#  Robust 2-D DOA Estimation in a Polarization Sensitive Single User
Environment

Md Imrul Hasan, Mohammad Saquib  Md Imrul Hasan, and Mohammad Saquib are with
the Department of Electrical and Computer Engineering, The University of Texas
at Dallas, Texas, USA (e-mail<EMAIL_ADDRESS>and
saquib@utdallas.edu).

###### Abstract

Apart from the conventional parameters (such as signal-to-noise ratio, array
geometry and size, sample size), several other factors (e.g. alignment of the
antenna elements, polarization parameters) influence the performance of
direction of arrival (DOA) estimating algorithms. When all the antenna
elements are identically aligned, the polarization parameters do not affect
the steering vectors, which is the underlying assumption of all the
conventional DOA algorithms. Unfortunately, in this case, for a given set of
DOA angles there exists a range of polarization parameters which could result
in a very low signal-to-noise ratio (SNR) across all the antenna elements in
the array. To avoid this type of catastrophic event, different antenna element
needs to be aligned differently. However, this fact will make almost all
commonly used DOA estimation algorithms non-operable, since the steering
vectors are contaminated by the polarization parameters. To the best of our
knowledge, no work in the literature addresses this issue even for a single
user environment. In this paper, that line of inquiry is pursued. We consider
a circular array with the minimum number of antenna elements and propose an
antenna alignment scheme to ensure that at any given point no more than one
element will suffer from significantly low SNR due to the contribution of
polarization. A low complexity algorithm that estimates the DOA angles in a
closed-form manner is developed. We treat MUSIC as the baseline algorithm and
demonstrate how it can reliably operate in all possible DOA and polarization
environments. Finally, a thorough performance and complexity analysis are
illustrated for the above two algorithms.

###### Index Terms:

Single source, DOA estimation, polarization, closed-form estimation, MUSIC,
antenna alignment, antenna mapping.

##  I Introduction

Direction of arrival (DOA) is a key issue in various important applications,
such as sonar, radar, medical sector, astronomy, defense operations,
navigation, geophysics, acoustic tracking, and so on [1, 2, 3, 4, 5].
Nowadays, owing to the exceptional development of modern technology and smart
devices, DOA estimation techniques have been widely used in wireless
communication and the internet of things (IoT) [6, 7, 8, 9, 10]. The
localization of a single narrow-band source by a passive sensor array has also
attracted tremendous interest in the literature due to its numerous
applications [11, 12, 13, 14, 15, 16, 17, 18]. Over several decades, extensive
studies have been performed, and numerous algorithms have been developed to
estimate the DOAs, i.e., multiple signal classification (MUSIC) [19], maximum-
likelihood (ML) [20], Capon [21], estimation of signal parameters via
rotational invariance techniques (ESPRIT) [22], Min-Norm [23], etc. Among all
these methods, the MUSIC algorithm is especially noteworthy due to its easy
implementation with different array structures.

Antenna array geometry plays an important role while implementing the DOA
estimation algorithms. To estimate the DOAs, linear array geometry is
extensively studied in the literature [24, 25, 26]. However, the important
drawback of a linear antenna array is the 1-D angle scanning. The circular
antenna array (UCA) is offered to overcome the problem owing to its advantage
of providing 360-degree azimuthal coverage as well as the elevation
information of the DOAs [27, 28, 29, 30, 31]. The polarization sensitivity of
the array to incident signals needs to be considered in the DOA estimation.
The received signal power is greatly affected by polarization while forming a
transmitter-receiver pair. When the polarization of the receiver antenna
matches with the transmitter antenna’s polarization, the receiver collects the
signal with the maximum possible power. On the other hand, a polarization
mismatch between the transmitter-receiver pair can result in severe
degradation of the received signal power. The polarization of the transmitter
can be previously known; however, the polarization state of the transmitted
signal can change when the electromagnetic wave scatters from a target.
Therefore, keeping all the antenna elements in the same direction to match the
polarization of the transmitter can often cause to receive the signal with a
very low signal-to-noise ratio (SNR). Contrarily, using different directions
for different antennas will affect the elements of the steering vector
differently. This could lead the regular DOA estimation algorithms to exhibit
unreliable performance.

Two kinds of strategies have been used in the existing literature to eliminate
the effects of polarization for DOA estimation algorithms. The works in the
first category [32, 33, 34] handle the problem utilizing complex hardware. In
[32], a uniform linear array with crossed dipoles was used for ESPRIT [22] to
jointly estimate the DOAs and the polarization parameters. The MUSIC algorithm
of joint polarization-DOA estimation based on the polarization-sensitive
circular array with a crossed dipole is discussed in [33]. A uniform linear
crossed tripole array is introduced for a dimension-reduction-based MUSIC
algorithm in [34]. The works in the second category [35] managed the
polarization issue by facing all the antenna elements towards the same
direction which can cause them to suffer from low SNR due to the contribution
of the polarization. A handful of works adopted both the categories at the
same time, e.g. [36, 37] have performed the estimation using a circular vector
sensor array comprising of co-centered orthogonal loop and dipole (COLD) pairs
placing all of them in the same direction.

In this paper, we present how to localize a single narrowband source using a
UCA of simple short dipole antenna elements just employing signal processing
techniques. Here, we want to utilize the minimum number of antenna elements,
and at the same time, ensure that no more than one antenna element suffers
from significantly low received power or SNR owing to the contribution of
polarization. This fact leads us to mathematically develop an antenna
alignment scheme, which also helps us to cancel the effects of polarization
from the non-signal subspace while implementing the popular MUSIC algorithm.
The entire algorithm is referred to as C-MUSIC. While cleansing the non-signal
subspace for MUSIC, we develop a reduced complexity algorithm, namely CF, that
estimates the DOA angles in a closed-form manner. The other major
contributions of this paper can be outlined as follows:

  1. 1.

As claimed, under the proposed antenna element alignment scheme, no more than
one element will suffer from very low received power due to the polarization
contribution. A decision threshold $K$ is introduced to decide whether the
antenna element with the smallest received power should be considered in the
process of DOA estimation or not.

    1. (a)

Two methods are developed to design the threshold $K$. The
${1}^{\underline{\text{st}}}$ method is based on the well-known Neyman-Pearson
lemma and the ${2}^{\underline{\text{nd}}}$ one is based on the central limit
theorem (CLT).

    2. (b)

An extensive probabilistic analysis of $K$ is performed in order to provide
significant insights into the impact of $K$ on the system performance.

  2. 2.

Two non-signal subspace cleansing methods are demonstrated for C-MUSIC.

    1. (a)

One of the methods can be employed only when the smallest received power among
all the antenna elements is below the threshold $K$. The other one applies to
all possible received power scenarios.

  3. 3.

A through performance and complexity study is performed between C-MUSIC and CF
algorithms.

The rest of the paper is organized as follows: Section II presents the System
Model. In Section III, the problem statement is formulated and the DOA
estimation algorithms (C-MUSIC and CF) are developed in Section IV assuming no
background noise. Implementations of those algorithms in a noisy arbitrarily
polarized environment are discussed in Section V with analysis. In Section VI,
the numerical results are presented, and finally, the concluding remarks of
this paper are given in Section VII.

Notations: We use lowercase and uppercase bold letters to denote vectors and
matrices, respectively. Lowercase letters in italics are used to represent
scalars. The notation * refers to the complex conjugate of a scalar,
$[\cdot]^{T}$ refers to transpose, and $[\cdot]^{\dagger}$ denotes the
Hermitian of a matrix.

##  II System Model

Let’s consider a UCA of radius $r$ composed of arbitrarily aligned $N$
identical short dipole antennas. The elements are positioned on the
circumference of the circle on the $xy-$plane where the first element is
located on the $x-$axis; see Fig. 1. These surface antenna elements are low
profile, easily mountable, and very robust on a rigid surface [38].

Figure 1: Uniform circular array geometry.

The incident signal at the array is a complex unknown narrow-band signal
$s(m)$ coming from a far field source. The azimuth angle $\phi$ of this signal
is measured counterclockwise from the $x$ axis, and the elevation angle
$\theta$ is measured downward from the $z-$axis; where
$0^{\circ}\leq\phi<360^{\circ}$ and $0^{\circ}\leq\theta\leq 90^{\circ}$. The
signal is arbitrarily polarized, where $0^{\circ}\leq\eta<360^{\circ}$, and
$0^{\circ}\leq\gamma<90^{\circ}$ represent the polarization phase difference,
and the auxiliary polarization angle, respectively. These terms satisfy
$\tan\gamma={A_{y}}/{A_{x}}$, and $\eta=\phi_{y}-\phi_{x}$, where, $A_{y}$ and
$\phi_{y}$ are the electric field amplitude and the electric field phase along
the $y-$axis, respectively, and $A_{x}$, and $\phi_{x}$ are defined
accordingly for $x-$axis [39].

The output voltage from each short dipole is proportional to the electric
field component along the dipole axis and the length of the dipole [40].
Hence, the outputs of these identical elements parallel to the $x$, and $y$
axes will be proportional to the $x$, and $y$ components of the electric
field, respectively. According to [40, 37], the electric fields along $x$, and
$y$ axes are given by

$\begin{bmatrix}e_{\mathrm{x}}\\\
e_{\mathrm{y}}\end{bmatrix}=\begin{bmatrix}e^{j\eta}\sin\gamma\cos\theta\cos\phi-\sin\phi\cos\gamma\\\
e^{j\eta}\sin\gamma\cos\theta\sin\phi+\cos\phi\cos\gamma\end{bmatrix}\,.$ (1)

At the ${n}^{\underline{\text{th}}}$ dipole antenna element, the output
voltage is proportional to the electric field

$\displaystyle e_{n}$
$\displaystyle=e^{j\eta}\sin\gamma\cos\theta\cos(\phi-\zeta_{n})-\sin(\phi-\zeta_{n})\cos\gamma$
$\displaystyle=e_{\mathrm{x}}\cos\zeta_{n}+e_{\mathrm{y}}\sin\zeta_{n}\,,$ (2)

where $\zeta_{n}$ is the alignment angle of that element w.r.t. the $x-$axis;
see Fig. 2. Notice in (II) that the electric field components
$e_{\mathrm{x}}$, and $e_{\mathrm{y}}$ of (1) are special cases of $e_{n}$ for
$\zeta_{n}=0^{\circ}$, and $\zeta_{n}=90^{\circ}$, respectively. Equation (II)
also implies that for $\theta=90^{\circ}$, and $\gamma=90^{\circ}$,
irrespective of any antenna alignment angle $\zeta_{n}$, the received signal
power (RSP) = 0 at all the antenna elements due to the electric field
$e_{n}=0$. If that is the case, we decide the source is along the $xy$-plane,
since $\theta=90^{\circ}$, and the value of $\phi$ can not be determined. This
situation can be overcome by having one or more short dipole antenna elements
along the $z-$axis at the cost of losing low profile and very robust
properties of the surface antenna elements [38], which is beyond the scope of
this paper. Throughout the rest of this paper, we consider $\theta\neq
90^{\circ}$ or $\gamma\neq 90^{\circ}$ which is equivalent to $e_{x}\neq 0$ or
$e_{y}\neq 0$.

Figure 2: Electric field in short dipole antenna.

The steering term for the $n^{\underline{\mathrm{th}}}$ element shown in Fig.
1 relates to the source’s DOA angles $(\theta,\phi)$ as

${a_{\mathrm{s}_{n}}}=e^{j\frac{2\pi}{\lambda}r\sin\theta\cos(\phi-\beta_{n})},$
(3)

where $\beta_{n}={2\pi(n-1)}/{N}$; $n=1,2,3,...,N$, and $\lambda$ denotes the
wavelength of the target signal. The array manifold term for the
$n^{\underline{\mathrm{th}}}$ antenna element can be written as the product of
the associated electric field response and the steering terms [36, 37, 39] as

$a_{n}=e_{n}a_{\mathrm{s}_{n}}.$ (4)

At the $m^{\underline{\mathrm{th}}}$ snapshot, the output of element $n$ is

${{{{x}}}_{n}}(m)={a_{n}}s(m)+w_{n}(m)\,,$ (5)

where $w_{n}(m)$ is zero-mean white complex Gaussian noise with average power
$\sigma^{2}$, and spatially and temporally independent of $s(m)$.

Now collecting outputs from all $N$ antenna elements, we form:

$\mathbf{{x}}(m)=\mathbf{a}{s(m)}+\mathbf{w}(m),$ (6)

where

$\displaystyle\mathbf{a}=\begin{bmatrix}a_{{1}}\\\ a_{{2}}\\\ \vdots\\\
a_{{N}}\end{bmatrix};\mathbf{{x}}(m)=\begin{bmatrix}{x}_{1}(m)\\\
{x}_{2}(m)\\\ \vdots\\\
{x}_{N}(m)\end{bmatrix};\mathbf{w}(m)=\begin{bmatrix}{w}_{1}(m)\\\
{w}_{2}(m)\\\ \vdots\\\ {w}_{N}(m)\end{bmatrix}.$ (7)

Equation (6) will be processed to extract the desired DOA information using
C-MUSIC and CF algorithms. Both the algorithms will be developed soon assuming
no background noise (i.e., $\sigma=0$). Subsequently, we will show how those
algorithms will be implemented in real-life scenarios in the presence of
background noise. Before we do so, the importance of mitigating the
polarization contribution will be signified in the context of the conventional
MUSIC algorithm.

##  III Problem Formulation: Limitations of MUSIC

Let’s begin with a short description of the subspace based 2-D MUSIC
algorithm. It exploits the eigen structure of the auto-correlation matrix of
the received signal (6), which is

$\displaystyle\mathbf{R}$
$\displaystyle=\operatorname{\mathbb{E}}\\{\mathbf{{x}}(m)\mathbf{{x}}(m)^{\dagger}\\}$
$\displaystyle=\mathbf{a}s(m)s(m)^{\dagger}\mathbf{a}^{\dagger}+\operatorname{\mathbb{E}}\\{\mathbf{w}(m)\mathbf{w}(m)^{\dagger}\\}$
$\displaystyle=\sigma^{2}_{\mathrm{s},m}\mathbf{a}\mathbf{a}^{\dagger}+\sigma^{2}\mathbf{I}\,,$
(8)

where $\sigma^{2}_{\mathrm{s},m}=|s(m)|^{2}$ ,
$\mathbb{E}\\{\mathbf{w}(m)\mathbf{w}(m)^{\dagger}\\}=\sigma^{2}\mathbf{I}$,
and $\mathbf{I}$ is an $N\times N$ identity matrix. In practice, the above
auto-correlation matrix will be replaced by the sample auto-correlation matrix
averaged over $M$ time samples (or snapshots). Since a single source is
considered, the largest eigenvalue of this autocorrelation matrix is
corresponding to the signal subspace. The others are corresponding to the non-
signal (or noise) subspace which is defined as

$\mathbf{E}=[\mathbf{v}_{2},\mathbf{v}_{3},\cdot\cdot\cdot,\mathbf{v}_{N}]\,,$
(9)

where $\\{\mathbf{v}_{l}\\}_{l=1}^{N}$ denotes the eigenvector corresponding
to the real eigenvalue $\\{\lambda_{l}\\}_{l=1}^{N}$ of $\mathbf{R}$. These
eigenvalues are sorted in the descending order. Using the noise subspace
$\mathbf{E}$ (9), the MUSIC spectrum is defined as [41]

${S_{\mathrm{MUSIC}}}=\frac{1}{\mathbf{a}^{\dagger}\mathbf{E}{\mathbf{E}}^{\dagger}\mathbf{a}}\,.$
(10)

Next, we discuss the implementation requirement of the MUSIC algorithm in an
arbitrarily polarized environment. MUSIC needs at least two eigenvectors in
the noise subspace in order to estimate two DOA angles (azimuthal and
elevation) unambiguously. Therefore, the minimum required number of antenna
elements is three. Due to the polarization and DOA angles, the compound
steering element $|a_{n}|$ may be approximately $0$ in (4) at one (or more)
antenna depending on the alignment resulting in very low signal power.
Following is an example of this catastrophic situation.

Let’s assume all three antennas are aligned along the $x-$ axis, and the
incoming wave arrives with the azimuthal angle $\phi=90^{\circ}$, and the
auxiliary polarization angle $\gamma=90^{\circ}$. This set of parameters
yields received electric field $e_{n}=e_{\rm{x}}=0$, hence $a_{n}=0$, for all
$n$ which leads the received signal in (7) to be zero in the noiseless
scenario. In this situation, no regular DOA estimating algorithm will be able
to operate. To resolve this issue, one must use at least four element UCA with
different alignments for different elements such that the above harmful event
(i.e. $|a_{n}|\approx 0$) can’t affect more than one antenna element.

Figure 3: MUSIC spectrum in a polarization sensitive environment.

Unfortunately, a four element UCA with different alignments is not even enough
for the conventional MUSIC to operate. It is due to the fact that the
conventional MUSIC assumes $e_{n}=e_{n^{\prime}}$ for $n\neq n^{\prime}$ in
(II), that results the non-signal subspace free from the contribution of the
polarization parameters. It is not true when the antenna elements are aligned
differently. At the $n^{\underline{\mathrm{th}}}$ element, the compound term
in (4) can be written as

$a_{n}=e_{n}a_{s_{n}}=|{e_{n}}|e^{j\delta_{n}}|a_{s_{n}}|e^{j\psi_{n}}\,,$
(11)

where $|a_{s_{n}}|=1$. Terms $\delta_{n}$ and $\psi_{n}$ are the phases of
voltage $e_{n}$, and steering element $a_{s_{n}}$, respectively. The element
$(p,q)$ of the auto-correlation matrix $\mathbf{R}$ is
$\sigma^{2}_{\mathrm{s},m}|e_{p}||e_{q}|e^{j(\delta_{p,q}+\psi_{p,q})}$, where
$\delta_{p,q}=\delta_{p}-\delta_{q}$, and $\psi_{p,q}=\psi_{p}-\psi_{q}$.
Since all the antenna elements have different alignments, $e_{p}\neq e_{q}$
for $p\neq q$ and therefore, different elements of the auto-correlation matrix
$\mathbf{R}$ are affected by the polarization parameters differently. This
forces the associated non-signal subspace to deviate from the desired one,
which is free from the polarization parameters. This could cause failure of
the MUSIC algorithm; see Fig. 3, where MUSIC spectrum is plotted using the DOA
angle pair, $\theta=30^{\circ}$, $\phi=60^{\circ}$, and the polarization
parameter pair, $\gamma=45^{\circ}$, $\eta=45^{\circ}$, and alignment angle of
antenna element $n$, $\zeta_{n}=(n-1)\times 30^{\circ}$, where $n=1,2,3,4$.
More discussion and analysis on this issue will be found in the following
section. Note that MUSIC performs perfectly for the above set of DOA and
polarization angles, if all the antenna elements are aligned in the same
direction (i.e., $\zeta_{n}=\zeta_{n^{\prime}}$ for $n\neq n^{\prime}$).
However, this same alignment could result the catastrophic incident for a
different set of DOA and polarization angles as demonstrated in the earlier
example.

The above two examples jointly suggest that in a single user arbitrarily
polarized environment to find the DOA angles, different alignment must be used
for different elements, and the implementation of MUSIC requires cleansing of
the polarization parameters from the non-signal subspace. Such schemes will be
developed next considering all possible system conditions. Thus, in this work
MUSIC is referred to as C-MUSIC, where the alphabet ‘C’ stands for the
additional signal processing task (cleansing operation of the polarization
contribution) performed prior to implementing the conventional MUSIC
algorithm. In the following section, we will also demonstrate that after
cleansing, how one can estimate the DOA angles in a closed-form manner using
the CF algorithm.

##  IV Estimation of the DOA Angles: C-MUSIC and CF

Before start elaborating upon the C-MUSIC and CF algorithms, let’s derive the
antenna alignment guideline to ensure that the effects of the polarization
should not be able to hurt more than one antenna element in terms of RSP as
long as $e_{x}\neq 0$ or $e_{y}\neq 0$. In that regard, the following theorem
would be useful.

######  Theorem 1.

If antenna elements $n$ and $n^{\prime}\neq n$ are aligned such that
$\zeta_{n^{\prime}}\neq\zeta_{n}$ or $\zeta_{n^{\prime}}\neq\zeta_{n}\pm
180^{\circ}$, the RSP can’t be simultaneously zero at antenna $n$ and
$n^{\prime}$ when $e_{x}\neq 0$ or $e_{y}\neq 0$.

###### Proof:

Let’s assume RSP = 0 at antenna element $n$ and equivalently from (II), we get
the received electric field at antenna $n$

$e_{n}=e_{x}\cos\zeta_{n}+e_{y}\sin\zeta_{n}=0\,.$ (12)

Without loss of generality, we can write the alignment angle of antenna
$n^{\prime}\neq n$ as

$\zeta_{n^{\prime}}=\zeta_{n}+\zeta\,,$ (13)

where $0\leq\zeta<360^{\circ}$. Applying the above equation and (12) to (II),
and after simplification, the received electric voltage at antenna
$n^{\prime}$ becomes

$e_{n^{\prime}}=\sin\zeta\left({e_{x}\sin\zeta_{n}-e_{y}\cos\zeta_{n}}\right)\,.$
(14)

If $e_{x}\neq 0$ or $e_{y}\neq 0$, (12) implies

$e_{x}\sin\zeta_{n}-e_{y}\cos\zeta_{n}\neq 0\,,$

equivalently

$e_{n^{\prime}}\neq 0\,,$

unless $\zeta\neq 0$ (i.e., $\zeta_{n^{\prime}}\neq\zeta_{n}$ ) or $\pm
180^{\circ}$ (i.e., $\zeta_{n^{\prime}}\neq\zeta_{n}\pm 180^{\circ}$) in (13).
This proves the theorem. ∎

Condition $\zeta_{n^{\prime}}\neq\zeta_{n}$ for $n\neq n^{\prime}$ in Theorem
1 implies that all 4 antenna elements must be aligned differently. In
addition, if we choose the antenna element alignment such that
$0^{\circ}\leq\zeta_{n}<180^{\circ}$ for all $n$, the other requirement
$\zeta_{n^{\prime}}\neq\zeta_{n}\pm 180^{\circ}$ in Theorem 1 will be met. One
such array that meets both criteria of Theorem 1 is depicted in Fig. 4. Here,
it can be seen that antenna 1 and 3 are aligned along the $x$\- and $y$\-
axis, respectively. Antenna 2 and 4 are aligned by $45^{\circ}$, and
$135^{\circ}$, respectively, with respect to the $x$-axis. C-MUSIC and CF
algorithms now will be developed using the above antenna element alignment
scheme.

Figure 4: Proposed antenna alignment scheme for a UCA with $N=4$.

###  IV-A C-MUSIC

Theorem 1 and the antenna alignment scheme given in Fig. 4 jointly result in
five different RSP scenarios with the presence of a target. We categorize
those scenarios into two primary events. In Event 1, $a_{n}=0$ only at antenna
element $n$; since $n=1,2,3,4$, there are four sub-events. Term $\Omega_{1,n}$
represents the ${{n}^{\underline{\text{th}}}}$ sub-event of Event 1. In Event
2 (denoted by $\Omega_{2}$), $a_{n}\neq 0$ at all the antenna elements. In
practice, Event 2 occurs much more frequently than Event 1, since the former
to happen requires a special combination of polarization parameters ($\eta$
and $\gamma$) for a given pair of DOA angles ($\phi$ and $\theta$).
Nevertheless, the DOA estimation algorithms must operate efficiently in each
of the above possible events. Structures of the signal and non-signal subspace
will vary from one event to another. This fact requires a dedicated
polarization suppression method for each of the five effective events. As a
result, to apply the right method, the event that has occurred must be
correctly identified. For the purpose of the demonstration, we assume the
perfect detection of the events given the RSP scenarios at all the antennas.
Later, we will present two threshold-based techniques to operate the algorithm
in a real-life scenario. Let’s begin with Event 1 and demonstrate how to
implement C-MUSIC.

####  IV-A1 Event 1

Firstly, we consider the first sub-event of event 1, i.e., $\Omega_{1,1}$. At
the ${m}^{\underline{\text{th}}}$ sample, for $\beta_{n}={2\pi(n-1)}/{N}$ and
$\zeta_{n}=(n-1)\times 45^{\circ}$, the signal component of the received
signal (6) is

$\mathbf{{x}}(m)=\mathbf{s}_{\mathrm{t}}s(m)=\begin{bmatrix}0\\\
\frac{1}{\sqrt{2}}e_{y}e^{j\kappa_{2}}\\\ e_{y}e^{-j\kappa_{1}}\\\
\frac{1}{\sqrt{2}}e_{y}e^{-j\kappa_{2}}\end{bmatrix}s(m)\,,$ (15)

where $\kappa_{1}=\kappa\cos\phi$, $\kappa_{2}=\kappa\sin\phi$ and
$\kappa=2\pi r\sin\theta/\lambda$. Removing the first element (i.e., $0$) from
the original received signal vector, we form a new received signal vector as

$\mathbf{{\tilde{x}}}(m)=e_{y}\mathbf{\tilde{s}}_{\mathrm{t}}s(m),\,$ (16)

where
$\mathbf{\tilde{s}}_{\mathrm{t}}=\begin{bmatrix}\frac{1}{\sqrt{2}}e^{j\kappa_{2}}\quad
e^{-j\kappa_{1}}\quad\frac{1}{\sqrt{2}}e^{-j\kappa_{2}}\end{bmatrix}^{\top}\,.$
Note that in the newly formed received signal vector (16), the polarization
contributes two multiplicative terms, one is $e_{y}$, and the other one is
$1/\sqrt{2}$ (in the ${1}^{\underline{\text{st}}}$ and
${3}^{\underline{\text{rd}}}$ elements of vector
$\mathbf{\tilde{s}}_{\mathrm{t}}$). Vector $\mathbf{\tilde{s}}_{\mathrm{t}}$
carries useful information pertaining to the DOA. In event 1, C-MUSIC can be
implemented in two ways, the first method, namely Method 1, executes by
cleaning the non-signal subspace of the auto-correlation matrix of the
received signal (16) and, the second method, namely Method 2, operates
estimating $\kappa_{1}$ and $\kappa_{1}$. Next we describe Method 1.

##### Method 1

We can write the auto-correlation matrix of the received signal as

$\mathbf{{\tilde{R}}}=|e_{y}|^{2}\sigma_{\mathrm{s},m}^{2}\mathbf{\tilde{s}}_{\mathrm{t}}\mathbf{\tilde{s}}_{\mathrm{t}}^{\dagger}=|e_{y}|^{2}\sigma_{\mathrm{s},m}^{2}\mathbf{{\tilde{R}}}_{\mathrm{s}},$
(17)

where

$\mathbf{{\tilde{R}}}_{\mathrm{s}}=\begin{bmatrix}\frac{1}{2}&\frac{1}{\sqrt{2}}e^{j(\kappa_{2}+\kappa_{1})}&\frac{1}{2}e^{2j\kappa_{2}}\\\
\frac{1}{\sqrt{2}}e^{-j(\kappa_{1}+\kappa_{2})}&1&\frac{1}{\sqrt{2}}e^{-j(\kappa_{1}-\kappa_{2})}\\\
\frac{1}{2}e^{-2j\kappa_{2}}&\frac{1}{\sqrt{2}}e^{j(\kappa_{1}-\kappa_{2})}&\frac{1}{2}\end{bmatrix}\,.$

To develop C-MUSIC, we derive a subspace from $\mathbf{{\tilde{R}}}$ identical
to the non-signal subspace of

$\mathbf{{\tilde{R}}}_{\mathrm{s,c}}=\mathbf{\tilde{s}}_{\mathrm{t,c}}\mathbf{\tilde{s}}_{\mathrm{t,c}}^{\dagger},$

where $\mathbf{\tilde{s}}_{\mathrm{t,c}}=\begin{bmatrix}e^{j\kappa_{2}}\quad
e^{-j\kappa_{1}}\quad e^{-j\kappa_{2}}\end{bmatrix}^{\top}$ is the clean
version of the compounded steering vector $\mathbf{\tilde{s}}_{\mathrm{t}}$ in
(16). The above auto-correlation matrix $\mathbf{{\tilde{R}}}_{\mathrm{s,c}}$
can be shown as

$\mathbf{{\tilde{R}}}_{\mathrm{s,c}}=\begin{bmatrix}1&e^{j(\kappa_{2}+\kappa_{1})}&e^{2j\kappa_{2}}\\\
e^{-j(\kappa_{1}+\kappa_{2})}&1&e^{-j(\kappa_{1}-\kappa_{2})}\\\
e^{-2j\kappa_{2}}&e^{j(\kappa_{1}-\kappa_{2})}&1\end{bmatrix}\,.$ (18)

We now prove the following theorem to derive the non-signal subspace of
$\mathbf{{\tilde{R}}}_{\mathrm{s,c}}$ from $\mathbf{{\tilde{R}}}$ of (17).

######  Theorem 2.

The non-signal subspace of $\mathbf{{\tilde{R}}}_{\mathrm{s,c}}$ in (18) is
the null space of $\mathbf{{\tilde{R}}}\mathbf{{F}}_{1}$, where
$\mathbf{F}_{1}=\mathrm{diag}{\left[{1\quad\frac{1}{\sqrt{2}}\quad
1}\right]}$.

###### Proof:

Let $\mathbf{{v}}=\begin{bmatrix}v_{1}\quad v_{2}\quad
v_{3}\end{bmatrix}^{\top}$ be a vector belonging to the non-signal subspace of
$\mathbf{{\tilde{R}}}_{s,c}$, then

$\begin{bmatrix}1&e^{j(\kappa_{2}+\kappa_{1})}&e^{2j\kappa_{2}}\\\
e^{j(\kappa_{1}+\kappa_{2})}&1&e^{-j(\kappa_{1}-\kappa_{2})}\\\
e^{-2j\kappa_{2}}&e^{j(\kappa_{1}-\kappa_{2})}&1\end{bmatrix}\begin{bmatrix}v_{1}\\\
v_{2}\\\ v_{3}\end{bmatrix}=\begin{bmatrix}0\\\ 0\\\ 0\end{bmatrix}\,.$ (19)

Since $\mathbf{{\tilde{R}}}\mathbf{{F}}_{1}\mathbf{{v}}$ equals to

$|e_{y}|^{2}\sigma_{\mathrm{s},m}^{2}\begin{bmatrix}\frac{1}{2}&\frac{1}{2}e^{j(\kappa_{2}+\kappa_{1})}&\frac{1}{2}e^{2j\kappa_{2}}\\\
\frac{1}{\sqrt{2}}e^{-j(\kappa_{1}+\kappa_{2})}&\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}e^{-j(\kappa_{1}-\kappa_{2})}\\\
\frac{1}{2}e^{-2j\kappa_{2}}&\frac{1}{2}e^{j(\kappa_{1}-\kappa_{2})}&\frac{1}{2}\end{bmatrix}\begin{bmatrix}v_{1}\\\
v_{2}\\\ v_{3}\end{bmatrix}\,,$

applying (19) to the above expression, we get

$\mathbf{{\tilde{R}}}\mathbf{{F}}_{1}\mathbf{{v}}=\begin{bmatrix}0\quad 0\quad
0\end{bmatrix}^{\top},$

and we prove the theorem. ∎

Next, we find the angles corresponding to the maximum spectrum value in (10)
utilizing the null space of $\mathbf{{\tilde{R}}}\mathbf{{F}}_{1}$ and use
those as the DOA estimates. Theorem 2 guides us how to obtain the non-signal
subspace for implementing C-MUSIC when $a_{n}=0$ for $n=1$. Similarly, when
$a_{n}=0$ for $n\neq 1$, the auto-correlation matrix $\mathbf{{\tilde{R}}}$
will be formed discarding the received signal samples from that antenna
element. As a result, the structure of $\mathbf{{\tilde{R}}}$ changes with $n$
that change the structure of $\mathbf{F}_{n}$ as follows:

$\mathbf{F}_{2}=\mathrm{diag}{\left[{1\quad-1\quad-\frac{1}{\sqrt{2}}}\right]};\quad\mathbf{F}_{3}=\mathrm{diag}{\left[{1\quad\sqrt{2}\quad-\sqrt{2}}\right]};$

and

$\mathbf{F}_{4}=\mathbf{F}_{1}\,.$

The derivations of $\\{\mathbf{{F}}_{n}\\}_{n=2}^{4}$ are similar to
$\mathbf{F}_{1}$ and omitted for conciseness. Let’s focus again on antenna
when RSP = 0 at $n=1$, and elaborate upon Method 2.

##### Method 2

The following lemma, which can be straightforwardly obtained from (15), forms
the basis for this method.

######  Lemma 1.

The phase parameters $\\{\kappa_{i}\\}_{i=1}^{2}$ of the steering elements can
be derived from each sample of the received signal vector
$\mathbf{{\tilde{x}}}(m)$ in (15) as

$\kappa_{2}=\frac{1}{2}\angle\\{{x}_{2}(m){x}^{*}_{4}(m)\\},$

and

$\kappa_{1}=\angle\\{{x}_{2}(m){x}^{*}_{3}(m)\\}-\kappa_{2}\,.$

In a noisy environment, applying Lemma 1 to each sampled received signal, and
then averaging those over $M$ samples, we get the estimates of $\kappa_{1}$
and $\kappa_{2}$ which are denoted by $\hat{\kappa}_{1}$ and
$\hat{\kappa}_{2}$, respectively. The auto-correlation matrix
$\mathbf{{\tilde{R}}}_{\mathrm{s,c}}$ (18) can be estimated using those
estimates. After that MUSIC algorithm can be readily applied.111Note that an
auto-correlation matrix of size $4\times 4$ can be formed by padding
$e^{j\hat{\kappa}_{1}}$ at the top of the newly estimated steering vector
$\mathbf{\tilde{s}}_{\mathrm{t,c}}$. However, this padding will add complexity
to the MUSIC algorithm.

When $a_{n}=0$ for $n\neq 1$, the estimation procedure of $\kappa_{1}$ and
$\kappa_{2}$ changes as given in the following lemma:

######  Lemma 2.

For event $\Omega_{1,2}$ (i.e., when $a_{n}=0$ for $n=2$), the phase
parameters $\\{\kappa_{i}\\}_{i=1}^{2}$ satisfy

$\kappa_{1}=\frac{1}{2}\angle\\{-{x}_{1}(m){x}^{*}_{3}(m)\\},$

and

$\kappa_{2}=\angle\\{-{x}_{1}(m){x}^{*}_{4}(m)\\}-\kappa_{1}\,.$

For event $\Omega_{1,3}$ (i.e., when $a_{n}=0$ at $n=3$), the phase parameters
$\\{\kappa_{i}\\}_{i=1}^{2}$ satisfiy

$\kappa_{2}=\frac{1}{2}\angle\\{-{x}_{2}(m){x}^{*}_{4}(m)\\},$

and

$\kappa_{1}=\angle\\{{x}_{1}(m){x}^{*}_{2}(m)\\}+\kappa_{2}\,.$

For event $\Omega_{1,4}$ (i.e., when $a_{n}=0$ at $n=4$), the phase parameters
$\\{\kappa_{i}\\}_{i=1}^{2}$ satisfy

$\kappa_{1}=\frac{1}{2}\angle\\{{x}_{1}(m){x}^{*}_{3}(m)\\},$

and

$\kappa_{2}=\angle\\{{x}_{2}(m){x}^{*}_{1}(m)\\}+\kappa_{1}\,.$

####  IV-A2 Event 2

Now, we consider the event where $a_{n}\neq 0$ for all $n$, i.e.,
$\Omega_{2}$. The contribution of the target in the received signal (6) at the
${m}^{\underline{\text{th}}}$ sample is

$\mathbf{{x}}(m)=\mathbf{s}_{\mathrm{t}}s(m),$ (20)

where the joint contribution of the steering vector and the polarization
parameters is embedded in

$\mathbf{s}_{\mathrm{t}}=\begin{bmatrix}e_{x}e^{j\kappa_{1}}\\\
\frac{1}{\sqrt{2}}(e_{x}+e_{y})e^{j\kappa_{2}}\\\ e_{y}e^{-j\kappa_{1}}\\\
\frac{1}{\sqrt{2}}(-e_{x}+e_{y})e^{-j\kappa_{2}}\end{bmatrix}\,.$ (21)

We want vector $\mathbf{s}_{\mathrm{t}}$ (21) to be free from terms $e_{x}$
and $e_{y}$ as follows:

$\mathbf{s}_{\mathrm{t,c}}=\begin{bmatrix}e^{j\kappa_{1}}\\\
e^{j\kappa_{2}}\\\ e^{-j\kappa_{1}}\\\ e^{-j\kappa_{2}}\ \end{bmatrix}\,.$
(22)

Using (21), it can be shown that unlike Event 1, the auto-correlation matrix
of $\mathbf{{x}}(m)$ is a joint function of $e_{x}$ and $e_{y}$ and the
implementation of Method 1 in this case requires the knowledge of their ratio.
This fact makes Method 1 very difficult to implement for Event 2, whereas, the
principle of implementation of Method 2 remains unchanged. Similar to Event 1,
Method 2 operates estimating $\\{\kappa_{i}\\}_{i=1}^{2}$. However, the
estimation technique is different now, since the received signal vector
$\mathbf{{x}}(m)$ in (20) is different than that in Event 1. After estimating
$\\{\kappa_{i}\\}_{i=1}^{2}$, a clean steering vector
$\mathbf{s}_{\mathrm{t,c}}$ will be formed (22). The non-signal subspace of
its auto-correlation matrix will be used for C-MUSIC.

To estimate $\\{\kappa_{i}\\}_{i=1}^{2}$, we need to prove the following two
lemmas.

######  Lemma 3.

If

$c_{1}(m)=x_{1}(m)x^{*}_{2}(m)+x^{*}_{3}(m)x_{4}(m)\,,$

where $x_{i}(m)$ is the ${i}^{\underline{\text{th}}}$ element of
$\mathbf{{x}}(m)$ (20), then

$\angle c_{1}(m)=\kappa_{1}-\kappa_{2}.$ (23)

###### Proof:

Equation (20) yields

$x_{1}(m)x^{*}_{2}(m)=\frac{1}{\sqrt{2}}\left({|e_{x}|^{2}+e_{x}e^{*}_{y}}\right)e^{j(\kappa_{1}-\kappa_{2})}\sigma_{s,m}^{2},$

and

$x^{*}_{3}(m)x_{4}(m)=\frac{1}{\sqrt{2}}\left({-e_{x}e^{*}_{y}+|e_{y}|^{2}}\right)e^{j(\kappa_{1}-\kappa_{2})}\sigma_{s,m}^{2}.$

Now we add the above two equations to prove the lemma. ∎

######  Lemma 4.

If

$c_{2}(m)=-x_{1}(m)x^{*}_{4}(m)+x^{*}_{3}(m)x_{2}(m)\,,$

then

$\angle c_{2}(m)=\kappa_{1}+\kappa_{2}\,.$ (24)

The proof of the above lemma is similar to that of Lemma 23 and due to the
conciseness, it is omitted. The above two lemmas straightforwardly yield the
estimates of $\kappa_{1}$ and $\kappa_{2}$ for each sample $m$. After
processing $M$ samples, the auto-correlation matrix for the steering vector
$\mathbf{s}_{\mathrm{t,c}}$ can be formed to implement MUSIC.222To estimate
both the DOA angles, we need at least a steering vector of length 3. Therefore
an auto-correlation matrix of size $3\times 3$ can be formed by ignoring one
of the elements of the newly estimated steering vector
$\mathbf{s}_{\mathrm{t,c}}$ and it will save some computational complexity.
Now, we will show how the CF algorithm will be implemented using the estimates
of $\kappa_{1}$ and $\kappa_{2}$.

###  IV-B CF

Recall that

$\kappa_{1}=\kappa\cos\phi;\quad\kappa_{2}=\kappa\sin\phi;$

and

$\kappa=\sqrt{\kappa_{1}^{2}+\kappa_{2}^{2}}=2\pi r\sin\theta/\lambda\,.$ (25)

In (25), parameter $\kappa=0$ or $\kappa\neq 0$.

  1. 1.

When $\kappa=0$, the elevation angle $\theta=0^{\circ}$ indicating that the
target is along the $z-$axis and the azimuthal angle has no significance.

  2. 2.

When $\kappa\neq 0$, parameter $\kappa_{1}$ and $\kappa_{2}$ can’t be
simultaneously zero due to the elevation angle $0<\theta\leq 90^{\circ}$ in
(25). The above 3 expressions will be used to estimate DOA angles in a closed-
form manner considering following 3 combinations of the estimated $\kappa_{1}$
and $\kappa_{2}$.

    1. (a)

If ${\kappa}_{1}\neq 0$ and ${\kappa}_{2}\neq 0$, we estimate the azimuthal
angle

${\phi}=\arctan\\{{\kappa}_{2}/{\kappa}_{1}\\}\,,$

and since $\kappa>0$, the sign of ${\kappa}_{1}$ or ${\kappa}_{2}$ will help
to estimate $\phi$ without ambiguity. We now use (25) to estimate of the
elevation angle as

${\theta}=\arcsin\left({{\kappa}\lambda/2\pi r}\right)\,.$

    2. (b)

If ${\kappa}_{1}=0$ and ${\kappa}_{2}\neq 0$, the estimate of the azimuthal
angle is either $90^{\circ}$ (for ${\kappa}_{2}>0$) or $270^{\circ}$ (for
${\kappa}_{2}<0$). The estimate of term $\kappa$ is $|{\kappa_{2}}|$ which
yields the estimate of the elevation angle as

${\theta}=\arcsin\left({|{\kappa}_{2}|\lambda/2\pi r}\right)\,.$

    3. (c)

If ${\kappa}_{1}\neq 0$ and ${\kappa}_{2}=0$, the estimate of the azimuthal
angle is either $0^{\circ}$ (for ${\kappa}_{1}>0$) or $180^{\circ}$ (for
${\kappa}_{1}<0$). The estimate of term $\kappa$ is $|{\kappa_{1}}|$ which
yields the estimate of the elevation angle as

${\theta}=\arcsin\left({|{\kappa}_{1}|\lambda/2\pi r}\right)\,.$

Next, we demonstrate how the above two algorithms (C-MUSIC and CF) will be
implemented in the presence of background noise. Here, the estimates of
$\kappa_{1}$ and $\kappa_{2}$ may not be zero and the CF algorithm will
execute following the above conditions given in 2a.

Figure 5: Normalized threshold $K$ vs number of samples $M$.

##  V Implementation of the Algorithms and Analysis

In the above section, the working principles of the C-MUSIC and CF algorithms
were described using the following two assumptions:

  1. 1.

Assumption 1: The system is noise free (i.e., $\sigma^{2}=0$).

  2. 2.

Assumption 2: Due to the contribution of the signal polarization, under the
proposed antenna element assignment only one antenna element could suffer in
terms of zero RSP, and that antenna element is known.

As said, now we want to make the desired algorithms practically realizable. As
a result, we first relax Assumption 1 by introducing background noise in the
system with known average noise power $\sigma^{2}>0$. However, for an unknown
environment, this power can be easily estimated [42, 43]. To relax Assumption
2, we bring in a decision threshold $K$ to check whether the received power at
antenna $n$ is below or above this threshold. In particular, after receiving
$M$ samples of the received signal $x_{n}(m)$, the RSP at the antenna element
$n$ can be estimated as

$P_{n}=\frac{1}{M}\sum_{m=1}^{M}|{x}_{n}(m)|^{2}.$ (26)

In the presence of a target, if $P_{n}<K$, we decide in the favor of the
hypothesis

  * •

$H_{0}$: signal is not present at antenna element $n$ due to the compound
steering element $a_{n}=0$ in (11),

otherwise, we conclude hypothesis

  * •

$H_{1}$: signal is present at antenna element $n$, and the compound steering
element $a_{n}\neq 0$.

Figure 6: Probability of correctly identifying Event 1.

Note that in low received signal-to-noise ratio (RSNR) (i.e.,
$\frac{|a_{n}|^{2}}{M}\sum_{m=1}^{M}\sigma^{2}_{\mathrm{s},m}/\sigma^{2}$)
scenarios, RSP tests at multiple antennas may favor hypothesis $H_{0}$ for
more than one antenna. If that is the case, our proposed antenna element
alignment in Fig. 4 dictates us to discard the antenna output with the lowest
measured RSP. Our analysis will shortly reveal that such a scenario occurs
with negligible probability within the desired operating regime of RSNR.

Now, we discuss how to find the decision threshold $K$. A popular way of
designing the threshold $K$ is based on the Neyman-Pearson lemma. Usually, a
prefixed significance level $\alpha$ is set (e.g., $\alpha=0.001$) to restrict
the probability of making a Type I error [i.e.,
$1-\mathrm{P}\left({H_{0}|\Omega_{1,n}}\right)$] to a certain percentage (in
this case, 0.1%). Next, a test is chosen to minimize Type II error [i.e.,
$1-\mathrm{P}\left({H_{1}|a_{n}\neq 0}\right)$]. Due to the monotonic nature
of Type II error w.r.t. $K$, the value of $K$ will be determined by solving
$\mathrm{P}\left({H_{0}|\Omega_{1,n}}\right)=1-\alpha$. Next, we present the
derivation of $\mathrm{P}\left({H_{0}|\Omega_{1,n}}\right)$.

###  V-A Derivation of $\mathrm{P}\left({H_{0}|\Omega_{1,n}}\right)$

We propose two different techniques to derive
$\mathrm{P}\left({H_{0}|\Omega_{1,n}}\right)$ in order to find $K$. The first
technique, namely Technique 1, is based on the probability density function
(PDF) of $P_{n}$ and the second technique, namely Technique 2, applies the
central limit theorem (CLT) to (26) and models $P_{n}$ as a Gaussian random
variable. Now, we elaborate upon Technique 1.

####  V-A1 Technique 1

Given $\Omega_{1,n}$, the received signal samples only contain noise. As a
result, $|{x}_{n}(m)|^{2}$ in (26) is a sum of two squared independent
identically distributed (i.i.d) Gaussian random variables each of which has a
mean of 0 and a variance of $\sigma^{2}/2$. Thus, $|{x}_{n}(m)|^{2}$ is a
central chi-square random variable with degrees of freedom $\rm{(DoF)}=2$. As
the received power $P_{n}$ in (26) is the average of $|{x}_{n}(m)|^{2}$ for
$M$ samples, it also follows a central chi-square distribution with
$\mathrm{DoF}=2M$. The probability density function (PDF) of a central chi-
square random variable with DoF = $\nu$ is [44]

$f_{\nu}(w)=\left\\{\begin{array}[c]{ll}0&w\leq 0\,,\\\
\frac{w^{\frac{\nu}{2}-1}e^{-\frac{w}{2}}}{2^{\frac{\nu}{2}}\Gamma(\frac{\nu}{2})}&w>0,\\\
\end{array}\right.$ (27)

Figure 7: Probability of correctly identifying Event 2.

and the corresponding cumulative distribution function (CDF) is

$F_{\nu}(u)=\int_{0}^{u}f_{\nu}(w)\,dw=\left\\{\begin{array}[c]{ll}0&u\leq
0\,,\\\
\frac{\Gamma_{u}(\frac{\nu}{2},\frac{u}{2})}{\Gamma(\frac{\nu}{2})}&u>0\,,\end{array}\right.$
(28)

where $\Gamma(\frac{\nu}{2})=\int_{0}^{\infty}t^{(\frac{\nu}{2}-1)}e^{-t}dt$
denotes the gamma function, and
$\Gamma_{u}(\frac{\nu}{2},\frac{u}{2})=\int_{0}^{\frac{u}{2}}t^{(\frac{\nu}{2}-1)}e^{-t}\,dt$
represents the lower incomplete gamma function. Now, we use (27) to write the
conditional PDF of the measured power $P_{n}$ given Event $\Omega_{1,n}$ as

$f_{P_{n}|\Omega_{1,n}}(w)=\frac{1}{\tilde{\sigma}^{2}}f_{\nu}\left({\frac{w}{\tilde{\sigma}^{2}}}\right)\,,$
(29)

where $\tilde{\sigma}^{2}=\sigma^{2}/2M$. Our goal was to find
$\mathrm{P}\left({H_{0}|\Omega_{1,n}}\right)=\mathrm{P}\left({P_{n}\leq
K|\Omega_{1,n}}\right)$, which is

$\int_{0}^{K}f_{P_{n}|\Omega_{1,n}}(w)\,dw=\int_{0}^{K}\frac{1}{\tilde{\sigma}^{2}}f_{\nu}\left({\frac{w}{\tilde{\sigma}^{2}}}\right)dw=F_{\nu}\left({\frac{K}{\tilde{\sigma}^{2}}}\right)\,.$
(30)

Last equality of (30) is obtained from (28). Now, we numerically solve
$F_{\nu}\left({\frac{K}{\tilde{\sigma}^{2}}}\right)=1-\alpha$ to find the
decision threshold $K$. For fixed $M$ and $\sigma^{2}$, the value of $K$ can
be precomputed. However, this method could be computationally complex and time
consuming specially for a time-varying noise environment. That environment may
demand a quick method and the following technique (i.e., Technique 2) could be
found more useful.

####  V-A2 Technique 2

To obtain a quick value of $K$, we apply CLT to (26). The chi-square
distribution given in (29) yields the conditional mean of $P_{n}$ as
$\mu_{P_{n}|\Omega_{1,n}}=\sigma^{2}$, and its variance
$\sigma^{2}_{P_{n}|\Omega_{1,n}}={\sigma^{4}}/M$. The desired probability
$\mathrm{P}\left({H_{0}|\Omega_{1,n}}\right)=\mathrm{P}\left({P_{n}\leq
K|\Omega_{1,n}}\right)$ which is

$\mathrm{P}\left({H_{0}|\Omega_{1,n}}\right)=1-\mathrm{P}\left({P_{n}>K|\Omega_{1,n}}\right),$

and under CLT, it simplifies to

$\mathrm{P}\left({H_{0}|\Omega_{1,n}}\right)=1-Q\left({\frac{M\left({K-\sigma^{2}}\right)}{\sigma^{4}}}\right)\,,$

where $Q(\cdot)$ is the tail distribution function of the standard normal
distribution. Now, we solve
$Q\left({\frac{M\left({K-\sigma^{2}}\right)}{\sigma^{4}}}\right)=\alpha$ to
get the value of $K$.

Figure 8:  $\rm{RMSE}_{\phi}$ vs average SNR (received) for Event 1.

###  V-B Analysis

Recall that in a noiseless polarized sensitive environment, under the proposed
antenna element assignment scheme two events can occur; Event 1 ($a_{n}=0$,
where $n=1$ or 2 or 3 or 4), and Event 2 ($a_{n}\neq 0$ for all $n$). Both
C-MUSIC and CF operate differently in each of the above events. Therefore, the
performance of the algorithms will depend on how accurately those events can
be identified after processing $M$ samples of the received signal. Our next
objective is to analyze $\mathrm{P}\left({I|\Omega_{i}}\right)$, which is the
probability of accurately identifying Event $i$ given Event $i$ occurred for a
given set of source parameters, where $i=1,2$. Since Event 1 has 4 sub-events,
and $\mathrm{P}\left({I|\Omega_{1}}\right)$ is different for each of those
sub-cases, we need to add another condition (i.e., $a_{n}=0$) in the
derivation of the desired probability. Thus, for Event 1, we analyze
$\mathrm{P}\left({I|\Omega_{1,n}}\right)$. In the derivation of the desired
probability for Event 1, we will exploit the following two lemmas. The
${1}^{\underline{\text{st}}}$ one is straightforwardly obtained from (28).

######  Lemma 5.

Given Event 1 due to $a_{n}=0$ (i.e., $\Omega_{1,n}$), the conditional
probability of the measured received power at antenna $n$ (i.e., $P_{n}$)
below the decision threshold $K$ is

$\mathrm{P}\left({P_{n}\leq
K|\Omega_{1,n}}\right)=F_{\nu}\left({\frac{K}{\tilde{\sigma}^{2}}}\right)\,.$
(31)

Given event $\Omega_{1,n}$, the conditional PDF of the measured received power
at antenna $n^{\prime}\neq n$ follows a non-central chi-square distribution
defined as [45, 46]

$f_{P_{n^{\prime}}|\Omega_{1,n}}(w)=\frac{1}{\tilde{\sigma}^{2}}f_{\nu,\,n^{\prime}}\left({\frac{w}{\tilde{\sigma}^{2}}}\right)\,,$
(32)

where

$f_{\nu,\,n^{\prime}}(u)=\left\\{\begin{array}[c]{ll}\exp\left({-\frac{\Delta_{n^{\prime}}}{2}}\right)\sum_{j=0}^{+\infty}\frac{\Delta^{j}_{n^{\prime}}}{2^{j}j!}f_{2j+\nu}(u)&u>0\,,\\\
0&\text{otherwise.}\end{array}\right.$ (33)

In (33),
$\Delta_{n^{\prime}}=\sum_{m=1}^{M}2\mu^{2}_{m,\,n^{\prime}}/\sigma^{2}$ is
the non-centrality parameter where
$\mu^{2}_{m,\,n^{\prime}}=|{a}_{n^{\prime}}|\sigma^{2}_{\mathrm{s},m}$ at the
${n^{\prime}}^{\underline{\text{th}}}$ element, and $f_{2j+\nu}(u)$ follows
the central chi-square distribution (27) with $\mathrm{DoF}=2j+\nu$.
Integrating both sides of (33) from $K$ to $\infty$, we get

$\bar{F}_{\nu,\,n^{\prime}}(K)=\exp\left({-\frac{\Delta_{n^{\prime}}}{2}}\right)\sum_{j=0}^{+\infty}\frac{\Delta^{j}_{n^{\prime}}}{2^{j}j!}\bar{F}_{2j+\nu}(K)\,,$
(34)

where $\bar{F}_{2j+\nu}(K)=1-F_{2j+\nu}(K)$ is the complement of the CDF given
in (28). Now, we are ready to state the other lemma to get the desired
probability $\mathrm{P}\left({I|\Omega_{1,n}}\right)$.

Figure 9:  $\rm{RMSE}_{\theta}$ vs average SNR (received) for Event 1.

######  Lemma 6.

Given Event 1 due to $a_{n}=0$ (i.e., $\Omega_{1,n}$), the conditional
probability of the received measured power at antenna $n^{\prime}\neq n$
(i.e., $P_{n^{\prime}}$) above the decision threshold $K$ is

$\mathrm{P}\left({P_{n^{\prime}}>K|\Omega_{1,n}}\right)=\bar{F}_{\nu,\,n^{\prime}}\left({\frac{K}{\tilde{\sigma}^{2}}}\right)\,.$
(35)

Since the received noise vector at antenna $n$ is independent of that at
antenna $n^{\prime}\neq n$ for a given set of source parameters, we get the
desired conditional probability $\mathrm{P}\left({I|\Omega_{1,n}}\right)$ as
stated in the following theorem.

######  Theorem 3.

Given Event 1 due to $a_{n}=0$ (i.e. $\Omega_{1,n}$), the conditional
probability of accurately identifying Event 1 is

$\displaystyle\mathrm{P}\left({I|\Omega_{1,n}}\right)$ $\displaystyle=$
$\displaystyle\mathrm{P}\left({P_{n}\leq K|\Omega_{1,n}}\right)$
$\displaystyle\times\prod_{n^{\prime}=1,\,\neq
n}^{4}\mathrm{P}\left({P_{n^{\prime}}>K|\Omega_{1,n}}\right)$ $\displaystyle=$
$\displaystyle
F_{\nu}\left({\frac{K}{{\tilde{\sigma}^{2}}}}\right)\prod_{n^{\prime}=1,\,\neq
n}^{4}\bar{F}_{\nu,\,n^{\prime}}\left({\frac{K}{\tilde{\sigma}^{2}}}\right)\,.$

Figure 10: $\rm{RMSE_{\phi}}$ vs average SNR (received) for Event 2.

Reader must note that the probability in Theorem 3 will change from one sub-
event to another in Event 1. However, since
$\bar{F}_{\nu,\,n^{\prime}}\left({\frac{K}{\tilde{\sigma}^{2}}}\right)\leq 1$
and $F_{\nu}\left({\frac{K}{{\tilde{\sigma}^{2}}}}\right)=1-\alpha$, the
probability in Theorem 3 will have the same upper bound for all $n$ as given
in the following corollary.

######  Corollary 1.

Given Event 1 due to $a_{n}=0$ (i.e. $\Omega_{1,n}$), the conditional
probability of accurately identifying Event 1 is upper bounded by

$\displaystyle\mathrm{P}\left({I|\Omega_{1,n}}\right)\leq
1-\alpha;\quad\forall n\,.$

In Event 2, $a_{n}\neq 0$ $\forall n$, and the conditional PDF of the measured
received power $f_{P_{n}|\Omega_{2}}(w)$ follows a non-central chi-square
distribution as shown in (32). This distribution coupling with the fact that
the received noise vector at antenna $n$ is independent of that at antenna
$n^{\prime}\neq n$ gives us the following theorem.

######  Theorem 4.

Given Event 2 (i.e., $\Omega_{2}$), the conditional probability of accurately
identifying Event 2 is

$\displaystyle\mathrm{P}\left({I|\Omega_{2}}\right)$ $\displaystyle=$
$\displaystyle\prod_{n=1}^{4}\mathrm{P}\left({P_{n}>K|\Omega_{2}}\right)=\bar{F}_{\nu,\,n}\left({\frac{K}{\tilde{\sigma}^{2}}}\right)\,.$

##  VI Numerical Results

In this section, our objectives are 1) to provide insights into the system
performance through numerical examples based on our analytical results, and 2)
to perform a thorough comparative study between C-MUSIC and CF algorithms. In
all numerical examples, i) a 4-element UCA is considered with the antenna
alignment scheme proposed in Figure 4, ii) the number of samples $M=50$, iii)
the significance level $\alpha=0.001$, iv) received waveform samples modelled
as i.i.d complex Gaussian random variables with mean = 0 and variance = 1/2
per dimension, unless otherwise specified.

Figure 11: $\rm{RMSE_{\theta}}$ vs average SNR (received) for Event 2.

Recall that identification of each of the five effective events is performed
employing the decision threshold $K$. It can be determined by setting Type I
Error (i.e., $1-\mathrm{P}\left({H_{0}|\Omega_{1,n}}\right)$) equals to
$\alpha$ using either Technique 1 or Technique 2. In Figure 5, the decision
threshold normalized by the average noise power is plotted by setting (Type I
Error) $\alpha=0.001$. Here, it can be noticed that Technique 1 (which is
based on the exact PDF of the decision statistics) yields a slightly higher
value of $K$ than Technique 2 (which is based on the CLT). This higher value
of $K$ will result in higher Type II error and its effects will be
demonstrated shortly.

In the next example, we evaluate our derived expression of
$\mathrm{P}\left({I|\Omega_{1,n}}\right)$ (i.e. the conditional probability of
accurately identifying Event 1) given in Theorem 3 as a function of average
RSNR using both Technique 1 and Technique 2; see Figure 6. Here, Event 1
occurs due to the DOA $\phi=45^{\circ}$, $\theta=10^{\circ}$, and polarization
parameters $\eta=90^{\circ}$, $\gamma=45^{\circ}$. These parameters cause
$a_{n}=0$ at $n=1$. In Figure 6, it can be noticed that Technique 2 slightly
performs better than Technique 1 especially at low RSNR due to the use of a
higher value of $K$. We also see that when the average RSNR is above or equal
to 4 dB, both techniques exhibit the maximum achievable performance as given
by the Corollary 1. In Figure 7, we plot
$\mathrm{P}\left({I|\Omega_{2}}\right)$ (i.e. the conditional probability of
accurately identifying Event 2) given in Theorem 4 as a function of average
RSNR for both Technique 1 and Technique 2. Here, the DOA and polarization
parameters are $\phi=45^{\circ}$, $\theta=10^{\circ}$, $\eta=90^{\circ}$ and
$\gamma=45^{\circ}$, respectively. Similar to Figure 6, the results in Figure
7 suggest that both techniques are equally capable of identifying Event 2
almost perfectly when the average RSNR $\geq$ 4 dB. From now on, Technique 2
will be employed to design the decision threshold $K$.

Next, we would like to compare the performance of the CF algorithm against
that of the C-MUSIC for Event 1 and Event 2. We first consider Event 1 due to
$a_{1}=0$ (i.e., $\Omega_{1,1}$), where $\phi=30^{\circ}$,
$\theta=70.529^{\circ}$, $\eta=0^{\circ}$ and $\gamma=60^{\circ}$; see Figure
8 and 9, where RMSE (Root Mean Square Error) is plotted as a function of the
average RSNR. C-MUSIC has been implemented using both Method 1 and Method 2
where the former is observed to exhibit slightly better performance than the
latter. Here, it can also be noticed that as expected, at low SNR (close to
$5$ dB) the performance difference between C-MUSIC and CF algorithms is
somewhat noticeable. However, as the average RSNR increases that performance
difference starts to diminish. At RSNR $\geq$ 10 dB, both proposed CF and
C-MUSIC algorithms exhibit almost identical performance. Similar observations
are made in Event 2 for $\phi=45^{\circ}$, $\theta=10^{\circ}$,
$\eta=90^{\circ}$ and $\gamma=45^{\circ}$; see Figure 10 and 11.

In the following numerical study, we would like to demonstrate the performance
of both the algorithms as the system transits from one event to another due to
the change in the azimuthal angle. Please see Figure 12, and 13, where we plot
RMSE by varying the azimuth angle $\phi$ (with the increment of $\pm
0.5^{\circ}$). Here, we use SNR = $20$ dB, $\theta=70.529^{\circ}$,
$\eta=0^{\circ}$ and $\gamma=60^{\circ}$. Note that when $\phi=30^{\circ}$,
Event 1 occurs due to $a_{1}=0$ and otherwise, we have Event 2. Using the
above two figures, we make the following set of observations: 1) the designed
decision threshold $K$ is capable of efficiently differentiating between Event
1 and Event 2, 2) the RMSE of Event 2 is slightly higher than Event 1 since as
$\phi$ approaches $30^{\circ}$, compound steering element $a_{1}$ approaches
0. Thus, it is better to discard the output of antenna element 1, 3) as
expected, Event 1 occurs for two different azimuthal angles which are
$180^{\circ}$ apart from each other, and 4) both the CF and C-MUSIC algorithms
not only exhibit almost identical performance but also robust to the
transition from Event 1 to Event 2 or vice versa. Similar results are also
observed when we vary the elevation angles for a given azimuthal angle. For
the conciseness, figures related to those observations are omitted.

Figure 12: $\rm{RMSE_{\phi}}$ as a function of the azimuthal angle $\phi$.
Figure 13:  $\rm{RMSE_{\theta}}$ as a function of the azimuthal angle $\phi$.

Finally, we compare the complexity of the CF and C-MUSIC algorithms. In our
calculation, we consider the complexity incurred after identifying Event $i$,
where $i=1,2$. The complexities of the algorithms are measured by the number
of real-time multiplications associated with the major operations. Note that
one complex multiplication is equivalent to four real multiplications. In
C-MUSIC (Method 1), which is only applicable for Event 1, a new matrix is
formed which requires $4(N-1)^{2}M+2(N-1)^{2}-(N-1)+2(N-1)^{2}-(N-1)$ real
multiplications. Here, $4(N-1)^{2}M+2(N-1)^{2}-(N-1)$ is due to the estimation
of the autocorrelation matrix and the additional $2(N-1)^{2}-(N-1)$ is an
upper bound on the number of multiplications between the autocorrelation
matrix and the diagonal matrix $\mathbf{{F}}_{n}$; $n=1,2,3,4$. As previously
discussed, both C-MUSIC (Method 2) and CF operate by estimating the phases of
the steering elements, $\kappa_{1}$, and $\kappa_{2}$. The cost associated
with this estimation is $8M+2\log_{2}p+1$ real multiplications for Event 1,
and $16M+2\log_{2}p+1$ real multiplications for Event 2, where $p$ refers to
the number of digits of precision [47, 48]. It is known that the MUSIC
algorithm performs eigenvalue decomposition (EVD) which often is obtained from
a singular value decomposition (SVD). As per [49], this complexity associated
with an SVD is $12N^{3}$. As C-MUSIC (Method 1 and 2) operates on the 2D MUSIC
algorithm, the DOA angle search using the null space requires
$12N^{3}+N_{\theta}N_{\phi}\\{4N(N-1)+2(N-1)+1\\}$ real multiplications, where
$N_{\theta}$ and $N_{\phi}$ represent the searching point number on the
azimuthal and elevation planes, respectively. On the other hand, the required
cost to estimate the DOA angles is upper bounded by
$1+\log_{2}p+3+1+\log_{2}p$ from the estimates of $\kappa_{1}$, and
$\kappa_{2}$ using the CF algorithm. Here, the first two terms are related to
the estimation of the azimuthal angle $\phi$, the third term is to calculate
the $\kappa$, and the rest of the terms are associated with the estimation of
the elevation angle $\theta$. All the corresponding costs are added and
tabulated in Table I for complexity comparison. According to this table, the
cubic order of the array size, the number of samples, and the product of the
searching points dominate the complexity of the C-MUSIC algorithm, whereas,
the complexity of the CF algorithm is primarily dictated by the number of
samples.

TABLE I: Complexity analysis Algorithm | Event 1 | Event 2
---|---|---
MUSIC | Method 1 | $(N-1)^{2}(4M+4)$ | N/A
$-2(N-1)$
$+12(N-1)^{3}$
$+N_{\theta}N_{\phi}\times$
$\\{4N^{2}-10N+5\\}$
Method 2 | $8M+2\log_{2}p+1$ | $16M+2\log_{2}p+1$
$+12(N-1)^{3}$ | $+12N^{3}$
$+N_{\theta}N_{\phi}\times$ | $+N_{\theta}N_{\phi}\times$
$\\{4N^{2}-10N+5\\}$ | $\\{4N^{2}-2N-1\\}$
CF | $8M+6$ | $16M+6$
$+4\log_{2}p$ | $+4\log_{2}p$
Figure 14: Complexity vs number of samples $M$ for Event 1. Figure 15:
Complexity vs number of samples $M$ for Event 2.

Now, we use Table I to demonstrate the complexities of the DOA estimating
algorithms by using numerical examples. Please see Fig. 14, and 15, where the
complexities of C-MUSIC (w.r.t the CF algorithm) are plotted as functions of
the number of samples $M$ for Event 1 and Event 2, respectively. Here, the
search in MUSIC is conducted with $1^{\circ}$ precision of the DOA angles. The
results in those figures jointly suggest that the CF algorithm offers
significant complexity gain over the C-MUSIC. For instance, this gain is
$33.29$ dB in Event 1 under Method 1, $33.28$ dB in Event 1 under Method 2 and
$33.29$ dB in Event 2 for $M=50$. This gain increases as the number of samples
$M$ decreases.

##  VII Conclusions

In this paper, we addressed the problem of localizing a single narrowband
source in all possible polarization scenarios just employing simple (short
dipole) antenna elements and signal processing techniques. Depending on the
antenna alignment, the contribution of the polarization and DOA angles could
result in poor received signal power at one or more antenna elements. To
overcome this issue, an antenna alignment scheme was mathematically developed
for a UCA that operates with the minimum number of required antenna elements.
Under this scheme, antenna elements are aligned in such a way that not more
than one element will suffer from low received signal power due to the joint
effects of polarization and DOA angles. A decision threshold was developed to
decide whether the antenna element with the smallest received power should be
considered in the process of DOA estimation or not. We demonstrated how the
polarization contribution can be suppressed from the non-signal subspace in
order for the popular MUSIC algorithm to operate in all the polarization
scenarios. During the process of cleaning the non-signal subspace, we designed
an algorithm that estimates the DOA angles in a closed-form manner. Of course,
the latter is significantly less complex than the former. Our numerical
results demonstrated that depending on the system condition, that gain could
be up to 32.29 dB without sacrificing any performance as long as the received
SNR is 10 dB or more.

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