{I})\\\
=&\text{tr}(\mathbf{C}-\mathbf{I})-\log|\mathbf{C}|=\sum^{m-1}_{k=0}\lambda_{k}-1-\log\lambda_{k}.\end{split}$
(98)

###  B-C Jensen-Shannon Divergence $\mathcal{D}_{\text{JS}}$

$\begin{split}&\varphi_{\mathcal{D}_{\text{JS}}}(\mathcal{P}(\mathbf{C}_{1},\mathbf{C}_{2}))=\mathcal{D}_{\text{JS}}(\mathbf{C}_{1},\mathbf{C}_{2})=\mathcal{D}_{\text{JS}}(\mathbf{C},\mathbf{I})\\\
=&\frac{1}{2}\left\\{\text{tr}\left[\mathbf{C}\left(\frac{\mathbf{C}+\mathbf{I}}{2}\right)^{-1}-\mathbf{I}\right]-\log\left|\mathbf{C}\left(\frac{\mathbf{C}+\mathbf{I}}{2}\right)^{-1}\right|\right.\\\
&+\left.\text{tr}\left[\left(\frac{\mathbf{C}+\mathbf{I}}{2}\right)^{-1}-\mathbf{I}\right]-\log\left|\left(\frac{\mathbf{C}+\mathbf{I}}{2}\right)^{-1}\right|\right\\}\\\
=&\text{tr}\left[\left(\frac{\mathbf{C}+\mathbf{I}}{2}\right)\left(\frac{\mathbf{C}+\mathbf{I}}{2}\right)^{-1}-\mathbf{I}\right]-\frac{1}{2}\log\left|\mathbf{C}\left(\frac{\mathbf{C}+\mathbf{I}}{2}\right)^{-2}\right|\\\
=&\sum^{m-1}_{k=0}\log\frac{\lambda_{k}+1}{2\sqrt{\lambda_{k}}}.\end{split}$
(99)

###  B-D log-determinant Divergence $\mathcal{D}_{\text{LD}}$

$\begin{split}&\varphi_{\mathcal{D}_{\text{LD}}}(\mathcal{P}(\mathbf{C}_{1},\mathbf{C}_{2}))=\mathcal{D}_{\text{LD}}(\mathbf{C}_{1},\mathbf{C}_{2})=\mathcal{D}_{\text{LD}}(\mathbf{C},\mathbf{I})\\\
=&\log\left|\frac{\mathbf{C}+\mathbf{I}}{2}\right|-\log\sqrt{\left|\mathbf{C}\right|}=\sum^{m-1}_{k=0}\log\frac{\lambda_{k}+1}{2\sqrt{\lambda_{k}}}.\end{split}$
(100)

## References

  * [1] O. Zeitouni, J. Ziv, and N. Merhav, “When is the generalized likelihood ratio test optimal?” _IEEE Transactions on Information Theory_ , vol. 38, no. 5, pp. 1597–1602, Sep. 1992.
  * [2] P. Stoica, Jianhua Liu, and Jian Li, “Maximum-likelihood double differential detection clarified,” _IEEE Transactions on Information Theory_ , vol. 50, no. 3, pp. 572–576, 2004.
  * [3] H. Chen, B. Chen, and P. K. Varshney, “Further results on the optimality of the likelihood-ratio test for local sensor decision rules in the presence of nonideal channels,” _IEEE Transactions on Information Theory_ , vol. 55, no. 2, pp. 828–832, 2009.
  * [4] G. V. Moustakides, G. H. Jajamovich, A. Tajer, and X. Wang, “Joint detection and estimation: Optimum tests and applications,” _IEEE Transactions on Information Theory_ , vol. 58, no. 7, pp. 4215–4229, 2012.
  * [5] F. Pascal, P. Forster, J. Ovarlez, and P. Larzabal, “Performance analysis of covariance matrix estimates in impulsive noise,” _IEEE Transactions on Signal Processing_ , vol. 56, no. 6, pp. 2206–2217, 2008.
  * [6] C. Y. Chong, F. Pascal, J. Ovarlez, and M. Lesturgie, “Mimo radar detection in non-gaussian and heterogeneous clutter,” _IEEE Journal of Selected Topics in Signal Processing_ , vol. 4, no. 1, pp. 115–126, 2010.
  * [7] G. Pailloux, P. Forster, J. Ovarlez, and F. Pascal, “Persymmetric adaptive radar detectors,” _IEEE Transactions on Aerospace and Electronic Systems_ , vol. 47, no. 4, pp. 2376–2390, 2011.
  * [8] Y. Rong, A. Aubry, A. De Maio, and M. Tang, “Adaptive radar detection in low-rank heterogeneous clutter via invariance theory,” _IEEE Transactions on Signal Processing_ , vol. 69, pp. 1492–1506, 2021.
  * [9] S.-i. Amari, _Information Geometry and Its Applications_ , 1st ed. Springer Publishing Company, Incorporated, 2016\.
  * [10] M. Arnaudon, F. Barbaresco, and L. Yang, “Riemannian medians and means with applications to radar signal processing,” _IEEE Journal of Selected Topics in Signal Processing_ , vol. 7, no. 4, pp. 595–604, 2013.
  * [11] W. Zhao, W. Liu, and M. Jin, “Spectral norm based mean matrix estimation and its application to radar target cfar detection,” _IEEE Transactions on Signal Processing_ , vol. 67, no. 22, pp. 5746–5760, 2019.
  * [12] J. Zhang, X. Zhang, W. Deng, L. Ye, and Q. Yang, “A geometric barycenter-based clutter suppression method for ship detection in hf mixed-mode surface wave radar,” _Remote Sensing_ , vol. 11, no. 9, 2019.
  * [13] X. Hua and L. Peng, “Mig median detectors with manifold filter,” _Signal Processing_ , p. 108176, 2021.
  * [14] S. Said, L. Bombrun, Y. Berthoumieu, and J. H. Manton, “Riemannian gaussian distributions on the space of symmetric positive definite matrices,” _IEEE Transactions on Information Theory_ , vol. 63, no. 4, pp. 2153–2170, 2017.
  * [15] S. Said, H. Hajri, L. Bombrun, and B. C. Vemuri, “Gaussian distributions on riemannian symmetric spaces: Statistical learning with structured covariance matrices,” _IEEE Transactions on Information Theory_ , vol. 64, no. 2, pp. 752–772, 2018.
  * [16] J. Lapuyade-Lahorgue and F. Barbaresco, “Radar detection using siegel distance between autoregressive processes, application to HF and X-band radar,” in _Radar Conference, 2008. RADAR ’08. IEEE_ , 2008, pp. 1–6.
  * [17] A. Decurninge and F. Barbaresco, “Robust burg estimation of radar scatter matrix for autoregressive structured sirv based on fréchet medians,” _IET Radar, Sonar & Navigation_, vol. 11, no. 1, pp. 78–89, 2017.
  * [18] Frédéric, “Radar monitoring of a wake vortex: Electromagnetic reflection of wake turbulence in clear air,” _Comptes Rendus Physique_ , vol. 11, no. 1, pp. 54–67, 2010, propagation and remote sensing.
  * [19] Z. Liu and F. Barbaresco, _Doppler Information Geometry for Wake Turbulence Monitoring_. Berlin, Heidelberg: Springer Berlin Heidelberg, 2013, pp. 277–290. [Online]. Available: https://doi.org/10.1007/978-3-642-30232-9_11
  * [20] X. Hua, Y. Cheng, H. Wang, Y. Qin, Y. Li, and W. Zhang, “Matrix CFAR detectors based on symmetrized Kullback-Leibler and total Kullback-Leibler divergences,” _Digital Signal Processing_ , vol. 69, pp. 106–116, 2017\.
  * [21] X. Hua, H. Fan, Y. Cheng, H. Wang, and Y. Qin, “Information geometry for radar target detection with total Jensen-Bregman divergence,” _Entropy_ , vol. 20, no. 4, p. 256, 2018.
  * [22] A. Cherian, S. Sra, A. Banerjee, and N. Papanikolopoulos, “Jensen-bregman logdet divergence with application to efficient similarity search for covariance matrices,” _IEEE Transactions on Pattern Analysis and Machine Intelligence_ , vol. 35, no. 9, pp. 2161–2174, 2013.
  * [23] X. Hua, Y. Cheng, H. Wang, Y. Qin, and D. Chen, “Geometric target detection based on total Bregman divergence,” _Digital Signal Processing_ , vol. 75, pp. 232–241, 2018.
  * [24] X. Hua, Y. Cheng, H. Wang, and Y. Qin, “Information geometry for covariance estimation in heterogeneous clutter with total Bregman divergence,” _Entropy_ , vol. 20, no. 4, p. 258, 2018.
  * [25] X. Hua, Y. Cheng, H. Wang, Y. Qin, and Y. Li, “Geometric means and medians with applications to target detection,” _Iet Signal Processing_ , vol. 11, no. 6, pp. 711–720, 2017.
  * [26] W. Zhao, C. Liu, W. Liu, and M. Jin, “Maximum eigenvalue based target detection for the k distributed clutter environment,” _IET Radar, Sonar and Navigation_ , vol. 12, 08 2018.
  * [27] X. Hua, Y. Ono, L. Peng, Y. Cheng, and H. Wang, “Target detection within nonhomogeneous clutter via total bregman divergence-based matrix information geometry detectors,” _IEEE Transactions on Signal Processing_ , vol. 69, pp. 4326–4340, 2021.
  * [28] K. M. Wong, J.-K. Zhang, J. Liang, and H. Jiang, “Mean and median of psd matrices on a riemannian manifold: Application to detection of narrow-band sonar signals,” _IEEE Transactions on Signal Processing_ , vol. 65, no. 24, pp. 6536–6550, 2017.
  * [29] F. Barbaresco, “Coding statistical characterization of radar signal fluctuation for lie group machine learning,” in _2019 International Radar Conference (RADAR)_ , 2019, pp. 1–6.
  * [30] ——, “Innovative tools for radar signal processing based on cartan’s geometry of spd matrices information geometry,” in _2008 IEEE Radar Conference_ , 2008, pp. 1–6.
  * [31] X. Pennec, “Intrinsic statistics on riemannian manifolds: Basic tools for geometric measurements,” _Journal of Mathematical Imaging and Vision_ , vol. 25, pp. 127–154, 07 2006.
  * [32] R. M. Gray, “Toeplitz and circulant matrices: a review,” _Foundations and Trends in Communications and Information Theory_ , vol. 2, no. 3, pp. 155–239, 2005.
  * [33] M. Harandi, M. Salzmann, and R. Hartley, “Dimensionality reduction on spd manifolds: The emergence of geometry-aware methods,” _IEEE Transactions on Pattern Analysis and Machine Intelligence_ , vol. 40, no. 1, pp. 48–62, 2018\.
  * [34] Z. Yang, Y. Cheng, H. Wu, and H. Wang, “Enhanced matrix cfar detection with dimensionality reduction of riemannian manifold,” _IEEE Signal Processing Letters_ , vol. 27, pp. 2084–2088, 2020.
  * [35] S.-G. Hwang, “Cauchy’s interlace theorem for eigenvalues of hermitian matrices,” _American Mathematical Monthly_ , vol. 111, 02 2004.