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Dynamic interference for uplink SCMA in largescale wireless networks without coordination
EURASIP Journal on Wireless Communications and Networking volume 2018, Article number: 213 (2018)
Abstract
Fast varying active transmitter sets are a key feature of wireless communication networks with very short length transmissions arising in communications for the Internet of Things. As a consequence, the interference is dynamic, leading to nonGaussian statistics. At the same time, the very high density of devices is motivating nonorthogonal multiple access (NOMA) techniques, such as sparse code multiple access (SCMA). In this paper, we study the statistics of the dynamic interference from devices using SCMA. In particular, we show that the interference is αstable with nontrivial dependence structure for largescale networks modeled via Poisson point processes. Moreover, the interference on each frequency band is shown to be subGaussian αstable in the special case of disjoint SCMA codebooks. We investigate the impact of the αstable interference on achievable rates and on the optimal density of devices. Our analysis suggests that ultra dense networks are desirable even with αstable interference.
1 Introduction
Mordern wireless communication networks are increasingly heterogeneous. This heterogeneity stems from nonuniform placement of access points and variations in transmit power constraints and a characteristic of network employing small cells and ad hoc networks. Another form of heterogeneity is due to differences in the services that networks provide. For instance, there are key differences in quantity and type of data, as well as transmission protocols between networks supporting standard cellular or WLAN communication and machinetomachine (M2M) communications, which are increasingly ubiquitous with the rapid development of the Internet of Things (IoT) [1].
In standard cellular services, data transmissions typically vary between 1 KB and 2 MB per transmission for text and image transfers and up to 3 GB for video transmission [2]. On the other hand, in M2M communications, data transmission is of the order of 1 MB per month [3]. Transmissions in M2M are therefore very short in time. As a consequence, the active set of transmitting devices at each time can change rapidly. A natural question is how devices should operate in this setting. In particular, what are the statistical properties of the resulting interference (is it even Gaussian?), and what interference mitigation strategies are appropriate?
In the uplink—where a large number of devices connect to one or more access points—one interference mitigation strategy is to ensure that device transmissions are nearly orthogonal. That is, transmisssions overlap on different frequencies, different times, different spatial dimensions, or different power levels. This approach falls under the class of nonorthogonal multiple access (NOMA) transmission strategies [4].
One promising NOMA strategy is sparse code multiple access (SCMA) for OFDM systems [5]. In this strategy, users can transmit on a sparse subset of all frequency bands, analogous to CDMA where in contrast, the coding is performed over time slots. We remark that SCMA falls in the class of codebased NOMA as opposed to the pure power NOMA strategy, which exploits differences in received power levels to discriminate between users via successive interference cancellation [6].
In this paper, we assess the impact of rapidly changing active transmitter sets—or dynamic interference—in largescale wireless networks with SCMA, such as in M2M. We consider a worstcase scenario where the network is interferencelimited, uncoordinated, and the locations of interferers are governed by a homogeneous Poisson point process. This setup is relevant for networks supporting the Internet of Things and in largescale sensor networks, where transmitting devices are very simple and have limited ability to coordinate. In particular, limited coordination arises in the grant free scenarios recently considered in the context of NOMA [6], where devices randomly access timefrequency resources. In many cases, such random access is necessary due to the use of lowpower, lowcost devices. A coordinated channel has high power usage due to the need for control signals. This is not realistic with ultralow power devices.
The study of interference in wireless networks has a long history. An early significant contribution was due to Middleton [7, 8] where it was established that the electromagnetic interference probability density function is represented via an infinite series, leading to the Middleton class A and class B models. Reduced to two terms to simplify derivations, the resulting distribution is called the BernoulliGaussian model [9, 10]. Around the year 2000, work on Ultra Wide Band (UWB) communications also gave rise to many empirical modeling approaches. These works are relevant because the topologies are similar and impulsive radio transmissions gave rise to dynamic interference. The empirical models are often based on pragmatic choices, which provide a good fit with simulated data and analytical solutions for the maximum likelihood detector [11–13].
Dynamic interference in wireless networks without SCMA was introduced in [14, 15] by considering fastvarying (symbolbysymbol) active transmitter sets, with locations governed by a homogeneous Poisson point process. In particular, it was shown that the interfering signal power in each timeslot follows the αstable law, which is widely used to model impulsive noise signals [16]. Further analysis of these models via stochastic geometry [17, 18] has established that the αstable interference model is a good approximation of the true interference distribution when the radius of the network is large and there are no guard zones [19–21].
The class of αstable random variables are wellknown to model impulsive signals; however, unlike Gaussian models, αstable models are challenging to study due to the absence of a closedform probability density function [22]. Although an expression without a closedform for the error probability in the presence of αstable interference was derived in [14], little is known about achievable rates and the optimal density of devices in this scenario and in particular for networks using SCMA.
1.1 Methods and overview of contributions
In this paper, we study the statistics of dynamic interference in largescale networks exploiting SCMA. Unlike existing work on SCMA, we consider the presence of dynamic interference. Accounting for the impact of dynamic interference is critical as the resulting high amplitude interference is known to have an important impact on key network performance indicators (e.g., bit error rate and capacity [23]). As Gaussian interference models underestimate the probability of such high amplitude interference, the resulting design can be highly suboptimal.
A key focus, unlike existing work on dynamic interference, is not only on the statistics of interference but also on the dependence between the signals on different bands. Our main result is to establish that when devices randomly and independently select frequency bands to transmit on—i.e., the nonzero elements of the SCMA codebook—the resulting interference is an αstable random vector. However, the interference on each band is not in general independent. To study this dependence structure, we focus on a particular class of SCMA codebooks where devices transmit on a restricted set of bands. For this class of SCMA codebooks detailed in the sequel, we show that the resulting dynamic interference has subGaussian αstable dependence structure.
We then study achievable rates in the presence of isotropic αstable noise, which is a special case of subGaussian αstable noise for which rate bounds are not currently known. Based on the achievable rate, we study consequences for network design and in particular the impact of device density on the area spectral efficiency. This provides new insights into the design of networks in the presence of dynamic interference by characterizing the optimal device density. Moreover, the characterization of the achievable rate provides a basis for network optimization, e.g., power control. Finally, we discuss the general problem of characterizing the dependence structure induced by general SCMA codebooks. In particular, we propose an approach based on copulas and investigate consequences for signal detection.
The paper is organized as follows. In Section 2, the setup is formalized for largescale SCMAbased wireless networks with dynamic interference. In Section 3, the statistics of the dynamic interference are characterized and shown to follow the law of an αstable random vector. In Section 4, we study achievable rates in αstable noise and the impact on device density. In Section 5, we discuss our results and in Section 8, we conclude.
2 Problem formalization
Consider an uplink singlecell network in which the simultaneous transmissions from a large number of devices are received by a single access point at the origin. The device transmissions are over a subset of orthogonal bands \(\mathcal {B} = \{1,2,\ldots,K\}\).
At each time slot t, each device independently transmits with probability p≪1. The subset of transmitting devices is governed by a Poisson point process Φ_{t} with intensity λ. Since p≪1, the distance from devices in Φ_{t}, denoted by r_{j}, and devices in \(\Phi _{t^{\prime }}\) corresponding to another time t≠t^{′}, denoted by \(r_{j^{\prime }}\), to the access point are modeled as independent random variables. That is, for a device j in Φ_{t} and a device j^{′} in \(\Phi _{t^{\prime }}\) we have \(\text {Pr}\left (r_{j} \leq Rr_{j^{\prime }}\right) = \text {Pr}(r_{j} \leq R)\).
In a given time slot, each device j seeks to transmit a message W_{j} uniformly drawn from the set \(\mathcal {W}_{j} = \{1,2,\ldots,M\}\) over n channel uses. M is the number of different messages that the device can transmit. We denote by m the number of bands that each device can use to transmit their data. Each device is also equipped with an encoder \(\mathcal {E}_{j}: \mathcal {W}_{j} \rightarrow \mathbb {C}^{K\times n}\), which maps each message to n SCMA codewords in \(\mathbb {C}^{K}\), allowing for vector codewords^{Footnote 1}. One SCMA codeword is composed of m nonzero elements selected from the set of all subsets \(\mathcal {B}_{m}\), which consists of all subsets of {1,2,…,K} with size m.
We are interested in two classes of SCMA codebooks. In the first class, any set in \(\mathcal {B}_{m}\) can be selected by a device, where the nonzero elements of each codeword are assumed to be independently chosen for each device. This first case is referred to as the general random SCMA codebook.
In the second class of SCMA codebooks, only disjoint elements of \(\mathcal {B}_{m}\) can be selected. In this case, there are \(\left \lfloor \frac {K}{m} \right \rfloor \) codewords. If two devices choose different codebooks (i.e., different elements of \(\mathcal {B}_{m}\)), then there is no interference. On the other hand, if two devices choose the same element of \(\mathcal {B}_{m}\), then they will interfere on all bands with a nonzero signal. This second case is referred to as the disjoint random SCMA codebook. These codebooks are detailed further in Section 3.
Accounting for the contributions of all transmitting devices at time t, the received signal at the access point on band k in the interference limited regime is given by:
where \(h_{j,k} \sim \mathcal {CN}(0,1)\) is the Rayleigh fading coefficient of the jth device on band k, r_{j}(t) is the distance from the device j to the access point, and η>2 is the pathloss exponent. We remark that Rayleigh fading models are appropriate for IoT applications indoor or in dense environments such as those that arise in city centers [24]. Although we do not treat other fading models further in this paper, we remark that if the input signal x_{j,k}(t) is isotropic and has finite power, then our results also apply to Rician or Nakagami fading.
We can now formalize the key question that we address in the remainder of this paper:

What are the statistics of the interference in SCMA networks with fastvarying active transmitter sets? In particular, what is the distribution of theKdimensional random vector Z(t), t=1,2,…with elements
$$\begin{array}{*{20}l} Z_{k}(t) = \sum\limits_{j \in \Phi_{t}} h_{j,k}(t)r_{j}(t)^{\eta/2}x_{j,k}(t) \end{array} $$(2)corresponding to the interference from the devices on band k at time t?
3 Interference characterization
In this section, we investigate the effect of rapidly changing active transmitter sets on the interference statistics, that is, we characterize the distribution of dynamic interference. A key feature of dynamic interference is its impulsive nature, which we formally establish by showing that the interference on each band follows the αstable distribution under the assumptions in Section 2. We also study the dependence structure of the random vector arising from the interference on each band.
3.1 Preliminaries
Before characterizing the interference statistics, we review definitions and important properties of αstable random variables and vectors. The αstable random variables are a key class of random variables with heavytailed probability density functions, which have been widely used to model impulsive signals [16, 22]. The probability density function of an αstable random variable is parameterized by four parameters: the exponent 0≤α≤2; the scale parameter \(\gamma \in \mathbb {R}_{+}\); the skew parameter β∈[−1,1]; and the shift parameter \(\delta \in \mathbb {R}\). As such, a common notation for an αstable random variable X is X∼S_{α}(γ,β,δ). In the case β=δ=0, X is said to be a symmetric αstable random variable.
In general, αstable random variables do not have closedform probability density functions. Instead, they are usually represented by their characteristic function, given by ([22], Eq. 1.1.6)
In addition to the characteristic function, symmetric αstable random variables admit a series representation. This is known as the LePage series and is detailed in the following theorem (a proof is available in ([22], Theorem 1.4.2), which will play a key role in our analysis of dynamic interference.
Theorem 1
Let 0<α<2, (Γ_{i}, i=1,2,…) be a Poisson process of rate 1 and W_{1},W_{2},… be a sequence of independent and identically distributed symmetric^{Footnote 2} random variables. Then, the sum \({\sum \nolimits }_{i=1}^{\infty } \Gamma _{i}^{1/\alpha }W_{i}\) converges almost surely to a random variable X whose distribution is \(S_{\alpha }\left (\left (C_{\alpha }^{1}\mathbb {E}\left [\left (W_{1}^{\alpha }\right)^{1/\alpha }\right ]\right),0,0\right)\), where
It is possible to extend the notion of an αstable random variable to the multivariate setting. A sufficient condition for a random vector X to be a symmetric αstable random vector is that the marginal distributions of the elements of X are symmetric αstable^{Footnote 3}. In general, ddimensional symmetric αstable random vectors are represented via their characteristic function, given by [22]
where Γ is the unique symmetric measure on the ddimensional unit sphere \(\mathbb {S}^{d1}\). A particular class of αstable random vectors are an instance of the subGaussian αstable random vectors^{Footnote 4}, defined as follows.
Definition 1
Any vector X distributed as X=(A^{1/2}G_{1},…,A^{1/2}G_{d}), where
and \(\mathbf {G} = \left [G_{1},\ldots,G_{d}\right ]^{T} \sim \mathcal {N}\left (\mathbf {0},\sigma ^{2}\mathbf {I}\right)\) is called a subGaussian αstable random vector in \(\mathbb {R}^{d}\) with underlying Gaussian vector G.
SubGaussian random vectors are typically characterized by either the scalemixture representation in Definition 1 or via their characteristic function ([22], Proposition 2.5.5) as detailed in the following proposition.
Proposition 1
Let X be a symmetric αstable random vector in \(\mathbb {R}^{d}\). Then, the following statements are equivalent:

1.
X is subGaussian αstable with an underlying Gaussian vector having i.i.d. \(\mathcal {N}\left (0,\sigma ^{2}\right)\) components.

2.
The characteristic function of X is of the form
$$\begin{array}{*{20}l} \mathbb{E}\left[e^{i\boldsymbol{\theta} \cdot \mathbf{X}}\right] = \exp\left(2^{\alpha/2}\sigma^{\alpha}\\boldsymbol{\theta}\^{\alpha}\right). \end{array} $$(7)
SubGaussian αstable random vectors are preserved under orthogonal transformations. The following proposition provides further characterizations of subGaussian αstable random vectors via their relationship to orthogonal transformations^{Footnote 5}.
Proposition 2
Let \(\mathcal {O}(d)\) be the set of real orthogonal matrices and \(\mathbf {U} \in \mathcal {O}(d)\). Then, \(\mathbf {Z} \overset {d}{=} \mathbf {U}\mathbf {Z}\) holds for all \(\mathbf {U} \in \mathcal {O}(d)\) if and only if Z is a subGaussian αstable random vector with an underlying Gaussian vector having i.i.d. \(\mathcal {N}\left (0,\sigma ^{2}\right)\) components.
Proof
See Section 6. □
SubGaussian αstable random vectors also play an important role in studying complex αstable random variables, that is, a random variable with αstable distributed real and imaginary components. In particular, the generalization of subGaussian αstable random variables to the complex case is known as the class of isotropic αstable random variables, defined as follows.
Definition 2
Let Z_{1},Z_{2} be two symmetric αstable random variables. The complex αstable random variable Z=Z_{1}+iZ_{2} is isotropic if it satisfies the condition:

\(\mathbf {C1:} e^{i\phi }Z \overset {(d)}{=} Z\) for any ϕ∈ [ 0,2π).
The random vector Z is said to be induced by the isotropic αstable random variable Z. Due to the fact that baseband signals are typically complex, isotropic αstable random variables will play an important role in the interference characterization.
The following proposition ([22], Corollary 2.6.4) highlights the link between isotropic αstable random variables and subGaussian αstable random vectors.
Proposition 3
Let 0<α<2. A complex αstable random variable Z=Z_{1}+iZ_{2} is isotropic if and only if there are two independent and identically distributed zeromean Gaussian random variables G_{1},G_{2} with variance σ^{2} and a random variable A∼S_{α/2}((cos(πα/4))^{2/α},1,0) independent of (G_{1},G_{2})^{T} such that (Z_{1},Z_{2})^{T}=A^{1/2}(G_{1},G_{2})^{T}. That is, (Z_{1},Z_{2})^{T} is a subGaussian αstable random vector.
We remark that isotropic complex αstable random variables are closely related to subGaussian random vectors as can be observed from a comparison with Definition 1. Moreover, unlike the isotropic (or circularly symmetric) Gaussian case (α=2), isotropic αstable random variables with α<2 do not have independent real and imaginary components. This dependence arises from the characterization in Proposition 3 through the dependence of the αstable random variable A in both the real and imaginary components.
3.2 Interference for general random SCMA codebooks
We now study the statistics of the interference from devices under the general SCMA codebook setup detailed in Section 2. We first consider the distribution of the interference on each band. In particular, we characterize the distribution of:
Consider the scenario of a general SCMA codebook, where each device selects the m bands that it transmits on from a distribution P_{m} on \(\mathcal {B}_{m}\), independently of the other devices. For example, suppose that device j selects bands uniformly from \(\mathcal {B}_{m}\) then the probability that it transmits on band k at time t is \(q = \frac {\stackrel {K 1}{m1}}{\stackrel {K}{m}}\). As such, the devices that transmit on band k also form a homogeneous Poisson point process with rate λq. In this case, the following theorem holds.
Theorem 2
Suppose that
Further, suppose that each device selects m bands from a given distribution P_{m} on \(\mathcal {B}_{m}\), independently of the other devices. Then, Z_{k}(t)converges almost surely to an isotropic \(\frac {4}{\eta }\)stable random variable.
Proof
Using the mapping theorem for PPPs [25], it follows that \(\{r_{l}^{2}\}_{l}\) is a onedimensional Poisson point process with intensity λπ ([21], Theorem 1). Since the codebooks for each device are independent, the hypothesis of the theorem implies that the LePage representation in Theorem 1 holds. It then follows that the interference has real and imaginary parts that are almost surely symmetric \(\frac {4}{\eta }\)stable random variables. Since we consider Rayleigh fading, h_{l,n}x_{l,n} is isotropic and condition C1 for isotropic αstable random variables is also satisfied, as required. □
A consequence of Theorem 2 is that the interference random vector Z(t) is an αstable random vector when devices use general SCMA codebooks. This is in sharp contrast to standard models for nondynamic interference, which are typically Gaussian.
Although Z(t) is an αstable random vector, the dependence structure remains unspecified. We note that unlike Gaussian interference models, the dependence structure is not characterized through the covariance matrix. In fact, the covariance of αstable random variables is either infinite or undefined, depending on the value of 0<α<2. Next, we study the dependence structure for the special case of disjoint random SCMA codebooks, where each device transmits on one of a family of disjoint sets consisting of m bands.
3.3 Interference for disjoint random SCMA codebooks
We now consider the case where devices transmit using disjoint random SCMA codebooks. Given K bands it is possible to obtain ⌊K/m⌋ disjoint subsets. Devices are said to use disjoint random SCMA codebooks if they are restricted to transmitting on one of these ⌊K/m⌋ disjoint subsets of \(\mathcal {B}\), and the set of m bands is selected uniformly. In this section, we characterize the statistics of the interference vector arising from the use of disjoint random SCMA codebooks.
To begin, consider the case where devices use SCMA codewords that have nonzero values on disjoint sets of bands. By the independent thinning theorem for Poisson point processes [25], the interference on bands in different disjoint subsets in \(\mathcal {B}_{m}\) are independent. Therefore, the key challenge is to establish the dependence of interference on bands within the same disjoint subset in \(\mathcal {B}_{m}\).
Consider a given disjoint subset in \(\mathcal {B}_{m}\) and denote the interference on these m bands as Z_{m}(t). In this case, the received signal vector at the access point on the m bands under consideration can then be written as:
where ∘ is the Hadamard elementwise product and
The statistics of the interference random vector Z_{m} are given in the following theorem. The proof is provided in Section 3.5.
Theorem 3
The interference Z_{m} induced by disjoint random SCMA codebooks follows the subGaussian αstable distribution with an underlying Gaussian vector having i.i.d. \(\mathcal {N}\left (0,\sigma _{\mathbf {Z}}^{2}\right)\) components, α=4/η and parameter
where \(q_{D} = \frac {1}{\lfloor K/m \rfloor }\) and \(C_{\frac {4}{\eta }}\) as defined in Theorem 1.
3.4 Numerical results
To illustrate the dependence structure resulting from subGaussian αstable interference. Interference samples are generated from a network of devices distributed uniformly on a disc of radius R=500 m and number of access points drawn from a Poisson distribution with parameter πR^{2}λ with λ=0.1. These samples approximate a realization of a Poisson point process with rate λ [25].
Figure 1 plots samples obtained from the real and imaginary parts of interference on a single band. Observe that the samples are distributed isotropically as expected from Theorem 2. Figure 2 plots the real parts of interference samples from two bands b_{1},b_{2} in a disjoint SCMA codebook with b_{1},b_{2} in the same disjoint subset. Observe that the samples exhibit the same dependence structure as in Fig. 1, which is consistent with Theorem 3.
Figure 3 plots the real parts of interference samples from two bands b_{1},b_{2} in a general SCMA codebook with b_{1} selected with probability 1/2 and b_{2} selected with probability 1/3 if b_{1} is selected and probability 2/3 if b_{1} is not selected. This scenario can occur in codebooks constructed to reduce decoding complexity. Observe that the dependence structure differs from Fig. 2, which demonstrates that SCMA codebooks can have dependence structures that do not arise from subGaussian random αstable random vectors. In Section 5, we discuss methods to model this dependence in a tractable manner.
3.5 Proof of Theorem 3
We first establish that the elements of Z_{m} are complex αstable random variables. Without loss of generality, consider the first element of Z_{m}, denoted by Z_{1}, given by:
Giving each element of Φ an index in \(\mathbb {N}\), we write
Let z_{j}=h_{j,1}x_{j,1} and denote the real an imaginary parts as z_{j,r} and z_{j,i}, respectively. We then have
Now, recall that the distances \(\{r_{j}\}_{j=1}^{\infty }\) are from points in a homogeneous PPP to the origin. Using the mapping theorem, it follows that \(\left \{r_{j}^{2}\right \}_{j=1}^{\infty }\) with intensity q_{D}λπ. It then follows from the LePage series representation in Theorem 1 that Z_{1} converges almost surely to Z_{r}+iZ_{i}, where Z_{r} and Z_{i} are symmetric 4/ηstable random variables. Morever, \(e^{i\theta }Z_{1} \overset {d}{=} Z_{1}\) since for each j, \(e^{i\theta }h_{j,1}x_{j,1} \overset {d}{=} h_{j,1}x_{j,1}\) by the fact that the Rayleigh fading coefficient is isotropic. Therefore, by Proposition 3, it follows that an equivalent representation of Z_{1} is a twodimensional real subGaussian αstable random vector.
By applying the same argument to each element of Z_{m}, it follows that it is a complex 4/ηstable random vector. We now need to show that stacking the real valued representation of each element of Z_{m} yields a subGaussian αstable random vector. Let Z^{′} be the vector obtained by stacking the real and imaginary parts of Z_{m}.
To proceed, we will apply Proposition 2. In particular, let \(\mathbf {U} \in \mathcal {O}(2m)\) (recall that \(\mathcal {O}(2m)\) is the set of real orthogonal matrices of dimensions 2m×2m). Further, let \(Z^{\prime }_{j,l}\) represent the real or imaginary part of the interference on a band k due to device j, which is given by
in the case that \(Z^{\prime }_{j,l}\) corresponds to a real part. A similar expression can be also obtained for the case of an imaginary part. For both cases, \(Z^{\prime }_{j,l} = r_{j}^{\eta /2}\left (f_{j,k}\text {Re}\left (x_{j,k}\right) + g_{j,k}\text {Im}\left (x_{j,k}\right)\right)\), where f_{j,k} and g_{j,k} are Gaussian with variance 1/2 due to the fact that h_{j,k} is a Rayleigh fading coefficient. As such, the vector of interference from device j can be written as \(\mathbf {Z}^{\prime }_{j} = r_{j}^{\eta /2}(\mathbf {f} \cdot \text {Re}(\mathbf {x}_{j}) + \mathbf {g} \cdot \text {Im}(\mathbf {x}_{j}))\). Since U is orthogonal it then follows that \(\mathbf {U}\mathbf {f} \overset {d}{=} \mathbf {f}\) and \(\mathbf {U}\mathbf {g} \overset {d}{=} \mathbf {g}\), which implies that \(\mathbf {UZ^{\prime }} \overset {d}{=} \mathbf {Z}^{\prime }\). Since the choice of \(\mathbf {U} \in \mathcal {O}(2m)\) is arbitrary, the result then follows by applying Proposition 2.
4 Achievable rates with dynamic interference
At present, the capacity of dynamic interference channels has not been established even for single frequency systems. To this end, in this section, we study achievable rates for dynamic interference channels. Although our focus is on single frequency systems, it is also applicable to SCMAbased systems when each band is treated separately.
As shown in Theorem 2, the interference on each band is isotropic αstable distributed. To study achievable rates in this interference, we introduce the additive isotropic αstable noise (AIαSN) channel. In particular, the output of the AIαSN channel is given by
where r_{d} is the distance from a device to its serving access point located at the origin, \(h_{d} \sim \mathcal {CN}(0,1)\) is the Rayleigh fading coefficient and X_{d} is the baseband emission for the device under consideration, and Z is the isotropic αstable distributed interference. Unlike the capacity of the power constrained Gaussian noise channel, tractable expressions are not known for the power constrained AIαSN channel. For this reason, it is desirable to consider alternative constraints.
One choice of constraints is the combination of amplitude and fractional moment constraints. In particular, the input signal X_{d} in (17) is required to satisfy
where 0<ζ<α. Note that the presence of the amplitude constraint ensures that the input has finite moments, including power.
To characterize the capacity of the AIαSN channel in (17) subject to the constraints in (18), we proceed in two steps. First, we relax the amplitude constraints and consider the optimization problem given by
where \(\mathcal {P}\) is the set of probability measures on \(\mathbb {C}\) and 0<ζ<α. The unique (see [26]) solution to (19) is lower bounded in the following theorem.
Theorem 4
For fixed r_{d} and h_{d}, the capacity of the additive isotropic \(\frac {4}{\eta }\)stable noise channel in (17) subject to the fractional moment constraints in (19) is lower bounded by:
where Γ(·) is the Gamma function and
Proof
We consider the case that X_{d} is an isotropic αstable random variable satisfying the constraints in (19). By Theorem 5 in [26], the mutual information of the channel Y=X_{d}+Z is derived using the stability property; i.e., Y is an isotropic αstable random variable since X_{d} and Z are. The mutual information is then given by
The result then follows by observing that \(r_{d}^{\frac {\eta }{2}}h_{d}X_{d}\) is also an isotropic \(\frac {4}{\eta }\)stable random variable with parameter \(r_{d}^{\frac {\eta }{2}}h_{d}\sigma _{\mathbf {N}}\) using the fact that X_{d} is isotropic and ([22], Property 1.2.3). □
Remark 1
Note that the achievable rate derived in Theorem 4 bears similarities to the capacity of additive white Gaussian noise channels, but is not the same. A key difference is the absence of the noise variance, which is in fact infinite for αstable noise models. Instead, the parameter σ_{N} characterizes the statistics of the noise.
The achievable rates from Theorem 4 are obtained by using input signals that are isotropic αstable random variables, which do not satisfy the amplitude constraints in (18). The second step in characterizing the capacity of the AIαSN channel subject to (18) is therefore to consider a truncated isotropic αstable input. This guarantees the amplitude constraints are satisfied and, as we will show, yields a mutual information in the AIαSN channel that is well approximated by Theorem 4 for a sufficiently large truncation level T.
The truncated isotropic αstable random variables are defined as follows. Let X be an isotropic αstable random variable, with real part X_{r} and imaginary part X_{i}. The truncation of X, denoted by X_{T}, is given by
Using the truncated isotropic αstable input, an achievable rate of the amplitude and fractional moment constrained AIαSN channel is obtained by evaluating the mutual information I(y;X_{T}), where y is the output of the channel in (17). In fact, using a similar argument to that for the power constrained Gaussian noise channel [27], it is straightforward to show that all rates R<I(y;X_{T}) are achievable by using a codebook consisting of 2^{nR} codewords W^{n}(1),…,W^{n}(2^{nR}) with W_{i}(w), i=1,2,…,n, w=1,2,…,2^{nR} independent truncated isotropic αstable random variables.
Unfortunately, truncated isotropic αstable inputs do not lead to a closedform mutual information for the channel in (17). In fact, only scaling laws for the capacity have been recently derived for realvalued inputs [28, 29]. In order to characterize the achievable rates in the presence of dynamic interference, we therefore approximate I(X_{T};y) by the lower bound in Theorem 4.
To verify that this approximation is indeed accurate, we numerically compute the mutual information I(X_{T};y) and compare it with the result in Theorem 4 in Figs. 4 and 5 for α=1.7 and α=1.3, respectively. Observe that for a sufficiently large truncation level, the approximation based on Theorem 4 is in good agreement with I(X_{T};y). Moreover, the achievable rate is significantly larger than the case of a Gaussian input. This suggests that Gaussian signaling is not necessarily desirable in the presence of dynamic interference with the constraints in (18).
In light of the validity of the achievable rate approximation based on Theorem 4, we now turn to characterizing the effect of device density in largescale networks with dynamic interference.
4.1 Area spectral efficiency analysis
In this section, we investigate the effect of device density λ on network performance. In particular, we study the area spectral efficiency, which is defined as the expected total rate per square meter. We focus on the setting where each device is associated to its own access point and the other devices introduce interference.
The area spectral efficiency captures the tradeoff between the distance between each device and its base station as well as the increasing interference as the device density increases. Formally, let A_{1}⊂A_{2}⊂⋯ be a sequence of discs such that Area(A_{n})→∞ as n→∞. The area spectral efficiency is then given by
where Φ(A_{n}) is the PPP Φ restricted to the disc A_{n} and R_{i}(A_{n}) corresponds to the achievable rate with a truncated isotropic αstable input and devices in Φ(A_{n}).
The area spectral efficiency for this model is given in the following theorem.
Theorem 5
The area spectral efficiency with device locations governed by a homogeneous PPP Φ with rate λ and truncated isotropic αstable inputs is given by
where R_{i} is the achievable rate of the AIαSN channel with a truncated isotropic αstable input and devices in Φ.
Proof
See Section 7. □
As observed in Section 4, R_{i}=I(y_{i};X_{T}) does not have a closedform expression which makes characterizing the area spectral efficiency ζ challenging. To proceed, we exploit the approximation of I(y_{i};X_{T}) based on Theorem 4. In particular, we obtain the following approximation for the area spectral efficiency:
which is tight when the truncation level for the input T is sufficiently large. Further insight into the approximation error can be obtained numerically, such as in Figs. 4 and 5.
The expression in (26) provides insight into the effect of the device density λ. In particular, consider a function of the form
which captures the dependence of the spatial rate density approximation in (26) on the device density λ. Since we are interested in studying the impact of device density, to apply Leibniz’s rule in (26). it is sufficient to consider (27). Evaluating the derivative yields \(f^{\prime }(\lambda) = \log \left (1 + \frac {1}{\lambda }\right)  \frac {1}{1 + \lambda }\).
Since \(\log x > 1  \frac {1}{x}\) for x>1, it follows that \(\log \left (1 + \frac {1}{\lambda }\right) > \frac {1}{1 + \lambda }\) and hence for λ>0, f^{′}(λ)>0. This implies that the area spectral efficiency ζ is an increasing function of the density λ (illustrated in Fig. 6). We therefore conclude that dense networks maximize the area spectral efficiency. We remark that dense networks are also desirable for slowly varying active interferer sets [30]. This implies that although the optimal signaling strategy for each link is no longer Gaussian, the basic network structure is the same both for dynamic interference and interference arising from a slowly varying active interferer set.
5 Discussion
In Section 3, we established that the interference is αstable in networks with SCMA and fastvarying active transmitter sets. Moreover, in the special case of disjoint random SCMA codebooks, the interference vector on bands within a disjoint set is subGaussian αstable while bands in different disjoint sets are independent. A natural question is therefore how the dependence structure for general SCMA codebooks can be characterized. One approach to resolving this problem is to exploit a copula representation of the dependence structure, which has been proposed in [31].
In particular, let \(\phantom {\dot {i}\!}F_{\mathbf {Z}_{2K}}\) be the joint distribution of the real interference random vector Z_{2K} obtained by stacking real and imaginary parts as detailed in Section 3. We seek a simple and general characterization of the joint distribution. To this end, let C_{Z} be a copula, which is a function [0,1]^{2K}→[0,1]. Since the elements of Z_{2k} are αstable random variables, it follows that the cumulative distribution function (CDF) of each element is continuous, denoted by F. As a consequence, there exists a unique copula C_{Z} such that the joint distribution of Z_{2k} can be written as
by Sklar’s theorem [32].
Note that the copula provides a representation of the dependence structure, independent of the marginal distributions. That is, random vectors can be characterized in terms of the marginal distributions and the copula. In particular, the joint distribution of Z_{2K} can be characterized by the univariate αstable distributions of each z_{k,r} and z_{k,i} and the copula C_{Z}. In this case, the probability distribution density is given by:
where c_{Z} is the density function of the copula C_{Z} and has such a form that \(c_{Z}(u_{1},\ldots,u_{n})=\frac {{\partial }^{n} C_{Z}(u_{1},\ldots,u_{n})}{\partial u_{1},\ldots,\partial u_{n}}\)
This divides the density into two components, the copula component and the independent component consisting of the marginal distributions, which provides a basis for optimizing receiver structures and other system components. In particular, we desire a tractable copula C for which detection rules can be derived.
As an example, consider the standard Archimedean copula
where the generator ϕ:[0,1]→[0,∞] is a continuous and completely monotonic function such that ϕ(1)=0. In this case, the density function is given by:
where ψ=ϕ^{−1} is the inverse generator.
This representation of the dependence structure for the interference forms a basis for maximum likelihood decoding schemes at the receiver, which are otherwise intractable. Indeed, the loglikelihood ratio (LLR) Λ(Z_{2K}) for BPSK transmissions and additive vector symmetric αstable noise can be written as [33, 34]:
where Λ_{⊥}(Z_{2K}) is the independent component of the LLR arising from the marginal distributions and Λ_{c}(I_{2K}) is the component of the LLR depending on the copula and represents the dependence structure. This representation provides a clear separation between the independent and the dependent components, which may be useful in the design of efficient receivers.
However, there are still some challenging issues. As shown in (30), Archimedean copulæ assume the same dependence structure of each pair (u_{k},u_{j}). On the other hand, the dependence of the pair (F(z_{k,r}),F(z_{k,i})) arises from the subGaussian αstable distribution by Theorem 2 and in general will be different from the distribution of the pairs (F(z_{k,r}),F(z_{j,r})),k≠j or (F(z_{k,r}),F(z_{j,i})),k≠j. Hence, even if it allows for more tractable expressions [33, 34], the Archimedean family may not be an appropriate choice. It remains an open question to select useful general copula representations for dynamic interence with dynamic interference and SCMA codebooks.
6 Proof of Proposition 2
The proof follows a similar argument as for isotropic complex αstable random variables ([22], Theorem 2.6.3). We first show that if Z is a ddimensional subGaussian αstable random vector, then UZ is also subGaussian αstable. Since Z is subGaussian αstable, then it admits a scale mixture representation A^{1/2}G, where \(\mathbf {G} \sim \mathcal {N}\left (0,\sigma ^{2} \mathbf {I}\right)\). Therefore, we can write UZ=A^{1/2}UG, which implies that UZ is subGaussian αstable if \(\mathbf {U}\mathbf {G} \sim \mathcal {N}\left (0,\sigma _{U}^{2}\mathbf {I}\right)\) for some σ_{U}>0. This holds since the covariance matrix of UG is given by \(\Sigma _{\mathbf {U}} = \mathbb {E}\left [\mathbf {U}\mathbf {G}\mathbf {G}^{T}\mathbf {U}^{T}\right ] = \sigma ^{2}\mathbf {I}\) by the orthogonality of U.
We now show that \(\mathbf {Z} \overset {d}{=} \mathbf {U}\mathbf {Z}\) for all \(\mathbf {U} \in \mathcal {O}(d)\) implies that Z is subGaussian αstable. The characteristic function of Z is given by:
where (a) holds since orthogonal transformations preserve the magnitude of the inner product in \(\mathbb {R}^{d}\). It then follows that for any Borel set \(B \subset \mathbb {R}^{d}\), we have Γ_{U}(B)=Γ(UB) by the uniqueness of the spectral measure. Since \(\mathbf {U} \in \mathcal {O}(d)\) is arbitrary, it follows that Γ_{U}=Γ for all \(\mathbf {U} \in \mathcal {O}(d)\). As such, Γ is uniform on \(\mathbb {S}^{d1}\), which by ([22], Theorem 2.5.5) implies that Z is subGaussian αstable.
7 Proof of Theorem 5
In order to compute the area spectral efficiency ζ, observe that for a given A_{n}, the random variables R_{i}(A_{n}) are identically distributed (but not independent) since the distances r_{d} are independent and identically distributed, and the locations of the devices are independently and uniformly distributed in A_{n} conditioned on the number of devices N(A_{n}) in A_{n} [25]. By the strong law of large numbers for PPPs [35], \(\frac {N(A_{n})}{\text {Area}(A_{n})} \cong \lambda ~\mathrm {a.s.}\) as n→∞. Let ε>0, it then follows that
A direct consequence of the strong law of large numbers of PPPs is that as n→∞, Pr(λ_{1}∈[λ−ε,λ+ε])→1.
Next, for sufficiently large n select A_{n} such that λArea(A_{n}) is an integer and ε>0 sufficiently small such that λArea(A_{n}) is the only integer in [λ−ε,λ+ε]. It then follows that
To evaluate \({\lim }_{n \rightarrow \infty } \mathbb {E}[R_{i}(A_{n})]\), let \(y_{i,A_{n}}\) be the received signal at the access point served by the ith device in Φ(A_{n}). For fixed r_{d},h_{d}, \(\phantom {\dot {i}\!}R_{i}(A_{n}) = I(y_{i,A_{n}};X_{T})\). From the LePage series representation of the interference in Theorem 1, it follows that the signal received by the access point served by the ith device in Φ satisfies \(y_{i} \overset {(d)}{=} r_{d}^{\frac {\eta }{2}}h_{d}X_{T} + I,~\mathrm {a.s.}\) as n→∞.
Since the conditions in ([36], Theorem 1) hold, it follows that for fixed r_{d},h_{d} we have \(I(y_{i,A_{n}};X_{T}) \rightarrow I(y_{i};X_{T})\) as n→∞. As R_{i}(A_{n}) is positive and R_{i}(A_{n})→R_{i} as n→∞, we then obtain the desired result.
8 Conclusions
An important feature of wireless communication networks supporting the IoT is that devices can transmit very small amounts of data leading to dynamic interference. In this paper, we have studied the problem of dynamic interference when devices exploit the NOMA strategy SCMA. In particular, we established that the interference is characterized by a multivariate αstable distribution. This work also considered the impact of αstable interference on network design. We have derived achievable rates for a class of NOMA networks and shown that the area spectral efficiency increases with the density of devices.
An avenue of future research is the study of dynamic interference in networks using SCMA codebooks that have reduced decoding complexity. The design of such codebooks motivates the study of a general class of vector additive αstable noise channels for which fundamental limits on data rates are not known.
Notes
A codeword corresponds to the signal sent by the transmitter consisting of n symbols. On the other hand, a SCMA codeword corresponds to one symbol of the codeword.
A random variable X with support on \(\mathbb {R}\) is said to be symmetric if for all Borel sets \(A \subset \mathbb {R}\), Pr(X∈A)=Pr(−X∈A).
A formal definition of the general class of αstable random vectors is given in [22]; however, for the purposes of this paper this definition is sufficient as we only consider special cases and the general case is not required. We remark that for α<1 there are important subtleties for general αstable random vectors.
There exist also subGaussian αstable random variables that allow for more general dependence structure [22], but they are not necessary for the purposes of this paper.
Recall that an orthogonal transformation in \(\mathcal {O}(d)\) is a matrix \(\mathbf {U} \in \mathbb {R}^{d \times d}\) such that UU^{T}=U^{T}U=I.
Abbreviations
 A I α S N :

Additive isotropic αstable noise
 CDF:

Cumulative distribution function
 IoT:

Internet of Things
 LLR:

Log likelihood ratio
 M2M:

Machinetomachine
 NOMA:

Nonorthogonal multiple access
 SCMA:

Sparse code multiple access
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Acknowledgements
The authors would like to acknowledge useful discussions with Gareth W. Peters, Samir M. Perlaza, and Vyacheslav Kungurtsev. Fruitful discussions in the framework of the COST ACTION CA15104, IRACON and in IRCICA, USR CNRS 3380 have also contributed to this work.
Funding
This work has been (partly) funded by the French National Agency for Research (ANR) under grant ANR16CE250001  ARBURST.
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ME, LC, and JMG conceived the project. ME and MF carried out the derivations. ME, CZ, and LC carried out the simulations. All authors contributed to writing the paper. All authors read and approved the final manuscript.
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Egan, M., Clavier, L., Zheng, C. et al. Dynamic interference for uplink SCMA in largescale wireless networks without coordination. J Wireless Com Network 2018, 213 (2018). https://doi.org/10.1186/s136380181225z
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DOI: https://doi.org/10.1186/s136380181225z