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Ergodic Spectral Efficiency in MIMO Cellular Networks
Abstract
This paper shows how the application of stochastic geometry to the analysis of wireless networks is greatly facilitated by (i) a clear separation of time scales, (ii) abstraction of small-scale effects via ergodicity, and (iii) an interference model that reflects the receiver's lack of knowledge of how each individual interference term is faded. These procedures render the analysis both simpler and more precise and more amenable to the incorporation of subsequent features. In particular, the paper presents analytical characterizations of the ergodic spectral efficiency of cellular networks with single-user MIMO and sectorization. These characterizations, in the form of easy-to-evaluate expressions, encompass both the distribution of spectral efficiency over the network locations as well as the average thereof.
... The achievability of the ergodic rate in systems with timediversity or frequency diversity is discussed in detail in the literature (e.g., [10]- [12]). With a sufficient delay, the ergodic rate can be achieved with no outage. ...
... This lack of effect occurs because the MAC decisions (i.e., when to transmit) are independent of the routing decisions. Hence, the optimization of the routing function for the probe receiving node is independent of the routing functions of all other transmitting nodes and can be solved directly from (12). ...
... as its threshold zone, A Z,i (see Fig. 1), and denote by N Z,i the number of nodes in the intersection between the threshold zone, A Z,i , and the routing zone (with radius r A ). The function G(i, M ) in (12) can be lower bounded using only the known local knowledge, M 0 , and the system parameters by: Fig. 1: Local neighborhood of the probe transmitting node: each dot represents a node in the network. The triangle is the probe transmitting node and the star is the tested relay (node i in this example). ...
In this paper we consider routing in random wireless-adhoc-networks (WANETs), where each node is equipped with a single antenna. Our analysis uses a proper model of the physical layer together with an abstraction of higher communication layers. We assume that the nodes are distributed according to a Poisson-point-process and consider routing schemes that select the next relay based on the geographical locations, the channel gains of its neighbor nodes and the statistical characterization of all other nodes. While many routing problems are formulated as optimization problems, the optimal distributed solution is rarely accessible. In this work, we present the exact optimal solution for the scenario analyzed. The optimal routing is given as a maximization of a routing metric which depends solely on the known partial channel state information (CSI) and includes an expectation with respect to the interference statistics. The optimal routing scheme is important because it gives an upper bound on the performance of any other routing scheme. We also present sub-optimal routing schemes that only use part of the available knowledge and require much lower computational complexity. Numerical results demonstrate that the performance of the low complexity schemes is close to optimal and outperforms other tested routing schemes.
... Furthermore, the average interfering signal power from each interfering BS is by definition smaller than the average signal power from the serving BS, which simplifies the derivations. These advantages make the downlink cellular network model thoroughly studied [8]- [12] and easier to be combined with many types of emerging techniques, e.g., cooperative transmission [13]- [15], MIMO [16]- [18], and D2D communication [19]- [21]. ...
... The Paley-Zygmund bound can be used to roughly quantify the fraction of links that can achieve a certain fraction of the average performance. Fig. 9 shows the meta distribution from both the simulation result and the analytical expression in (18). It verifies the accuracy of the approximation of the moments given in Theorem 1. ...
... The analytical meta distribution(18), the simulated curve, the Markov bounds(19) for b ∈[4], the Chebyshev bounds(20) and(21) and the Paley-Zygmund bound(22) for α = 4, θ = 0 dB and ǫ = 0.5 in the uplink. ...
The meta distribution of the signal-to-interference ratio (SIR) provides fine-grained information about the performance of individual links in a wireless network. This paper focuses on the analysis of the meta distribution of the SIR for both the cellular network uplink and downlink with fractional power control. For the uplink scenario, an approximation of the interfering user point process with a non-homogeneous Poisson point process is used. The moments of the meta distribution for both scenarios are calculated. Some bounds, the analytical expression, the mean local delay, and the beta approximation of the meta distribution are provided. The results give interesting insights into the effect of the power control in both the uplink and downlink. Detailed simulations show that the approximations made in the analysis are well justified.
... This imposed inherent limitations to the applicability of those expressions to complex optimization problems. In order to overcome this problem the authors of [19] provided for the first time in the literature expressions for the DL ergodic rate in the interference limited case, for a fully loaded scenario (i.e. for λ U E ≫ λ) that did not involve any numerical integration. In this direction, the authors of [19] provided lookup tables and employed the Meijer-G function to provide a tight approximation of the DL ergodic rate in the fully loaded case. ...
... In order to overcome this problem the authors of [19] provided for the first time in the literature expressions for the DL ergodic rate in the interference limited case, for a fully loaded scenario (i.e. for λ U E ≫ λ) that did not involve any numerical integration. In this direction, the authors of [19] provided lookup tables and employed the Meijer-G function to provide a tight approximation of the DL ergodic rate in the fully loaded case. In particular, the approximate ergodic rate of [19] is given by: ...
... In this direction, the authors of [19] provided lookup tables and employed the Meijer-G function to provide a tight approximation of the DL ergodic rate in the fully loaded case. In particular, the approximate ergodic rate of [19] is given by: ...
The employment of stochastic geometry for the analysis and design of ultra dense networks (UDNs) has provided significant insights into network densification. In addition to the characterization of the network performance and behavior, these tools can also be exploited toward solving complex optimization problems that could maximize the capacity benefits arising in UDNs. However, this is preconditioned on the existence of tractable closed form expressions for the considered figures of merit. In this course, the present paper introduces an accurate approximation for the moment generating function (MGF) of the aggregate other-cell interference created by base stations whose positions follow a Poisson point process of given spatial density. Given the pivotal role of the MGF of the aggregate interference in stochastic geometry and the tractability of the derived MGF, the latter can be employed to substantially simplify ensuing stochastic geometry analyses. Subsequently, the present paper employs the introduced MGF to provide closed form expressions for the downlink ergodic capacity for the interference limited case, and validates the accuracy of these expressions by the use of extensive Monte Carlo simulations. The derived expressions depend on the density of users and base stations, setting out a densification road map for network operators and designers of significant value.
... An alternative approach assumes that each link achieves the ergodic rate (e.g., the Asymptotic-Density-of-Rate-Progress, ADORP, [18]). The ergodic rate is higher than the outage rate (e.g., [24,25]) but typically requires a longer delay. The achievability of the ergodic rate in MIMO WANETs was recently discussed in detail [25]. ...
... The ergodic rate is higher than the outage rate (e.g., [24,25]) but typically requires a longer delay. The achievability of the ergodic rate in MIMO WANETs was recently discussed in detail [25]. Conveniently, the ergodic rate also lends itself better to analysis. ...
... Lemma 1 Consider the decoding of the k-th data stream from transmitter j, at node i, where the transmitter uses EBF and the receiver uses PZF according to (25). The distribution of Y i,j,k is the same for all k, and depends only on K j and ...
In this paper we consider opportunistic routing in multiple-input-multiple-output (MIMO) random wireless ad-hoc networks (WANETs). Our analysis uses a proper model of the physical layer together with an abstraction of the higher communication layers. We assume that the nodes are distributed according to a Poisson-Point-Process and consider a routing scheme that opportunistically selects the next relay and the number of spatially multiplexed data streams. The routing decisions are based on geographic locations, the channel gains of the neighbor nodes and the statistical characterization of all other nodes. Unlike the single antenna case, the optimal routing scheme cannot be explicitly expressed. Hence, we propose a smart-routing scheme for MIMO that adapts the number of data streams per user to the channel conditions. Numerical results demonstrate that this scheme outperforms all previously published schemes for this scenario. The findings highlight the importance of channel state information for efficient routing, and the need for an adaptive selection of the number of data streams at each transmitter.
... This has led to unexpected observations in specific scenarios, as well as to divergent or even contrasting conclusions on the fundamental limits of network densification. Furthermore, the standard assumption of exponential distributed channel power lack the flexibility to adapt to different fading behaviors pertaining to advanced communication and signal processing technique e.g., massive MIMO, coordinated multipoint (CoMP) and mmWave communications [18]- [20]. Although some works investigated the effect of pathloss singularity [21], [22], [23] or boundedness [2], [8], incorporating meaningfully the combined effect of path-loss and generalized channel power models is usually ignored. ...
... Our general framework enables us to model any channel power distribution, including multipath, shadowing and channel gains due to antenna pattern and beamforming, etc., that can be observed in current wireless communications and networking. In particular, using general channel power distributions, we provide the ability to capture the impact of deploying multiple antenna BSs on the network performance [15]- [20], [24]. Remarkably, this paper provides a unified framework to analyze multi-antenna networks by easily transplanting the developed framework for SISO networks under Fox's H fading channel. ...
... (20) Interestingly, it follows that when it comes to multi-antenna networks, (19) is compatible with any forms of η(ξ). Compared with existing approaches in [15]- [20] which requires the calculation of N t k − 1, derivatives of η(ξ) when g x k is gamma distributed as Gamma(N t k , 1), Proposition 1 generalizes to multi-antenna networks with no additional computational complexity and thus preserves the tractability. Note that assuming a Gamma distribution for the interferers' power gain, i.e. g xj ∼ Gamma(χ j , φ j ), j ∈ {1, . . . ...
In this paper, we develop an innovative approach to quantitatively characterize the performance of ultra-dense wireless networks in a plethora of propagation environments. The proposed framework has the potential of significantly simplifying the cumbersome procedure of analyzing the coverage probability and allowing the remarkable unification of single- and multi-antenna networks through compact representations. By harnessing this key feature, we develop a novel statistical machinery to study the scaling laws of wireless network densification considering general channel power distributions including the entire space of multipath and shadowing models as well as associated beamforming gain due to the use of multiple antenna. We further formulate the relationship between network density, antenna height, antenna array seize and carrier frequency showing how the coverage probability can be maintained with ultra-densification. From a system design perspective, we present a new innovative theoretical discovery stipulating that if multiple antenna BS are deployed and moved to higher frequencies, then monotonically increasing the coverage probability by means of ultra-densification is possible, and this without lowering the antenna height. Such findings are completely different from the conclusions in existing works, who suggest to lower the BS height as to leverage the potential of network densification. Simulation results substantiate performance trends leveraging network densification and antenna deployment and configuration against path loss models and signal-to-noise plus interference (SINR) thresholds.
... This imposed inherent limitations to the applicability of those expressions to complex optimization problems. In order to overcome this problem, the authors of [19] provided for the first time in the literature expressions for the DL ergodic rate in the interference limited case, for a fully loaded scenario (i.e., for λ UE λ) that did not involve any numerical integration. In this direction, the authors of [19] provided lookup tables and employed the Meijer-G function to provide a tight approximation of the DL ergodic rate in the fully loaded case. ...
... In order to overcome this problem, the authors of [19] provided for the first time in the literature expressions for the DL ergodic rate in the interference limited case, for a fully loaded scenario (i.e., for λ UE λ) that did not involve any numerical integration. In this direction, the authors of [19] provided lookup tables and employed the Meijer-G function to provide a tight approximation of the DL ergodic rate in the fully loaded case. In particular, the approximate ergodic rate of [19] is given by: ...
... In this direction, the authors of [19] provided lookup tables and employed the Meijer-G function to provide a tight approximation of the DL ergodic rate in the fully loaded case. In particular, the approximate ergodic rate of [19] is given by: ...
Abstract The employment of stochastic geometry for the analysis and design of ultra dense networks (UDNs) has provided significant insights into network densification. In addition to the characterization of the network performance and behavior, these tools can also be exploited toward solving complex optimization problems that could maximize the capacity benefits arising in UDNs. However, this is preconditioned on the existence of tractable closed form expressions for the considered figures of merit. In this course, the present paper introduces an accurate approximation for the moment generating function (MGF) of the aggregate other-cell interference created by base stations whose positions follow a Poisson point process of given spatial density. Given the pivotal role of the MGF of the aggregate interference in stochastic geometry and the tractability of the derived MGF, the latter can be employed to substantially simplify ensuing stochastic geometry analyses. Subsequently, the present paper employs the introduced MGF to provide closed form expressions for the downlink ergodic capacity for the interference limited case, and validates the accuracy of these expressions by the use of extensive Monte Carlo simulations. The derived expressions depend on the density of users and base stations, setting out a densification road map for network operators and designers of significant value.
... This reinforces the interest in analytical solutions, and such is the subject of this paper. To embark upon the analysis of massive MIMO settings, we invoke tools from stochastic geometry that have been successfully applied already in non-MIMO [12]- [18] and in MIMO contexts [19]- [22]. ...
... being the local-average SIR in single-user transmission [19]. Hence, (21) can be rewritten as ...
... To characterize this distribution, we capitalize on results derived for the typical user in a PPP-distributed network of BSs [18], [19], in accordance with the PPP convergence exposed in Section II-A. Specifically, ρ 0 , . . . ...
This paper presents analytical expressions for the signal-to-interference ratio (SIR) and the spectral efficiency in macrocellular networks with massive MIMO conjugate beamforming, both with a uniform and a channel-dependent power allocation. These expressions, which apply to very general network geometries, are asymptotic in the strength of the shadowing. Through Monte-Carlo simulation, we verify their accuracy for relevant network topologies and shadowing strengths. Also, since the analysis does not include pilot contamination, we further gauge through Monte-Carlo simulation the deviation that this phenomenon causes with respect to our results, and hence the scope of the analysis.
... Motivated by the above observations, in this paper we focus on a spatially extended network with non-symmetric users due to the fact that the pathloss between each user and the ENs is different, and depends on the users and ENs locations. As extensively discussed in [22,23], in this context it is meaningful to consider the per-user ergodic rate where averaging (ergodic behavior) is with respect to the small-scale fading and conditioning is with respect to the random realization of the pathloss, which in turn depends on the random placement of users and ENs on the plane. Since such conditional ergodic rate is a function of the random network geometry, it is a random variable itself. ...
... APPENDIX A PROOF OF (22), (23) The Laplace transform of 1/ρ k, , for s ∈ C, is given by ...
The coded caching scheme proposed by Maddah-Ali and Niesen (MAN) critically hinges on the ability of the system to deliver a common coded multicast message from a server to all users in the system at a fixed rate, independent of the number of users. In order to apply this paradigm to a spatially distributed wireless network, it is important to make sure that such common multicast rate does not vanish as the number of users in the network and/or the network area increase. This paper starts from a variant of the MAN scheme successively proposed for the so-called combination network, where the multicast message is further encoded by a Maximum Distance Separable (MDS) code and the MDS-coded blocks are sent to multiple spatially distributed single-antenna Edge Nodes (ENs), transmitting at a fixed rate with no channel state information. The users have multiple antennas. They obtain receiver channel state information from standard downlink pilots and can select to decode a desired number of EN transmissions, while either nulling or treating as noise the others. The system is reminiscent of the so-called evolved Multimedia Broadcast Multicast Service (eMBMS), since the fundamental underlying transmission mechanism is multipoint multicasting, where each user can independently (in a user-centric manner) decide which EN to decode, without any explicit association of users to ENs. We study the performance of the proposed system when users and ENs are distributed according to homogeneous Poisson Point Processes in the plane and the propagation is affected by Rayleigh fading and distance dependent pathloss. Our analysis allows the optimization of the PHY parameters (PHY coding rate at the ENs and MDS coding rate) for given MAN scheme parameters. The proposed scheme achieves full A short version of this paper was presented in the 22nd International ITG Workshop on Smart Antennas (WSA) 2018 [1].). Ratheesh K. Mungara is with Ericsson AB, Torshamnsgatan 23, 164 83 Stockholm, Sweden. (email: ratheesh.mungara@gmail.com). 2 spatial scalability in the following sense: for an extended network with arbitrary constant ratio of users per EN and area A = O(N E), where N E denotes the number of ENs, the system achieves a per-user delivery rate that does not vanish as N E → ∞.
... An alternative approach assumes that each link achieves the ergodic rate (e.g., the Asymptotic-Density-of-Rate-Progress, ADORP, [18]). The ergodic rate is higher than the outage rate (e.g., [24,25]) but typically requires a longer delay. The achievability of the ergodic rate in MIMO WANETs was recently discussed in detail [25]. ...
... The ergodic rate is higher than the outage rate (e.g., [24,25]) but typically requires a longer delay. The achievability of the ergodic rate in MIMO WANETs was recently discussed in detail [25]. Conveniently, the ergodic rate also lends itself better to analysis. ...
In this paper we consider opportunistic routing in multiple-input–multiple-output (MIMO) random wireless ad-hoc networks (WANETs). Our analysis uses a proper model of the physical layer together with an abstraction of the higher communication layers. We assume that the nodes are distributed according to a Poisson point process and consider a routing scheme that opportunistically selects the next relay and the number of spatially multiplexed data streams. The routing decisions are based on geographic locations, the channel gains of the neighbor nodes, and the statistical characterization of all other nodes. Unlike the single antenna case, the optimal routing scheme cannot be explicitly expressed. Hence, we propose a smart-routing scheme for MIMO that adapts the number of data streams per user to the channel conditions. The numerical results demonstrate that this scheme outperforms all previously published schemes for this scenario. The findings highlight the importance of channel state information for efficient routing, and the need for an adaptive selection of the number of data streams at each transmitter.
... In addition, work [11] analyses the effect of retransmissions on the multi-antenna-enabled network performance. In [12], the ergodic spectral efficiency in multi-antenna cellular networks with sectorization is investigated. In [13], the power control algorithm is applied in the cellular network performance analysis with multiple antennas. ...
... Unfortunately, it is difficult to apply any numerical optimization method for the optimal bias in the result (12) as it is difficult to calculate its differential result. Therefore, we apply an approximation of the Gauss hypergeometry function, ...
The 1000 fold capacity enhancement is one of the key requirements in the future 5G networks, stimulating the interest in jointly adopting several advanced techniques (e.g. multiple antennas and heterogeneous networks (HetNets)). Analysis of the performance of the HetNets jointly with multiple antennas becomes crucial. In this paper, we analyse the K-tier multi-antenna HetNets from a downlink coverage perspective. The coverage probability is derived using the Gil-Pelaez inversion theorem under the stochastic geometry framework. Moreover, a closed form approximated result is obtained for observing the influence of the normalized range bias (NRB). The result shows that our proposed result closely match the Monte Carlo simulation, and the approximation result is effective for searching the optimal NRB.
... Provided the AP locations are agnostic to the radio propagation, shadowing has been shown to make such locations seem Poisson-distributed from the vantage of any user [9]. This approximation sharpens as the shadowing strengthens, being highly precise for values of interest [9], [10]. Leveraging this result, we place the APs and users randomly over the network, such that their locations conform to respective (mutually independent) binomial point processes; as the network grows, these converge to Poisson point processes. ...
... In turn, the projection of y k on G n,k h * n,k f cb n,k − G n,k E[h * n,k f cb n,k ] is self-interference. Combining (6)- (10), the observation at user k can be written as (2) atop the page, from which the SINR emerges as ...
We present a modification of conjugate beamforming for the forward link of cell-free massive MIMO networks. This modification eliminates the self-interference and yields a performance that, without forward pilots, closely approaches what would be achieved with such pilots in place. The simplicity of conjugate beamforming is preserved, with no need for matrix inversions, at the expense of fading-rate coordination among the access points.
... Therefore, the inter D2D interference in this section is specified for one subcarrier n r ∈ N kr . It is shown in [30] that in a dense D2D deployed system, the cumulative inter D2D interference signal can be modelled as a complex Gaussian random variable with zero mean and variance I v kr,nr . Then, we have I c kr,nr ≈ I v kr,nr = l ∈Lactive\{lr} P k l ,nr · d kr,k l −α , (31) where P k l ,nr is the transmit power of D2D k l on subcarrier n r and d kr,k l is the distance between D2D k l and D2D kr . ...
... The upper bound of theC T OT is shown in (47) at the top of this page, by substituting (30) and (44) into (29). Since the sub-cell resource allocation is independent with each other and the fading channels among D2D pairs are i.i.d., the integrations and summations in (47) can be interchanged, andC u T OT can be further simplified according to Theorem 1 as, ...
Resource allocation of cellular-assisted device-todevice (D2D) communication is very challenging when frequency reuse is considered among multiple D2D pairs within a cell, as intense inter D2D interference is difficult to tackle and generally causes extremely large signaling overhead for channel state information (CSI) acquisition. In this paper, a novel resource allocation framework for cellular-assisted D2D communication is developed with low signaling overhead while maintaining high system capacity. By utilizing the spatial dispersion property of D2D pairs, a geography-based sub-cell division strategy is proposed to divide the cell into multiple sub-cells and D2D pairs within one sub-cell are formed into one group. Then, sub-cell resource allocation is performed independently among sub-cells without the need of any prior knowledge of inter D2D interference. Under the proposed resource allocation framework, a tractable approximation for the inter D2D interference modelling is obtained and a computationally efficient expression for the average ergodic sum capacity of the cell is derived. The expression further allows us to obtain the optimal number of sub-cells, which is an important parameter for maximizing the average ergodic sum capacity of the cell. It is shown that with small CSI feedback, system capacity can be improved significantly by adopting the proposed resource allocation framework, especially in dense D2D deployed systems.
... ̅ is E { , } or average user rate in terms of the subcarriers. Problem (24) can be easily solved by using the search algorithms. ...
... Based on the method in [7], it is possible to decompose (7) into subproblems, related to each user and each BS in order to find the maximum user rate by obtaining the best . ∀ , (25c) Equations (24) and (25) are mixed integer linear problems (MILP), thus it can be solved easily by pattern search techniques such as brand-and-branch algorithm [33]. ...
Heterogeneous ultra-dense network (HUDN) is a promising network structure, which increases network efficiency in 4G and 5G networks. However, it faces the new challenges regarding interference and mobility management. To overcome these challenges, joint of resource allocation (RA) and mobility management is necessary, which, to the best of our knowledge, has not been sufficiently investigated. This paper represents a solution to this issue. First, analytical investigation and numerical analysis are carried out to model and justify the behavior of expectation of handover (HO) success rate (
) versus coverage probability. This paper provides more insight and tools for mobility-based RA researches and design of the network as well. Then, a new approach of hybrid cell-resource allocation is introduced. It is noteworthy that this is a practical structure that is adaptable to dynamic network changes in parameters, such as traffic distribution, mobility pattern, network topology, and different tiers’ acceptable signal-to-interference-plus-noise ratio. The advantage of this new proposed approach is demonstrated by a numerical analysis. The results are compared with traditional approaches with and without HO priority consideration called hybrid-partial CRA (HP-CRA) and traditional CRA (T-CRA), respectively. The results show a considerable improvement of
about 20% and 80% compared with HP-CRA and T-CRA, respectively, under the loaded situations, while the network sum rate is kept near the optimal solution.
... From application perspective, the results can be extended to the analysis of cache enabled network to study metrics like total hit probability and caching throughput; see [48], [63]. Finally, this framework can be extended to the analysis of other key performance metrics such as ergodic spectral efficiency [64] and bit error rate. ...
This paper develops a new approach to the modeling and analysis of HetNets that accurately incorporates coupling across the locations of users and base stations, which exists due to the deployment of small cell base stations (SBSs) at the places of high user density (termed user hotspots in this paper). Modeling the locations of the geographical centers of user hotspots as a homogeneous Poisson Point Process (PPP), we assume that the users and SBSs are clustered around each user hotspot center independently with two different distributions. The macrocell base station (BS) locations are modeled by an independent PPP. This model is consistent with the user and SBS configurations considered by 3GPP. Using this model, we study the performance of a typical user in terms of coverage probability and throughput for two association policies: i) Policy 1, under which a typical user is served by the open-access BS that provides maximum averaged received power, and ii) Policy 2, under which the typical user is served by the small cell tier if the maximum averaged received power from the open-access SBSs is greater than a certain power threshold; and macro tier otherwise. A key intermediate step in our analysis is the derivation of distance distributions from a typical user to the open-access and closed-access interfering SBSs. Our analysis demonstrates that as the number of SBSs reusing the same resource block increases, coverage probability decreases whereas throughput increases. Therefore, contrary to the usual assumption of orthogonal channelization, it is reasonable to assign the same resource block to multiple SBSs in a given cluster as long as the coverage probability remains acceptable. This approach to HetNet modeling and analysis significantly generalizes the state-of-the-art approaches that are based on modeling the locations of BSs and users by independent PPPs.
... Note that the given interpretation of the definition of E(λ) assumes that the channel is constant over each transmitted codeword. However, this quantity serves as an upper bound on the achievable rate if the transmitted messages suffer from multiple small-scale fluctuations during each codeword as indicated in[41]. ...
Distance-based attenuation is a critical aspect of wireless communications. As opposed to the ubiquitous power-law path loss model, this paper proposes a stretched exponential path loss model that is suitable for short-range communication. In this model, the signal power attenuates over a distance r as , where are tunable parameters. Using experimental propagation measurements, we show that the proposed model is accurate for short to moderate distances in the range meters and so is a suitable model for dense and ultradense networks. We integrate this path loss model into a downlink cellular network with base stations modeled by a Poisson point process, and derive expressions for the coverage probability, potential throughput, and area spectral efficiency. Although the most general result for coverage probability has a double integral, several special cases are given where the coverage probability has a compact or even closed form. We then show that the potential throughput is maximized for a particular BS density and then collapses to zero for high densities, assuming a fixed SINR threshold. We next prove that the area spectral efficiency, which assumes an adaptive SINR threshold, is non-decreasing with the BS density and converges to a constant for high densities.
... Note that this is an upper bound on the actual spectral efficiency, since this formulation assumes that the transmitter knows all the fading coefficients between all other transmitters and the receiver, which would be a very generous assumption. A tighter lower bound could be found using the approach described in[33]. ...
This paper develops a stochastic geometry-based approach for the modeling and analysis of single- and multi-cluster wireless networks. We first define finite homogeneous Poisson point processes to model the number and locations of the transmitters in a confined region as a single-cluster wireless network. We study the coverage probability for a reference receiver for two strategies; closest-selection, where the receiver is served by the closest transmitter among all transmitters, and uniform-selection, where the serving transmitter is selected randomly with uniform distribution. Second, using Matern cluster processes, we extend our model and analysis to multi-cluster wireless networks. Here, the receivers are modeled in two types, namely, closed- and open-access. Closed-access receivers are distributed around the cluster centers of the transmitters according to a symmetric normal distribution and can be served only by the transmitters of their corresponding clusters. Open-access receivers, on the other hand, are placed independently of the transmitters and can be served by all transmitters. In all cases, the link distance distribution and the Laplace transform (LT) of the interference are derived. We also derive closed-form lower bounds on the LT of the interference for single-cluster wireless networks. The impact of different parameters on the performance is also investigated.
... The results of this paper are in line with this view and provide insight as to how such an architecture will perform in cellular networks with FD capabilities. To derive the outage probability, i.e. the cumulative distribution function of the SINR, we take the expectation over both small-and large-scale fading [29], [32]; decoupling the two is left as a potential future direction [43]. Therefore, conditioning on the nearest BS being at a distance r we have, ...
In this paper, we consider two fundamental full-duplex (FD) architectures, two-node and three-node, in the context of cellular networks where the terminals employ directional antennas. The simultaneous transmission and reception of data in non-orthogonal channels makes FD radio a potential solution for the currently limited spectrum. However, its implementation generates high levels of interference either in the form of loopback interference (LI) from the output to the input antenna of a transceiver or in the form of co-channel interference in large-scale multicell networks due to the large number of active links. Using a stochastic geometry model, we investigate how directional antennas can control and mitigate the co-channel interference. Furthermore, we provide a model which characterizes the way directional antennas manage the LI in order to passively suppress it. Our results show that both architectures can benefit significantly by the employment of directional antennas. Finally, we consider the case where both architectures are employed in the network and derive the optimal values for the density fraction of each architecture which maximize the success probability and the network throughput.
... Using the previous two expressions to take the expectation of the conditional CDF given in (60) with respect to R K+1 , yields (21). ...
We consider a user-centric co-operative cellular network, where base stations (BSs) close to a mobile co-operate to detect its signal using a (joint) linear minimum-mean-square-error receiver. The BSs are at arbitrary positions and mobiles are modeled as a planar Poisson Point Process (PPP). Combining stochastic geometry and infinite-random-matrix theory, we derive a simple expression for the spectral efficiency of this complex system as the number of antennas grows large. This framework is applied to BS locations from PPPs and hexagonal grids, and are validated through Monte Carlo simulations. The results reveal the influence of tangible system parameters such as mobile and base-station densities, number of antennas per BS, and number of co-operating BSs on achievable spectral efficiencies. Among other insights, we find that for a given BS density and a constraint on the total number of co-operating antennas, all co-operating antennas should be located at a single BS. On the other hand, in our asymptotic regime, for the same number of co-operating antennas, if the network is limited by the area density of antennas, then the number of co-operating BSs should be increased with fewer antennas per BS.
... Such an approach can enable system designers to tradeoff various parameters such as BS and mobile densities, number of antennas, number of cooperating base stations and spectral efficiency. These approaches have been observed to "lead to remarkably precise characterizations" [24]. Further, analysis of cellular networks that explicitly considers the BS and mobile distribution in space is quite challenging, even for single antenna systems [25]. ...
We consider the downlink of a cooperative cellular communications system, where several base-stations around each mobile cooperate and perform zero-forcing to reduce the received interference at the mobile. We derive, for the first time, closed-form expressions for the asymptotic performance of the network as the number of antennas per base station grows large. These expressions capture the tradeoffs between various system parameters, and characterize the joint effect of noise and interference (where either noise or interference is asymptotically dominant and where both are asymptotically relevant). The presented analysis is significantly more challenging than the uplink analysis due to the dependence between beamforming vectors of nearby base stations. This statistical dependence is handled by introducing novel bounds on marked shot-noise point processes with dependent marks, which are also useful in other contexts. The asymptotic results are verified using Monte Carlo simulations, which indicate that they are useful even when the number of antennas per base station is only moderately large. Based on these expressions, we present a novel power allocation algorithm that is asymptotically optimal while significantly reducing the coordination overhead between base stations.
... Multi-Input Multi-Output (MIMO) communication systems play a vital role in achieving reliable high data rate transmission while improving spectrum efficiency and channel capacity [1]. In order to mitigate channel distortions and separate source signals, MIMO systems employ an increased number of training sequences, usually referred to as pilots, which do not carry any useful data. ...
Minimizing training overhead in channel estimation is a crucial challenge in wireless communication systems. This paper presents an extension of the traditional blind algorithm, called “Mutually referenced equalizers” (MRE), specifically designed for MIMO systems. Additionally, we propose a novel semi-blind method, SB-MRE, which combines the benefits of pilot-based and MRE approaches to achieve enhanced performance while utilizing a reduced number of pilot symbols. Moreover, the SB-MRE algorithm helps to minimize complexity and training overhead and to remove the ambiguities inherent to blind processing. The simulation results demonstrated that SB-MRE outperforms other linear algorithms, i.e., MMSE, ZF, and MRE, in terms of training overhead symbols and complexity, thereby offering a promising solution to address the challenge of minimizing training overhead in channel estimation for wireless communication systems.
... 1) URDC-UL: The maximum achievable transmission rate is defined by the ergodic capacity C = ζ W E[log 2 (1 + SINR)] bits/s. Operating at this rate requires instantaneous knowledge of the SINR [65]. However, this is not applicable to large-scale IoT networks. ...
This paper proposes an ultra-reliable device-centric uplink (URDC-UL) communication scheme for airborne networks. In particular, base stations (BSs) are mounted on unmanned aerial vehicles (UAVs) that travel to schedule UL transmissions and collect data from devices. To attain an ultra-reliable unified device-centric performance, the UL connection is established when the UAV-BS is hovering at the nearest possible distance from the scheduled device. The performance of the proposed URDC-UL scheme is benchmarked against a stationary UAV-centric uplink (SUC-UL) scheme where the devices are scheduled to communicate to UAV-BSs that are continuously hovering at static locations. Utilizing stochastic geometry and queueing theory, novel spatiotemporal mathematical models are developed, which account for the UAV-BS spatial densities, mobility, altitude, antenna directivity, ground-to-air channel, and temporal traffic, among other factors. The results demonstrate the sensitivity of the URDC-UL scheme to the ratio between hovering and traveling time. In particular, the hovering to traveling time ratio should be carefully adjusted to maximize the harvested performance gains for the URDC-UL scheme in terms of link reliability, transmission rate, energy efficiency, and delay. Exploiting the URDC-UL scheme allows IoT devices to minimize transmission power while maintaining unified reliable transmission. This preserves the device's battery and addresses a critical IoT design challenge.
... Treating interference as noise, the maximum achievable Tx rate is defined by the ergodic capacity C = ζ×W ×E{log 2 (1+SINR)} bits/sec. To achieve C bits/sec, the transmitting devices require instantaneous knowledge (i.e., every time slot) of the SINR realization in order to adapt their Tx rate and alleviate outages [32]. Indeed this is infeasible for large-scale IoT networks. ...
This paper studies data aggregation in large-scale regularly deployed Internet of Things (IoT) networks, where devices generate synchronized time-triggered traffic (e.g., measurements or updates). The data granularity, in terms of information content and temporal resolution, is parameterized by the sizes of the generated packets and the duty cycle of packet generation. The generated data packets at the devices are aggregated through static terrestrial gateways. Universal frequency reuse is adopted across all gateways and randomized scheduling is utilized for the IoT devices associated with each gateway. Such network model finds applications in environmental sensing, precision agriculture, and geological seismic sensing to name a few. To this end, we develop a novel spatiotemporal mathematical model to characterize the interplay between data granularity, transmission reliability, and delay. The developed model accounts for several IoT design parameters, which include packet sizes, generation duty cycle, devices and gateways spatial densities, transmission rate adaptation, power control, and antenna directivity. For tractable analysis, we propose two accurate approximations, based on the Poisson point process, to characterize the signal-to-interference-plus-noise-ratio (SINR) based transmission reliability. For the delay analysis, we propose a phase-type arrival/departure (PH/PH/1) queueing model that accounts for packet generation, transmission scheduling, and rate-sensitive SINR-based packet departure. The developed model is utilized to obtain the optimal transmission rate for the IoT devices that minimizes delay. The numerical results delineate the joint feasibility range of packet sizes and inter-arrival times for data aggregation and reveal significant gains when deploying directional antennas.
... In stochastic geometry, generally the location of the nodes in the wireless network is modeled as random, fol-lowing for example a poisson point process. In stochastic geometry based methods [111,112], the location of the users being random, their geographic distribution then induces a certain probability distribution for the channel attenuation. This leads to results on the coverage probability, the capacity, the outage probability and other fundamental limits in wireless networks. ...
Multiple antennas at the base station side can be used to enhance the spectral efficiency and energy efficiency of the next generation wireless technologies. Indeed, massive multi-input multi-output (MIMO) is seen as one promising technology to bring the aforementioned benefits for fifth generation wireless standard, commonly known as 5G New Radio (5G NR). In this monograph, we will explore a wide range of potential topics in multi-userMIMO (MU-MIMO) relevant to 5G NR,• Sum rate maximizing beamforming (BF) design and robustness to partial channel stateinformation at the transmitter (CSIT)• Asymptotic analysis of the various BF techniques in massive MIMO and• Bayesian channel estimation methods using sparse Bayesian learning.One of the potential techniques proposed in the literature to circumvent the hardware complexity and power consumption in massive MIMO is hybrid beamforming. We propose a globally optimal analog phasor design using the technique of deterministic annealing, which won us the best student paper award. Further, in order to analyze the large system behaviour of the massive MIMO systems, we utilized techniques from random matrix theory and obtained simplified sum rate expressions. Finally, we also looked at Bayesian sparse signal recovery problem using the technique called sparse Bayesian learning (SBL). We proposed low complexity SBL algorithms using a combination of approximate inference techniques such as belief propagation (BP), expectation propagation and mean field (MF) variational Bayes. We proposed an optimal partitioning of the different parameters (in the factor graph) into either MF or BP nodes based on Fisher information matrix analysis.
... We consider a network with non-cooperating transmitters; e.g., MIMO cellular networks [16]- [20], or MIMO ad-hoc networks [21]- [24]. We also assume that each receiver treats the unknown interference as additional (spatially colored) noise. ...
This work sheds light on the effects of spatially multiplexed interference on multiple-input-multiple-output (MIMO) networks. In particular, any increase in the number of interfering data streams (while keeping the total interference power constant) is shown to degrade the quality of the interfered link. Although this statement appears very intuitive, it has yet to be proven. In this work, we first give a mathematical definition of the intuitive notion of 'increasing the number of streams' leads to proof that the achievable rate of a link decreases when any of its interferers increases its number of data streams. The achievable rate is measured by the mutual information of the link or by the spectral efficiency of the optimal linear Minimum-Mean-Square (MMSE) receiver. Correlatively, we show that the worst power allocation for an interferer is the equal power allocation for all the streams. This result highlights the importance of the optimization of the number of data streams at each transmitter in MIMO networks.
... In recent years, stochastic geometry as a tractable tool has been widely used to model and analyse the cellular networks. Most of these literatures focused on downlink (DL) and rapidly extend different hot spot fields, such as further enhanced inter cell interference coordination (FeICIC) [9], coordinated multipoint transmission (CoMP) [10], mm-Wave communication [11], multi-input multi-output (MIMO) cellular networks [12], device-to-device (D2D) communication [13] and so on. ...
The low-power wide area network (LPWAN) is designed for low-power, wide area, light load, high latency applications. In many use-case applications of traffic being usually less than 1k of bytes transmitted data per day, it is desirable for a user equipment (UE) to work for 10 years, powered by a primary battery. There is neither real test data nor mathematical models to validate a 10 years battery lifetime. Furthermore, the energy consumption is affected by many factors and is very different in diverse networks. In this paper, we consider two types of LPWAN: LoRa wide area network (LoRaWAN) and narrow-band Internet of Things (NBIoT) network. We first propose a framework to calculate the average number of retransmissions in LoRaWAN networks and NBIoT networks based on stochastic geometry. Combining the average number of retransmissions, we give an approximate method to calculate both networks’ energy efficiency. Utilizing the energy efficiency we can estimate the battery lifetime in LoRaWAN networks and NBIoT networks. The numerical results show that the battery lifetime is mainly influenced by the number of active UEs and the spreading factor in LoRaWAN networks and sleeping mode in NBIoT networks, when the data size transmitted each day is fixed. In NBIoT networks, the UEs can work for much longer with power saving mode (PSM) than with extended idle-mode discontinuous reception cycle (eDRX), even exceeding LoRaWAN networks in some cases though the transmitting power is higher and protocol is more complex in NBIoT networks. Finally, in LoRaWAN networks, smaller spreading factors can achieve longer battery lifetime, and increasing the number of base stations also extends the battery lifetime, which is not the case for NBIoT networks.
... Specifically, ergodic SE and SE bounds of MIMO-based cellular networks were derived in [10] and [11], respectively. In [12], the authors studied EE for MIMO broadcasting chan-nel (BC) and proposed an energy efficient iterative waterfilling transmission scheme. ...
Energy efficiency (EE) optimization is investigated for a multi-user cognitive radio network (CRN) over multiple-input-multiple-output (MIMO) interference channels (ICs). To reduce the system overhead due to information exchange among the secondary CR uses (SUs), the EE optimization problem is formulated as a non-cooperative game, where each SU transmitter competes against the other SU pairs by optimizing its transmit covariance matrix. Specifically, each multi-antenna SU maximizes locally its energy efficiency in terms of the number of bits transmitted per unit energy consumption, subject to the per-SU transmit power constraint and the primary user (PU) perceived total interference constraint. It is proved that the formulated non-cooperative game admits at least one Nash equilibrium (NE), and the sufficient condition for a unique NE is derived subsequently. Primal decomposition is employed in the local EE optimization problem to relax the coupling PU perceived interference constraint such that fully distributed operation is allowed. A distributed iterative EE optimization algorithm (DIEEOA) is then proposed to obtain the unique NE, which is shown to converge to the global optimum. Linear precoding techniques are employed to mitigate the impacts of multi-user interference and imperfect channel state information (CSI). Through numerical simulations, effectiveness of the proposed scheme is validated and the system setting parameters’ impacts on the performance are studied.
... There are results on the temporal correlation of interference in Poisson networks (see [6], [8], [9], [12]- [14]) and for the stochastic dependency of interference, typically expressed as the joint outage probability of multiple transmissions (see [4]- [6], [15]- [19]). In all these publications, however, the node locations are the sole source of correlation and mobility is not considered. ...
This article takes an analytical approach to investigate the temporal dynamics of interference in wireless networks. We propose a framework to calculate the autocorrelation of interference in Poisson networks and derive closed-form expressions for the case of Nakagami fading. The framework takes three correlation sources into account: the location of interferers, the wireless channel, and the data traffic. We introduce the interference coherence time -- in analogy to the well-established channel coherence time -- and show how its basic qualitative behavior depends on the source of correlation. The insights gained can be useful in the design of medium access control and retransmission protocols.
... However, the resulting deterministic analysis is quite cumbersome and does not allow much analytical insight. In stochastic geometry based methods [13], the location of the users is assumed to be random, their geographic distribution then inducing a certain probability distribution for the channel attenuations. Whereas most stochastic geometry work focuses on the distribution of the attenuations, here we consider an extension to multi-antenna systems. ...
This work deals with coordinated beamforming (BF) for the Multi-Input Single-Output (MISO) Interfering Broadcast Channel (IBC), i.e. the MISO MultiUser Multi-Cell downlink (DL). The novel beamformers are here optimized for the Expected Weighted Sum Rate (EWSR) for the case of Partial Channel State Information at the Transmitters (CSIT). Gaussian (pos-terior) partial CSIT can optimally combine channel estimate and channel covariance information. With Gaussian partial CSIT, the beamformers only depend on the means (estimates) and (error) covariances of the channels. We extend a recently introduced large system analysis for optimized beamformers with partial CSIT, by a stochastic geometry inspired randomization of the channel covariance eigen spaces, leading to much simpler analytical results, which depend only on some essential channel characteristics. In the Massive MISO (MaMISO) limit, we obtain deterministic approximations of the signal and interference plus noise powers at the receivers, which are tight as the number of antennas and number of users M, K → ∞ at fixed ratio. Simulation results exhibit the correctness of the large system results and the performance superiority of optimal BF designs based on both the MaMISO limit of the EWSR and using Linear Minimum Mean Squared Error (LMMSE) channel estimates.
Heterogeneous cellular networks (HCNs) constitute a necessary step in the evolution of cellular networks. In this paper, we apply the signal-to-interference ratio (SIR) meta distribution framework for a refined SIR performance analysis of HCNs, focusing on K-tier heterogeneous cellular networks based on the homogeneous independent Poisson point process (HIP) model, with range expansion bias (offloading bias) in each tier. Expressions for the b-th moment of the conditional success probability for both the entire network and each tier are derived, based on which the exact meta distributions and the beta approximations are evaluated and compared. Key performance metrics including the mean success probability, the variance of the conditional success probability, the mean local delay and the asymptotic SIR gains of each tier are obtained. The results show that the biases are detrimental to the overall mean success probability of the whole network and that the b-th moment curve (versus the SIR threshold) of the conditional success probability of each tier can be excellently approximated by the horizontal shifted versions of the first moment curve of the single-tier PPP network. We also provide lower bounds for the region of the active probabilities of the base stations to keep the mean local delay of each tier finite.
Long Term Evolution (LTE) frequency bands are being considered for use in 5G due to their coverage advantage. However, LTE usage is expected to be non-negligible for a while, so the amount of spectrum resources for it should be carefully estimated. In this letter, we propose a criterion for evaluating the adequacy of LTE spectrum resources in terms of edge user Quality of Service (QoS). For resource allocation to the edge user, we consider cooperative schemes that reflects the joint resource utilization of the associated and neighbor base stations. Based on stochastic geometry model, we derive semi-closed expressions of lower and upper bounds for the ergodic spectral efficiency at the edge user and devise a QoS satisfaction evaluation method through the decision region determined from them. Numerical analysis validates our derivations and shows the effect of joint resource utilization and cooperative allocations on QoS satisfaction.
This paper proposes an ultra-reliable device-centric uplink (URDC-UL) communication scheme for airborne networks. In particular, base stations (BSs) are mounted on unmanned aerial vehicles (UAVs) that travel to schedule UL transmissions and collect data from devices. To attain an ultra-reliable unified device-centric performance, the UL connection is established when the UAV-BS is hovering at the nearest possible distance from the scheduled device. The performance of the proposed URDC-UL scheme is benchmarked against a stationary UAV-centric uplink (SUC-UL) scheme where the devices are scheduled to communicate to UAV-BSs that are continuously hovering at static locations. Utilizing stochastic geometry and queueing theory, novel spatiotemporal mathematical models are developed, which account for the UAV-BS spatial densities, mobility, altitude, antenna directivity, ground-to-air channel, and temporal traffic, among other factors. The results demonstrate the sensitivity of the URDC-UL scheme to the ratio between hovering and traveling time. In particular, the hovering to traveling time ratio should be carefully adjusted to maximize the harvested performance gains for the URDC-UL scheme in terms of link reliability, transmission rate, energy efficiency, and delay. Exploiting the URDC-UL scheme allows IoT devices to minimize transmission power while maintaining unified reliable transmission. This preserves the device's battery and addresses a critical IoT design challenge.
This article studies data aggregation in large-scale regularly deployed Internet of Things (IoT) networks. The data granularity, in terms of information content and temporal resolution, is parameterized by the sizes of the generated packets and the average interpacket generation time. The generated data packets at the devices are aggregated through static terrestrial gateways. Universal frequency reuse is adopted across all gateways and randomized scheduling is utilized for the IoT devices associated with each gateway. Such network model finds applications in environmental sensing, precision agriculture, and geological seismic sensing to name a few. To this end, we develop a novel spatiotemporal mathematical model to characterize the interplay between data granularity, transmission reliability, and delay. The developed model accounts for several IoT design parameters, which include packet sizes, average generation duty cycle, devices and gateways spatial densities, transmission rate adaptation, power control, and antenna directivity. For tractable analysis, we propose two accurate approximations, based on the Poisson point process (PPP), to characterize the signal-to-interference-plus-noise-ratio (SINR)-based transmission reliability. For the delay analysis, we propose a phase-type arrival/departure (PH/PH/1) queueing model that accounts for packet generation, transmission scheduling, and rate-sensitive SINR-based packet departure. The developed model is utilized to obtain the optimal transmission rate for the IoT devices that minimizes delay. The numerical results delineate the joint feasibility range of packet sizes and interarrival times for data aggregation and reveal significant gains when deploying directional antennas.
Attaining maximum spectral efficiency using massive MIMO in a cell network system is a promising method to expand the efficiency in the system. By means of arranging arrays of antennas at the base stations in terms of large number of dynamic components and carry out the processing of coherent transceiver techniques are improving the system performance. General guideline is that these frameworks ought to have a significant degree of magnitudes of massive N number of antennas are required in contrast to the active premeditated L number of users. In support of this reason that the users channels are probably orthogonal to each other with N/L greater than 10. The proposed work, investigate the L* number of planned users, relies upon N and some of the parameters mentioned in this framework. The spectral efficiency articulations are determined to empower the framework level assessment with power management , reuse of random generation pilot, and user arbitrary positions. The estimation of L* number of active user in the massive N system be determined inside the close structure. But experimental simulations are utilized limited number of N to demonstrate at various interference situations, with various pilot reuse issues, and for various handling process. The transmission is capable of half of block is devoted for pilots signalling and the best possible N/L is below 10 within several situations of convenient relevance for functional significance and L* relies emphatically upon the processing scheme.
In this paper, we develop an innovative approach to quantitatively characterize the performance of ultra-dense wireless networks in a plethora of propagation environments. The proposed framework has the potential of simplifying the cumbersome procedure of analyzing the coverage probability and allowing the unification of single- and multi-antenna networks through compact analytical representations. By harnessing this key feature, we develop a novel statistical machinery to study the scaling laws of wireless networks densification considering general channel power distributions including small-scale fading and shadowing as well as associated beamforming and array gains due to the use of multiple antenna. We further formulate the relationship between network density, antenna height, antenna array seize and carrier frequency showing how the coverage probability can be maintained with ultra-densification. From a system design perspective, we show that, if multiple antenna base stations are deployed at higher frequencies, monotonically increasing the coverage probability by means of ultra-densification is possible, and this without lowering the antenna height. Simulation results substantiate performance trends leveraging network densification and antenna deployment and configuration against path loss models and signal-to-noise plus interference thresholds.
This letter applies a feedforward neural network trained in an unsupervised fashion to the problem of optimizing the transmit powers in cellular wireless systems. Both uplink and downlink are considered, with either centralized or distributed power control. Various objectives are entertained, all of them such that the problem can be cast in convex form. The performance of the proposed procedure is very satisfactory and, in terms of computational cost, the scalability with the system dimensionality is markedly superior to that of convex solvers. Moreover, the optimization relies on directly measurable channel gains, with no need for user location information.
Drawing on the notion of parallel interference cancellation, this paper formulates a one-shot linear receiver for the uplink of centralized, possibly cloud-based, radio access networks (C-RANs) operating in a cell-free fashion. This receiver exhibits substantial interference rejection abilities, yet it does not involve any matrix inversions; rather, its structure hinges on the pairwise projections of the users’ channel vectors. Its performance is markedly superior to that of matched-filter beamforming, while the computational cost is decidedly inferior to that of an MMSE filter, altogether constituting an attractive alternative in terms of performance vs cost. Furthermore, with a proper sparsification of the channel matrix that it estimates and processes, the proposed receiver can be rendered scalable in the sense of its computational cost per access point not growing with size of the network. Uplink power control is also readily accommodated.
This paper formulates linear MMSE receivers that are both network- and user-centric for the uplink of cell-free wireless networks with centralized processing. Precisely, every user’s reception involves a distinct subset of access points (APs) while every AP participates in the reception of a distinct subset of users, hence the moniker
subset MMSE receivers
. These subsets, defined on the basis of the large-scale channel gains between users and APs, capture the most relevant signal and interference contributions while disregarding those whose processing is cost-ineffective and whose associated channel estimations would incur unnecessary overheads. With that, subset reception approaches the performance of network-wide MMSE reception, offering a multiple-fold improvement over cellular and matched-filtering counterparts, while being scalable in terms of cost and channel estimation. Moreover, because the subsets overlap considerably, they can sometimes be advantageously combined and the computation of the corresponding receivers can share a hefty amount of processing.
This paper proposes respective policies for uplink power control and downlink power allocation in cell-free wireless networks. Both policies rely only on large-scale quantities and are expressed in closed form, being therefore scalable. The uplink policy, which generalizes the fractional power control employed extensively in cellular networks, features a single parameter; by adjusting this parameter, the SIR distribution experienced by the users can be compressed or expanded, trading average performance for fairness. The downlink policy dualizes the uplink solution, featuring two parameters that again allow effecting a tradeoff between average performance and fairness.
This paper presents analytical expressions for the signal-to-interference ratio (SIR) and the spectral efficiency in macrocellular networks with massive MIMO conjugate beamforming, both with a uniform and a channel-dependent power allocation. These expressions, which apply to very general network geometries, are asymptotic in the strength of the shadowing. Through Monte-Carlo simulation, we verify their accuracy for relevant network topologies and shadowing strengths. Also, since the analysis does not include pilot contamination, we further gauge through Monte-Carlo simulation the deviation that this phenomenon causes with respect to our results, and hence the scope of the analysis.
Cambridge Core - Communications and Signal Processing - Foundations of MIMO Communication - by Robert W. Heath Jr
The coded caching scheme proposed by Maddah-Ali and Niesen (MAN) critically hinges on the ability of the system to deliver a common coded multicast message from a server to all users in the system at a fixed rate, independent of the number of users. In order to apply this paradigm to a spatially distributed wireless network, it is important to make sure that such a common multicast rate does not vanish, as the number of users in the network and/or the network area increase. This paper starts from a variant of the MAN scheme successively proposed for the so-called
combination network
, where the multicast message is further encoded by a
maximum distance separable
(MDS) code, and the MDS-coded blocks are sent to multiple spatially distributed single-antenna
edge nodes
(ENs), transmitting at a fixed rate with no channel state information. The users have multiple antennas. They obtain
receiver
channel state information from the standard downlink pilots and can select to decode a desired number of EN transmissions while either nulling or treating as noise the others. The system is reminiscent of the so-called
evolved Multimedia Broadcast Multicast Service
, since the fundamental underlying transmission mechanism is
multipoint multicasting
, where each user can independently (in a user-centric manner) decide which EN to decode, without any explicit association of users with ENs. We study the performance of the proposed system when users and ENs are distributed according to homogeneous Poisson point processes in the plane, and the propagation is affected by Rayleigh fading and distance-dependent pathloss. Our analysis allows the optimization of the PHY parameters (PHY coding rate at the ENs and MDS coding rate) for given MAN scheme parameters. The proposed scheme achieves full spatial scalability in the following sense: for an extended network with arbitrary constant ratio of users per EN and area
, where
denotes the number of ENs, the system achieves a per-user delivery rate that does not vanish as
.
We consider the number of users associating with each base station in a cellular network. Extending and unifying the characterizations for certain settings available in the literature, we derive a result that is asymptotic in the strength of the shadowing, yet otherwise universally valid: it holds for every network geometry and shadowing distribution. We then illustrate how this result provides excellent representations in various classes of networks and with realistic shadowing strengths, evidencing broad applicability.
Densifying networks and deploying more antennas at each access point are two principal ways to boost the capacity of wireless networks. However, the complicated distributions of the signal power and the accumulated interference power, largely induced by various space–time processing techniques, make it highly challenging to quantitatively characterize the performance of multi-antenna networks. In this paper, using tools from stochastic geometry, a unified framework is developed for the analysis of such networks. The major results are two innovative representations of the coverage probability, which make the analysis of multi-antenna networks almost as tractable as the single-antenna case. One is expressed as an
-induced norm of a Toeplitz matrix, and the other is given in a finite sum form. With a compact representation, the former incorporates many existing analytical results on single- and multi-antenna networks as special cases and leads to tractable expressions for evaluating the coverage probability in both ad hoc and cellular networks. While the latter is more complicated for numerical evaluation, it helps analytically gain key design insights. In particular, it helps prove that the coverage probability of ad hoc networks is a monotonically decreasing convex function of the transmitter density and that there exists a peak value of the coverage improvement when increasing the number of transmit antennas. On the other hand, in multi-antenna cellular networks, it is shown that the coverage probability is independent of the transmitter density and that the outage probability decreases exponentially as the number of transmit antennas increases.
This paper presents a tutorial on stochastic geometry (SG) based analysis for cellular networks. This tutorial is distinguished by its depth with respect to wireless communication details and its focus on cellular networks. The paper starts by modeling and analyzing the baseband interference in a basic cellular network model. Then, it characterizes signal-to-interference-plus-noise-ratio (SINR) and its related performance metrics. In particular, a unified approach to conduct error probability, outage probability, and rate analysis is presented. Although the main focus of the paper is on cellular networks, the presented unified approach applies for other types of wireless networks that impose interference protection around receivers. The paper then extends the baseline unified approach to capture cellular network characteristics (e.g., frequency reuse, multiple antenna, power control, etc.). It also presents numerical examples associated with demonstrations and discussions. Finally, we point out future research directions.
This paper derives Gaussian approximation bounds for the standardized aggregate wireless interference (AWI) in the downlink of K-tier heterogeneous cellular networks when base stations in each tier are distributed over the plane according to a (possibly non-homogeneous) Poisson process. The proposed methodology is general enough to account for general bounded path-loss models and fading statistics. The deviations of the distribution of the standardized AWI from the standard normal distribution are measured in terms of the Kolmogorov-Smirnov distance. An explicit expression bounding the Kolmogorov-Smirnov distance between these two distributions is obtained as a function of a broad range of network parameters such as per-tier transmission power levels, base station locations, fading statistics and the path-loss model. A simulation study is performed to corroborate the analytical results. In particular, a good statistical match between the standardized AWI distribution and its normal approximation occurs even for moderately dense heterogeneous cellular networks. These results are expected to have important ramifications on the characterization of performance upper and lower bounds for emerging 5G network architectures.
We derive a general and closed-form result for the success probability in
multiple-antenna (MIMO) heterogeneous cellular networks (HetNets), utilizing a
novel Toeplitz matrix representation. This main result, which is equivalently
the signal-to-interference ratio (SIR) distribution, includes multiuser MIMO,
single-user MIMO and per-tier biasing for K different tiers of randomly
placed base stations (BSs). The large SIR limit of this result admits a simple
closed form that is accurate at moderate SIRs, e.g., above 5-10 dB. These
results reveal that the SIR-invariance property of SISO HetNets does not hold
for MIMO HetNets; instead the success probability may decrease as the network
density increases. We prove that the maximum success probability is achieved by
activating only one tier of BSs, while the maximum area spectral efficiency
(ASE) is achieved by activating all the BSs. This reveals a unique tradeoff
between the ASE and link reliability in multiuser MIMO HetNets. To achieve the
maximum ASE while guaranteeing a certain link reliability, we develop efficient
algorithms to find the optimal BS densities. It is shown that as the link
reliability requirement increases, more BSs and more tiers should be
deactivated.
This paper aims to validate the -Ginibre point process as a model for
the distribution of base station locations in a cellular network. The
-Ginibre is a repulsive point process in which repulsion is controlled
by the parameter. When tends to zero, the point process
converges in law towards a Poisson point process. If equals to one it
becomes a Ginibre point process. Simulations on real data collected in Paris
(France) show that base station locations can be fitted with a -Ginibre
point process. Moreover we prove that their superposition tends to a Poisson
point process as it can be seen from real data. Qualitative interpretations on
deployment strategies are derived from the model fitting of the raw data.
It has recently been observed that the SIR distributions of a variety of
cellular network models and transmission techniques look very similar in shape.
As a result, they are well approximated by a simple horizontal shift (or gain)
of the distribution of the most tractable model, the Poisson point process
(PPP). To study and explain this behavior, this paper focuses on general
single-tier network models with nearest-base station association and studies
the asymptotic gain both at 0 and at infinity.
We show that the gain at 0 is determined by the so-called mean
interference-to-signal ratio (MISR) between the PPP and the network model under
consideration, while the gain at infinity is determined by the expected
fading-to-interference ratio (EFIR).
The analysis of the MISR is based on a novel type of point process, the
so-called relative distance process, which is a one-dimensional point process
on the unit interval [0,1] that fully determines the SIR. A comparison of the
gains at 0 and infinity shows that the gain at 0 indeed provides an excellent
approximation for the entire SIR distribution. Moreover, the gain is mostly a
function of the network geometry and barely depends on the path loss exponent
and the fading. The results are illustrated using several examples of repulsive
point processes.
Capitalizing on the analytical potency of stochastic geometry and on some new ideas to model intercell interference, this paper presents analytical expressions that enable quantifying the spectral efficiency of interference alignment (IA) in cellular networks without the need for simulation. From these expressions, the benefits of IA are characterized. Even under favorable assumptions, IA is found to be beneficial only in very specific and relatively infrequent network situations, and a blanket utilization of IA is found to be altogether detrimental. Applied only in the appropriate situations, IA does bring about benefits that are significant for the users involved but relatively small in terms of average spectral efficiency for the entire system.
We model and analyze heterogeneous cellular networks with multiple antenna BSs (multi-antenna HetNets) with K classes or tiers of base stations (BSs), which may differ in terms of transmit power, deployment density, number of transmit antennas, number of users served, transmission scheme, and path loss exponent. We show that the cell selection rules in multi-antenna HetNets may differ significantly from the single-antenna HetNets due to the possible differences in multi-antenna transmission schemes across tiers. While it is challenging to derive exact cell selection rules even for maximizing signal-to-interference-plus-noise-ratio (SINR) at the receiver, we show that adding an appropriately chosen tier-dependent cell selection bias in the received power yields a close approximation. Assuming arbitrary selection bias for each tier, simple expressions for downlink coverage and rate are derived. For coverage maximization, the required selection bias for each tier is given in closed form. Due to this connection with biasing, multi-antenna HetNets may balance load more naturally across tiers in certain regimes compared to single-antenna HetNets, where a large cell selection bias is often needed to offload traffic to small cells.
We study multiple-input multiple-output (MIMO) based downlink heterogeneous
cellular network (HetNets) with joint transmit-receive diversity using
orthogonal space-time block codes at the base stations (BSs) and maximal-ratio
combining (MRC) at the users. MIMO diversity with MRC is especially appealing
in cellular networks due to the relatively low hardware complexity at both the
BS and user device. Using stochastic geometry, we develop a tractable
stochastic model for analyzing such HetNets taking into account the irregular
and multi-tier BS deployment. We derive the coverage probability for both
interference-blind (IB) and interference-aware (IA) MRC as a function of the
relevant tier-specific system parameters such as BS density and power, path
loss law, and number of transmit (Tx) antennas. Important insights arising from
our analysis for typical HetNets are for instance: (i) IA-MRC becomes less
favorable than IB-MRC with Tx diversity due to the smaller interference
variance and increased interference correlation across Rx antennas; (ii)
ignoring spatial interference correlation significantly overestimates the
performance of IA-MRC; (iii) for small number of Rx antennas, selection
combining may offer a better complexity-performance trade-off than MRC.
We introduce a simple yet powerful and versatile analytical framework to
approximate the SIR distribution in the downlink of cellular systems. It is
based on the mean interference-to-signal ratio and yields the horizontal gap
(SIR gain) between the SIR distribution in question and a reference SIR
distribution. As applications, we determine the SIR gain for base station
silencing, cooperation, and lattice deployment over a baseline architecture
that is based on a Poisson deployment of base stations and strongest-base
station association. The applications demonstrate that the proposed approach
unifies several recent results and provides a convenient framework for the
analysis and comparison of future network architectures and transmission
schemes, including amorphous networks where a user is served by multiple base
stations and, consequently, (hard) cell association becomes obsolete.
What will 5G be? What it will not be is an incremental advance on 4G. The
previous four generations of cellular technology have each been a major
paradigm shift that has broken backwards compatibility. And indeed, 5G will
need to be a paradigm shift that includes very high carrier frequencies with
massive bandwidths, extreme base station and device densities and unprecedented
numbers of antennas. But unlike the previous four generations, it will also be
highly integrative: tying any new 5G air interface and spectrum together with
LTE and WiFi to provide universal high-rate coverage and a seamless user
experience. To support this, the core network will also have to reach
unprecedented levels of flexibility and intelligence, spectrum regulation will
need to be rethought and improved, and energy and cost efficiencies will become
even more critical considerations. This paper discusses all of these topics,
identifying key challenges for future research and preliminary 5G
standardization activities, while providing a comprehensive overview of the
current literature, and in particular of the papers appearing in this special
issue.
In cellular network models, the base stations are usually assumed to form a
lattice or a Poisson point process (PPP). In reality, however, they are
deployed neither fully regularly nor completely randomly. Accordingly, in this
paper, we consider the very general class of motion-invariant models and
analyze the behavior of the outage probability (the probability that the
signal-to-interference-plus-noise-ratio (SINR) is smaller than a threshold) as
the threshold goes to zero. We show that, remarkably, the slope of the outage
probability (in dB) as a function of the threshold (also in dB) is the same for
essentially all motion-invariant point processes. The slope merely depends on
the fading statistics. Using this result, we introduce the notion of the
asymptotic deployment gain (ADG), which characterizes the horizontal gap
between the success probabilities of the PPP and another point process in the
high-reliability regime (where the success probability is near 1). To
demonstrate the usefulness of the ADG for the characterization of the SINR
distribution, we investigate the outage probabilities and the ADGs for
different point processes and fading statistics by simulations.
Cellular networks are in a major transition from a carefully planned set of large tower-mounted base-stations (BSs) to an irregular deployment of heterogeneous infrastructure elements that often additionally includes micro, pico, and femtocells, as well as distributed antennas. In this paper, we develop a tractable, flexible, and accurate model for a downlink heterogeneous cellular network (HCN) consisting of K tiers of randomly located BSs, where each tier may differ in terms of average transmit power, supported data rate and BS density. Assuming a mobile user connects to the strongest candidate BS, the resulting Signal-to-Interference-plus-Noise-Ratio (SINR) is greater than 1 when in coverage, Rayleigh fading, we derive an expression for the probability of coverage (equivalently outage) over the entire network under both open and closed access, which assumes a strikingly simple closed-form in the high SINR regime and is accurate down to -4 dB even under weaker assumptions. For external validation, we compare against an actual LTE network (for tier 1) with the other K-1 tiers being modeled as independent Poisson Point Processes. In this case as well, our model is accurate to within 1-2 dB. We also derive the average rate achieved by a randomly located mobile and the average load on each tier of BSs. One interesting observation for interference-limited open access networks is that at a given sinr, adding more tiers and/or BSs neither increases nor decreases the probability of coverage or outage when all the tiers have the same target-SINR.
Cellular networks are usually modeled by placing the base stations on a grid, with mobile users either randomly scattered or placed deterministically. These models have been used extensively but suffer from being both highly idealized and not very tractable, so complex system-level simulations are used to evaluate coverage/outage probability and rate. More tractable models have long been desirable. We develop new general models for the multi-cell signal-to-interference-plus-noise ratio (SINR) using stochastic geometry. Under very general assumptions, the resulting expressions for the downlink SINR CCDF (equivalent to the coverage probability) involve quickly computable integrals, and in some practical special cases can be simplified to common integrals (e.g., the Q-function) or even to simple closed-form expressions. We also derive the mean rate, and then the coverage gain (and mean rate loss) from static frequency reuse. We compare our coverage predictions to the grid model and an actual base station deployment, and observe that the proposed model is pessimistic (a lower bound on coverage) whereas the grid model is optimistic, and that both are about equally accurate. In addition to being more tractable, the proposed model may better capture the increasingly opportunistic and dense placement of base stations in future networks.
Inter-cell interference coordination (ICIC) and intra-cell diversity (ICD)
play important roles in improving cellular downlink coverage. Modeling cellular
base stations (BSs) as a homogeneous Poisson point process (PPP), this paper
provides explicit finite-integral expressions for the coverage probability with
ICIC and ICD, taking into account the temporal/spectral correlation of the
signal and interference. In addition, we show that in the high-reliability
regime, where the user outage probability goes to zero, ICIC and ICD affect the
network coverage in drastically different ways: ICD can provide order gain
while ICIC only offers linear gain. In the high-spectral efficiency regime
where the SIR threshold goes to infinity, the order difference in the coverage
probability does not exist, however the linear difference makes ICIC a better
scheme than ICD for realistic path loss exponents. Consequently, depending on
the SIR requirements, different combinations of ICIC and ICD optimize the
coverage probability.
The spatial structure of transmitters in wireless networks plays a key role
in evaluating the mutual interference and hence the performance. Although the
Poisson point process (PPP) has been widely used to model the spatial
configuration of wireless networks, it is not suitable for networks with
repulsion. The Ginibre point process (GPP) is one of the main examples of
determinantal point processes that can be used to model random phenomena where
repulsion is observed. Considering the accuracy, tractability and
practicability tradeoffs, we introduce and promote the -GPP, an
intermediate class between the PPP and the GPP, as a model for wireless
networks when the nodes exhibit repulsion. To show that the model leads to
analytically tractable results in several cases of interest, we derive the mean
and variance of the interference using two different approaches: the Palm
measure approach and the reduced second moment approach, and then provide
approximations of the interference distribution by three known probability
density functions. Besides, to show that the model is relevant for cellular
systems, we derive the coverage probability of the typical user and also find
that the fitted -GPP can closely model the deployment of actual base
stations in terms of the coverage probability and other statistics.
New research directions will lead to fundamental changes in the design of future fifth generation (5G) cellular networks. This article describes five technologies that could lead to both architectural and component disruptive design changes: device-centric architectures, millimeter wave, massive MIMO, smarter devices, and native support for machine-to-machine communications. The key ideas for each technology are described, along with their potential impact on 5G and the research challenges that remain.
This paper presents a tractable model for analyzing non-coherent
joint-transmission base station (BS) cooperation, taking into account the
irregular BS deployment typically encountered in practice. Besides
cellular-network specific aspects such as BS density, channel fading, average
path loss and interference, the model also captures relevant cooperation
mechanisms including user-centric BS clustering and channel-dependent
scheduling. The locations of all BSs are modeled by a Poisson point process.
Using tools from stochastic geometry, the signal-to-interference-plus-noise
ratio (SINR) distribution with cooperation is precisely characterized in a
generality-preserving form. The result is then applied to practical design
problems of recent interest. We find that increasing the network-wide BS
density improves the SINR, while the gains increase with the path loss
exponent. For pilot-based channel estimation, the averaged spectral efficiency
saturates at cluster sizes of around 7 BSs for typical values, irrespective of
backhaul quality. Finally, it is shown that intra-cluster frequency reuse is
favorable in moderately-loaded cells with generous triggering of a
joint-transmission, while intra-cluster coordinated scheduling may be better in
lightly-loaded cells with conservative triggering of a joint transmission.
In this paper, we examine the benefits of multiple antenna communication in random wireless networks, the topology of which is modeled by stochastic geometry. The setting is that of the Poisson bipolar model introduced in [1], which is a natural model for ad-hoc and device-to-device (D2D) networks. The primary finding is that, with knowledge of channel state information between a receiver and its associated transmitter, by zero-forcing successive interference cancellation, and for appropriate antenna configurations, the ergodic spectral efficiency can be made to scale linearly with both 1) the minimum of the number of transmit and receive antennas, 2) the density of nodes and 3) the path-loss exponent. This linear gain is achieved by using the transmit antennas to send multiple data streams (e.g. through an open-loop transmission method) and by exploiting the receive antennas to cancel interference. Furthermore, when a receiver is able to learn channel state information from a certain number of near interferers, higher scaling gains can be achieved when using a successive interference cancellation method. A major implication of the derived scaling laws is that spatial multiplexing transmission methods are essential for obtaining better and eventually optimal scaling laws in multiple antenna random wireless networks. Simulation results support this analysis.
One of the principal underlying assumptions of current approaches to the analysis of heterogeneous cellular networks (HetNets) with random spatial models is the uniform distribution of users independent of the base station (BS) locations. This assumption is not quite accurate, especially for user-centric capacity-driven small cell deployments where low-power BSs are deployed in the areas of high user density, thus inducing a natural correlation in the BS and user locations. In order to capture this correlation, we enrich the existing K-tier Poisson Point Process (PPP) HetNet model by considering user locations as Poisson Cluster Process (PCP) with the BSs at the cluster centers. In particular, we provide the formal analysis of the downlink coverage probability in terms of a general density functions describing the locations of users around the BSs. The derived results are specialized for two cases of interest: (i) Thomas cluster process, where the locations of the users around BSs are Gaussian distributed, and (ii) Mat\'ern cluster process, where the users are uniformly distributed inside a disc of a given radius. Tight closed-form bounds for the coverage probability in these two cases are also derived. Our results demonstrate that the coverage probability decreases as the size of user clusters around BSs increases, ultimately collapsing to the result obtained under the assumption of PPP distribution of users independent of the BS locations when the cluster size goes to infinity. Using these results, we also handle mixed user distributions consisting of two types of users: (i) uniformly distributed, and (ii) clustered around certain tiers.
We consider the point process of signal strengths emitted from transmitters in a wireless network and observed at a fixed position. In our model, transmitters are placed deterministically or randomly according to a hard core or Poisson point process and signals are subjected to power law path loss and random propagation effects that may be correlated between transmitters. We provide bounds on the distance between the point process of signal strengths and a Poisson process with the same mean measure, assuming correlated log-normal shadowing. For "strong shadowing" and moderate correlations, we find that the signal strengths are close to a Poisson process, generalizing a recently shown analogous result for independent shadowing.
This paper presents a unified mathematical paradigm, based on stochastic geometry, for downlink cellular networks with multiple-input-multiple-output (MIMO) base stations (BSs). The developed paradigm accounts for signal retransmission upon decoding errors, in which the temporal correlation among the signal-to-interference plus-noise-ratio (SINR) of the original and retransmitted signals is captured. In addition to modeling the effect of retransmission on the network performance, the developed mathematical model presents twofold analysis unification for MIMO cellular networks literature. First, it integrates the tangible decoding error probability and the abstracted (i.e., modulation scheme and receiver type agnostic) outage probability analysis, which are largely disjoint in the literature. Second, it unifies the analysis for different MIMO configurations. The unified MIMO analysis is achieved by abstracting unnecessary information conveyed within the interfering signals by Gaussian signaling approximation along with an equivalent SISO representation for the per-data stream SINR in MIMO cellular networks. We show that the proposed unification simplifies the analysis without sacrificing the model accuracy. To this end, we discuss the diversity-multiplexing tradeoff imposed by different MIMO schemes and shed light on the diversity loss due to the temporal correlation among the SINRs of the original and retransmitted signals. Finally, several design insights are highlighted.
This paper combines Poisson Cluster Process (PCP) with a Poisson Hole Process (PHP) to develop a new spatial model for an inband device-to-device (D2D) communications network, where D2D and cellular transmissions share the same spectrum. The locations of the devices engaging in D2D communications are modeled by a modified Thomas cluster process in which the cluster centers are modeled by a PHP instead of more popular homogeneous Poisson Point Process (PPP). While the clusters capture the inherent proximity in the devices engaging in D2D communications, the holes model exclusion zones where D2D communication is prohibited in order to protect cellular transmissions. For this setup, we characterize network performance in terms of coverage probability and area spectral efficiency.
In this paper, mathematical frameworks for system-level analysis and design of uplink heterogeneous cellular networks with multiple antennas at the base station (BS) are introduced. Maximum ratio combining (MRC) and optimum combining (OC) at the BSs are studied and compared. A generalized cell association criterion and fractional power control scheme are considered. The locations of all tiers of BSs are modeled as points of homogeneous and independent Poisson point processes. With the aid of stochastic geometry, coverage probability and average rate are formulated in integral but mathematically and computationally tractable expressions. Based on them, performance trends for small-and large-scale multiple-antenna BSs are discussed. Coverage and rate are shown to highly depend on several parameters, including the path-loss exponent, the fractional power control compensation factor, and the maximum transmit power of the mobile terminals. The gain of OC compared with MRC is proved to increase, if a more aggressive power control is used and if the number of BS antennas increases but is finite. For the same number of BS antennas, OC is shown to reach the noise-limited asymptote faster than MRC. All findings are validated via Monte Carlo simulations.
Explicit derivation of interferences in hexagonal wireless networks has been widely considered intractable and requires extensive computations with system level simulations. In this paper, we fundamentally tackle this problem and explicitly evaluate the downlink interference-to-signal ratio (ISR) for any mobile location m in a hexagonal wireless network, whether composed of omni-directional or tri-sectorized sites. The explicit formula of ISR is a very convergent series on m and involves the use of Gauss hypergeometric and Hurwitz Riemann zeta functions. Besides, we establish simple identities that well approximate this convergent series and turn out quite useful compared to other approximations in literature. The derived expression of ISR is easily extended to any frequency reuse pattern. Moreover, it is also exploited in the derivation of an explicit form of SINR distribution for any arbitrary distribution of mobile user locations, reflecting the spatial traffic density in the network. Knowing explicitly about interferences and SINR distribution is very useful information in capacity and coverage planning of wireless cellular networks and particularly for macro-cells' layer that forms almost a regular point pattern.
A model of cellular networks where the base station locations constitute a Poisson point process and each base station is equipped with three sectorial antennas is proposed. This model permits studying the spatial distribution of the signal-to-interference-and-noise ratio (SINR) in the downlink. In particular, this distribution is shown to be insensitive to the distribution of antenna azimuths. Moreover, the effect of horizontal sectorization is shown to be equivalent to that of shadowing. Assuming ideal vertical antenna pattern, an explicit expression of the Laplace transform of the inverse of SINR is given. The model is validated by comparing its results to measurements in an operational network. It is observed numerically that, in the case of dense urban regions where interference is preponderant, one may neglect the effect of the vertical sectorization when calculating the distribution of the SINR, which provides considerable tractability. Combined with queuing theory results, the SINR's distribution permits to express the user's quality of service as function of the traffic demand. This permits in particular to operators to predict the required investments to face the continual increase of traffic demand.
Due to its tractability, a multitier model of mutually independent Poisson point processes (PPPs) for heterogeneous cellular networks (HCNs) has recently been attracting much attention. However, in reality, the locations of the BSs, within each tier and across tiers, are not fully independent. Accordingly, in this paper, we propose two HCN models with inter-tier dependence (Case 1) and intra-tier dependence (Case 2), respectively. In Case 1, the macro-base station (MBS) and the pico-base station (PBS) deployments follow a Poisson point process (PPP) and a Poisson hole process (PHP), respectively. Under this setup and conditioning on a fixed serving distance (distance between a user and its nearest serving BS), we derive bounds on the outage probabilities of both macro and pico users. We also use a fitted Poisson cluster process to approximate the PHP, which is shown to provide a good approximation of the interference and outage statistics. In Case 2, the MBSs and the PBSs follow a PPP and an independent Matern cluster process, respectively. Explicit expressions of the interference and the outage probability are derived first for fixed serving distance and second with random distance, and we derive the outage performance, the per-user capacity, and the area spectral efficiency (ASE) for both cases. It turns out that the proposed Case 2 model is a more appropriate and accurate model for a HCN with hotspot regions than the multitier independent PPP model since the latter underestimates some key performance metrics, such as the per-user capacity and the ASE, by a factor of 1.5 to 2. Overall, the two models proposed provide good tradeoffs between the accuracy, tractability, and practicability.
The equivalent-in-distribution (EiD)-based approach to the analysis of single-input-single-output (SISO) cellular networks for transmission over Rayleigh fading channels has recently been introduced [1]. Its rationale relies upon formulating the aggregate other-cell interference in terms of an infinite summation of independent and conditionally distributed Gaussian random variables (RVs). This approach leads to exact integral expressions of the error probability for arbitrary bi-dimensional modulations. In this paper, the EiD-based approach is generalized to the performance analysis of multiple-input-multiple-output (MIMO) cellular networks for transmission over Rayleigh fading channels. The proposed mathematical formulation allows us to study a large number of MIMO arrangements, including receive-diversity, spatial-multiplexing, orthogonal space-time block coding, zero-forcing reception and zero-forcing precoding. Depending on the MIMO setup, either exact or approximate integral expressions of the error probability are provided. In the presence of other-cell interference and noise, the error probability is formulated in terms of a two-fold integral. In interference-limited cellular networks, the mathematical framework simplifies to a single integral expression. As a byproduct, the proposed approach enables us to study SISO cellular networks for transmission over Nakagami- m fading channels. The mathematical analysis is substantiated with the aid of extensive Monte Carlo simulations.
We consider the point process of signal strengths from transmitters in a
wireless network observed from a fixed position under models with general
signal path loss and random propagation effects. We show via coupling arguments
that under general conditions this point process of signal strengths can be
well-approximated by an inhomogeneous Poisson or a Cox point processes on the
positive real line. We also provide some bounds on the total variation distance
between the laws of these point processes and both Poisson and Cox point
processes. Under appropriate conditions, these results support the use of a
spatial Poisson point process for the underlying positioning of transmitters in
models of wireless networks, even if in reality the positioning does not appear
Poisson. We apply the results to a number of models with popular choices for
positioning of transmitters, path loss functions, and distributions of
propagation effects.
Geographic locations of cellular base stations sometimes can be well fitted
with spatial homogeneous Poisson point processes. In this paper we make a
complementary observation: In the presence of the log-normal shadowing of
sufficiently high variance, the statistics of the propagation loss of a single
user with respect to different network stations are invariant with respect to
their geographic positioning, whether regular or not, for a wide class of
empirically homogeneous networks. Even in perfectly hexagonal case they appear
as though they were realized in a Poisson network model, i.e., form an
inhomogeneous Poisson point process on the positive half-line with a power-law
density characterized by the path-loss exponent. At the same time, the
conditional distances to the corresponding base stations become independent and
log-normally distributed, which can be seen as a decoupling between the real
and model geometry. The result applies also to Suzuki (Rayleigh-log-normal)
propagation model. We use Kolmogorov-Smirnov test to empirically study the
quality of the Poisson approximation and use it to build a linear-regression
method for the statistical estimation of the value of the path-loss exponent.
This paper presents an analytical framework that enables characterizing
analytically the spectral efficiency achievable by D2D (device-to-device)
communication links integrated within a cellular network. This framework is
based on a stochastic geometry formulation with a novel approach to the
modeling and spatial averaging of interference, which facilitates obtaining
compact expressions, and with the added possibility of incorporating exclusion
regions to protect cellular users from excessive interference from active D2D
transmitters. To illustrate the potential of the framework, a number of
examples are provided. These examples confirm the hefty potential of D2D
communication in situations of strong traffic locality as well as the
effectiveness of properly sized exclusion regions.
In this Letter, the Equivalent-in-Distribution (EiD)-based approach to the analysis of cellular networks is introduced. It is based upon finding EiD representations of the aggregate other-cell interference of cellular networks, which lead to tractable and exact mathematical formulations of the Average Symbol Error Probability (ASEP) for arbitrary bi-dimensional modulations. As a byproduct, a new lemma is introduced, which provides a single-integral expression of the ASEP in terms of the Complementary Cumulative Distribution Function (CCDF) of the Signal-to-Interference-plus-Noise-Ratio (SINR).
An almost ubiquitous assumption made in the stochastic-analytic approach to study of the quality of user-service in cellular networks is Poisson distribution of base stations, often completed by some specific assumption regarding the distribution of the fading (e.g. Rayleigh). The former (Poisson) assumption is usually (vaguely) justified in the context of cellular networks, by various irregularities in the real placement of base stations, which ideally should form a lattice (e.g. hexagonal) pattern. In the first part of this paper we provide a different and rigorous argument justifying the Poisson assumption under sufficiently strong lognormal shadowing observed in the network, in the evaluation of a natural class of the typical-user service-characteristics (including path-loss, interference, signal-to-interference ratio, spectral efficiency). Namely, we present a Poisson-convergence result for a broad range of stationary (including lattice) networks subject to log-normal shadowing of increasing variance. We show also for the Poisson model that the distribution of all these typical-user service characteristics does not depend on the particular form of the additional fading distribution. Our approach involves a mapping of 2D network model to 1D image of it “perceived” by the typical user. For this image we prove our Poisson convergence result and the invariance of the Poisson limit with respect to the distribution of the additional shadowing or fading. Moreover, in the second part of the paper we present some new results for Poisson model allowing one to calculate the distribution function of the SINR in its whole domain. We use them to study and optimize the mean energy efficiency in cellular networks.
For more than three decades, stochastic geometry has been used to model large-scale ad hoc wireless networks, and it has succeeded to develop tractable models to characterize and better understand the performance of these networks. Recently, stochastic geometry models have been shown to provide tractable yet accurate performance bounds for multi-tier and cognitive cellular wireless networks. Given the need for interference characterization in multi-tier cellular networks, stochastic geometry models provide high potential to simplify their modeling and provide insights into their design. Hence, a new research area dealing with the modeling and analysis of multi-tier and cognitive cellular wireless networks is increasingly attracting the attention of the research community. In this article, we present a comprehensive survey on the literature related to stochastic geometry models for single-tier as well as multi-tier and cognitive cellular wireless networks. A taxonomy based on the target network model, the point process used, and the performance evaluation technique is also presented. To conclude, we discuss the open research challenges and future research directions.
The spatial structure of base stations (BSs) in cellular networks plays a key role in evaluating the downlink performance. In this paper, different spatial stochastic models (the Poisson point process (PPP), the Poisson hard-core process (PHCP), the Strauss process (SP), and the perturbed triangular lattice) are used to model the structure by fitting them to the locations of BSs in real cellular networks obtained from a public database. We provide two general approaches for fitting. One is fitting by the method of maximum pseudolikelihood. As for the fitted models, it is not sufficient to distinguish them conclusively by some classical statistics. We propose the coverage probability as the criterion for the goodness-of-fit. In terms of coverage, the SP provides a better fit than the PPP and the PHCP. The other approach is fitting by the method of minimum contrast that minimizes the average squared error of the coverage probability. This way, fitted models are obtained whose coverage performance matches that of the given data set very accurately. Furthermore, we introduce a novel metric, the deployment gain, and we demonstrate how it can be used to estimate the coverage performance and average rate achieved by a data set.
The purpose of this article is to examine some of the models commonly used to represent fading as well as the information-theoretic metrics most commonly used to evaluate performance over those models. We raise the question of whether these models and metrics remain meaningful in light of the advances that wireless systems have undergone over the last two decades. A number of weaknesses are pointed out, and ideas on possible fixes are put forth. Some of the identified weaknesses have to do with models that, over time, have become grossly inadequate. Other weaknesses have to do with changes in the operating conditions of wireless systems, and others with the coarse and asymptotic nature of some of the most popular performance metrics ("diversity" and "multiplexing").
Pushing data traffic from cellular to WiFi is an example of inter radio access technology (RAT) offloading. While this clearly alleviates congestion on the over-loaded cellular network, the ultimate potential of such offloading and its effect on overall system performance is not well understood. To address this, we develop a general and tractable model that consists of M different RATs, each deploying up to K different tiers of access points (APs), where each tier differs in transmit power, path loss exponent, deployment density and bandwidth. Each class of APs is modeled as an independent Poisson point process (PPP), with mobile user locations modeled as another independent PPP, all channels further consisting of i.i.d. Rayleigh fading. The distribution of rate over the entire network is then derived for a weighted association strategy, where such weights can be tuned to optimize a particular objective. We show that the optimum fraction of traffic offloaded to maximize SINR coverage is not in general the same as the one that maximizes rate coverage, defined as the fraction of users achieving a given rate.
We develop a general downlink model for multi-antenna heterogeneous cellular networks (HetNets), where base stations (BSs) across tiers may differ in terms of transmit power, target signal-to-interference-ratio (SIR), deployment density, number of transmit antennas and the type of multi-antenna transmission. In particular, we consider and compare space division multiple access (SDMA), single user beamforming (SU-BF), and baseline single-input single-output (SISO) transmission. For this general model, the main contributions are: (i) ordering results for both coverage probability and per user rate in closed form for any BS distribution for the three considered techniques, using novel tools from stochastic orders, (ii) upper bounds on the coverage probability assuming a Poisson BS distribution, and (iii) a comparison of the area spectral efficiency (ASE). The analysis concretely demonstrates, for example, that for a given total number of transmit antennas in the network, it is preferable to spread them across many single-antenna BSs vs. fewer multi-antenna BSs. Another observation is that SU-BF provides higher coverage and per user data rate than SDMA, but SDMA is in some cases better in terms of ASE.
Cooperation is viewed as a key ingredient for interference management in wireless networks. This paper shows that cooperation has fundamental limitations. First, it is established that in systems that rely on pilot-assisted channel estimation, the spectral efficiency is upper-bounded by a quantity that does not depend on the transmit powers; in this framework, cooperation is possible only within clusters of limited size, which are subject to out-of-cluster interference whose power scales with that of the in-cluster signals. Second, an upper bound is also shown to exist if the cooperation extends to an entire (large) system operating as a single cluster; here, pilot-assisted transmission is necessarily transcended. Altogether, it is concluded that cooperation cannot in general change an interference-limited network to a noise-limited one. Consequently, the existing literature that routinely assumes that the high-power spectral efficiency scales with the log-scale transmit power provides only a partial characterization. The complete characterization proposed in this paper subdivides the high-power regime into a degree-of-freedom regime, where the scaling with the log-scale transmit power holds approximately, and a saturation regime, where the spectral efficiency hits a ceiling that is independent of the power. Using a cellular system as an example, it is demonstrated that the spectral efficiency saturates at power levels of operational relevance.
Based on a stationary Poisson point process, a wireless network model with
random propagation effects (shadowing and/or fading) is considered in order to
examine the process formed by the signal-to-interference-plus-noise ratio
(SINR) values experienced by a typical user with respect to all base stations
in the down-link channel. This SINR process is completely characterized by
deriving its factorial moment measures, which involve numerically tractable,
explicit integral expressions. This novel framework naturally leads to
expressions for the k-coverage probability, including the case of random SINR
threshold values considered in multi-tier network models. While the k-coverage
probabilities correspond to the marginal distributions of the order statistics
of the SINR process, a more general relation is presented connecting the
factorial moment measures of the SINR process to the joint densities of these
order statistics. This gives a way for calculating exact values of the coverage
probabilities arising in a general scenario of signal combination and
interference cancellation between base stations. The presented framework
consisting of mathematical representations of SINR characteristics with respect
to the factorial moment measures holds for the whole domain of SINR and is
amenable to considerable model extension.
In this book modern algorithmic techniques for summation, most of which have been introduced within the last decade, are developed and carefully implemented in the computer algebra system Maple. The algorithms of Gosper, Zeilberger and Petkovsek on hypergeometric summation and recurrence equations and their q-analogues are covered, and similar algorithms on differential equations are considered. An equivalent theory of hyperexponential integration due to Almkvist and Zeilberger completes the book.
The combination of all results considered gives work with orthogonal polynomials and (hypergeometric type) special functions a solid algorithmic foundation. Hence, many examples from this very active field are given.
Consider a cognitive radio network with two types of users: primary users (PUs) and cognitive users (CUs), whose locations follow two independent Poisson point processes. The cognitive users follow the policy that a cognitive transmitter is active only when it is outside the primary user exclusion regions. We found that under this setup the active cognitive users form a point process called the Poisson hole process. Due to the interaction between the primary users and the cognitive users through exclusion regions, an exact calculation of the interference and the outage probability seems unfeasible. Instead, two different approaches are taken to tackle this problem. First, bounds for the interference (in the form of Laplace transforms) and the outage probability are derived, and second, it is shown how to use a Poisson cluster process to model the interference in this kind of network. Furthermore, the bipolar network model with different exclusion region settings is analyzed.
From the Publisher:
IEEE Press is pleased to bring back into print this definitive text and reference covering all aspects of microwave mobile systems design. Encompassing ten years of advanced research in the field, this invaluable resource reviews basic microwave theory, explains how cellular systems work, and presents useful techniques for effective systems development. The return of this classic volume should be welcomed by all those seeking the original authoritative and complete source of information on this emerging technology. An in-depth and practical guide, Microwave Mobile Communications will provide you with a solid understanding of the microwave propagation techniques essential to the design of effective cellular systems.
Recent results [1], [2] on the distribution of the downlink SINR in heterogeneous wireless networks assume that the serving base station (BS) for a given user (UE) location is either (a) the BS that is geographically nearest to the UE location [1], or (b) the one that has the highest received power at the UE location [2]. For (a), the distribution of the downlink SINR at an arbitrary UE location can be derived exactly. For (b), the best result for the cumulative distribution function (CDF) of the downlink SINR [2] is exact only for arguments that exceed unity. In this paper, we extend the results in [2] to derive an exact expression for the CDF of the downlink SINR at an arbitrary UE location in a multi-tier heterogeneous network when the serving BS is chosen according to (b). We then explore some interesting implications of the result for coverage probabilities.
Covering point process theory, random geometric graphs and coverage processes, this rigorous introduction to stochastic geometry will enable you to obtain powerful, general estimates and bounds of wireless network performance and make good design choices for future wireless architectures and protocols that efficiently manage interference effects. Practical engineering applications are integrated with mathematical theory, with an understanding of probability the only prerequisite. At the same time, stochastic geometry is connected to percolation theory and the theory of random geometric graphs and accompanied by a brief introduction to the R statistical computing language. Combining theory and hands-on analytical techniques with practical examples and exercises, this is a comprehensive guide to the spatial stochastic models essential for modelling and analysis of wireless network performance.