# On the Design of Super Constellations

Thrassos K. Oikonomou, Dimitrios Tyrovolas,
Sotiris A. Tegos, Panagiotis D. Diamantoulakis,
Panagiotis Sarigiannidis, and George K. Karagiannidis T. K. Oikonomou, D.
Tyrovolas, and P. D. Diamantoulakis are with the Department of Electrical and
Computer Engineering, Aristotle University of Thessaloniki, 54124
Thessaloniki, Greece (e-mails: {toikonom, tyrovolas, padiaman}@auth.gr).S. A.
Tegos is with the Department of Electrical and Computer Engineering, Aristotle
University of Thessaloniki, 54124 Thessaloniki, Greece and also with the
Department of Electrical and Computer Engineering, University of Western
Macedonia, 50100, Kozani, Greece (e-mail: sotiristegos@ieee.org).P. G.
Sarigiannidis is with the Department of Electrical and Computer Engineering,
University of Western Macedonia, 50100, Kozani, Greece (e-mail:
psarigiannidis@uowm.gr).G. K. Karagiannidis is with the Department of
Electrical and Computer Engineering, Aristotle University of Thessaloniki,
54124 Thessaloniki, Greece and also with the Artificial Intelligence & Cyber
Systems Research Center, Lebanese American University (LAU), Lebanon (e-mail:
geokarag@auth.gr).This work was funded from the Smart Networks and Services
Joint Undertaking (SNS JU) under European Union’s Horizon Europe research and
innovation programme (Grant Agreement No. 101096456 - NANCY).The work of G. K.
Karagiannidis was implemented in the framework of H.F.R.I call basic research
financing (horizontal support of all sciences) under the National Recovery and
Resilience Plan Greece 2.0 funded by the European Union NextGenerationEU
(H.F.R.I. Project Number: 15642).

###### Abstract

In the evolving landscape of sixth-generation (6G) wireless networks, which
demand ultra-high data rates, this study introduces the concept of super-
constellation communications. Also, we present super amplitude phase shift
keying (SAPSK), an innovative modulation technique designed to achieve these
ultra-high data rate demands. SAPSK is complemented by the generalized polar
distance detector (GPD-D), which approximates the optimal maximum likelihood
detector in channels with Gaussian phase noise (GPN). By leveraging the
decision regions formulated by GPD-D, a tight closed-form approximation for
the symbol error probability (SEP) of SAPSK constellations is derived, while a
detection algorithm with $\mathcal{O}(1)$ time complexity is developed to
ensure fast and efficient SAPSK symbol detection. Finally, the theoretical
performance of SAPSK and the efficiency of the proposed $\mathcal{O}(1)$
algorithm are validated by numerical simulations, highlighting both its
superiority in terms of SEP compared to various constellations and its
practical advantages in terms of fast and accurate symbol detection.

###### Index Terms:

super constellations, phase noise, ultra-high data rates, detection scheme,
symbol error probability

##  I Introduction

Due to the emergence of promising applications such as augmented reality,
holographic communication and super-high-definition live streaming, achieving
high data rates is crucial for future digital societies [1, 2]. In this
direction, research has focused on employing higher-order modulation schemes.
Notably, IEEE 802.11 standard included 4096-quadrature amplitude modulation
(QAM), aiming to achieve higher peak data rates. Concurrently, the
architectural design of 6G networks is being tailored to function as a multi-
band framework that provides wider bandwidth and improved signal-to-noise
ratio (SNR) at the receiver through novel technologies such as massive
multiple-input multiple-output (mMIMO), reconfigurable intelligent surfaces
(RIS) and higher frequency bands, e.g., free space optics (FSO) [3, 4]. These
technological advancements pave the way for a transformation in modulation
schemes, transcending traditional modulation orders to embrace higher-order
constellations that can significantly increase the achievable rates, enabling
the transmission of more bits per symbol. In essence, to meet the demands of
next-generation applications with an emphasis on environmental sustainability,
it is imperative to design advanced modulation schemes capable of delivering
ultra-high data rates in the most energy-efficient manner.

###  I-A State-of-the-Art

In recent years, increasing the constellation order has been investigated as a
means to increase achievable rates substantially. In particular, recent
advances in constellation design have focused on the development and
optimization of high-order QAM schemes, with significant progress made in the
design of 1024-QAM and 4096-QAM constellations [5, 6, 7, 8, 9, 10]. For
example, in [5], the authors optimized the shape of a 1024-QAM constellation
in a fiber channel to achieve performance close to Shannon capacity.
Similarly, in [6], the authors investigated a shaped 1024-QAM constellation
that significantly increases data rates in orthogonal frequency division
multiplexing systems while meeting peak-to-average power ratio and symbol
error probability (SEP) requirements. In addition, [7] and [8] designed non-
uniformly shaped 256-QAM and 4096-QAM, respectively, to maximize the mutual
information of input signals to achieve increased data rates in broadcasting
systems. Finally, both [9] and [10] have implemented constellation shaping for
1024-QAM together with advanced coding techniques to increase the capacity of
the system.

Regarding the novel technologies that will be incorporated within 6G networks,
such as mMIMO systems and high frequency bands, extensive research has been
conducted into how these key technologies can enable the use of higher-order
constellations to achieve ultra-high data rates. Specifically, in [11] and
[12], efficient and low-complexity mMIMO receivers were proposed when 256-QAM
constellations are deployed, while in [13] the rate capabilities of mMIMO
satellite network using the 256-PSK constellation were investigated. In
addition, the authors of [14] developed an mmWave testbed operating at 220 GHz
and using 256-QAM modulation, achieving a significant increase in data rates,
while also investigating the practical challenges of the system in detail.
Furthermore, in [15], the transmission of 4096-QAM OFDM operating at 117 GHz
is demonstrated, achieving a substantial increase in data rates compared to
the existing OFDM systems. Similarly, in [16], a 1024-QAM OFDM-based system
operating at 28 GHz using 5G mmWave phased array antenna, with the main
emphasis being given on the benefits in higher peak data rates when high-order
QAM are utilized in OFDM. While high-order, the constellations used in these
studies are conventional and not carefully designed for achieving ultra-high
data rates in an energy-efficient manner. In the context of energy-efficient
higher-order modulations, the authors of [17] performed a detailed theoretical
analysis of the SEP for hexagonal-QAM (HQAM) and emphasized that optimal
symbol allocation can significantly improve system performance by achieving
higher data rates with lower energy consumption. Similarly, in [18], different
modulation schemes such as 1024-HQAM, 1024-cross QAM (XQAM) and 1024-Star QAM
were compared and the superiority of HQAM in terms of energy efficiency in a
mixed RF/FSO system was shown. Finally, the authors in [19], highlighted the
reduction of system’s energy when HQAM is employed in RIS-assisted systems
compared to conventional QAM.

###  I-B Motivation & Contribution

In the aforementioned works, high-order constellations were used in response
to the increasing demands for ultra-high data rate transmissions. However, the
transition to the era of super-constellation communications, where very high-
order modulation designs are employed, requires an accurate characterization
of their inherent susceptibility to specific system imperfections. One of the
main impairments that can lead to significant performance degradation in super
constellations is the presence of phase noise (PN) due to local oscillator
instabilities and the lack of accurate channel state information [20, 21, 22].
This inherent susceptibility to PN distortions can be attributed to the
increased number of symbols that are far from the origin in super
constellations so that even small PN distortions can lead to symbol detection
errors [23]. Even though some previous works in the literature, such as [24]
and [25], attempted to optimize the constellation lattice to minimize the SEP
under certain GPN conditions, optimizing very high-order constellations poses
significant challenges. The unstructured lattices of these optimized
constellations necessitate extensive comparisons to detect the transmitted
symbols, particularly in very high-order scenarios. Therefore, it is crucial
to develop detection methods that maintain low complexity regardless of the
constellation order and perform effectively even under PN conditions. In this
direction, the authors of [26] proposed a structured amplitude phase shift
keying (APSK) constellation called polar QAM (PQAM), which is characterized by
increased PN resistance and low-complexity detection but is particularly
energy inefficient due to its suboptimal lattice packing. Considering the
challenges posed by these research efforts an urgent need is revealed for a
new class of modulation designs that balance PN resistance, low detection
complexity, and energy consumption. To be classified as super constellations,
these designs must fulfill three critical criteria: i) they must have an order
of more than 1024 symbols and be capable of achieving ultra-high data rates,
ii) they must have low complexity symbol detection and iii) they must exhibit
PN robustness. To the best of the authors’ knowledge, there is no work that
proposes a novel modulation design that comprehensively fulfills the rules of
super constellation communication.

In this paper, we propose for the first time in the existing literature a
novel modulation design, namely super APSK (SAPSK), which conforms to the
rules of super constellation communication. Specifically, the contribution of
our work is as follows.

  * •

We define the structure of SAPSK that preserves optimal symbol assignment in
terms of energy consumption and PN robustness.

  * •

We propose a novel detector, the generalized polar distance detector (GPD-D),
which expresses a weighted Euclidean distance in the polar plane and provides
an analytical framework for both theoretical analysis and low-complexity
detection algorithms in super constellations.

  * •

Based on the decision regions formulated by GPD-D in the polar plane, we
provide a tight closed-form approximation for the SEP of SAPSK constellations
under the influence of additive white Gaussian noise (AWGN) and GPN. Also, we
develop a detection algorithm with $\mathcal{O}(1)$ time complexity tailored
to SAPSK constellations, which is a significant step towards the practical use
of SAPSK in super constellation communication.

  * •

We provide simulation results to evaluate the performance of SAPSK in super-
constellation communications. These results demonstrate the substantial
superiority of the proposed SAPSK in terms of energy efficiency compared to
PQAM and conventional QAM.

###  I-C Structure

The remainder of this paper is organized as follows. The system model and some
preliminaries on GPN channels are described in Section II. In Section III,
SAPSK constellation, and the proposed low-complexity detector for super-
constellation communications are presented. In Section IV, the performance
analysis of SAPSK constellations when affected by AWGN and GPN is provided,
and an $O(1)$ time complexity algorithm is developed for SAPSKs. In Section
IV, simulation results are provided to verify the derived theoretical results.
Finally, Section V concludes the paper.

##  II System model and Preliminaries

We consider a communication system that transmits a symbol $s\in\mathbb{C}$
that belongs to an $M$-ary constellation denoted by $\mathcal{C}$, with
average symbol energy $E_{s}$. Specifically, every constellation $\mathcal{C}$
is composed of $M$ symbols that are appropriately allocated upon distinct
energy levels, described as circles of radius $\sqrt{E_{q}}$, where $E_{q}$
denotes the $q$-th energy level of $\mathcal{C}$. By taking into account an
AWGN channel impaired by PN, the received signal $r\in\mathbb{C}$ can be
expressed as

$r=\lvert s\rvert e^{j(\phi+\arg\\{s\\})}+n,$ (1)

where $\lvert\cdot\rvert$ and $\arg\\{\cdot\\}$ denote the amplitude and the
phase of a complex number, respectively, $n$ represents a complex Gaussian
random variable (RV) with zero-mean and variance $\sigma_{n}^{2}$, and
$\bar{\gamma}=E_{s}/N_{0}$ denotes the SNR, where $N_{0}=\sigma_{n}^{2}$
denotes the AWGN power spectral density. Moreover, $\phi$ expresses the PN
caused by imperfect carrier phase recovery, usually modeled as a Tikhonov
distributed RV with zero mean and standard deviation $\sigma_{\phi}^{2}$ [27].
However, in modern communication systems, we focus on PN with
$\sigma_{\phi}^{2}\ll 1$, and thus $\phi$ can be accurately described as a GPN
with zero mean and variance $\sigma_{\phi}^{2}$ [28]. In more detail, in
modern communication systems a strong PN scenario is described by
$\sigma_{\phi}^{2}=10^{-1}$, while a moderate and low PN scenarios are
described by $\sigma_{\phi}^{2}=10^{-2}$ and $\sigma_{\phi}^{2}=10^{-4}$ [29],
respectively.

Typically, in digital communication systems where GPN is neglected, the
Euclidean distance detector (EUC-D) is used for symbol detection, which is
tailored for the AWGN channel. In more detail, EUC-D decides based on the
minimum Euclidean distance from the received symbol $r$, and the estimated
symbol can be expressed as

$\displaystyle\hat{s}=\arg\min_{s\in\mathcal{C}}\hskip 4.015pt\lvert
r-s\rvert^{2},$ (2)

However, EUC-D performs sub-optimal in cases where GPN affects the
communication system performance. To this end, it becomes imperative to devise
a detection scheme that effectively addresses the impact of GPN. In this
direction, by taking into account (1), a detector tailored for GPN channels,
denoted as GAP-D, can be derived as in [30]

$\displaystyle\hat{s}=\arg\min_{s\in\mathcal{C}}$
$\displaystyle\frac{\big{(}\lvert r\rvert-\lvert
s\rvert\big{)}^{2}}{\frac{\sigma_{n}^{2}}{2}}+\frac{\big{(}\arg\\{r\\}-\arg\\{s\\}\big{)}^{2}}{\sigma_{\phi}^{2}+\frac{\sigma_{n}^{2}}{2\lvert
s\rvert^{2}}}$ (3)
$\displaystyle+\log\left(\sigma_{\phi}^{2}+\frac{\sigma_{n}^{2}}{2\lvert
s\rvert^{2}}\right).$

As it can be observed in (3), the performance of GAP-D is influenced by both
$\sigma_{n}^{2}$ and $\sigma_{\phi}^{2}$, while it intricately incorporates
the amplitude and phase of the received symbol $r$ and the constellation
symbols. Finally, it should be highlighted that the incorporation of amplitude
and phase in (3) translates into an analytical focus on the symbols’ polar
coordinates.

To meet the escalating demands for high data rates in next-generation
communication networks, super constellations can be employed, allowing each
transmitted symbol to contain more information bits. However, it is imperative
to tackle the significant challenges posed by GPN on super constellations
while prioritizing the development of energy-efficient designs that minimize
the SEP for certain system’s energy consumption. Specifically, when the
constellation order $M$ is large, i.e., $M\geq 1024$, and EUC-D is applied,
even minor GPN leads to significant performance degradation, as shown in Fig.
1. For example, when dealing with minor GPN, e.g.,
$\sigma_{\phi}^{2}=10^{-4}$, low-order QAM constellations exhibit satisfactory
SEP performance, a phenomenon that is not observed in the case of high-order
QAM, signifying that EUC-D can not be used in super-constellation
communications. Consequently, in super-constellation communications, it is
imperative to consider a detector that mitigates the GPN effect. In this
direction, as it can be observed in Fig. 1, GAP-D can significantly reduce the
SEP of high-order QAM constellations, emphasizing the importance of detectors
that effectively minimize the impact of GPN.

Although GAP-D minimizes the effect of GPN in the system, it does not ensure
its elimination, leading to the presence of an error floor as shown in Fig. 1,
which indicates that increasing the SNR will not always translate into SEP
improvement. To this end, the employment of GAP-D needs to be combined with an
efficient modulation design that can further reduce this error floor to
achieve robust super-constellation communications. In more detail, as
illustrated in Fig. 2b, when GAP-D is applied for a conventional
constellation, e.g., QAM, the system performance is severely affected by an
increased probability of erroneous detection, since the symbols in the polar
plane are not separated by large distances. Therefore, it is necessary to
design constellation structures with symbols appropriately allocated in the
polar plane, demonstrating inherent GPN resilience and a reduced error floor
when GAP-D is applied. Moreover, given the increased SNR requirements for
robust super constellations, it is crucial to allocate symbols in the polar
plane to not only ensure resilience against GPN but also to reduce system’s
energy needs. In conclusion, optimizing symbol allocation in the polar plane
is essential for reducing the error floor occurred by GPN, and thus, achieving
efficient super-constellation communications.

$0$$10$$20$$30$$40$$50$$60$$70$$80$$10^{-5}$$10^{-4}$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$\frac{E_{s}}{N_{0}}$
(dB)Symbol Error Probability32-QAM (EUC-D)32-QAM (GAP-D)4096-QAM
(EUC-D)4096-QAM (GAP-D) Figure 1: EUC-D vs GAP-D with PN variance
$\sigma^{2}_{\phi}=10^{-4}$

##  III A Novel Modulation Design for super constellation Communication

###  III-A SAPSK

Quadrature QIn-Phase I

Amplitude $\rho$Phase $\theta$

In-Phase IQuadrature Q

Amplitude $\rho$Phase $\theta$

In-Phase IQuadrature Q

Amplitude $\rho$Phase $\theta$

Figure 2: $32$-QAM in a) IQ plane, b) Polar plane, $32$-PQAM($8$) in c) IQ
plane, d) Polar plane and SAPSK($32$,$8$) in e) IQ plane, f) Polar plane

Building upon the need to develop modulation schemes for robust super-
constellation communications, it becomes crucial to examine constellation
lattices with appropriate formation in the polar plane. In this direction,
APSK constellation structures have appeared to offer enhanced performance in
the presence of GPN, in contrast to other well-established constellations like
QAM [26]. Specifically, APSK constellations are allocated on energy levels,
with neighboring symbols on the same ring maintaining a specific angular
distance between them. Additionally, this spatial arrangement allows for the
dynamic adjustment of the angular distances and the energy levels of the
constellations to compensate for the effect of both AWGN and GPN and, also,
improve the energy consumption of the constellation. In this context, a
specific APSK formation shown in Fig. 2c and referred to as PQAM, has been
proposed in [26], which adopts a rectangular lattice in the polar plane, as
illustrated in Fig. 2. However, the energy efficiency of the constellation can
be further improved by compactly allocating the symbols in the polar lattice.
In this direction, by leveraging the flexibility of APSK in symbol allocation
and the increased energy efficiency demonstrated by triangular QAM (TQAM)
constellations in AWGN channels, a novel modulation design featuring a
triangular structure in the polar domain, denoted as SAPSK can be designed for
robust super-constellation communications. Specifically, as illustrated in
Fig. 2e, an $M$-ary SAPSK consists of $M$ symbols arranged on $\Gamma$
concentric circles of radius $\rho_{q}$ which is given as

$\displaystyle\rho_{q}=\frac{\delta_{\rho}}{2}(2q-1),$ (4)

where $q\in\\{1,2,\dotsc,\Gamma\\}$ and $\rho_{1}$ corresponds to the radius
of the lowest energy level, while $\delta_{\rho}$ is the radial distance
between two adjacent concentric circles which is equal to

$\displaystyle\delta_{\rho}=\sqrt{\frac{12E_{s}}{4\Gamma^{2}-1}}.$ (5)

Therefore, each symbol on the $q$-th concentric circle of the SAPSK has energy
$E_{q}$ which can be calculated

$\displaystyle E_{q}=\frac{3E_{s}\left(2q-1\right)^{2}}{4\Gamma^{2}-1}$ (6)

Furthermore, each concentric circle of a PTQAM constellation contains
$\frac{M}{\Gamma}\in\mathbb{N}^{+}$ symbols equally spaced by angle
$\delta_{\theta}$ that can be expressed as

$\displaystyle\delta_{\theta}=\frac{2\pi\Gamma}{M}$ (7)

and the argument of $p$-th symbol located at the $q$-th energy level is given
by

$\displaystyle\arg\\{s_{pq}\\}=\frac{\delta_{\theta}}{2}(2p-1)\left(1+\mathrm{mod}(q,2)\frac{\delta_{\theta}}{2}\right),$
(8)

where $p\in\\{1,2,\dotsc,\frac{M}{\Gamma}\\}$, and
$\mathrm{mod}\left(\cdot,\cdot\right)$ is the modulo operation. Consequently,
an SAPSK$\left(M,\Gamma\right)$ constellation is defined by $M$ and $\Gamma$
and can be described as

$\displaystyle\mathcal{C}^{\prime}=\bigg{\\{}\frac{\delta_{\rho}}{2}(2q-1)\exp$
$\displaystyle\left(j\frac{\delta_{\theta}}{2}(2p-1)\left(1+\mathrm{mod}(q,2)\frac{\delta_{\theta}}{2}\right)\right)$
(9) $\displaystyle\bigg{|}1\leq q\leq\Gamma,1\leq
p\leq\frac{M}{\Gamma}\bigg{\\}}.$

Finally, by applying a polar transformation in the symbols of
$\mathcal{C}^{\prime}$ a triangular lattice is obtained in the polar plane, as
illustrated in Fig. 2f. To this end, the definition of
SAPSK$\left(M,\Gamma\right)$ constellation based on its polar lattice is as
follows:

###### Definition 1

SAPSK$\left(M,\Gamma\right)$ is defined as a constellation in the polar plane
with $M$ symbols equally distributed on $\Gamma$ columns that are spaced by
$\delta_{\rho}$, and between adjacent symbols isosceles triangles are formed.

As it can be observed in Fig. 2d and Fig. 2f, depicting the polar lattices of
PQAM and SAPSK, respectively, both PQAM and SAPSK maintain an equal distance
between symbols in the same column, ensuring GPN resilience. However, it is
evident from the figures that SAPSK exhibits enhanced robustness compared to
PQAM in horizontal movements, as its lattice doubles the distance between
symbols in the horizontal direction induced by the AWGN. In summary, the SAPSK
lattice allows for a more compact symbol allocation in the polar plane
compared to PQAM, achieving increased robustness against erroneous symbol
detection induced by both AWGN and GPN, while preserving the same average
constellation energy.

###  III-B Low-complexity Detector Design for super-constellation
communications in GPN

We need detection algorithms with low time complexity to make the
implementation of super constellations in practical systems feasible. To
achieve this, we can capitalize on the geometric characteristics of the
constellation decision regions to avoid comparing all the symbols with the
received one to detect which symbol was transmitted. However, considering the
sensitivity of super constellations to GPN and the GAP-D detector, the
detection process in super-constellation communications must account for the
decision regions formulated in the polar plane, by utilizing an appropriate
detector tailored for GPN mitigation.

Given the high-order of super constellations, developing detection schemes
that maintain low complexity and operate effectively under GPN conditions is
crucial. In this direction, while the detection metric offered by GAP-D is
applicable for optimal symbol detection in GPN scenarios, its optimality is
countered by the high time complexity and the complex decision regions it
introduces. Specifically, it becomes apparent in (3) that GAP-D requires $M$
calculations to determine $\hat{s}$, resulting in a complexity of
$\mathcal{O}(M)$, rendering it unsuitable for super constellations.
Additionally, the dependence of GAP-D on the magnitude of every constellation
symbol $s$ and the logarithmic factor signify that GAP-D defines decision
regions that cannot be utilized to produce low-complexity detection
algorithms. In more detail, the decision regions formulated by GAP-D are
different for each symbol, hence the composed decision region lattice has
unstructured and complex geometry. Therefore, it is imperative to develop a
detector that performs nearly optimal, while enabling the production of
decision regions with common geometry in the polar plane, e.g., polygons. To
this end, in the following proposition, we introduce GPD-D, which tightly
approximates the performance of GAP-D and expresses a weighted Euclidean
distance in the polar plane, enabling the construction of polygonic decision
regions.

###### Proposition 1

The transmitted symbol $s\in\mathcal{C}$ affected by both AWGN and GPN can be
reliably estimated through GPD-D which is a weighted polar Euclidean distance
detector, and is expressed as

$\displaystyle\hat{s}=\arg\min_{s\in\mathcal{C}}\hskip
4.015pt\frac{2\big{(}\lvert r\rvert-\lvert
s\rvert\big{)}^{2}}{\sigma_{n}^{2}}+\frac{\big{(}\arg\\{r\\}-\arg\\{s\\}\big{)}^{2}}{\sigma_{\phi}^{2}+\frac{\sigma_{n}^{2}}{2\lvert
r\rvert^{2}}},$ (10)

###### Proof:

The GAP-D metric results in symbol-dependent structures unsuitable for
developing time-efficient detection algorithms. In this direction, we can
substitute the factor $|s|^{2}$ in GAP-D by a constant term to produce
constant symbol-independent decision regions. In this context, it is
imperative to understand the effects of the variables on the term
$\sigma_{\phi}^{2}+\frac{\sigma_{n}^{2}}{2\lvert s\rvert^{2}}$, enabling the
substitution of $|s|^{2}$ with a constellation-dependent constant parameter.
First, the operating SNR in super-constellation communications must be high to
achieve satisfactory performance. Moreover, in the context of super
constellation designs, symbols must be allocated across numerous energy levels
to prevent errors arising from phase rotation due to GPN with the same energy
level, while keeping the energy levels as close as possible to the origin to
confine energy requirements. As a result, the SNR value of two consecutive
energy levels can be approximately considered equal, leading to the following
approximation

$\displaystyle\frac{\sigma_{n}^{2}}{E_{q}}\approx\frac{\sigma_{n}^{2}}{E_{q+1}}.$
(11)

Furthermore, using (11), the following approximation can be derived

$\displaystyle\sigma_{\phi}^{2}+\frac{\sigma_{n}^{2}}{2E_{q}}\approx\sigma_{\phi}^{2}+\frac{\sigma_{n}^{2}}{2E_{q+1}}.$
(12)

Consequently, by considering that the energy levels $E_{q}$ and $E_{q+1}$
enclose the received symbol $r$, i.e., $|r|^{2}\in[E_{q},E_{q+1}]$, then for
every $s\in\mathcal{P}$, where $\mathcal{P}=\big{\\{}s\in\mathcal{C}:\lvert
s\rvert=\sqrt{E_{q}}\lor\lvert s\rvert=\sqrt{E_{q+1}}\big{\\}}$, the following
approximation holds

$\displaystyle\sigma_{\phi}^{2}+\frac{\sigma_{n}^{2}}{2\lvert
s\rvert^{2}}\approx\sigma_{\phi}^{2}+\frac{\sigma_{n}^{2}}{2\lvert
r\rvert^{2}},\hskip 10.00002pt\forall s\in\mathcal{P},$ (13)

By substituting (13) in (3) and omitting the logarithmic term since its
argument becomes symbol-independent, (10) can be obtained, which concludes the
proof. ∎

$R_{d}$$R_{w}$$\alpha$$\delta_{\theta}^{\prime}$$\delta_{\rho}^{\prime}$$r$Amplitude
$\rho$Phase $\theta$

$\alpha$$\delta_{\theta}^{\prime}$$\delta_{\rho}^{\prime}$$R_{d}$$R_{w}$$r$Amplitude
$\rho$Phase $\theta$

Figure 3: Hexagonal decision regions formulated by GPD-D of a SAPSK($32$,$8$)
constellation oriented a) along $\rho$ axis, b) along $\theta$ axis

###### Remark 1

As it can be seen in (10), the GPD-D detection metric can be characterized a
weighted Euclidean distance in the polar domain which produces symbol-
independent and symmetrical decision regions based on the received symbol
energy. Thus, this detector provides a foundation for the development of time-
efficient nearest neighbor detection algorithms based on the constellation’s
geometry in the polar plane. This approach is aimed at addressing the demand
for low-complexity detection in super-constellation communications.
Consequently, in the context of SAPSK, by applying GPD-D hexagonal regions are
formulated in the polar plane, thus SAPSK symbols are allocated in the polar
plane in line with the optimal 2D sphere packing [17].

##  IV Error Analysis and Detection in SAPSK

As mentioned before, SAPSK stands out as a constellation with its symbols
being compactly positioned in the polar plane, thereby providing a solution
aligning with the criteria of super-constellation communications. Within this
framework, the introduction of GPD-D, enables the analytical investigation of
the SEP of SAPSK constellations and also lays the foundation for effective
detection algorithms. In the following subsections, we characterize how the
shape of the SAPSK decision regions is adjusted to the varying noise
conditions, and based on that, we provide a tight closed-form approximation
for their SEP and a novel detection algorithm with $\mathcal{O}\left(1\right)$
computational complexity.

###  IV-A Performance Analysis of SAPSK Constellations

To calculate the SEP of an SAPSK$\left(M,\Gamma\right)$ constellation under
AWGN and GPN conditions, the shape of the decision regions that are formulated
by GPD-D is required. As illustrated in Fig. 2f, the $M$ symbols of an
SAPSK$\left(M,\Gamma\right)$ constellation are arranged in $\Gamma$ columns in
the polar plane, particularly allocated to form a triangular lattice. In this
direction, as shown in Fig. 3, by applying GPD-D, which coincides with
applying nearest neighbour in the polar plane, the formulated decision regions
of an SAPSK are equivalent to irregular hexagons. However, by taking into
account (10), the shape of the decision regions is affected by the values of
AWGN power $\sigma_{n}^{2}$, the GPN level ${\sigma_{\phi}}^{2}$, and the
energy of the received symbol $r$, signifying the alteration of the decision
regions based on the prevailing noise conditions. In more detail, the polar
coordinates of the constellation symbols and the received symbol $r$ undergo a
scaling defined by the denominators in (10), which results in the hexagonal
decision regions being elongated along the $\rho$ and $\theta$ axes, and thus
reorienting the hexagons along these axes, as illustrated in Fig. 3. It should
be highlighted, that the decision region reorientation is a mechanism of GPD-D
to effectively mitigate the symbol shift induced by AWGN along $\rho$ and
$\theta$ axes, as well as the shift induced by GPN along the $\theta$ axis
depending on the energy of the received symbol $r$.

By taking into account the shape of the decision regions, a SEP approximation
for SAPSK constellations can be derived by decomposing the hexagonal regions
into a rectangular region along with two triangles whose areas are further
decomposed to a definite number of rectangles, as shown in Fig. 3. However,
based on the prevailing noise conditions the hexagons are correspondingly
extended either along $\rho$ or $\theta$ axis, which leads to the variation of
their shape. In more detail, in cases where the AWGN is greater than GPN, the
hexagons are extended along $\rho$ axis to mitigate symbol displacements
between two adjacent energy levels induced by AWGN, as illustrated in Fig. 3a.
Conversely, in cases where the GPN becomes the dominant distortion, the
decision regions start to extend along $\theta$ axis, as shown in Fig. 3b, to
mitigate the displacements that occur only along the $\theta$ axis due to GPN.

Building upon the prevailing noise conditions, the shape of the hexagonal
decision regions varies, and thus the location of the triangular regions that
will be decomposed to rectangles changes as well. Therefore, to approximate
the triangular regions with rectangles, we need to characterize the decision
regions’ orientation and calculate the received SNR threshold value
$\gamma_{q}$ above which the hexagons are reoriented from $\rho$ to $\theta$
axis.

###### Proposition 2

Assuming that $r$ lies between $q$-th and $\left(q+1\right)$-th energy levels
and that the decision regions are designed based on GPD-D, the orientation
$O_{h}$ of the hexagonal decision regions of an SAPSK$\left(M,\Gamma\right)$
constellation is given as

$\displaystyle O_{h}\equiv\begin{cases}\begin{split}\textbf{$\rho$
axis},\end{split}&\text{$\bar{\gamma}$}\leq\gamma_{q}\\\ \\\
\begin{split}\textbf{$\theta$
axis},\end{split}&\text{$\bar{\gamma}$}>\gamma_{q},\end{cases}$ (14)

where $\gamma_{q}$ is a received SNR threshold value given as

$\displaystyle\gamma_{q}=\frac{\left(2q-1\right)^{2}\pi^{2}\Gamma^{2}\left(4\Gamma^{2}-1\right)-4M^{2}\left(4\Gamma^{2}-1\right)}{24M^{2}\left(2q-1\right)^{2}\sigma_{\phi}^{2}}.$
(15)

###### Proof:

The proof is provided in Appendix A. ∎

###### Remark 2

While GPD-D aims to adaptively extend the decision regions to enhance GPN
resilience, the effectiveness of this adaptation is hindered under high GPN
conditions. Specifically, as GPN severity increases, this strategic extension
of the decision regions along the $\theta$ axis cannot reduce the error floor,
since the elevated level of GPN leads to a high probability of the received
symbol being displaced outside of its designated decision region, limiting the
efficiency of GPD-D in compensating the impact of GPN.

Having calculated the threshold $\gamma_{q}$ under which the hexagonal
decision regions reorient in the polar plane, the positions of the triangular
sub-areas can be defined, and thus their decomposition to rectangles can be
performed. To this end, in the following proposition, a tight closed-form
approximation for the SEP of an SAPSK$\left(M,\Gamma\right)$ constellation is
derived.

###### Proposition 3

By decomposing each hexagonal decision region with a main rectangle and two
triangles that are approximated by $2N$ rectangles, the SEP of an
SAPSK$\left(M,\Gamma\right)$ constellation affected by AWGN and GPN can be
tightly approximated as

$\displaystyle P_{e}\approx\frac{1}{\Gamma}\sum_{q=1}^{\Gamma}P_{q},$ (16)

where $P_{q}$ denotes the error probability of symbol $s_{q}$ that lies on the
$q$-th energy level and it is written as in (17) at the top of the next page,
and
$Q(x)=\frac{1}{\sqrt{2\pi}}\int_{x}^{\infty}\exp\left(-\frac{u^{2}}{2}\right)\,du$
is the Gaussian $Q$-function. Furthermore, $B_{z}$ is given as

$\displaystyle
P_{q}=\begin{cases}\begin{split}&1-\left(1-2Q\left(\sqrt{\frac{24\bar{\gamma}}{2\Gamma^{2}-1}}\right)\right)\left(1-2Q\left(\frac{\left(1-D_{q}\right)\pi\Gamma}{M\sqrt{\sigma_{\phi}^{2}+\frac{\sigma_{n}^{2}}{2E_{q}}}}\right)\right)\\\
&\quad-4\sum_{z=1}^{N}\left(Q\left(\frac{\left(1-D_{q}\right)\pi\Gamma}{M\sqrt{\sigma_{\phi}^{2}+\frac{\sigma_{n}^{2}}{2E_{q}}}}\right)-Q\left(\frac{B_{z}\pi\Gamma}{M\sqrt{\sigma_{\phi}^{2}+\frac{\sigma_{n}^{2}}{2E_{q}}}}\right)\right)\left(Q\left(\sqrt{\frac{24\bar{\gamma}(z-1)^{2}}{\left(4\Gamma^{2}-1\right)N^{2}}}\right)-Q\left(\sqrt{\frac{24\bar{\gamma}z^{2}}{\left(4\Gamma^{2}-1\right)N^{2}}}\right)\right),\end{split}&\text{$\bar{\gamma}$}\leq\gamma_{q}\\\
\begin{split}&1-\left(1-2Q\left(\sqrt{\frac{24\bar{\gamma}(1-D_{q})^{2}}{4\Gamma^{2}-1}}\right)\right)\left(1-2Q\left(\frac{\pi\Gamma}{M\sqrt{\sigma_{\phi}^{2}+\frac{\sigma_{n}^{2}}{2E_{q}}}}\right)\right)\\\
&\quad-4\sum_{z=1}^{N}\left(Q\left(\sqrt{\frac{24\bar{\gamma}(1-D_{q})^{2}}{4\Gamma^{2}-1}}\right)-Q\left(\sqrt{\frac{24\bar{\gamma}B_{z}}{4\Gamma^{2}-1}}\right)\right)\left(Q\left(\frac{(z-1)\pi\Gamma}{NM\sqrt{\sigma_{\phi}^{2}+\frac{\sigma_{n}^{2}}{2E_{q}}}}\right)-Q\left(\frac{z\pi\Gamma}{NM\sqrt{\sigma_{\phi}^{2}+\frac{\sigma_{n}^{2}}{2E_{q}}}}\right)\right),\end{split}&\text{$\bar{\gamma}$}>\gamma_{q}\end{cases}$
(17)

$\displaystyle B_{z}=\frac{\left(1-2D_{q}\right)z}{N}+D_{q},$ (18)

with $z\in\\{1,\dots,N\\}$, and $D_{q}$ is given by

$\displaystyle
D_{q}=\begin{cases}\begin{split}\frac{24M^{2}\bar{\gamma}\left(\sigma_{\phi}^{2}+\frac{4\Gamma^{2}-1}{6\bar{\gamma}(2q-1)^{2}}\right)+\pi^{2}\Gamma^{2}\left(4\Gamma^{2}-1\right)}{2\pi^{2}\Gamma^{2}\left(4\Gamma^{2}-1\right)},\end{split}&\text{$\bar{\gamma}$}\leq\gamma_{q}\\\
\\\
\begin{split}\frac{24M^{2}\bar{\gamma}\left(\sigma_{\phi}^{2}+\frac{4\Gamma^{2}-1}{6\bar{\gamma}(2q-1)^{2}}\right)+\pi^{2}\Gamma^{2}\left(4\Gamma^{2}-1\right)}{48M^{2}\bar{\gamma}\left(\sigma_{\phi}^{2}+\frac{4\Gamma^{2}-1}{6\bar{\gamma}(2q-1)^{2}}\right)},\end{split}&\text{$\bar{\gamma}$}>\gamma_{q}.\end{cases}$
(19)

###### Proof:

The proof is provided in Appendix B. ∎

###### Remark 3

Considering a fixed $M$, SNR, and $\sigma_{\phi}^{2}$, (16) provides a
straightforward method for obtaining the optimal design parameter $\Gamma$ of
an SAPSK$\left(M,\Gamma\right)$, which minimizes the SEP for a given energy
constraint, thereby enhancing the overall energy efficiency of the system.

###### Remark 4

As $\bar{\gamma}\rightarrow\infty$ the $P_{e}$ is equal to

$\displaystyle P_{e}=Q\left(\frac{2\pi\Gamma}{M\sigma_{\phi}}\right).$ (20)

###  IV-B SAPSK Detection Algorithm for practical implementation in super-
constellation communications

$r_{p}$Amplitude $\rho$Phase $\theta$ Figure 4: Detection scheme

In the context of an SAPSK constellation affected by AWGN and GPN, the
decision rule outlined in (3) stands as the optimal maximum likelihood
detection method, effectively minimizing symbol error probability. Despite its
theoretical optimality, this approach describes a linear traversal of all the
symbols in the constellation to detect the received signal, resulting in a
complexity of $\mathcal{O}(M)$. Such an algorithmic complexity deteriorates
the system performance in terms of time, especially in super constellations
where $M$ takes large values. Conversely, the GPD-D metric offers a promising
avenue for developing time-efficient algorithms with lower complexity, while
still preserving demodulation accuracy compared to the optimal detector.
Specifically, this decision rule can be conceptualized as a joint amplitude
and phase detector, entailing a weighted combination of amplitude and phase
differences between the received signal and the symbols of the constellation
within specific zones in the polar domain. To this end, in this section a
novel detection algorithm for SAPSK constellations of complexity
$\mathcal{O}(1)$ is presented.

Let the $i$-th symbol of constellation $\mathcal{C}$ be of the form
$s_{i}=x_{i}+jy_{i}$, where $j$ is the imaginary unit. Each symbol is written
in the polar form as $s_{i}=|s_{i}|e^{j\arg\\{s_{i}\\}}$ and it is one-to-one
mapped to a symbol in the polar domain of the form
$s_{\mathrm{p}i}=|s_{i}|+j\arg\\{s_{i}\\}$, where $|s_{i}|\in[0,\infty)$ and
$\arg\\{s_{i}\\}\in[-\pi,\pi]$. Furthermore, after receiving a symbol
$r=x_{r}+jy_{r}=\lvert r\rvert e^{j\arg{r}}$, it is transformed into the polar
domain as $r_{p}=|r_{p}|+j\arg\\{r_{p}\\}$, as depicted in Fig. 4. The key
idea of our algorithm lies in identifying potential constellation symbols
close to the received symbol $r_{p}$, thereby reducing the necessary number of
computed Euclidean distances between $r$ and the constellation symbols. Within
this framework, we define the regions $R_{1}=[\sqrt{E_{q}},\sqrt{E_{q+1}}]$
and $R_{2}=[\theta_{w},\theta_{w+1}]$, $q\in\\{1,\dots,\Gamma\\}$ and
$w\in\\{1,\dots,\frac{M}{\Gamma}\\}$. Furthermore, $E_{q}$, $E_{q+1}$,
$\theta_{w}$, and $\theta_{w+1}$ are the lines that are parallel to the axes,
as illustrated in Fig. 4, and they enclose the received symbol $r_{p}$ between
two consecutive energy levels and two consecutive angles of SAPSK
constellation, respectively. The detection process is completed by obtaining
the symbol $\hat{s}\in G=R_{1}\cap R_{2}$ which minimizes the metric, given by
(10). To accomplish this, it becomes evident that we need to identify the
constellation symbols that are contained in the region $G$. In this direction,
the $\Gamma$ distinct energy levels of the SAPSK constellation are stored in
ascending order in the array $S_{E}$ as

$S_{E}=[E_{1},\dots,E_{q},E_{q+1},\dots,E_{\Gamma}].$ (21)

Afterwards, for each $E_{q}\in S_{E}$ an adjacent list with all distinct
$\theta$ values in this amplitude is created as

$\displaystyle
A_{E_{q}}=[\theta_{q,1},\theta_{q,2},\dots,\theta_{q,\frac{M}{\Gamma}}],$ (22)

where $\theta$ values are sorted in ascending order.

To design an $\mathcal{O}(1)$ detection algorithm for SAPSK constellations, it
is essential to exploit their structure and, particularly, the arrays $S_{E}$
and $A_{E_{q}}$. In this direction, by noting that for an SAPSK constellation
the values in $S_{E}$ and $A_{E_{q}}$ increase with a constant rate, we can
formulate a linear interpolation function $f(\cdot)$ that returns the position
of its input in the arrays, and thus find the constellation symbols within $G$
in $\mathcal{O}(1)$ time complexity. In particular, when the input value $u$
of $f$ matches one of the array elements, $f$ returns an integer value
representing the position of $u$ in the array. Conversely, if $u$ is not an
array element, $f$ yields a non-integer value, which results in the position
of $S_{E}$ with the closest corresponding value to $u$ when rounded.
Therefore, given that the elements of $S_{E}$ are equispaced by
$\delta_{\rho}$ due to the SAPSK geometry, we can define the linear
interpolation function $f_{E}:\mathbb{R}\rightarrow\mathbb{R}$, which is given
by

$\displaystyle
f_{E}(u)=\frac{1}{\delta_{\rho}}v+1-\frac{\sqrt{E_{1}}}{\delta_{\rho}},$ (23)

where $E_{1}$ is the first element of $S_{E}$. To this end, by utilizing (23),
we can obtain the consecutive positions $q$ and $q+1$ in $S_{E}$ that satisfy
$\sqrt{E_{q}}\leq\lvert r\rvert\leq\sqrt{E_{q+1}}$, which are given as

$\displaystyle q=\left\lfloor f_{E}(|r|)\right\rfloor$ (24) $\displaystyle
q+1=\left\lceil f_{E}(|r|)\right\rceil,$

where $\left\lceil\cdot\right\rceil$ and $\left\lfloor\cdot\right\rfloor$ are
the ceil function. Similarly, given that the elements of $A_{E_{q}}$ are
equispaced by $\delta_{\theta}$ due to the SAPSK geometry, we can also define
the linear interpolation function
$f_{\theta,q}:\mathbb{R}\rightarrow\mathbb{R}$, which is given by

$\displaystyle
f_{\theta,q}(v)=\frac{1}{\delta_{\theta}}v+1-\frac{\theta_{q,1}}{\delta_{\theta}},$
(25)

where $\theta_{q,1}$ is the first element of $A_{E_{q}}$. Therefore, the
consecutive positions $\left(w,w+1\right)$ and $\left(q,q+1\right)$ of
$A_{E_{q}}$ and $A_{E_{q+1}}$, respectively, that satisfy
$\theta_{q,w}\leq\arg\\{r\\}\leq\theta_{q,w+1}$ and
$\theta_{q+1,l}\leq\arg\\{r\\}\leq\theta_{q+1,l+1}$ can be obtained as

$\displaystyle w=\left\lfloor f_{\theta,q}(\arg\\{r\\})\right\rfloor$ (26)
$\displaystyle w+1=\left\lceil f_{\theta,q}(\arg\\{r\\})\right\rceil$
$\displaystyle l=\left\lfloor f_{\theta,q+1}(\arg\\{r\\})\right\rfloor$
$\displaystyle l+1=\left\lceil f_{\theta,q+1}(\arg\\{r\\})\right\rceil.$

Consequently, we can obtain the set $S_{c}$ that contains the symbols of SAPSK
on region $G$ which can be expressed in IQ coordinates as

$\displaystyle
S_{c}=\begin{cases}\begin{split}\left\\{\sqrt{E_{q}}e^{j\theta_{q,w}},\sqrt{E_{q+1}}e^{j\theta_{q+1,l+1}}\right\\},\end{split}&\text{if
$q$ even}\\\ \\\
\begin{split}\left\\{\sqrt{E_{q}}e^{j\theta_{q,w+1}},\sqrt{E_{q+1}}e^{j\theta_{q+1,l}}\right\\},\end{split}&\text{if
$q$ odd}.\end{cases}$ (27)

Finally, the symbol $\hat{s}$ that GPD-D determines as the most likelihood to
have been sent can be obtained as

$\displaystyle\hat{s}=\arg\min_{s\in S_{c}}\hskip 4.015pt\frac{\big{(}\lvert
r\rvert-\lvert
s\rvert\big{)}^{2}}{\sigma_{n}^{2}}+\frac{\big{(}\arg\\{r\\}-\arg\\{s\\}\big{)}^{2}}{\sigma_{\phi}^{2}+\frac{\sigma_{n}^{2}}{2\lvert
r\rvert^{2}}},$ (28)

It should be highlighted that for any $M$-ary SAPSK constellation, it holds
that $|S_{c}|\leq 2$. Consequently, instead of comparing $M$ symbols as
required by the optimal detector in the presence of GPN, known as GAP-D, we
only need to perform two calculations to determine which symbol was
transmitted, resulting in $\mathcal{O}(1)$ complexity.

Algorithm 1 SAPSK Constellation Detection Algorithm

0: Coordinates of the received symbol $r=x_{r}+jy_{r}$

0: Detected symbol $\hat{s}$

1: Transform $r$ into polar coordinates: $r_{p}=\lvert r\rvert+j\arg\\{r\\}$

2: Initialize arrays $S_{E}$ for storing energy levels

3: Initialize arrays $A_{E_{q}}$, $\forall E_{q}\in S_{E}$, for storing the
symbol arguments on each energy level

4: Calculate $q$-th and $\left(q+1\right)$-th energy levels using linear
interpolation: $q=\left\lfloor f_{E}(\lvert r\rvert)\right\rfloor$,
$q+1=\left\lceil f_{E}(\lvert r\rvert)\right\rceil$

5: Calculate $w$-th and $\left(w+1\right)$-th symbol arguments of the $q$-th
energy level using linear interpolation within $A_{E_{q}}$: $w=\left\lfloor
f_{\theta,q}(\arg\\{r\\})\right\rfloor$, $w+1=\left\lceil
f_{\theta,q}(\arg\\{r\\})\right\rceil$

6: Calculate $l$-th and $\left(l+1\right)$-th symbol arguments of the
$\left(q+1\right)$-th energy level using linear interpolation within
$A_{E_{q+1}}$: $l=\left\lfloor f_{\theta,q+1}(\arg\\{r\\})\right\rfloor$,
$l+1=\left\lceil f_{\theta,q+1}(\arg\\{r\\})\right\rceil$

7: Construct the candidate set  $\displaystyle
S_{c}=\begin{cases}\begin{split}\left\\{\sqrt{E_{q}}e^{j\theta_{q,w}},\sqrt{E_{q+1}}e^{j\theta_{q+1,l+1}}\right\\},\end{split}&\textbf{$q$
even}\\\ \\\
\begin{split}\left\\{\sqrt{E_{q}}e^{j\theta_{q,w+1}},\sqrt{E_{q+1}}e^{j\theta_{q+1,l}}\right\\},\end{split}&\textbf{$q$
odd}.\end{cases}$

8: Find the symbol $\hat{s}$ in $S_{c}$ that minimizes the metric:
$\hat{s}=\arg\min_{s\in S_{c}}\frac{(\lvert r\rvert-\lvert
s\rvert)^{2}}{\sigma_{n}^{2}}+\frac{(\arg\\{r\\}-\arg\\{s\\})^{2}}{\sigma_{\phi}^{2}+\frac{\sigma_{n}^{2}}{2\lvert
r\rvert^{2}}}$

9: return $\hat{s}$

##  V Numerical Results

In this section, we verify the effectiveness of the proposed GPD-D in channels
affected by both AWGN and GPN, as well as the accuracy of the proposed SEP
approximation for an SAPSK$\left(M,\Gamma\right)$ constellation, where the
trade-off involving SEP, SNR, $\sigma_{\phi}^{2}$, $M$, and $\Gamma$ is
illustrated. Furthermore, we show the gain in terms of energy efficiency that
we obtain when SAPSK is used over PQAM in super-constellation communications.
Finally, we validate the detection accuracy of the proposed detection
algorithm for SAPSK constellations whose time complexity is independent of $M$
and achieves a detection performance equivalent to GAP-D.

In Figs. 5a and 5b, it is illustrated that GPD-D achieves equivalent
performance to the optimal GAP-D metric under two GPN scenarios for different
constellation designs. In particular, it can be seen that in both considered
GPN scenarios, i.e., $\sigma_{\phi}^{2}=10^{-4}$ and
$\sigma_{\phi}^{2}=10^{-2}$, the performance of GPD-D is identical to GAP-D,
indicating its effectiveness for different GPN conditions. In addition,
similar to GAP-D as shown in Fig. 1, the system performance is severely
affected by a high error floor when GPD-D and QAM are used, highlighting a
significant degradation in super-constellation communications when
constellations not tailored to the existence of GPN are used. However, when
modulation schemes such as PQAM and SAPSK are employed along with GPD-D, the
system exhibits significantly improved performance, highlighting the
importance of novel GPN-resilient modulation schemes for super-constellation
communications, even at relatively low GPN levels, to reduce the incurred
error floor and thus ensure reliable SEP performance.

$30$$40$$50$$60$$70$$80$$10^{-6}$$10^{-5}$$10^{-4}$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$\frac{E_{s}}{N_{0}}$
(dB)Symbol Error Probability4096-PQAM(512) (10)SAPSK(4096,512) (10)4096-QAM
(10)GAPD (a) $\sigma^{2}_{\phi}=10^{-4}$

$30$$40$$50$$60$$70$$80$$10^{-5}$$10^{-4}$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$\frac{E_{s}}{N_{0}}$
(dB)Symbol Error Probability4096-PQAM(512) (10)SAPSK(4096,512) (10)4096-QAM
(10)GAPD (b) $\sigma^{2}_{\phi}=10^{-2}$

Figure 5: Performance of GPD-D metric for various constellation design

Figs. 6a and 6b validate the proposed SEP approximation for the SAPSK
constellation for the case where $\sigma_{\phi}^{2}=10^{-4}$ and
$\sigma_{\phi}^{2}=10^{-2}$, respectively. Specifically, the proposed SEP
approximation is compared with the simulated SEP for the case of a 4096-SAPSK
in the presence of GPN, for various $\Gamma$ values. As it can be seen, the
tightness of the proposed approximation is confirmed for all investigated
$\Gamma$ values, indicating the effectiveness of our approximation.
Furthermore, in the low-SNR regime, where AWGN is the dominant distortion,
smaller values of $\Gamma$ exhibit better SEP performance due to their
property to mitigate the distortions caused by AWGN. However, as the SNR
increases and the GPN becomes the dominant distortion, small values of
$\Gamma$ show a performance degradation compared to higher values of $\Gamma$,
because higher $\Gamma$ translates into assigning fewer symbols per energy
level, thus increasing the robustness to the GPN. To this end, it can be
deduced that there exists an optimal $\Gamma$ value according to the
prevailing noise conditions and the order of the SAPSK constellation,
highlighting the existence of a unique trade-off between energy efficiency and
reliability in super-constellation communications.

$30$$40$$50$$60$$70$$10^{-5}$$10^{-4}$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$\frac{E_{s}}{N_{0}}$
(dB)Symbol Error ProbabilitySAPSK(4096,256) (sim.)SAPSK(4096,512)
(sim.)SAPSK(4096,1024) (sim.)SAPSK(4096,2048) (sim.)Analytical (16) (a)
$\sigma^{2}_{\phi}=10^{-4}$

$30$$40$$50$$60$$70$$10^{-5}$$10^{-4}$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$\frac{E_{s}}{N_{0}}$
(dB)Symbol Error ProbabilitySAPSK(4096,256) (sim.)SAPSK(4096,512)
(sim.)SAPSK(4096,1024) (sim.)SAPSK(4096,2048) (sim.)Analytical (16) (b)
$\sigma^{2}_{\phi}=10^{-2}$

Figure 6: SAPSK SEP approximation

Building on the previous trade-off for the $\Gamma$ value, Figs. 7a, 7b, and
7c illustrate how the optimal value of $\Gamma$ varies with respect to the
prevailing noise conditions, the modulation scheme employed, and the order $M$
of the constellation. These figures clearly show that the optimal $\Gamma$
value for SAPSK is always greater than or equal to that of PQAM, which means
that SAPSK symbols are arranged in a way that increases robustness to GPN,
since an increased $\Gamma$ translates into fewer symbols per energy level.
However, this increased GPN robustness should not come at the cost of
sacrificing the system’s energy efficiency and robustness to AWGN. To this
end, using the optimal values of $\Gamma$ for the cases where $M=4096$ and
$M=16384$, we show in Figs. 8a and 8b the SEP of SAPSK and PQAM for
$\sigma_{\phi}^{2}=10^{-4}$ and $\sigma_{\phi}^{2}=10^{-2}$, respectively,
according to the analytical expression given in (16). As it can be observed,
SAPSK has a higher energy efficiency compared to PQAM, offering an SNR gain
ranging from 1dB to 3dB over the presented SNR range and for the studied GPN
levels. Therefore, Fig. 8 illustrates SAPSK as a more energy-efficient
modulation scheme compared to PQAM that also maintains a balance between GPN
and AWGN robustness. However, it should be emphasized that although the
adaptation of $\Gamma$ based on the prevailing SNR and GPN levels can optimize
the performance, it requires an increased system complexity since the
transmitter needs to accurately know both the received SNR and GPN levels at
the receiver side, thus requiring complex feedback mechanisms to obtain
perfect channel state information. Consequently, to meet the system
requirements, the used $\Gamma$ value must be predetermined to strike a
balance between improved SEP performance and system complexity.

$30$$40$$50$$60$$70$$80$$0$$1{,}000$$2{,}000$$3{,}000$$4{,}000$$\frac{E_{s}}{N_{0}}$
(dB)$\Gamma$16384-SAPSK16384-PQAM (a) $\sigma^{2}_{\phi}=10^{-4}$

$30$$40$$50$$60$$70$$80$$0$$500$$1{,}000$$1{,}500$$2{,}000$$\frac{E_{s}}{N_{0}}$
(dB)$\Gamma$4096-SAPSK4096-PQAM (b) $\sigma^{2}_{\phi}=10^{-4}$

$30$$40$$50$$60$$70$$80$$0$$500$$1{,}000$$1{,}500$$2{,}000$$\frac{E_{s}}{N_{0}}$
(dB)$\Gamma$4096-SAPSK4096-PQAM (c) $\sigma^{2}_{\phi}=10^{-2}$

Figure 7: Optimal $\Gamma$ vs SNR

$30$$40$$50$$60$$70$$80$$10^{-5}$$10^{-4}$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$\frac{E_{s}}{N_{0}}$
(dB)Symbol Error Probability4096-SAPSK (sim.)16384-SAPSK (sim.)$M$-SAPSK
(Analytical)$M$-PQAM (a) $\sigma^{2}_{\phi}=10^{-4}$

$30$$40$$50$$60$$70$$80$$10^{-5}$$10^{-4}$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$\frac{E_{s}}{N_{0}}$
(dB)Symbol Error Probability4096-SAPSK (sim.)16384-SAPSK (sim.)$M$-SAPSK
(Analytical)$M$-PQAM (b) $\sigma^{2}_{\phi}=10^{-2}$

Figure 8: SAPSK vs PQAM for optimal $\Gamma$

To illustrate the adaptive scheme described for continuously varying the
$\Gamma$ of SAPSK to achieve optimal SEP over varying SNRs, Figs. 9a and 9b
illustrate the SEP performance of an SAPSK and a PQAM constellation with
$M=4096$ and $\Gamma$ set to 256, 1024 and 4096, considering GPN levels of
$\sigma_{\phi}^{2}=10^{-4}$ and $\sigma_{\phi}^{2}=10^{-2}$. Comparing these
variations, we can observe that SAPSK consistently outperforms PQAM in terms
of SEP, achieving a gain of 5 dB over PQAM for $\Gamma\in\\{1024,2048\\}$ for
the GPN scenarios considered. In particular, for $\sigma_{\phi}^{2}=10^{-4}$,
SAPSK($4096$, $256$) shows significant performance gains compared to
$4096$-PQAM($256$) up to SNR equal to 55dB, where the SEP reaches $10^{-3}$,
while as GPN becomes the dominant distortion, the performance of both
constellations is equivalent. In more detail, SAPSK and PQAM have the same
angular space between symbols on the same energy level, thus when GPN is
dominant compared to AWGN, the performance of SAPSK and PQAM is similar. It
should also be noted that the SNR value where GPN is the dominant distortion
depends on the value of $\Gamma$ and the prevailing noise condition.
Therefore, as shown in Fig.9a, for larger values of $\Gamma$, the performance
of SAPSK and PQAM has not converged even for SNR values larger than 70dB.
Moreover, as shown in Fig. 9b, the performance of SAPSK($4096$, $256$) and
$4096$-PQAM($256$) becomes equivalent for very small values of SNR because the
GPN level is increased to $\sigma_{\phi}^{2}=10^{-2}$, reducing the value of
SNR where GPN is dominant over AWGN. Finally, it can be observed that in the
case where the $4096$-SAPSK constellation is used, with a fixed $\Gamma$
value, $\Gamma=1024$ can be chosen as a compromise between energy consumption
and resilience to the presence of GPN and AWGN.

$30$$40$$50$$60$$70$$80$$10^{-5}$$10^{-4}$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$\frac{E_{s}}{N_{0}}$
(dB)Symbol Error
ProbabilitySAPSK(4096,256)SAPSK(4096,1024)4096-SAPSK(4096)$M$-PQAM($\Gamma$)
(a) $\sigma^{2}_{\phi}=10^{-4}$

$30$$40$$50$$60$$70$$80$$10^{-5}$$10^{-4}$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$\frac{E_{s}}{N_{0}}$
(dB)Symbol Error
ProbabilitySAPSK(4096,256)SAPSK(4096,1024)4096-SAPSK(4096)$M$-PQAM($\Gamma$)
(b) $\sigma^{2}_{\phi}=10^{-2}$

Figure 9: SAPSK vs PQAM for fixed $\Gamma$

Finally, Fig. 10 illustrates a comparison between the detection efficiency of
the proposed detection algorithm for SAPSK constellations, outlined in
Algorithm 1, and the performance achieved by the optimal detector GAP-D, as
defined by (3). Using GAP-D for symbol-by-symbol detection in the presence of
GPN has a complexity of $\mathcal{O}(M)$, since it requires $M$ computations
to obtain $\hat{s}$. However, using the decision regions of SAPSK formulated
in the polar plane by the proposed GPD-D metric, we have designed Algorithm 1
with a complexity of $\mathcal{O}(1)$. This complexity remains constant,
requiring only two computations to obtain $\hat{s}$ regardless of the
constellation order $M$, thus making the adaptation of super constellations in
communication systems feasible in terms of detection time complexity.

$30$$40$$50$$60$$70$$10^{-5}$$10^{-4}$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$\frac{E_{s}}{N_{0}}$
(dB)Symbol Error ProbabilitySAPSK(4096,256) (3)SAPSK(4096,512)
(3)SAPSK(4096,1024) (3)SAPSK(4096,2048) (3)Algorithm 3 Figure 10: Proposed
detection algorithm for SAPSK when $\sigma^{2}_{\phi}=10^{-2}$

##  VI Conclusion

In this paper, we introduced a novel modulation scheme, namely SAPSK, that is
consistent with the rules of robust super-constellation communications.
Specifically, we proposed GPD-D, a novel detector that closely approximates
the performance of the optimal maximum likelihood detector GAP-D in the
presence of both AWGN and GPN, which describes a weighted Euclidean distance
detector in the polar plane. According to our results, we verified the
effectiveness of GPD-D, which allows the derivation of a tight, closed-form
approximation for the SEP of SAPSK in channels affected by both AWGN and GPN.
Furthermore, we demonstrated the superiority of SAPSK over PQAM in terms of
energy efficiency in both adaptive and fixed communication links. Finally, we
illustrate the detection efficiency of the proposed $\mathcal{O}(1)$ algorithm
for SAPSK constellations, paving the way for their practical implementation.

##  Appendix A Proof of Proposition 2

Initially, to obtain the decision regions of an SAPSK constellation, it is
imperative to characterize the scaling that will be induced to the
constellation symbols due to the existing noise conditions. In this direction,
considering that a symbol $r$ is received such that it satisfies
$E_{q}\leq|r|^{2}\leq E_{q+1}$ and using (11) and (13), the altered vertical
and horizontal distances $\delta_{\theta}^{\prime}$ and
$\delta_{\rho}^{\prime}$ between two neighboring symbols in the polar plane
after the application of GPD-D as illustrated in Fig. 3, can be expressed as

$\displaystyle\delta_{\rho}^{\prime}=\frac{\delta_{\rho}}{\sigma_{n}},$ (29)

and

$\displaystyle\delta_{\theta}^{\prime}=\frac{\delta_{\theta}}{\sqrt{\sigma_{\phi}^{2}+\frac{\sigma_{n}^{2}}{2E_{q}}}}.$
(30)

For the case where the AWGN is greater than GPN, then the hexagons are
oriented along the $\rho$ axis to mitigate the symbol displacements caused by
AWGN across consecutive energy levels. However, as GPN increases, the hexagons
start to elongate along the $\theta$ axis to compensate for the effect of GPN
on the constellation symbols. Furthermore, the elongation of the hexagons
along the $\theta$ axis is upper limited by the half distance
$\frac{\delta_{\theta}^{\prime}}{2}$ between two adjacent symbols on the same
column due to the NN nature of GPD-D metric. Thus, when the distance between
the symbol and the peak of the hexagons along $\theta$ axis equals
$\frac{\delta_{\theta}^{\prime}}{2}$, the orientation $O_{h}$ changes from
$\rho$ to $\theta$ axis undergoing a $90^{\degree}$ rotation. In this
direction, we can determine that the hexagonal regions’ orientation has
changed from $\rho$ axis to $\theta$ axis when the distance $R_{d}$ which
describes the distance between each symbol and the hexagon peak for a hexagon
oriented to $\rho$ axis becomes equal to $\frac{\delta_{\theta}^{\prime}}{2}$.
Therefore, according to the decision regions’ orientation, the distance
$R_{d}$ is given as

$\displaystyle
R_{d}=\begin{cases}\begin{split}\frac{\sqrt{\delta_{\rho}^{\prime
2}+\left(\frac{\delta_{\theta}^{\prime}}{2}\right)^{2}}}{2\cos{\left(90^{\degree}-\alpha\right)}},\end{split}&\textbf{$O_{h}\equiv\rho$
axis}\\\ \\\ \begin{split}\frac{\sqrt{\delta_{\rho}^{\prime
2}+\left(\frac{\delta_{\theta}^{\prime}}{2}\right)^{2}}}{2\cos{\alpha}},\end{split}&\textbf{$O_{h}\equiv\theta$
axis}\end{cases}$ (31)

where angle $\alpha$ is depicted in Fig. 3. Additionally, by utilizing
$\tan{\alpha}=\frac{\delta_{\theta}^{\prime}}{2\delta_{\rho}^{\prime}}$ and
$\cos\alpha=\frac{1}{\sqrt{1+\tan^{2}{\alpha}}}$, (31) can be rewritten as

$\displaystyle
R_{d}=\begin{cases}\begin{split}\frac{\delta_{\theta}^{\prime}}{2}\left(\frac{\tan^{2}{\alpha}+1}{2\tan^{2}{\alpha}}\right),\end{split}&\textbf{$O_{h}\equiv\rho$
axis}\\\ \\\
\begin{split}\delta_{\rho}^{\prime}\left(\frac{\tan^{2}{\alpha}+1}{2}\right),\end{split}&\textbf{$O_{h}\equiv\theta$
axis}\end{cases}$ (32)

Moreover, by substituting (5) in (29), as well as, (6) and (7) in (30), then
it holds that

$\displaystyle\tan{\alpha}=\frac{\pi^{2}\Gamma^{2}\left(4\Gamma^{2}-1\right)}{M^{2}\sqrt{24\bar{\gamma}\left(\sigma_{\phi}^{2}+\frac{4\Gamma^{2}-1}{6\bar{\gamma}(2q-1)^{2}}\right)}},$
(33)

and by substituting (33) in (32), $R_{d}$ can be rewritten as

$\displaystyle
R_{d}=\begin{cases}\begin{split}\frac{\delta_{\theta}^{\prime}}{2}\underbrace{\left(\frac{24\bar{\gamma}\left(\sigma_{\phi}^{2}+\frac{4\Gamma^{2}-1}{6\bar{\gamma}(2q-1)^{2}}\right)+A^{2}(M,\Gamma)}{2A^{2}(M,\Gamma)}\right)}_{D_{1,q}},\end{split}&\textbf{$O_{h}\equiv\rho$
axis}\\\ \\\
\begin{split}\delta_{\rho}^{\prime}\underbrace{\left(\frac{24\bar{\gamma}\left(\sigma_{\phi}^{2}+\frac{4\Gamma^{2}-1}{6\bar{\gamma}(2q-1)^{2}}\right)+A^{2}(M,\Gamma)}{48\bar{\gamma}\left(\sigma_{\phi}^{2}+\frac{4\Gamma^{2}-1}{6\bar{\gamma}(2q-1)^{2}}\right)}\right)}_{D_{2,q}},\end{split}&\textbf{$O_{h}\equiv\theta$
axis}\end{cases}$ (34)

where $A\left(M,\Gamma\right)=\frac{\pi\Gamma\sqrt{4\Gamma^{2}-1}}{M}$. Thus,
to determine that the hexagonal regions’ orientation has changed from $\rho$
axis to $\theta$ axis, i.e., the distance $R_{d}$ for a hexagon oriented to
$\rho$ axis becomes equal to $\frac{\delta_{\theta}^{\prime}}{2}$, $D_{1,q}$
needs to be equal to 1. To this end, by setting $D_{1,q}=1$, we can obtain the
threshold value $\gamma_{q}$ as

$\displaystyle\gamma_{q}=\frac{\left(2q-1\right)^{2}\pi^{2}\Gamma^{2}\left(4\Gamma^{2}-1\right)-4M^{2}\left(4\Gamma^{2}-1\right)}{24M^{2}\left(2q-1\right)^{2}\sigma_{\phi}^{2}},$
(35)

which concludes the proof.

##  Appendix B Proof of Proposition 3

As it can be seen in Fig. 3, the hexagonal decision regions can be decomposed
by a main rectangular region and two triangles. Thus, for the case where GPD-D
is applied and according to the decision regions orientation, the probability
of erroneous detection of a symbol $s_{q}$ on the $q$-th energy level of an
SAPSK constellation is given as

$P_{q}=1-\left(P_{q,A}+2P_{q,B}\right),$ (36)

where $P_{q,A}$ is the probability to be within the main rectangle, and
$P_{q,B}$ is the probability to lie within a triangle of the hexagonal
decision region. To this end, to calculate the SEP for an SAPSK symbol, we
need to derive $P_{q,A}$ and $P_{q,B}$.

Initially, the form of the main rectangle and the triangles is affected by the
prevailing noise conditions. Therefore, according to the orientation $O_{h}$
of the decision region and the scaling occurred due to the GPD-D, assuming
that $s_{q}$ coincides with the origin $O=(0,0)$ since it is located in the
center of the decision region and that $n_{\rho}$ and $n_{\theta}$ are
independent RVs, the probability $P_{q,A}$ can be written as

$\displaystyle
P_{q,A}=\begin{cases}\begin{split}&P\left(-\delta_{\rho}^{\prime}\leq\frac{n_{\rho}}{\sqrt{\frac{\sigma_{n}^{2}}{2}}}\leq\delta_{\rho}^{\prime}\right)\\\
&\times
P\left(-\frac{R_{w}}{2}\leq\frac{n_{\theta}}{\sqrt{\sigma_{\phi}^{2}+\frac{\sigma_{n}^{2}}{2E_{q}}}}\leq\frac{R_{w}}{2}\right),\end{split}&\text{$\bar{\gamma}$}\leq\gamma_{q}\\\
\\\
\begin{split}&P\left(-\frac{R_{w}}{2}\leq\frac{n_{\rho}}{\sqrt{\frac{\sigma_{n}^{2}}{2}}}\leq\frac{R_{w}}{2}\right)\\\
&\times
P\left(-\frac{\delta_{\theta}^{\prime}}{2}\leq\frac{n_{\theta}}{\sqrt{\sigma_{\phi}^{2}+\frac{\sigma_{n}^{2}}{2E_{q}}}}\leq\frac{\delta_{\theta}^{\prime}}{2}\right),\end{split}&\text{$\bar{\gamma}$}>\gamma_{q},\end{cases}$
(37)

where $R_{w}$ denotes the width of the main rectangle according to its
orientation $O_{h}$, as shown in Fig. 3, and it is given as

$\displaystyle
R_{w}=\begin{cases}\begin{split}\delta_{\theta}^{\prime}-2R_{d},\end{split}&\text{$\bar{\gamma}$}\leq\gamma_{q}\\\
\\\
\begin{split}2\delta_{\rho}^{\prime}-2R_{d},\end{split}&\text{$\bar{\gamma}$}>\gamma_{q}\end{cases}$
(38)

Additionally, by utilizing (34), (38) can be rewritten as

$\displaystyle
R_{w}=\begin{cases}\begin{split}\delta_{\theta}^{\prime}\left(1-D_{1,q}\right),\end{split}&\text{$\bar{\gamma}$}\leq\gamma_{q}\\\
\\\
\begin{split}2\delta_{\rho}^{\prime}\left(1-D_{2,q}\right),\end{split}&\text{$\bar{\gamma}$}>\gamma_{q}\end{cases}$
(39)

Finally, to calculate (37), the cumulative distribution function (CDF) of RVs
$n_{\rho}$ and $n_{\theta}$ is required. In this direction, by taking into
account [30], it holds that
$n_{\rho}\sim\mathcal{N}\left(0,\frac{\sigma_{n}^{2}}{2}\right)$ and
$n_{\theta}\sim\mathcal{N}\left(0,\sigma_{\phi}^{2}+\frac{\sigma_{n}^{2}}{2E_{q}}\right)$,
thus (37) can be rewritten as

$\displaystyle
P_{q,A}=\begin{cases}\begin{split}\left(1-2Q\left(\delta_{\rho}^{\prime}\right)\right)\left(1-2Q\left(\frac{\delta_{\theta}^{\prime}}{2}\left(1-D_{1,q}\right)\right)\right),\end{split}&\text{$\bar{\gamma}$}\leq\gamma_{q}\\\
\\\
\begin{split}\left(1-2Q\left(\frac{\delta_{\theta}^{\prime}}{2}\right)\right)\left(1-2Q\left(\delta_{\rho}^{\prime}\left(1-D_{2,q}\right)\right)\right),\end{split}&\text{$\bar{\gamma}$}>\gamma_{q},\end{cases}$
(40)

which concludes the calculation of $P_{q,A}$.

Regarding the calculation of $P_{q,B}$, we can split each triangle into two
orthogonal triangles, calculate the probability $P_{t}$ for the received
symbol $r$ to lie within one of these orthogonal triangles, and then multiply
it by four. However, the probability of $s_{q}$ lying inside a triangle does
not have a closed-form expression due to the complexity of the bivariate
normal distribution and the constraints imposed by the triangle’s boundaries.
Thus, the area of each triangle can be tightly approximated by $N$ rectangles,
as shown in Fig. 3, whose dimensions depend on the decision region’s
orientation. Therefore, considering $O_{h}$, $P_{t}$ can be approximated as

$\displaystyle
P_{t}\approx\begin{cases}\begin{split}\sum_{z=1}^{N}&P\left(\frac{(z-1)\delta_{\rho}^{\prime}}{N}\leq\frac{n_{\rho}}{\sqrt{\frac{\sigma_{n}^{2}}{2}}}\leq\frac{z\delta_{\rho}^{\prime}}{N}\right)\\\
&\times
P\left(\frac{R_{w}}{2}\leq\frac{n_{\theta}}{\sqrt{\sigma_{\phi}^{2}+\frac{\sigma_{n}^{2}}{2E_{q}}}}\leq\frac{B_{z,1}\delta_{\theta}}{2}\right),\end{split}&\text{$\bar{\gamma}$}\leq\gamma_{q}\\\
\\\
\begin{split}\sum_{z=1}^{N}&P\left(\frac{\left(z-1\right)\delta_{\theta}^{\prime}}{2N}\leq\frac{n_{\rho}}{\sqrt{\frac{\sigma_{n}^{2}}{2}}}\leq\frac{z\delta_{\theta}^{\prime}}{2N}\right)\\\
&\times
P\left(\frac{R_{w}}{2}\leq\frac{n_{\theta}}{\sqrt{\sigma_{\phi}^{2}+\frac{\sigma_{n}^{2}}{2E_{q}}}}\leq
B_{z,2}\delta_{\rho}^{\prime}\right),\end{split}&\text{$\bar{\gamma}$}>\gamma_{q},\end{cases}$
(41)

where $B_{z,1}=\frac{\left(1-2D_{1,q}\right)z}{N}+D_{1,q}$, and
$B_{z,2}=\frac{\left(1-2D_{2,q}\right)z}{N}+D_{2,q}$. Finally, by leveraging
the CDFs of $n_{\rho}$ and $n_{\theta}$, $P_{t}$ can be derived, thus
$P_{q,B}$ can be calculated as $P_{q,B}=4P_{t}$, which concludes the proof.

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