Energy-efficient multidimensional Hellinger modulation for SPAD-based optical wireless communications

Ling-Han Si-Ma; Jian Zhang; Bin-Qiang Wang; Yan-Yu Zhang

doi:10.1364/OE.25.022178

1. Introduction

Optical wireless communication (OWC) has been viewed as an attractive technique for the future wireless communication, due to its unique advantages of high security, freedom from spectral licensing issue and large potential bandwidth [1–7]. For such system, the ubiquitously installed light emitting diodes (LEDs) and photodiodes (PDs) are natural transmitters and receivers using simple intensity modulation and direct detection (IM/DD) [8–10]. Recently, single photon avalanche diode (SPAD), as a new type of detector for OWC [11,12], has drawn much attention due to its higher sensitivity than the commonly used PD [13–15] and its potential applications to underwater communications [16, 17]. For underwater environment, the light signal suffers three main impairments including absorption, scattering and turbulence. The influnece of these impairments can be described by various models such as double-gamma function [18] and log-normal function [19]. After long propagation, the received signal is usually weak, thus high energy-efficiency of transmitted signal is essential to ensure a reliable communication link. In this paper, we consider the energy-efficient constellation design for the SPAD OWC systems.

However, compared with the design criterion and techniques of radio frequency (RF) communication [20–27] and PD OWC, SPAD OWC system has two significant differences. The first difference is that the signal of SPAD OWC or PD OWC is required to be nonnegative which is different from that of the RF communications [28,29]. Therefore, the well-developed radio frequency techniques [28–33] can not be applied to SPAD OWC systems in a straightforward manner. Since the output of SPAD is influenced by photon-counting-induced additive Poisson noise (APN) [13–15,34–38]. For SPAD OWC, the second significant difference is that the signal design criterion and techniques over APN cannot directly follow those over AWGN channels. For example, for PD OWC, pulse amplitude modulation (PAM) is the most energy-efficient scalar constellation based on the Euclidian distance criterion. Unfortunately, according to the results in [39–41], this is no longer true for our considered SPAD OWC, which is under Poisson regime. For SPAD OWC, the received optical signal is usually weak and thus, to ensure energy-efficient transmission, multidimensional signal design is required. However, the error performance analysis of the maximum likelihood (ML) detection in Poisson regime is a challenging task. More recently, the authors in [39] and [42] proposed Anscombe root (AR) receiver and further established a Hellinger distance criterion for signal design. It is also noticed that the designed Hellinger-distance-optimal squared PAM (SPAM) [39,42] is optimal only within one-dimensional integer space. Up to now, the energy-efficient multidimensional constellation design remains unsolved due to the challenge of solving the corresponding max-min optimization problems.

The above factors indeed motivate us to attack this open problem. In this paper, we devise an energy-efficient multidimensional integer constellation by maximizing the Hellinger distance under a normalized average optical power budget. Comprehensive simulations indicate that our designed constellation significantly outperforms the traditional PAM and SPAM schemes for SPAD OWC systems.

2. System model with AR receiver

In this section, we introduce the channel model of SPAD OWC and the previously proposed AR receiver for such system.

2.1. Transmitter

Let us consider an SPAD OWC system over an ideal Poisson channel. Let $x \in 𝒳 \subseteq ℝ_{+}^{L}$ denote the transmitted signal over L time slots. $ℝ_{+}^{L}$ is the set of all L × 1 vectors whose entries are nonnegative real-valued numbers to satisfy the nonnegativity requirement of intensity modulation. Then, the output vector of the SPADs, r, is given by [43]

r = α x + β 1_{L \times 1} + p

where α and β respectively represent the power gain caused by SPAD and the counted photon number caused by the dark current over a symbol interval. Since p is an L-dimensional Poisson noise vector, the entries of r are identically independently Poisson-distributed. Then, the probability density functions (PDFs) of r and p are respectively:

{Pr}_{r} (r_{ℓ} = n) = e^{- λ_{ℓ}} \frac{λ_{ℓ}^{n}}{n!}, {Pr}_{p} (p_{ℓ} = \hat{n}) = e^{- λ_{ℓ}} \frac{λ_{ℓ}^{\hat{n} + λ_{ℓ}}}{(\hat{n} + λ_{ℓ})!}

where n and n̂ + λ_ℓ are nonnegative integers, and λ_ℓ is the mean of Poisson distribution, which can be expressed as λ_ℓ = αx_ℓ + β for ℓ = 1, 2, · · · , L [39,43].

2.2. Anscombe root (AR) receiver

A vector-version AR receiver was proposed in [42]. For presentation convenience, now we would like to introduce this receiver in the following.

We first use the existing Anscombe root (AR) operation to transform the entries of r as

r_{ℓ}^{(AR)} = 2 \sqrt{r_{ℓ} + 3 / 8}, ℓ = 1, 2, \dots, L .

Then, when the value of r_ℓ is sufficiently large [42], the output of AR transform,

r_{ℓ}^{(AR)}

, is approximately distributed with a Gaussian distribution. At the same time, the proof in [44] tells the fact that

E (r_{ℓ}^{(AR)}) \approx 2 \sqrt{E (r_{ℓ}) + 3 / 8}

where E(.) is the expectation of a random statistic. Therefore, we can finally approximate

r_{ℓ}^{(AR)}

by

r_{ℓ}^{(AR)} \approx {\bar{x}}_{ℓ} + ξ_{ℓ}

where

{\bar{x}}_{ℓ} = 2 \sqrt{α x_{ℓ} + β + 3 / 8}

and ξ_ℓ is additive Gaussian noise with zero mean and variance being one. In [42], the scalar-version expression (3) was extended to a vector version. For notation simplicity, we define

r^{(AR)} = {(r_{1}^{(AR)}, \dots, r_{L}^{(AR)})}^{T}

and x̄ = (x̄₁, · · · , x̄_L)^T. The vector-version expression for the resulting Gaussian channel is expressed as

r^{(AR)} \approx \bar{x} + ξ .

Now, the vector-version AR receiver proposed in [42] for our considered system is given below.

AR Receiver: Given the L-dimensional received signal r defined by (1), the estimation of the transmitted x, x̂, at the AR receiver is determined by the following two successive steps:

AR transform: $r^{(AR)} = {(r_{1}^{(AR)}, \dots, r_{L}^{(AR)})}^{T}$ , where $r_{ℓ}^{(AR)}$ is given by (2).
Minimum Distance Decision: $\hat{x} = {arg min}_{x \in 𝒳} {‖ r^{(AR)} - \bar{x} ‖}_{2}^{2}$ , where 𝒳 is the constellation of transmitted signal.

In this paper, our main task is to design an energy-efficient L-dimensional constellation 𝒳 for the above-mentioned AR receiver.

3. Energy-efficient multidimensional Hellinger constellation

In this section, we formulate the constellation design problem and then present the main result.

3.1. Problem formulation

In this subsection, our main task is to formulate our design problem for the AR receiver. Here, a natural question comes up: what decides the error performance of the constellation design at the AR receiver? According to the introduction of AR receiver in Subsection 2.2, we know that, when the optical power of the transmitted signal x is sufficiently high, the error performance for our considered channel is dominated by the minimum Hellinger distance between two distinct signal vectors [42]. For convenience, we define HD(𝒳) as the minimum Hellinger distance of the transmitted constellation 𝒳, say

HD (𝒳) = min_{x \neq \hat{x}, x, \hat{x} \in 𝒳} \sum_{ℓ = 1}^{L} {| \sqrt{x_{ℓ}} - \sqrt{{\hat{x}}_{ℓ}} |}^{2} .

This observation provides us with an explicit Hellinger-distance-based criterion for energy-efficient signal design for AR receiver. In general, for AR receiver, the Hellinger-distance-optimal multidimensional constellation design is as hard as solving a parallel problem of designing the optimal multidimensional constellation over AWGN channels, which is a long-standing and open problem in RF digital communication [20,24,25,45]. Since the optimal design problem for AR receiver is not easy to be transformed into a tractable one, we constrain our discussion into an integer set. Therefore, our design problem is formulated below:

Problem 1 Let $x^{\frac{1}{2}} = {(x_{1}^{\frac{1}{2}}, x_{2}^{\frac{1}{2}}, \dots, x_{L}^{\frac{1}{2}})}^{T}$ . $ℤ_{+}^{L}$ denotes the set of L × 1 vectors with all L entries being nonnegative integers. Then, for arbitrarily given positive integers L and K, devise an L-dimensional size-2^K constellation $𝒳 = {x : x^{〈 \frac{1}{2} 〉} \in ℤ_{+}^{L}}$ such that the average transmitted optical power $\frac{1}{2^{K}} \sum_{x \in 𝒳} 1^{T} x$ is minimized subject to HD(𝒳) = 1, where HD(𝒳) is defined in (5).

3.2. Main result

In this subsection, we present the solution to Problem 1. To make our presentation as clear as possible, let us first introduce some existing mathematical theories related to solving Problem 1. The solution to Problem 1 is related to a Diophantine equation ${‖ z ‖}_{2}^{2} = n$ with respect to z. Solving this equation is equivalent to answering the following question: “Under what condition, an integer can be decomposed into the sum of integer squares ?” For such question, there are several classical theorems for specific L. For example, when L = 2, Fermat’s two-square theorem asserts that an odd prime number z can be expressed as z = x² + y² for integer x and y, if and only if z has the form 4n + 1, n ∈ ℤ₊ [46,47]. Moreover, Legendre’s three-square theorem and Lagrange’s four-square theorem were proved by [48] and [49] respectively. Unfortunately, up to now, the solution to ${‖ z ‖}_{2}^{2} = n$ for a general L still remains unsolved [50]. We are interested in the fact that the total number of the solutions to ${‖ z ‖}_{2}^{2} = n$ is finite.

For presentation convenience, we denote $𝒩 = {n : {‖ z ‖}_{2}^{2} = n, z \in ℤ_{+}^{L}}$ and define R_n by the cardinality of ${z : {‖ z ‖}_{2}^{2} = n, z \in ℤ_{+}^{L}, n \in 𝒩}$ . Furthermore, we let N̄ be defined by the smallest positive integer satisfying $\sum_{n \in 𝒩, n \leq \bar{N}} R_{n} \geq 2^{K}$ . By our definition of 𝒩, we denote number of all the elements of 𝒩 not larger than N̄ by I. Then, without loss of generality, we arrange all these I numbers in an increasing order as n₁ < · · · < n_I with n₁ = 0 and n_I = N̄.

Now, we formally state our main result in this paper.

Theorem 1 An optimal solution to Problem 1 is given by

\tilde{𝒳} = \cup_{i = 1}^{i = I - 1} {\tilde{𝒳}}_{n_{i}} \cup {\bar{𝒳}}_{n_{I}},

where

{\tilde{𝒳}}_{n_{i}} = {x : 1^{T} x = n_{i}, x^{〈 \frac{1}{2} 〉} \in ℤ_{+}^{L}}

for 1 ≤ i ≤ I − 1 and

{\tilde{𝒳}}_{n_{I}} \subseteq {x : 1^{T} x = n_{I}, x^{\frac{1}{2}} \in ℤ_{+}^{L}}

where

| {\bar{𝒳}}_{n_{I}} | = 2^{K} - \sum_{i = 1}^{I - 1} R_{n_{i}}

.

On Theorem 1, whose proof is provided in the appendix, we would like to make the following remarks:

In this theorem, we show that our proposed design is closely related to a well-known mathematical problem. By using the fact that the total number of the solutions to ${‖ z ‖}_{2}^{2} = n$ is finite, we attain the energy-efficient structure within a nonnegative integer set for AR receiver. Our proposed constellation is given for a general dimension number L. It is observed that, when L = 1, the constellation defined by (6) is reduced to ${k^{2}}_{k = 0}^{k = 2^{K} - 1}$ , which is exactly the so-called SPAM, developed in [39].
To show the significant difference of our proposed constellation from SPAM and commonly used PAM, we illustrate these constellations in Figs. 1 and 2 for L = 2 and L = 3, respectively. As illustrated by these two figures, PAM generated by adding proper direct current to the unipolar version is equally spaced in each dimension, while each dimension of SPAM is the square of modified PAM. By Figs. 1 and 2, our proposed constellation can be viewed as a subset curved from a larger-size SPAM since the energy-efficiency of SPAM is remarkably influenced by the signal points with the largest energy. Therefore, our proposed scheme is the most energy-efficient compared with PAM and SPAM for a fixed Hellinger distance.

Fig. 1 PAM, SPAM and our proposed constellations with L = 2 and different K.

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Fig. 2 Proposed constellations for L = 3 and different K.

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4. Simulation results

In this section, we carry out simulations to examine the performance of AR receiver and constellations. Because a proper bit-mapping is still unavailable for the proposed scheme and the relevant study is out of the scope of this paper, average codeword error rate is used as the performance metrics in our simulations, which is defined by the ratio of the number of estimated codewords in error to the total number of transmitted codewords. A codeword error occurs when x ≠ x̂, where x and x̂ are the transmitted codeword and the estimation of x respectively. In addition, we would like to compare the following three schemes.

Pulse amplitude modulation (PAM): The transmitted signal based on PAM is given by $x = \frac{2 N}{\sum_{ℓ = 1}^{L} (2^{K_{ℓ}} - 1)}$ (x₁, x₂, · · · , x_L)^T where $x_{ℓ} \in {k}_{k = 0}^{k = 2^{K_{ℓ}} - 1}$ for 1 ≤ ℓ ≤ L and $\sum_{ℓ = 1}^{L} K_{ℓ} = K$ ;
Square pulse amplitude modulation (SPAM) [39]: For SPAM, the transmitted signal vector is determined as $x = \frac{6 N}{\sum_{ℓ = 1}^{L} (2^{K_{ℓ}} - 1) (2^{K_{ℓ} + 1} - 1)}$ (x₁, x₂, · · · , x_L)^T where $x_{ℓ} \in {k^{2}}_{k = 0}^{k = 2^{K_{ℓ}} - 1}$ for 1 ≤ ℓ ≤ L and $\sum_{ℓ = 1}^{L} K_{ℓ} = K$ .
Proposed constellation: The proposed constellation 𝒳̃ is defined in Theorem 1.

For comparison fairness, we assure that the above three kinds of constellation have the same bit rate and average transmitted optical power. In addition, the simulation parameters are adopted as α = 4.52 × 10¹⁴s/J, β = 7.27Hz [43] and more details are given below.

4.1. Performance of AR and ML receivers

In this subsection, we show the asymptotic statistic behavior of the AR receiver. Robust error performance of AR receiver only requires a reliable estimation of the transmitted symbol x. For AR receiver, the PDF difference between $r_{ℓ}^{(AR)}$ and xぃ_ℓ + ξ_ℓ decides the receiver performance. Thus, we first investigate the PDF of $r_{ℓ}^{(AR)}$ for various optical irradiance and then examine the performance of AR receiver compared with its ML counterpart.

1) PDF difference between $r_{ℓ}^{(AR)}$ and x̄_ℓ + ξ_ℓ: Without loss of generality, let us consider r̂_ℓ which is expressed as ${\hat{r}}_{ℓ} = r_{ℓ}^{(AR)} - {\hat{x}}_{ℓ}$ . Simulations are carried out to investigate whether the PDF of r̂_ℓ approaches that of r_ℓ which is a Gaussian distributed variable with zero mean and variance being 1. Results are shown in Fig. 3 with various power levels and indicate that the PDF of the AR transformed signals asymptotically approaches that of a Gaussian random variable with increasing optical irradiance.
2) Error performance comparisons of ML and AR receivers: To examine the performance of AR receiver by comparing with that of ML receiver for PAM, SPAM and our proposed constellation, we simulate for L = 2 and different modulation order and show the results in Fig. 4, which illustrates that the AR receiver has performance approaching the ML receiver for various modulation order when PAM, SPAM and our proposed multi-dimensional constellation are adopted. Specifically, in Fig. 4(a) the performance loss of AR receiver for PAM constellation are about 0.07dB, 0.05dB and 0.04dB for K = 3, 4, 5 respectively at the error rate of 10⁻⁴, while in Fig. 4(b) at the same target error rate, the respective power losses caused by AR receiver are about 0.23dB, 0.20dB and 0.10dB respectively when SPAM is adopted. Fig. 4(c) also indicates the performance gaps between AR receiver and its ML counterpart are respectively 0.20dB, 0.13dB and 0.15dB for K = 3, 4, 5 with the proposed constellation at the error rate of 10⁻⁴. These results indicate that our proposed AR receiver can be viewed as an effective alternative to the ML receiver and thus, our proposed Hellinger distance design criterion can be provide a useful insight into the signal design for SPAD optical wireless communications.

Fig. 3 Comparisons for the PDF estimation of ẑ_ℓ and Gaussian distribution with various optical irradiance

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Fig. 4 Receiver performance comparisons for L = 2, various K and different constellation.

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4.2. Performance of different constellations

Based on AR receiver, we compare the average codeword error rate of proposed constellation with those of PAM and SPAM for different modulation order with L = 2 and L = 3 respectively in Figs. 5 and 6. Notice that our proposed constellation and SPAM outperform PAM mainly for the reason that PAM is designed based on Euclidian distance and therefore is not energy-efficient any longer for our considered Poisson channels. In addition, the performance comparisons between SPAM and the proposed constellation are carried for L = 4 and L = 5 respectively in Figs. 7 and 8. Combining with the results in Figs. 5 and 6, we can observe that, for L = 2, K = 4, L = 3, K = 6 and L = 4, K = 8, SPAM and our proposed constellations have almost similar performance. It should be noticed that the attained gain by our proposed scheme is dependent on L and K. This phenomenon occurs when K can be exactly divided by L. The reason for this phenomenon is that, under such condition, the signal points of SPAM are evenly distributed on each axis which has been especially illustrated by Fig. 1(b), whose geometrical structure is similar to that of our proposed scheme. We also notice that our proposed constellation can obtain huge power gain over SPAM. Specifically, when K = 3, 5 in Fig. 5, the proposed constellation has respective power advantages of about 1.7 and 1.3 dB over SPAM at the error rate of 10⁻⁴. When L = 3, from Fig. 6, the attained power gains of the proposed constellation over SPAM are about 1.75 and 1.2 dB for K = 4, 5, respectively, at the target error rate of 10⁻⁴.

Fig. 5 Comparisons among PAM, SPAM and proposed constellation for AR receiver with L = 2 and different K.

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Fig. 6 Comparisons among PAM, SPAM and proposed constellation for AR receiver with L = 3 and different K.

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Fig. 7 Comparisons between SPAM and proposed constellation for AR receiver with L = 4 and different K.

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Fig. 8 Comparisons between SPAM and proposed constellation for AR receiver with L = 5 and different K.

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For L = 4, as shown in Fig. 7, the power gains by the proposed constellation compared with SPAM at the target error rate of 10⁻⁴ are 2.2, 2, 1.4 and 0.4 dB for K = 5, 6, 7, 8, respectively. While, for L = 5, the respective power advantages over SPAM for K = 6, 7, 8, 9 at the target error rate of 10⁻⁴ are 1.6, 1.5, 1.9 and 1.1 dB which is shown in Fig. 8. Now, we conclude that our designed constellation outperforms SPAM with the power gain due to its energy-efficient constellation structure.

5. Conclusion

In this paper, we have investigated the energy-efficient multidimensional constellation designs for the SPAD-based OWC systems. For AR receiver, we have designed an energy-efficient multi-dimensional constellation by minimizing the average optical power for a fixed minimum Hellinger distance of two distinct signals within the nonnegative integer set. Comprehensive simulations have shown that our proposed multi-dimensional constellation has significant performance gains over the currently available PAM and SPAM for this system.

This is the first systematic design of multidimensional signals for SPAD OWC systems and will provide a useful insight into the energy-efficient modulation for OWC under Poisson regimes. However, it should be noted that the AR detection complexity of our proposed constellation is exponential with respect to the set size. It is quite interesting to study how to reduce the complexity. In the future, we will attack this problem by utilizing the special structure of the proposed scheme. For example, the regular structure brought by Diophantine equation motivates us to demodulate the signal layer by layer. Moreover, notice that the proposed constellation is curved from a larger-size SPAM while an SPAM is the square of a modified PAM, after AR transform, the constellation is almost equally spaced in transform domain due to the square root operation, which inspires us to design a low-complexity receiver with equally spaced thresholds.

Appendix

Let us consider any size-2^K constellation 𝒳 satisfying HD(𝒳) = 1 and $x^{〈 \frac{1}{2} 〉} \in ℤ_{+}^{L}$ for any x ∈ 𝒳. It should be noted that 𝒳 can be always decomposed into I disjoint subsets such that $𝒳 = \cup_{i = 1}^{i = I} 𝒳_{i}$ where $𝒳_{i} = {x_{i, j}}_{j = 1}^{j = R_{n_{i}}}$ for 1 ≤ i ≤ I − 1 and $𝒳_{I} = {x_{I, j}}_{j = 1}^{j = {\bar{R}}_{n_{I}}}$ where ${\bar{R}}_{n_{I}} = 2^{K} - \sum_{i = 1}^{I - 1} R_{n_{i}}$ with the following assumptions,

1^T x_i,j ≤ 1^T x_i,j+1 for j = 1, · · · , R_{n_i} − 1,
1^T x_{i,R_{n_i}} ≤ 1^T x_i+1,1 for i = 1, · · · , I − 1,
1^T x_I,j ≤ 1^T x_I,j+1 for $j = 1, \dots, 2^{K} - \sum_{i = 1}^{I - 1} R_{n_{i}} - 1$ .

Then, we claim that 1^T x_i,j ≥ n_i for 1 ≤ i ≤ I and 1 ≤ j ≤ R_{n_i}. When i = 1, implying n₁ = 0, we have that 1^T x_1,1 ≥ 0 is indeed true. To prove our claim by induction, we assume that 1^T x_i,1 ≥ n_i also holds for i = J with 1 ≤ J ≤ I − 1. Then, for i = J +1, if 1^T x_i,1 ≥ n_J+1 is not satisfied, then, we can arrive at n_J ≤ 1^T x_J,1 ≤ 1^T x_J,2 ≤ · · · 1^T x_{J,R_{n_J}} ≤ 1^T x_J+1,1 < n_J+1. Combining this result with our assumption that for any x ∈ 𝒳,

x^{〈 \frac{1}{2} 〉} \in ℤ_{+}^{L}

, we can attain that 1^T x_J+1,1 = n_J, telling us that the Diophantine equation 1^T x = n_J with respect to

x^{〈 \frac{1}{2} 〉} \in ℤ_{+}^{L}

has at least R_{n_J} + 1 solutions in total. This result contradicts with the fact that there are at most R_{n_J} solutions to the equation 1^T x = n_J with respect to

x^{〈 \frac{1}{2} 〉} \in ℤ_{+}^{L}

. Therefore, 1^T x_i,1 ≥ n_i indeed holds for i = J + 1. Recalling our assumptions that 1) 1^T x_i,j ≤ 1^T x_i,j+1 for j = 1, · · · , R_{n_i} − 1; 2) 1^T x_{i,R_{n_i}} ≤ 1^T x_i+1,1 for i = 1, · · · , I − 1 and 3) 1^T x_I,j ≤ 1^T x_I,j+1 for

j = 1, \dots, 2^{K} - \sum_{i = 1}^{I - 1} R_{n_{i}} - 1

, we obtain

\sum_{x \in 𝒳} 1^{T} x \geq \sum_{i = 1}^{I} R_{n_{i}}

. By our definitions of 𝒩 and I, we know

\sum_{x \in \tilde{𝒳}} 1^{T} \tilde{x} = \sum_{i = 1}^{I} R_{n_{i}}

, where 𝒳̃ is defined by (6). Therefore,

\sum_{x \in 𝒳} 1^{T} x \geq \sum_{x \in \tilde{𝒳}} 1^{T} \tilde{x}

, giving us the desired and completing the proof of Theorem 1.

Funding

Henan Major Scientific and Technological Project (161100210200); National Youth Foundation of China (61701536).

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Energy-efficient multidimensional Hellinger modulation for SPAD-based optical wireless communications

Abstract

1. Introduction

2. System model with AR receiver

2.1. Transmitter

2.2. Anscombe root (AR) receiver

3. Energy-efficient multidimensional Hellinger constellation

3.1. Problem formulation

3.2. Main result

4. Simulation results

4.1. Performance of AR and ML receivers

4.2. Performance of different constellations

5. Conclusion

Appendix

Funding

References and links

Cited By

Figures (8)

Equations (7)

Optics Express