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Achievable rates optimization for broadcast channels using finite size constellations under transmission constraints
EURASIP Journal on Wireless Communications and Networking volume 2013, Article number: 254 (2013)
Abstract
In this paper, maximal achievable rate regions are derived for powerconstrained AWGN broadcast channel involving finite constellations and two users. The achievable rate region is studied for various transmission strategies including superposition coding and compared to standard schemes such as time sharing. The maximal achievable rates are obtained by optimizing over both the joint distribution of probability and over the constellation symbol positions. A numerical solution is proposed for solving this nonconvex optimization problem. Then, we consider several variations of the same problem by introducing various constraints on the optimization variables. The aim is to evaluate efficiency vs. complexity tradeoffs of several transmission strategies, some of which (the simplest ones) can be found in actual standards. The improvement for each scheme is evaluated in terms of SNR savings for target achievable rates or/and percentage of gain in achievable rates for one user compared to a reference scheme. As an application, two scenarios of coverage areas and user alphabets are considered. This study allows to evaluate with practical criteria the performance improvement brought by more advanced schemes.
1 Introduction
During the past few decades, information networks have witnessed tremendous and rapid advances, based on the important growth in the adoption of new wireless technologies, applications and services, first from cellular networks and more recently for computer networks (WLANs). Consequently, wireless networks are exposed to capacity and coverage problems, and the focus is now shifting towards capturing some of the aspects of realistic networks by studying natural network models such as models with broadcasting.
In 1972, achievable rate region is obtained by Cover in [1] for Gaussian broadcast channels with two outputs and generalized by Bergmans to broadcast channels with any number of outputs [2]. Roughly a year later, the optimality of the sets of achievable rates was established by Bergmans [3] and Gallager [4]. Superposition coding is a possible solution to achieve good rate regions in which information intended for highnoise receivers and information intended for lownoise receivers are superimposed and transmitted simultaneously on the same radio resource. The lownoise receivers can always decode messages intended for the highnoise receivers. Thus, they effectively cancel out the interference due to the signal intended for the highnoise receivers, and then decode their own message. The highnoise receivers decode their messages by treating the lownoise receivers message as noise. Superposition coding appears in several contexts in information theory and is closely related to multilevel coding and unequal error protection [5, 6]. Cover showed [1] that the superposition coding reaches the theoretical limit of the capacity region for two user Gaussian broadcast channel using an infinite Gaussian input alphabet for each user. A treatment of the case of multiple transmitter/receivers for the bandlimited additive white Gaussian noise channel is given by Bergmans and Cover in [7], where it is proved that superposition coding can achieve higherrate region than orthogonal schemes such as frequencydivision multiple access (FDMA) or time division multiple access (TDMA). However, in actual transmission systems, the channel input is constrained to a finite size alphabet with equal probability symbols. A wellknown practical implementation of superposition coding is hierarchical modulation, also called layered modulation, which uses constellations with nonuniformly spaced signal points creating different levels of error protection. Hierarchical modulation is used to mitigate the cliff effect in digital television broadcast and is included in various standards, such as Digital Video Broadcast for Terrestrial Television (DVBT) [8], DVB to Handhelds (DVBH), and DVB Satellite services to Handhelds (DVBSH) [9] standard proposal for mobile digital TV transmission. A study about the performance of hierarchical modulation and a comparison with time sharing strategy in terms of achievable rates can be found in [10].
The restriction imposed by practical systems in using finite signaling constellation and equiprobable symbols reduces the achievable rates and leads to a gap with the capacity region achieved with Gaussian input alphabets for AWGN broadcast channel. This gap can be reduced using a technique called constellation shaping. In fact, most results for constellation shaping with finite signal constellations consider only pointtopoint communication systems [11]. Then, the concept of constellation shaping has been adapted to most modern coding and modulation techniques as for example turbo coding and BICM schemes [12–19]. For broadcast channels, the achievable rate region for twouser AWGN broadcast channels with finite input alphabets is derived in [20] when superposition of modulated signal is used as transmission strategy. In their work, the authors assume a uniform distribution over the finite input set. To our knowledge, no study is available about the maximization of the achievable rate region for twouser AWGN broadcast channels with finite size constellations by optimizing over both the joint probability distribution and constellation symbol positions for a broadcast transmission strategy. This general framework encompasses hierarchical modulations as a special case. In this paper, maximal achievable rate regions are derived for powerconstrained AWGN broadcast channel of two users with Mpulse amplitude modulation (MPAM) constellations of M points using various transmission strategies. A numerical solution is proposed for solving this nonconcave optimization problem. In a typical broadcast system, there is a trade off between achievable rates and coverage areas. Therefore, we are interested in determining the transmission strategy which provides the best achievable rates or the maximal SNR gain for a given coverage scenario. The compromise between the simplicity of implementation and expected gains is also evaluated.
The organization of the paper is as follows. Section 2 recalls some information theory results on broadcast channels and degraded broadcast channels. In section 3, various transmission strategies for broadcast systems are described. Section 4 gives a formulation of the problem in terms of optimization for the various transmission strategies under consideration. Then, computational aspects are discussed. An iterative algorithm is proposed for the computation of maximal achievable rate regions using superposition coding (general case) and MPAM constellation or in the particular case of superposition modulation. The proposed algorithm can handle an optimization with respect to the joint distribution of probability or with respect to the positions of constellation symbols. Both variables can also be considered jointly. Obviously, the best results are obtained for the most general case. Our target is to (1) evaluate the loss experienced using simple schemes, (2) identify situations in which complex schemes (nonstandard) lead to significant improvements. As an application, we consider, in section 5, several scenarios of coverage areas and user alphabets, and we give conclusions about the transmission strategies which can provide the best trade off between efficiency and complexity of implementation.
2 AWGN broadcast channels
A tworeceiver (users) broadcast channel (BC) consists of an input alphabet , two output alphabets ${\mathcal{Y}}_{1}$ (user 1), ${\mathcal{Y}}_{2}$ (user 2), and a conditional pdf ${P}_{{Y}_{1}{Y}_{2}X}$ on ${\mathcal{Y}}_{1}\times {\mathcal{Y}}_{2}$. Let X, Y_{1}, and Y_{2} be random variables representing the input and outputs of the BC. Figure 1 depicts the two users BC with two independent messages W_{1} and W_{2}. The encoder generates a codeword x^{n}(w_{1},w_{2}) of length n based on these two messages. Each user receives, respectively, ${y}_{1}^{n}$ and ${y}_{2}^{n}$. A BC is said to be physically degraded if ${P}_{{Y}_{1}{Y}_{2}X}({y}_{1},{y}_{2}x)={P}_{{Y}_{1}X}({y}_{1}\leftx\right)\xb7{P}_{{Y}_{2}{Y}_{1}}\left({y}_{2}\right{y}_{1})$ (i.e., X→Y_{1}→Y_{2} form a Markov chain). A BC is said to be stochastically degraded or degraded if there exists a random variable $\stackrel{~}{{Y}_{1}}$ which has the same conditional pdf as Y_{1} given X, such that $X\to \stackrel{~}{{Y}_{1}}\to {Y}_{2}$ forms a Markov chain. We are interested in degraded BC because its capacity region is known, while it is not available for the general case.
In our system model, W_{1} denotes the private message intended for receiver 1 only, and W_{2} is a common message for both receivers. A typical example of this situation is digital TV broadcasting to two different groups of receivers, classified according to their channel conditions, where the basic signal (common signal) should be available to all receivers. The higher quality is realized by adding the basic signal with an incremental signal (private signal for receivers of good channel conditions) which carries TV signal with a high data rate, such as HDTV.
Let R_{1} and R_{2} be the rates at which the transmitter is sending W_{1} and W_{2}, respectively. Thus, user 1 achieves R_{1}+R_{2}, while user 2 achieves R_{2}. The capacity region of the degraded broadcast channel X→Y_{1}→Y_{2} in Figure 1 is the convex hull of the closure of rate pairs (R_{1}+R_{2},R_{2}) satisfying
for some joint distribution ${P}_{{\mathit{\text{UXY}}}_{1}{Y}_{2}}={P}_{\mathit{\text{UX}}}\xb7{P}_{{Y}_{1}X}\xb7{P}_{{Y}_{2}X}$ on $\{\mathcal{U}\times \mathcal{X}\times {\mathcal{Y}}_{1}\times {\mathcal{Y}}_{2}\}$[21]. ${P}_{{Y}_{1}X}$ and ${P}_{{Y}_{2}X}$ are conditional pdfs that depend on the channel model. P_{ UX } is the joint probability distribution of U and X, where the auxiliary random variable U has cardinality bounded by $\left\mathcal{U}\right\le min\left\{\right\mathcal{X},{\mathcal{Y}}_{1},{\mathcal{Y}}_{2}\left\right\}$. The capacity region is achieved using superposition coding, where U serves as the center of a cloud of codewords that can be distinguished by both receivers. Since the capacity region of a BC depends only on the conditional marginals, the capacity region of the stochastically degraded BC is equal to that of the corresponding physically degraded channel. Cover [1] showed that in the case of binary symmetric BC and AWGN BC, superposition coding expands the rate region beyond that achievable with time sharing.
Now, consider the Gaussian broadcast channel with two users. Without loss of generality, assume that Y_{1} is less noisy than Y_{2}. It can easily be shown that scalar Gaussian broadcast channels are equivalent to a degraded channel,
where ${Z}_{1}\sim \mathcal{N}(0,{\sigma}_{1}^{2}),{Z}_{2}\sim \mathcal{N}(0,{\sigma}_{2}^{2}),{Z}_{2}^{\prime}\sim \mathcal{N}(0,{\sigma}_{2}^{2}{\sigma}_{1}^{2})$, and ${Z}_{1},{Z}_{2}^{\prime}$ are independent. Thus, Gaussian BC is stochastically degraded. We assume an average power constraint on the transmitted power P defined as $\mathbb{E}\left[\phantom{\rule{0.3em}{0ex}}{X}^{2}\right]\le P$. The received signal to noise ratio for each user is ${\text{SNR}}_{i}=\frac{P}{{\sigma}_{i}^{2}}$, where SNR_{1}>SNR_{2}, and ${\sigma}_{i}^{2}$ is the variance of the noise Z_{ i }. The capacity region of the AWGNBC is the set of rate pairs (R_{1}+R_{2},R_{2}), such that
for all α∈ [ 0,1], where $C\left(x\right)=\frac{1}{2}\xb7\underset{2}{log}(1+x)$. The theoretical limit of twouser AWGN BC is achieved using signal superposition [1].
3 Broadcast transmission strategies
In this section, various transmission strategies for broadcast systems are described. The strategies are presented in ascending order of implementation complexity. Specifically, by moving from one strategy to another, we release some constraints on the system implementation to reach finally the most complex strategy that can be used to broadcast information for users. Obviously, since the simple schemes can be understood as adding constraints to the most general case, they are less efficient in terms of attainable rates.
3.1 Time sharing
Time sharing (TS) has been widely used in broadcast systems as broadcast transmission strategy. In time sharing scheme, a percentage of time is used to send one message, and the rest of the time is used to send another message. Thus, it is practical to implement because the rate pairs can be achieved by strategies used for pointtopoint channel and sharing the time between messages. As in previous works on broadcasting, this situation serves as a reference for the more advanced schemes.
In this work, a time sharing scheme with standard constellation MPAM (Figure 2) is considered when symbols are used with equal probability. A standard MPAM constellation is defined as a constellation with M real symbols belonging to $\mathcal{X}=\{M12\xb7(i1),\text{for}i=1,\dots ,M\}$. During the time slot dedicated to send a message, only one data stream is sent using the entire set of constellation points. In classical implementations of time sharing, the conventional MPAM symbols are equally spaced and used with equal probability.
3.2 Hierarchical Modulation (HM)
In twolayer hierarchical modulation, constellation symbols are used to transmit two data streams simultaneously for two users [22, 23]. Constellation symbols are usually chosen with the same probability but may be nonequally spaced. These symbols can be considered as the sum of two lowerorder modulations, one for each user. The modulation with higher power is used for the 'bad’ channel, the one with smallest power for the 'good’ channel. Hence, the encoding using hierarchical modulation can be separable for the two streams which is more practical.
This is explained here using 4PAM as an example. Figure 3 shows the constellation diagram of a hierarchical 4PAM with parameter ℓ=ℓ_{1}/ℓ_{2} used to determine the spacing between the groups of constellation points (clouds). ℓ is the ratio of the spacing between the groups to the spacing between individual points within a group. Standard values of ℓ are 1, 2, and 4. When ℓ increases, with a fixed total transmission power P, the two points from both sides of origin form a cloud. The location of a point within its cloud is regarded as the information for the 'good’ user. The other information, i.e., the number of the cloud in which the point is located is the information for the 'bad’ user. In this way, two separate data streams can be made available for transmission. Formally, we are still dealing with 4PAM, but in the hierarchical interpretation, it is viewed as the combination of two BPSK modulations which have different robustness to noise. In other words, the service coverage areas differ in size for both users. The betterprotected data stream is referred to as the highpriority (HP) stream which is mapped in Figure 3 to the most significant bit. The other one is referred to as the lowpriority (LP) stream (Figure 3) and mapped in Figure 3 to the least significant bit. Receivers with good reception conditions can receive both streams, while those with poorer reception conditions may only receive the high priority stream considering the LP stream as noise. This corresponds to a specific labeling of the modulation.
3.3 Superposition modulation
In superposition modulation (SM) [24], the M constellation points are used such that the labeling is separable, i.e., M=M_{1}M_{2}, and that the M points are obtained by adding (in ) two rv’s X_{1} and X_{2} of cardinality M_{1} and M_{2}, respectively (${M}_{1},{M}_{2}\in \mathbb{N}\setminus \{0,1\}$). Thus, this scheme is with an enlarged set of feasible labelings than in the previous case [25, 26]. This leads also to U≡X_{2} for superposition modulation because user 2 can distinguish only U.
This work studies several cases of superposition modulation. First, when the constellation symbols for each user are used with equal probability. This case will be denoted as ${\text{SM}}_{\mathcal{X},\overline{{P}_{\mathit{\text{UX}}}},\overline{{P}_{X}}}$. This is a practical case since the encoding of the messages is separable, and the symbols are used with equal probability as in real transmission systems. Then, the constraint of using equiprobable symbols is released and the symbols of user constellations can be dependent and used with nonequal probability (P_{ UX } nonuniform). Thus, the encoding here is done jointly for the two messages. This strategy will be denoted ${\text{SM}}_{\overline{\mathcal{X}},{P}_{\mathit{\text{UX}}},{P}_{X}}$ when the symbols take the values of a standard MPAM and ${\text{SM}}_{\mathcal{X},{P}_{\mathit{\text{UX}}},{P}_{X}}$, otherwise. In the latter case, the symbol positions can take arbitrary values and will be considered as variables to be optimized. The definition of superposition modulation can be generalized using more general form for P_{ UX } than the uniform case. In superposition modulation, ${2}^{{\mathit{\text{nR}}}_{2}}$ independent codewords u^{n}=x^{(2)n}(w_{2}) of length n are generated according to P_{ U }; for each of these codewords, ${2}^{{\mathit{\text{nR}}}_{1}}$ satellite codewords v^{n}=x^{(1)n}(w_{1}) are generated and added to form codewords x^{n}(w_{1},w_{2})=u^{n}+v^{n} according to P_{XU}. Thus, the fine information v^{n} is superimposed on the coarse information u^{n}.
Note that the capacity region of Gaussian broadcast channel is achieved using this coding scheme and successive cancellation decoding, where U (≡X_{2}) and V (≡X_{1}) are independent random variables following normal distributions. However, we do not assume here that U and V are independent. Consequently, for superposition modulation, P_{ UX } takes a specific expression. As an example, consider an 8PAM modulation. In that case, the transmitted signal at time k is the sum of the two users signals and is given by ${x}_{k}={x}_{k}^{\left(1\right)}+{x}_{k}^{\left(2\right)}$, where ${x}_{k}^{\left(1\right)}\in {\mathcal{X}}_{1}$ and ${x}_{k}^{\left(2\right)}\in {\mathcal{X}}_{2}$ with M_{1}·M_{2}=8. Two configurations are possible either M_{2}=4 (${\mathcal{X}}_{1}$ is a BPSK, and ${\mathcal{X}}_{2}$ is a 4PAM) or M_{2}=2 (${\mathcal{X}}_{1}$ is a 4PAM, and ${\mathcal{X}}_{2}$ is a BPSK). In both cases, P_{ UX } is a sparse matrix of size M_{2}×M with expression
where P_{ UX }[ i,j]=p_{i1,j1}= Pr{U=u_{i1},X=x_{j1}}. In both cases, the number of elements to be computed is 8.
Note also that P_{ UX } and (of cardinality M) determine the labeling of the input signal constellation for a fixed labeling for ${\mathcal{X}}_{1}$ and ${\mathcal{X}}_{2}$[25, 26]. Thus, the information can be distinguished using the labeling. Consider for example a label ${l}_{k}^{u}$ of ${\mathit{\text{log}}}_{2}\left(\right{\mathcal{X}}_{2}\left\right)$ binary labels for u_{ k } and ${l}_{j}^{v}$ of ${\mathit{\text{log}}}_{2}\left(\right{\mathcal{X}}_{1}\left\right)$ binary labels for v_{ j } with $k\in \{0,\mathrm{..},{\mathcal{X}}_{2}1\}$ and $j\in \{0,\mathrm{..},{\mathcal{X}}_{1}1\}$. Obviously, the M symbols ${x}_{i},\phantom{\rule{1em}{0ex}}i\in \{0,\mathrm{..},\mathcal{X}1\}$ carry log_{2}(M) binary labels which are the concatenations of the labels of u_{ k } and v_{ j } such as x_{ i }=u_{ k }+v_{ j }.
Part of this work on superposition modulation was presented in [25–27], where the achievable rate regions for ${\text{SM}}_{\mathcal{X},\overline{{P}_{\mathit{\text{UX}}}},\overline{{P}_{X}}}$ and ${\text{SM}}_{\mathcal{X},{P}_{\mathit{\text{UX}}},{P}_{X}}$ strategies are analyzed using a 4PAM constellation in [25, 26] and for {4,8,16}PAM constellations in [27]. In this work, the achievable rates are also derived for ${\text{SM}}_{\overline{\mathcal{X}},{P}_{\mathit{\text{UX}}},{P}_{X}}$ using {4,8,16}PAM constellations.
3.4 Superposition coding
Superposition coding (SC) is one of the basics of coding schemes in network information theory. This idea was first introduced by Cover in an information theoretic study of broadcast channels [1]. In superposition coding, the joint distribution of probability P_{ UX } can take a more general form than in the case of superposition modulation. In this case, the labeling cannot distinguish between the common information and the private information for user 1, a fact which increases the decoder complexity. Indeed, since the auxiliary random variable U has cardinality bounded by $\left\mathcal{U}\right\le min\left\{\right\mathcal{X},{\mathcal{Y}}_{1},{\mathcal{Y}}_{2}\left\right\}$, we use the name general superposition coding or superposition coding simply to describe the case, where $\left\mathcal{U}\right=min\left\{\right\mathcal{X},{\mathcal{Y}}_{1},{\mathcal{Y}}_{2}\left\right\}$. For superposition coding and with MPAM modulation, P_{ UX } is an M×M matrix with elements p_{i,j}.
The basics of superposition coding are briefly recalled below; a detailed description is given in [28]. In this scheme, ${2}^{{\mathit{\text{nR}}}_{2}}$ sequences ${u}^{n}\left({w}_{2}\right),{w}_{2}\in \phantom{\rule{0.3em}{0ex}}[\phantom{\rule{0.3em}{0ex}}1,{2}^{{\mathit{\text{nR}}}_{2}}]$ each i.i.d., are generated randomly and independently to represent the coarse message, each according to $\prod _{i=1}^{n}{p}_{U}\left({u}_{i}\right)$. For each auxiliary sequence, u^{n}(w_{2}) randomly, conditionally, and independently generates ${2}^{{\mathit{\text{nR}}}_{1}}$ sequences x^{n}(w_{1},w_{2}) and ${w}_{1}\in \phantom{\rule{0.3em}{0ex}}[\phantom{\rule{0.3em}{0ex}}1,{2}^{{\mathit{\text{nR}}}_{1}}]$, each according to $\prod _{i=1}^{n}{p}_{XU}\left({x}_{i}\right{u}_{i}\left({w}_{2}\right))$ to represent the fine message w_{1}. Thus, in superposition coding, the auxiliary random variable U serves as a cloud center for the information, distinguishable by both receivers. In this case, the decoding of information by users is based on large block joint typicality. This comes in contrast with the simpler cases where the message for user 2 was carried by the center of modulation clouds which imply a possible scalar detection.
The achievable rates for superposition coding will be studied for various strategies corresponding to different constraints on P_{ UX } and/or . An exhaustive list of all the strategies under consideration is given in Table 1, where redundant configurations are omitted.
4 Achievable rate regions
For a twouser Gaussian BC, the theoretical limit of the capacity region is achieved using Gaussian input alphabet for each user. However, practical implementation constraints impose the use of finite input alphabets, and the symbols are usually chosen with equal probability. These restrictions contribute to increase the gap between the capacity region achieved with infinite Gaussian inputs and the throughput obtained in practical situations. In this section, we are interested in computing the achievable rate region of powerconstrained AWGN BC when the transmitted signal is modulated using an MPAM constellation, under the various situations described above. Since the last case (superposition coding) encompasses all previous ones as special cases, the corresponding optimization problems can be solved with the same strategy, which is detailed in this section.
4.1 Problem formulation
Consider a twouser memoryless AWGN broadcast channel (SNR_{1}>SNR_{2}) with signal power constraint P. The channel input belongs to a finite set $\mathcal{X}=\{{x}_{0},\dots ,{x}_{M1}\}\subset \mathbb{R}$ represented by an MPAM constellation. Assume a symmetric input signal constellation with respect to the origin. Since has cardinality bounded by $\left\mathcal{U}\right\le min\left\{\right\mathcal{X},{\mathcal{Y}}_{1},{\mathcal{Y}}_{2}\left\right\}$, and the output alphabet cardinality for an AWGN channel is infinite, we have $\left\mathcal{U}\right\le \left\mathcal{X}\right$. Thus, $\left\mathcal{U}\right\le M$.
To determine the maximal achievable rate region using superposition coding, consider the case $\left\mathcal{U}\right=M$. For superposition modulation, we take into account the specificity on P_{ UX } given in section 3.3. We also consider within the same framework the problem of maximizing the achievable rates under additional constraints on optimization variables (P_{ UX } and ): standard MPAM symbols values, uniform distribution for P_{ UX }, uniform distribution for P_{ X }. The problem of maximizing the achievable rates under a specific situation is solved subject to a combination of constraints according to Table 1. We recall that in this work, message w_{2} is a common message to both receivers, and w_{1} is a private message to user 1. Thus, the achievable rate region (R_{2} vs. R_{1}+R_{2}) can be obtained by solving the weighted sum rate (θ·R_{1}+(1θ)·R_{2}) maximization for θ∈ [ 0,0.5]. Indeed, for θ=0, we maximize the common information rate R_{2}, and when θ=0.5, we maximize the rate achieved by user 1 (R_{1}+R_{2}). Using (1) and (2), the optimization problem under consideration is:
and subject to the constraint on the joint pdf P_{ UX } or on given in Table 1 for each strategy, where p_{ ij }= Pr{U=u_{ i },X=x_{ i }},j∈{0,..,M1}, and $i\in \{0,\mathrm{..},\mathcal{U}1\}$. The two mutual information I(X;Y_{1}U) and I(U;Y_{2}) can be written as follows:
where all logarithms are taken base 2. The AWGN channel for each user is characterized by the conditional pdf
When θ=0 or θ=1 and for $\left\mathcal{U}\right=M$ (which are referred in this paper as pointtopoint (PtP) channel case), the individual achievable rates R_{2} and R_{1} are maximized respectively. The problem (9) is equivalent to
where p_{ i }= Pr{X=x_{ i }},i∈{0,..,M1} is the input probability distribution, and k∈{1,2}. When θ=0 or 1, problem (13) is solved for k=2 and 1, respectively, with I(X;Y_{ k }) given by
For the time sharing scheme using standard constellation, the achievable rate pair (R_{1}+R_{2},R_{2}) is such that [1]
where $\overline{{R}_{1}}$ and $\overline{{R}_{2}}$ are achievable rates for PtP channel using standard MPAM constellation at SNR_{1} and SNR_{2}, respectively. Varying α from 0 to 1 yields achievable rate region.
Problem (9) is not convex; therefore, direct numerical optimization is inefficient. Clearly, an exhaustive search is not feasible as the complexity would be exponential in the total number of variables. An iterative method for solving (9) is proposed in the next section.
4.2 Numerical solution
Consider a regularized version of (9) as
where s is a regularization parameter. For a given value of s, the optimization problem in (16) is solved (for the most general case) with respect to P_{ UX } and to $\mathcal{X}=({x}_{0},{x}_{1},\dots ,{x}_{M1})$ alternately until convergence:
where ℓ is the iteration index, and denotes the set of constraints on P_{ UX } and can be defined either as $\mathcal{C}=\{{P}_{\mathit{\text{UX}}}\phantom{\rule{0.3em}{0ex}}:\phantom{\rule{0.3em}{0ex}}{p}_{\mathit{\text{ij}}}\ge 0,\sum _{i,j}{p}_{i,j}\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}1\}$ or as $\mathcal{C}=\{{P}_{\mathit{\text{UX}}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}:\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{p}_{\mathit{\text{ij}}}\ge 0,\sum _{i}{p}_{i,j}=\frac{1}{M}\}$ (equiprobable symbols). The optimization problem in (17) with constraint set $\mathcal{C}=\{{P}_{\mathit{\text{UX}}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}:\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{p}_{\mathit{\text{ij}}}\ge 0,\sum _{i,j}{p}_{i,j}=1\}$ can be handled by a modified 'BlahutArimoto’type algorithm [29]. Indeed, in order to take into account the regularization, we can show that the BlahutArimototype algorithm proposed in [30] for broadcast channels should be modified by replacing Equation (19) of Lemma 3 in [30] by ${q}^{\ast}(u,x)=\frac{\beta [Q,\stackrel{~}{Q},\stackrel{\u0304}{Q}](u,\phantom{\rule{0.3em}{0ex}}x)\xb7{e}^{s\frac{{x}^{2}}{1\theta}}}{\sum _{{u}^{\prime},{x}^{\prime}}\beta [Q,\stackrel{~}{Q},\stackrel{\u0304}{Q}]({u}^{\prime}\phantom{\rule{0.3em}{0ex}},\phantom{\rule{0.3em}{0ex}}{x}^{\prime})\xb7{e}^{s\frac{{x}^{\prime 2}}{1\theta}}}$ instead of ${q}^{\ast}(u,x)=\frac{\beta [Q,\stackrel{~}{Q},\stackrel{\u0304}{Q}](u,\phantom{\rule{0.3em}{0ex}}x)}{\sum _{{u}^{\prime},{x}^{\prime}}\beta [Q,\stackrel{~}{Q},\stackrel{\u0304}{Q}]({u}^{\prime}\phantom{\rule{0.3em}{0ex}},\phantom{\rule{0.3em}{0ex}}{x}^{\prime})}$, where $\beta [\phantom{\rule{0.3em}{0ex}}Q,\stackrel{~}{Q},\stackrel{\u0304}{Q}](u,x)$ is defined in Equation (19) of [30]. When there is an additional constraint on constellation symbols to be equiprobable, i.e., $\mathcal{C}=\{{P}_{\mathit{\text{UX}}}:{p}_{\mathit{\text{ij}}}\ge 0,\sum _{i,j}{p}_{i,j}=1\phantom{\rule{2.77626pt}{0ex}}\text{and}\phantom{\rule{2.77626pt}{0ex}}\sum _{i}{p}_{i,j}=\frac{1}{M}\}$, the BlahutArimototype algorithm in [30] should also be modified to take into account the additional constraint. In this case, Equation (19) of Lemma 3 in reference [30] should be replaced by ${q}^{\ast}(u,x)=\frac{1}{\left\mathcal{X}\right}\xb7\frac{\beta [Q,\stackrel{~}{Q},\stackrel{\u0304}{Q}](u,\phantom{\rule{0.3em}{0ex}}x)}{\sum _{u}\beta [Q,\stackrel{~}{Q},\stackrel{\u0304}{Q}](u,\phantom{\rule{0.3em}{0ex}}x)}$, which does not depend on s, where $\beta [\phantom{\rule{0.3em}{0ex}}Q,\stackrel{~}{Q},\stackrel{\u0304}{Q}](u,x)$ is defined in Equation (19) in this reference.
Now consider (18). The function $L({P}_{\mathit{\text{UX}}}^{\left(\ell \right)},{x}_{0},\mathrm{..},{x}_{M1},s)$ is not a concave function for all $\mathcal{X}\in {\mathbb{R}}^{M}$. However, we observed in our experiments that $L({P}_{\mathit{\text{UX}}}^{\left(\ell \right)},{x}_{0},\mathrm{..},{x}_{M1},s)$ is a concave function if $\mathcal{X}\in \mathcal{D}$, where $\mathcal{D}=\{\mathcal{X}\in {\mathbb{R}}^{M}:{x}_{i}{x}_{j}>d\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\forall i,j\in \{0,\mathrm{..},M1\}\phantom{\rule{2.77626pt}{0ex}}\text{and}\phantom{\rule{2.77626pt}{0ex}}i\ne j\}$, and d depends on the size of the constellation and on the SNR. Since we are interested in finding nondegenerated constellation, we restrict the optimization process to . Then, a simplex method is used to perform the optimization with initial value in .
The alternative maximization method can at least increase the objective function in each iteration. In the experiments, we have observed that this method converges at least to a local maximum (denoted ${p}_{i,j}^{\ast}\left(s\right),{x}_{j}^{\ast}\left(s\right),0\le j\le M1,0\le i\le \left\mathcal{U}\right1$). We discuss now the choice of s. Since we do not know a priori which value of s may correspond to the satisfaction of the equality power constraint, we propose to use an iterative process as follows:
where [.]^{+} is defined as [.]^{+}= max(.,0). The value of s is increased or decreased with the sign of $P\sum _{i=0}^{\left\mathcal{U}\right1}\sum _{j=0}^{M1}{p}_{\mathit{\text{ij}}}^{\ast}\left({s}^{\left(k\right)}\right)\xb7{\left({x}_{j}^{\ast}\right({s}^{\left(k\right)}\left)\right)}^{2}$. The process stops when the power constraint is fulfilled. The proposed algorithm is summarized in Table 2. Obviously, when constellation symbols are constrained to the values of a standard constellation, (P 2) which is defined in Table 2 will not be used. Similarly, when P_{ UX } is uniform, (P 1) is not used. An alternative interpretation of this algorithm is to recognize that L(P_{ UX },x_{0},..,x_{M1},s) is the Lagrangian dual of problem 9. Equations (17) and (18) are an iterative method for solving
The dual optimization problem mins.t. s≥0f(s) is solved in (19) with a gradienttype algorithm. Since f(s) is convex [31], a gradient search method is guaranteed to converge to a global optimum.
5 Result analysis
5.1 Point to point channel
We present in this section the results of maximizing achievable rates for PtP case using MPAM constellations with M=4,8,16 and for different values of SNR. To evaluate the contribution of constellation shaping, we compare, for a fixed SNR, the maximal achievable rate calculated by the algorithm proposed in the previous section to the 'standard constellation’ rate, whose symbols are used with equal probability, at the same SNR in terms of SNR saving (called SNR shaping gain). The SNR shaping gain depicted in (Figure 4) is the gain obtained with a fully optimized constellation (${P}_{\mathcal{X}}$ and ) compared to the standard MPAM constellation and when symbols are used with the same probability. To avoid the complexity of constructing nearly optimal input distribution codes, another method for doing constellation shaping is to optimize only the position of symbols in the constellation. Each signal point is assumed to be chosen with the same probability; however, the position of each point in the constellation is optimized. The corresponding shaping gain is given in (Figure 5). We observe the following: the shaping gain depends on the SNR and on the size of the constellation. The maximum gain is obtained for midrange SNR. The distribution of probability ${P}_{\mathcal{X}}$ (not reported) is very similar to the sampling of a Gaussian distribution. With the halfoptimized constellation ( only), a significant degradation is observed for midrange SNR compared to that for the fully optimized constellation. Hence, we can conclude that symbol pdf optimization is useless at low and high SNR, whereas the fully optimized constellation is efficient for midrange SNR, in which case the gain increases with the size of the constellation.
5.2 Broadcast channel
Current broadcast systems are using two practical transmission schemes for sending information to users: orthogonal schemes in which the time and/or frequency is split between the users, and superposition modulation schemes where the constellation for each user is fixed. In this section, a comparison is provided between these standard schemes and various (more complex) transmission strategies such as superposition coding. The effect of constellation shaping is evaluated by analyzing the achievable rate region curves obtained for an MPAM constellation (M=4,8,16) and for several pairs (SNR_{1},SNR_{2}). The following schemes are considered:

Time sharing using standard MPAM (TS).

SM  3 possible configurations (see Table 1)

SC  4 possible configurations (see Table 1)
In the following, we denote by the 'case 1’ of superposition modulation when M_{1}=2,M_{2}=4 and when M_{1}=2,M_{2}=8. 'Case 2’ is when M_{1}=4,M_{2}=2 and when M_{1}=4,M_{2}=4. 'Case 3’ refers to the case when M_{1}=8, M_{2}=2.
Achievable rate region curves are provided in Figures 6, 7, 8, 9, 10, and 11 for M=4,8,16. For each value of M, the display of the results is limited to two different pairs of SNR. In complement with the achievable rate region curves, comparisons are also conducted in terms of SNR savings for target achievable rates (maximum shaping gain) and in terms of maximum percentage of gain for user 1. These two quantities are defined below.
Definition 1
Consider two transmission strategies (A and B). The pair of rates (R_{1}+R_{2},R_{2}) is achieved for (SNR_{1},SNR_{2}) with A and for (SNR_{1}+Δ SNR,SNR_{2}+Δ SNR) with B. The shaping gain (with A compared to B) is Δ SNR. The maximum shaping gain is defined as
Definition 2
Consider two transmission strategies (A and B). For a given pair of SNR (SNR_{1},SNR_{2}) and a fixed value of R_{2}, the achievable pair of rates is$({R}_{1}^{A}+\phantom{\rule{0.3em}{0ex}}{R}_{2},{R}_{2})$ and$({R}_{1}^{B}+{R}_{2},{R}_{2})$ with A and B, respectively. The gain on the achievable rate for user 1 is given by
The maximum gain on the achievable rate for user 1 (with A compared to B) is given by
5.2.1 Superposition modulation
In this section, the three possible configurations of superposition modulation are compared. We can see from Figures 6, 7, 8, 9, 10, and 11 that${\text{SM}}_{\mathcal{X},\overline{{P}_{\mathit{\text{UX}}}},\overline{{P}_{X}}}$ (optimization of only) outperforms${\text{SM}}_{\overline{\mathcal{X}},{P}_{\mathit{\text{UX}}},{P}_{X}}$ (optimization of P_{ UX } only) in terms of maximal achievable rates per user when M=4. For M=8 and 16,${\text{SM}}_{\overline{\mathcal{X}},{P}_{\mathit{\text{UX}}},{P}_{X}}$ can achieve slightly higher rates than${\text{SM}}_{\mathcal{X},\overline{{P}_{\mathit{\text{UX}}}},\overline{{P}_{X}}}$. The implementation of a system with constellation symbols with nonstandard positions and generated with the same probability is less complex than the implementation of a system which generates symbols with nonuniform joint distribution of probability. Thus,${\text{SM}}_{\overline{\mathcal{X}},{P}_{\mathit{\text{UX}}},{P}_{X}}$ does not seem to be of interest since it is not very efficient in terms of achievable rates and is more complex to implement.
Figures of achievable rate region show that an improvement can be obtained with${\text{SM}}_{\mathcal{X},{P}_{\mathit{\text{UX}}},{P}_{X}}$ (full optimization) compared to${\text{SM}}_{\mathcal{X},\overline{{P}_{\mathit{\text{UX}}}},\overline{{P}_{X}}}$ (optimization of only) and depending on δ_{SNR}=SNR_{1}SNR_{ 2 }. Numerical values of the maximum gain in achievable rate (${\text{MG}}_{{R}_{1}}$) and of the maximum SNR savings (${\text{MG}}_{{\text{SNR}}_{\mathit{\text{dB}}}}$) are given in Table 3. We observe the following: a slight gain in terms of achievable rates can be translated into a noticeable gain in terms of SNR saving. The maximum shaping gain increases with the constellation size. Thus, constellation shaping for the SM strategy seems more useful for high values of M. The analysis of the optimal matrix P_{ UX } (results not reported) leads to the conclusion that X_{ 1 } and X_{ 2 } are not independent in general when using finitesize constellations. We observe also that the maximum shaping gain for${\text{SM}}_{\mathcal{X},{P}_{\mathit{\text{UX}}},{P}_{X}}$ versus${\text{SM}}_{\mathcal{X},\overline{{P}_{\mathit{\text{UX}}}},\overline{{P}_{X}}}$ increases when δ_{SNR} decreases, independently of M. In particular, full optimization (vs. optimization of the symbol position) does not provide significant improvement for large SNR gap in the SM strategy.
5.2.2 Time sharing or superposition modulation?
This section compares two strategies (TS and SM) classically considered in broadcast systems. In Figures 6 and 7 (M=4), we observe that the achievable rate region can be split into two parts. Indeed, for small and large values of R_{2}, TS is better than SM. On the contrary, SM is better than TS for middlerange values of R_{2}. Under a given rate requirement for one user, we can thus determine the best transmission strategy. We can also observe that the region in which SM is better than TS becomes small for larger values of SNR_{2}. With M=8 (Figures 8 and 9), the area in which SM is better than TS increases (compared to M=4) by considering the union of the two possible configurations for SM: M_{1}=2,M_{2}=4 (case 1) and M_{1}=4,M_{2}=2 (case 2). This is particularly true when δ_{SNR} increases. We also observe that TS can achieve higher rates than SM (case 1) for good SNR_{2} values. Indeed, the maximum rate of user 2 with SM is the maximum individual rate for a 4PAM constellation, whereas it is the individual user rate that achieved using standard 8PAM in the TS case. For low SNR_{2} values, optimized 4PAM may achieve higher rate than standard 8PAM; thus, SM becomes better in this interval. For a 16PAM constellation (Figures 10 and 11), SM is always better than TS for the studied pairs of (SNR_{1},SNR_{2}). Table 4 shows the maximum percentage of improvement in achievable rate of user 1 by TS when using${\text{SM}}_{\mathcal{X},{P}_{\mathit{\text{UX}}},{P}_{X}}$ (full optimization) strategy in the interval, where${\text{SM}}_{\mathcal{X},{P}_{\mathit{\text{UX}}},{P}_{X}}$ is better than TS. Clearly, the maximum percentage of improvement increases when δ_{SNR} increases, and an important gain is obtained for high values of δ_{SNR} as in the case of SNR_{1}=δ_{SNR}=10 dB for a 4PAM, where the percentage of gain on achievable rate of user 1 varies between 0% and 40.7%. For a 8PAM constellation, the percentage of gain on achievable rate of user 1 varies between 0% and 30.21% when SNR_{1}=16 dB and δ_{SNR}=8 dB. For a 16PAM, the percentages of improvements can be up to 35.08% when SNR_{1}=18 dB and δ_{SNR}=8 dB. We can conclude that SM is a better option than TS especially for large δ_{SNR} values. TS is optimal in the region, where we want to maximize the rate of user 2 for good values of SNR_{2} because the single user rate achieved by TS is the rate achieved using standard MPAM constellation (the constellation is split between users with SM). Thus, SM seems more gainful than TS when we want to serve users with very diverse SNRs.
5.2.3 Is superposition coding necessary?
For the three constellations under consideration (M=4,8,16), the maximal achievable rate region obtained by the optimal general case of superposition coding when we consider the general form of P_{ UX } (SC) can achieve, depending on M and user SNRs, a large region of rate pairs (R_{1}+R_{2},R_{2}) that cannot be achieved neither by TS nor by SM. Even when we fully optimize SM (${\text{SM}}_{\mathcal{X},{P}_{\mathit{\text{UX}}},{P}_{X}}$), we are far from maximal achievable rate region. Sometimes, the maximal achievable rate region curve is very close or even coincides with the${\text{SM}}_{\mathcal{X},{P}_{\mathit{\text{UX}}},{P}_{X}}$ achievable rate region in a pair of rates (${R}_{1}^{\ast}+{R}_{2}^{\ast},{R}_{2}^{\ast}$). This is the case when${\text{SM}}_{\mathcal{X},{P}_{\mathit{\text{UX}}},{P}_{X}}$ is the optimal superposition coding in terms of achievable rates. We can see for example in Figure 6 that the pair of rates (${R}_{1}^{\ast}+{R}_{2}^{\ast}=1.096,{R}_{2}^{\ast}=0.531$ which corresponds to the optimal rate pair when we optimize the general case of SC for θ=0.23) is an intersection point with${\text{SM}}_{\mathcal{X},{P}_{\mathit{\text{UX}}},{P}_{X}}$ achievable rate region.
We are interested now in the numerical evaluation of the gain in rate of user 1 (R_{1}+R_{2}) when we use${\text{SC}}_{\mathcal{X},{P}_{\mathit{\text{UX}}},{P}_{X}}$ (full optimization) compared to the best strategy between TS and SM. This gain (${\text{MG}}_{{R}_{1}}\left({\text{SC}}_{\mathcal{X},{P}_{\mathit{\text{UX}}},{P}_{X}}\right\mathit{\text{TS}}\bigcup {\text{SM}}_{\mathcal{X},{P}_{\mathit{\text{UX}}},{P}_{X}}$) calculated in % is the distance between the limit of the maximal achievable rate region and the limit of the union of achievable rate regions of TS and${\text{SM}}_{\mathcal{X},{P}_{\mathit{\text{UX}}},{P}_{X}}$.
The results are reported in Table 4. We observe that the part of the maximal achievable rate region which is unachievable by TS and SM is bigger when M is small because we observe that for the case of 4PAM, we have one configuration for SM. However, we have two configurations of SM for 8PAM constellation and three configurations for 16PAM constellation. Thus, when M increases, the union of achievable rates for all SM cases tends to the sets of achievable rates by the general superposition coding. Asymptotically, we know that when$M\to \infty ,{\text{SM}}_{\mathcal{X},{P}_{\mathit{\text{UX}}},{P}_{X}}$ is the optimal superposition coding scheme because it allows the capacity region for twouser AWGN BC using Gaussian alphabet for each user to be achieved. Thus, the maximum gain in user 1 rate decreases when the constellation order M increases. We observe also that the gain in achievable rates is high for high values of δ_{SNR}. On the other hand, the experiments show that by using the general superposition coding strategy with the constraint that symbols should be equiprobable (${\text{SC}}_{\mathcal{X},{P}_{\mathit{\text{UX}}},\overline{{P}_{X}}}$), the loss is limited compared to the full optimization (${\text{SC}}_{\mathcal{X},{P}_{\mathit{\text{UX}}},{P}_{X}}$), 4.84%, 7.66%, and 3.94% for the simulated pairs of (SNR_{1},SNR_{2}) when M=4, 8, and 16, respectively. This means that we can use equiprobable symbols with, in general, a small loss in achievable rates. However,${\text{SC}}_{\mathcal{X},{P}_{\mathit{\text{UX}}},\overline{{P}_{X}}}$ is not an interesting case when${\text{SM}}_{\mathcal{X},{P}_{\mathit{\text{UX}}},{P}_{X}}$ can achieve better rates since SM is less complex to implement than SC.
Moreover, with standard MPAM symbols, the two possible configurations (${\text{SC}}_{\overline{\mathcal{X}},{P}_{\mathit{\text{UX}}},{P}_{X}}$ (optimization of P_{ UX } and P_{ X }) and${\text{SC}}_{\overline{\mathcal{X}},{P}_{\mathit{\text{UX}}},\overline{{P}_{X}}}$ (optimization of P_{ UX } only)) give very similar results in most considered pairs of SNR. We also observe that the loss in maximum achievable rate experienced by user 1 with${\text{SC}}_{\overline{\mathcal{X}},{P}_{\mathit{\text{UX}}},{P}_{X}}$ is less than 10% under the rate experienced with${\text{SC}}_{\mathcal{X},{P}_{\mathit{\text{UX}}},{P}_{X}}$. Thus, we can use standard values of symbol positions without losing much on achievable rates.
In general, one can conclude that fixing constellations of users (i.e., assigning labels to the constellation so that we distinguish between the bits intended for each user) is not optimal for coding and may result in important loss in terms of rates for systems using finitesize constellations especially for loworder constellations. A better solution is to determine the optimal alphabet of the auxiliary alphabet U which is not necessarily a constellation and then to generate the codewords x^{n} which are not necessarily the sum of two codewords (see Section 3.4).
6 Application: coverage extension
We first consider a transmission over a broadcast channel with finite size input alphabet. For simplicity of the illustration and without loss of generality, let us assume that the existing user alphabet belongs initially to a standard constellation whose symbols are used with equal probability. We assume that the existing user is at distance d_{0} from the sender achieving a rate R_{0}. Some information is also to be transmitted to an upgraded layer of users. The sender can use up to 16 symbols, then several transmission schemes can be used. We are interested in comparing the transmission schemes to serve the new user under two scenarios: either the new user is closer to the transmitter than the existing user or the new user is farther than the existing one. For a target rate R_{0} that is fixed for the existing user and achievable using a standard MPAM and equiprobable symbols, we are interested in determining the variation of the coverage’s diameter ratio between the two layer of users as a function of the achievable rate by the upgraded user for various broadcast transmission strategies. We assume that$\text{SNR}\propto \frac{1}{{d}^{2}}$.
6.1 The sender can use up to 16 symbols
6.1.1 Scenario 1
In this scenario, the system consists initially of one layer of users. Now, assume that the data information is also to be transmitted to a second layer of users with higher SNR. In the following, we keep the notation from the preceding section, where the user with greater SNR is denoted by user 1. Thus, in this scenario, the legacy receivers are denoted by user 2 which is at a distance d_{2} from the transmitter and achieving a rate R_{0} when the data is modulated using standard 4PAM constellation and equiprobable symbols. The upgraded receivers are denoted by user 1 (SNR_{1}>SNR_{2}). We intend that the good user receives more throughput than user 2 via the use of 16PAM.
In this example SNR_{2} is fixed to 10 dB. Initially, user 2’s alphabet belongs to a 4PAM standard constellation (see section 3.1), and the rate transmitted to user 2 is R_{0}=1.582 bits/ch. use.
Now, a new layer of users called user 1 is introduced in the system with SNR_{1}>SNR_{2}. Our target is to provide the maximum bit rate to the new user without changing R_{0} or d_{0} and using a 16PAM. By enlarging the constellation and optimizing the symbol positions and probability distribution, we ensure that the rate of the initial user will not decrease after introducing a new user.
Consider now the results for the following strategies which can achieve a positive privatemessage rate for user 1: time sharing using standard 16PAM,${\text{SM}}_{\mathcal{X},\overline{{P}_{\mathit{\text{UX}}}},\overline{{P}_{X}}}{M}_{2}=8/{M}_{1}=2$ (optimization of only),${\text{SM}}_{\mathcal{X},{P}_{\mathit{\text{UX}}},{P}_{X}}{M}_{2}=8/{M}_{1}=2$ (full optimization) and${\text{SC}}_{\mathcal{X},{P}_{\mathit{\text{UX}}},{P}_{X}}$ (full optimization). Figure 12 illustrates the variation of d_{1}/d_{2}, which is the ratio of the diameter of the coverage area for user 1 over the diameter of the initial coverage area for user 2, as a function of the achievable rate for user 1 for a target rate R_{0}=1.582 for user 2.
Let us assume for example that the new user is midway between the transmitter and user 2 (d_{1}/d_{2}=0.5). Figure 12 shows that the most simple case of superposition modulation (${\text{SM}}_{\mathcal{X},\overline{{P}_{\mathit{\text{UX}}}},\overline{{P}_{X}}}{M}_{2}=8/{M}_{1}=2$) provides 16.3% more bit rate than time sharing for the new user. If we move immediately to a more complex case and optimize P_{ UX } (${\text{SM}}_{\mathcal{X},{P}_{\mathit{\text{UX}}},{P}_{X}}{M}_{2}=8/{M}_{1}=2$), a gain of 21% is obtained on the bit rate of user 1 compared to time sharing. This gain on achievable rate for the new user is equivalent to a gain of 1 dB on SNR_{1} compared to superposition modulation with uniform P_{UX x}. However, if we move to the most general case of superposition coding, it does not provide significant gain compared to superposition modulation.
Now, we assume that the new user is close to the transmitter such that d_{1}/d_{2}=0.2. We observe that the gain on the bit rate of user 1 using the simple case of superposition modulation increases to 45.7% compared to time sharing. By moving to a more complex case (${\text{SM}}_{\mathcal{X},{P}_{\mathit{\text{UX}}},{P}_{X}}{M}_{2}=8/{M}_{1}=2$), a gain of 47.8% is obtained on the bit rate of user 1 compared to time sharing. We observe also that it is relevant in this case to move to the most general case of superposition coding since it provides a gain of 61.8% on the bit rate of user 1 compared to time sharing.
Consequently, using superposition modulation provides always noticeable gain compared to time sharing. The general case of superposition coding${\text{SC}}_{\mathcal{X},{P}_{\mathit{\text{UX}}},{P}_{X}}$ is useful when user 1 is close to the transmitter, but not when it is close to user 2.
6.1.2 Scenario 2
Initially, consider a system of one layer of users, denoted by user 1, at a distance d_{1} from the transmitter and achieving a rate R_{0}. Moreover, the alphabet of user 1 belongs to a standard 8PAM constellation. In this example, SNR_{1} is fixed to 18 dB. Thus, user 1 can achieve a rate R_{0}=2.73 bits/ch. use in the initial situation. In this scenario, we want to serve a second layer of users denoted by user 2 which is farther to the transmitter than the existing user, i.e., SNR_{2}x<SNR_{1}.
Achievable rates for user 2 are obtained at different distance d_{2} from the transmitter and using various transmission strategies for a target rate of user 1 equal to R_{0} and a coverage diameter for user 1 fixed to d_{1}. Figure 13 illustrates the variation of d_{2}/d_{1}, which is the ratio of the diameter of the coverage area for user 2 over the diameter of the initial coverage area for user 1, as a function of the achievable rate for user 2 when a target rate for user 1 is fixed to R_{0}=2.73 bits/ch. use.
We observe in Figure 13 that superposition modulation can always achieve better rates for user 2 than time sharing using 16PAM. Let us assume first that we want to increase the diameter of the coverage area for the new user (user 2) such that d_{2}/d_{1}=4. Time sharing provides a bit rate less than 0.06 bits/ch. use. The most simple case of superposition modulation (${\text{SM}}_{\mathcal{X},{P}_{\mathit{\text{UX}}},{P}_{X}}{M}_{2}=2/{M}_{1}=8$) provides a significant improvement on the achievable rate for user 2 which is equal to 0.4 bits/ch. use in this case. If we increase the complexity by optimizing the joint probability distribution P_{ UX }, we obtain 35% more bit rate for user 2 comparing to superposition modulation with uniform P_{ UX }. If we move to the general case of superposition coding, we gain only 10% on the bit rate of the new user compared to superposition modulation (see Table 5). However, when the new layer of users is at distance d_{2}=2.25 d_{1}, the general case of superposition coding provides a significant gain of 41% on the achievable rate of user 2 comparing to superposition modulation.
Consequently, the general case of superposition coding can bring significant gains compared to superposition modulation, depending on the diameter of the coverage area for the new layer of users. For superposition modulation, optimizing the joint distribution of probability P_{ UX } provides often significant shaping gains.
6.2 The cardinality of the existing user alphabet is kept fixed
In this section, we study scenario 1 (and 2) supposing that the legacy receivers will continue working as in the initial situation, still using 4PAM (8PAM). The system consists initially one layer of users at distance d_{0} from the transmitter and achieves a rate R_{0}. Now, we want to change the transmitter, such that the upgraded receivers closer (farther) in range will be able to decode a refinement (coarse) layer and use a 16PAM constellation. Thus, only time sharing with M_{1}=M_{2}=4(M_{1}=8,M_{2}=2) and superposition modulation strategies can be used. We aim to study how small the reduction in legacy coverage can be made, depending on the rate of the refinement (coarse) information achieved by the upgraded users. Thus, suppose that the legacy coverage can be reduced from d_{0} to d_{2} (from d_{0} to d_{1}). We have studied this problem for SNR_{0}=12 dB and for SNR_{1}SNR_{2}=4 dB in scenario 1 (and for SNR_{0}=16 and SNR_{2}=14 dB in scenario 2). Figures 14 and 15 represent the reduction in coverage d_{2}/d_{0} (and d_{1}/d_{0} respectively) as a function of the rate of the refinement R_{1} (of the coarse R_{2}), while the rate achieved by the legacy receivers is kept fixed to its initial situation, i.e., R_{0}.
We observe in Figures 14 and 15 that the gain of superposition modulation strategies over time sharing becomes more important when d_{2}/d_{0} (d_{1}/d_{0}) is small. These figures show that using superposition modulation when both symbol positions and P_{ UX } are optimized, we gain around 5% from the initial coverage compared to the case of superposition modulation where symbols are used with equal probability. We can observe also that a reduction of only 10% and 20% in coverage area for the existing user can serve the upgraded user with a rate up to 20% and 35% (9% and 15%) from the rate achieved by the legacy users, using ${\text{SM}}_{\mathcal{X},\overline{{P}_{\mathit{\text{UX}}}},\overline{{P}_{X}}}$. Consequently, by using ${\text{SM}}_{\mathcal{X},\overline{{P}_{\mathit{\text{UX}}}},\overline{{P}_{X}}}$, the legacy receivers still use 4PAM (8PAM in scenario 2), and we can serve a new layer of users with an acceptable rate, a small reduction in coverage area, and with less complexity compared to ${\text{SM}}_{\mathcal{X},{P}_{\mathit{\text{UX}}},{P}_{X}}$.
7 Conclusion
In this work we considered the problem of maximizing the achievable rate region for powerconstrained AWGN broadcast channel of two users using MPAM constellations. The achievable rate region is given for various transmission strategies. Maximal achievable rate region for superposition coding and superposition modulation is obtained using constellation shaping. An iterative algorithm was proposed to solve this optimization problem. Then, the efficiency of several strategies are compared. For superposition modulation, the results showed that constellation shaping seems more useful for high values of M. Moreover, the gain in using a complex case of superposition modulation increases when the SNR gap between users decreases. We observed also that superposition modulation outperforms time sharing in a large part of the achievable rate region. On the other hand, it is shown that using the general case of superposition coding can bring important gains compared to classical schemes. We observed also that in the case of finite input alphabet, superposition modulation is not the optimal strategy as in the case of Gaussian input alphabets. Finally, in order to make clear that this paper provides useful tools for the system designer, we considered two scenarios of coverage areas and user alphabets where the systems served initially one layer of users. Then, we propose to serve a second layer of users, and we evaluate the achievable rate of the new layer depending on the broadcast strategy. To improve the system performance compared to time sharing, we can optimize the joint probability distribution and symbol positions of the superimposed modulations or consider the general case of superposition coding. In this work, we showed that the optimization of probabilities was often useful, but not always. However, superposition coding brings sometimes significant gains compared to superposition modulation, depending on the diameter of coverage area for the new layer of users.
This work can also be extended to twodimensional constellations like MQAM and other channel models. The maximization achievable rates using various transmission strategies can be performed also using the proposed algorithm based on alternative maximization with respect to symbol positions and the joint distribution of probability.
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Mheich, Z., Alberge, F. & Duhamel, P. Achievable rates optimization for broadcast channels using finite size constellations under transmission constraints. J Wireless Com Network 2013, 254 (2013). https://doi.org/10.1186/168714992013254
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Keywords
 AWGN broadcast channels
 Achievable rate region
 Hierarchical modulation
 Superposition modulation
 Superposition coding
 Constellation shaping
 Nonconvex optimization