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Massive MIMO Forward Link Analysis for Cellular Networks
Abstract
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.
... Full-duplex massive MIMO has been considered for cellular networks, millimeter wave (mmWave) IEEE 802.11ad and 802.11ay Wi-Fi standards, and 5G New Radio in 3GPP Release 15 [6], [7]. ...
... We denote h ,(l,k) ∼ N C (0, I) as the normalized reverse link N a × 1 small-scale fading between the k-th user located in cell l and the BS in cell and h * ,(l,k) as the forward link reciprocal, assuming time division duplexing (TDD) with perfect calibration [7]. In addition, we denote g ( ,n),(l,k) ∼ N C (0, σ 2 iui ) as 1 × 1 small-scale fading between the n-th user in cell and the k-th user in cell l. ...
... Proposition 3. For channel hardening without full-duplexing (hence no co-channel interference between users) and with full-resolution ADC/DACs, we retrieve the results derived for downlink users in multicell massive MIMO systems in [7]. ...
... We denote h ℓ,(l,k) ∼ N C (0, I) as the normalized reverse link N a × 1 small-scale fading between the k-th user located in cell l and the BS in cell ℓ and h * ℓ,(l,k) as the forward link reciprocal, assuming time division duplexing (TDD) with perfect calibration [18]. In addition, we denote by g (ℓ,n),(l,k) ∼ N C (0, 1) the 1 × 1 small-scale fading between the n-th user in cell ℓ and the k-th user in cell l [12]. ...
... Note that Proposition 2 entails the same result for forward link derived in [18]. ...
... System Parameters[18],[20]. ...
In this work, we consider a full-duplex (FD) massive multiple-input multiple-output (MIMO) cellular network with low resolution analog-to-digital converters (ADCs) and digital-to-analog converters (DACs). Our first contribution is to propose a unified framework for forward link analysis where matched filter precoders are applied at the FD base stations (BSs) under channel hardening. Second, we derive expressions for the signal-to-quantization-plus-interference-plus-noise ratio (SQINR) for general and special cases. Finally, we quantify effects of quantization error, pilot contamination, and full duplexing for a hexagonal cell lattice on spectral efficiency and cumulative distribution function (CDF) to show that FD outperforms half duplex (HD) in a wide variety of scenarios.
... Thanks to these benefits, massive MIMO FD has been considered for cellular networks, millimeter wave applications such as IEEE 802.11ad and 802.11ay Wi-Fi standards, and 5G New Radio (NR) in 3GPP Release 15 [4]- [6]. ...
... Theorem 1. The output SQINR of the k-th uplink user is (6). ...
... Proposition 3. Considering the channel hardening scenario, without full-duplexing (and hence no co-channel interference between users) and with full-resolution ADC/DAC, we retrieve the results derived for downlink users in multicell massive MIMO systems in [6]. ...
In this paper, we provide an analytical framework for full-duplex (FD) massive multiple-input multiple-output (MIMO) cellular networks with low resolution analog-to-digital and digital-to-analog converters (ADCs and DACs). Matched filters are employed at the FD base stations (BSs) at the transmit and receive sides. For both reverse and forward links, we derive the expressions of the signal-to-quantization-plus-interference-and-noise ratio (SQINR) for general and special cases. We further evaluate the outage probability and spectral efficiency for reverse and forward links, and quantify the effects of the quantization error, loopback self-interference and inter-user interference for cells arranged in a hexagonal lattice and Poisson Point Process (PPP) tessellations. Finally, we derive analytical expressions for spectral efficiency for asymptotic cases as well as for power scaling laws.
... Thanks to these benefits, massive MIMO FD has been considered for cellular networks, millimeter wave applications such as IEEE 802.11ad and 802.11ay Wi-Fi standards, and 5G New Radio (NR) in 3GPP Release 15 [4]- [6]. ...
... Theorem 1. The output SQINR of the k-th uplink user is (6). ...
... Proposition 3. Considering the channel hardening scenario, without full-duplexing (and hence no co-channel interference between users) and with full-resolution ADC/DAC, we retrieve the results derived for downlink users in multicell massive MIMO systems in [6]. ...
In this paper, we provide an analytical framework for full-duplex (FD) massive multiple-input multiple-output (MIMO) cellular networks with low resolution analog-to-digital and digital-to-analog converters (ADCs and DACs). Matched filters are employed at the FD base stations (BSs) at the transmit and receive sides. For both reverse and forward links, we derive the expressions of the signal-to-quantization-plus-interference-and-noise ratio (SQINR) for general and special cases. We further evaluate the outage probability and spectral efficiency for reverse and forward links, and quantify the effects of the quantization error, loopback self-interference and inter-user interference for cells arranged in a hexagonal lattice and Poisson Point Process (PPP) tessellations. Finally, we derive analytical expressions for spectral efficiency for asymptotic cases as well as for power scaling laws.
... We denote h ℓ,(l,k) ∼ N C (0, I) as the normalized reverse link N a × 1 small-scale fading between the k-th user located in cell l and the BS in cell ℓ and h * ℓ,(l,k) as the forward link reciprocal, assuming time division duplexing (TDD) with perfect calibration [18]. In addition, we denote by g (ℓ,n),(l,k) ∼ N C (0, 1) the 1 × 1 small-scale fading between the n-th user in cell ℓ and the k-th user in cell l [12]. ...
... Remark 2. Note that Proposition 2 entails the same result for forward link derived in [18]. ...
... We denote h ℓ,(l,k) ∼ N C (0, I) as the normalized reverse link N a × 1 small-scale fading between the k-th user located in cell l and the BS in cell ℓ [17]. We further denote H SI ∼ N C (0, µ 2 SI ) as the N a × N a SI channel matrix [13]. ...
... We denote h ℓ,(l,k) ∼ N C (0, I) as the normalized reverse link N a × 1 small-scale fading between the k-th user located in cell l and the BS in cell ℓ [17]. We further denote H SI ∼ N C (0, µ 2 SI ) as the N a × N a SI channel matrix [13]. ...
In this work, we consider a full-duplex (FD) massive multiple-input multiple-output (MIMO) cellular network with low-resolution analog-to-digital converters (ADCs) and digital-to-analog converter (DACs). Our first contribution is to provide a unified framework for reverse link analysis where matched filters are applied at the FD base stations (BSs) under channel hardening. Second, we derive the expressions of the signal-to-quantization-plus-interference-plus-noise ratio (SQINR) for general and special cases. Finally, we quantify effects of quantization error, pilot contamination, and full duplexing for a hexagonal cell lattice on spectral efficiency and cumulative distribution function (CDF) to show that FD outperforms half duplex (HD) in a wide variety of scenarios.
In this work, we provide a unified framework for full-duplex (FD) massive multiple-input multiple-output (MIMO) cellular networks with low-resolution analog-to-digital and digital-to-analog converters (ADCs and DACs). An objective of this work is to derive an accurate model to account for a wide variety of network irregularities and imperfections, including loopback self-interference (SI) that arises in full-duplex systems and quantization error from low-resolution data converters. Our contributions for forward and reverse links include (1) deriving signal-to-quantization-plus-interference-plus-noise ratio (SQINR) under pilot contamination, linear minimum mean square error (LMMSE) channel estimation and channel hardening; (2) deriving closed-form and approximate analytical expressions of spectral efficiency; (3) deriving asymptotic results and power scaling laws with respect to the number of quantization bits, base station antennas, and users, as well as base station and user equipment power budgets; and (4) analyzing outage probability and spectral efficiency vs. cell shape, shadowing, noise, cellular interference, pilot contamination, pilot overhead, and frequency reuse. Cell shapes include hexagonal, square, and Poisson Point Process (PPP) tessellations. In simulation, we quantify spectral and energy efficiency as well as the impact of SI power, inter-user interference, and cell shape on outage probability. We carry out the analysis for sub-7 GHz long term evolution (LTE) bands and then extend the framework to support millimeter wave (mmWave) bands.
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.
Massive multiple-input multiple-output (M-MIMO) is recognized as a promising technology for the next generation of wireless networks because of its potential to increase the spectral efficiency. In initial studies of M-MIMO, the system has been considered to be perfectly synchronized throughout the entire cells. However, perfect synchronization may be hard to attain in practice. Therefore, we study a M-MIMO system whose cells are not synchronous to each other, while transmissions in a cell are still synchronous. We analyze an asynchronous downlink M-MIMO system in terms of the coverage probability and the ergodic rate by means of the stochastic geometry tool. For comparison, we also obtain results for the corresponding synchronous system. In addition, we investigate the effect of the uplink power control and the number of pilot symbols on the downlink ergodic rate, and we observe that there is an optimal value for the number of pilot symbols maximizing the downlink ergodic rate of a cell. Our results also indicate that, compared to the synchronous system, the downlink ergodic rate is more sensitive to the uplink power control in the asynchronous mode.
Cambridge Core - Communications and Signal Processing - Foundations of MIMO Communication - by Robert W. Heath Jr
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.
Massive multiple-input multiple-output (MIMO) is one of the most promising technologies for the next generation of wireless communication networks because it has the potential to provide game-changing improvements in spectral efficiency (SE) and energy efficiency (EE). This monograph summarizes many years of research insights in a clear and self-contained way and provides the reader with the necessary knowledge and mathematical tools to carry out independent research in this area. Starting from a rigorous definition of Massive MIMO, the monograph covers the important aspects of channel estimation, SE, EE, hardware efficiency (HE), and various practical deployment considerations. From the beginning, a very general, yet tractable, canonical system model with spatial channel correlation is introduced. This model is used to realistically assess the SE and EE, and is later extended to also include the impact of hardware impairments. Owing to this rigorous modeling approach, a lot of classic "wisdom" about Massive MIMO, based on too simplistic system models, is shown to be questionable.
The spectral efficiency (SE) of cellular networks can be improved by the unprecedented array gain and spatial multiplexing offered by Massive MIMO. Since its inception, the coherent interference caused by pilot contamination has been believed to create a finite SE limit, as the number of antennas goes to infinity. In this paper, we prove that this is incorrect and an artifact from using simplistic channel models and suboptimal precoding/combining schemes. We show that with multicell MMSE precoding/combining and a tiny amount of spatial channel correlation or large-scale fading variations over the array, the SE increases without bound as the number of antennas increases, even under pilot contamination. More precisely, the result holds when the channel covariance matrices of the contaminating users are asymptotically linearly independent, which is generally the case. If also the diagonals of the covariance matrices are linearly independent, it is sufficient to know these diagonals (and not the full covariance matrices) to achieve an unlimited asymptotic capacity.
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 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 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.
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.
Assume that a multi-user multiple-input multiple-output (MIMO) system is designed from scratch to uniformly cover a given area with maximal energy efficiency (EE). What are the optimal number of antennas, active users, and transmit power? The aim of this paper is to answer this fundamental question. We consider jointly the uplink and downlink with different processing schemes at the base station and propose a new realistic power consumption model that reveals how the above parameters affect the EE. Closed-form expressions for the EE-optimal value of each parameter, when the other two are fixed, are provided for zero-forcing (ZF) processing in single-cell scenarios. These expressions prove how the parameters interact. For example, in sharp contrast to common belief, the transmit power is found to increase (not to decrease) with the number of antennas. This implies that energy-efficient systems can operate in high signal-to-noise ratio regimes in which interference-suppressing signal processing is mandatory. Numerical and analytical results show that the maximal EE is achieved by a massive MIMO setup wherein hundreds of antennas are deployed to serve a relatively large number of users using ZF processing. The numerical results show the same behavior under imperfect channel state information and in symmetric multi-cell scenarios.
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.
In a two-tier heterogeneous network (HetNet) where femto access points (FAPs) with lower transmission power coexist with macro base stations (BSs) with higher transmission power, the FAPs may suffer significant performance degradation due to inter-tier interference. Introducing cognition into the FAPs through the spectrum sensing (or carrier sensing) capability helps them avoiding severe interference from the macro BSs and enhance their performance. In this paper, we use stochastic geometry to model and analyze performance of HetNets composed of macro BSs and cognitive FAPs in a multichannel environment. The proposed model explicitly accounts for the spatial distribution of the macro BSs, FAPs, and users in a Rayleigh fading environment. We quantify the performance gain in outage probability obtained by introducing cognition into the femto-tier, provide design guidelines, and show the existence of an optimal spectrum sensing threshold for the cognitive FAPs, which depends on the HetNet parameters. We also show that looking into the overall performance of the HetNets is quite misleading in the scenarios where the majority of users are served by the macro BSs. Therefore, the performance of femto-tier needs to be explicitly accounted for and optimized.
Multi-user Multiple-Input Multiple-Output (MIMO) offers big advantages over
conventional point-to-point MIMO: it works with cheap single-antenna terminals,
a rich scattering environment is not required, and resource allocation is
simplified because every active terminal utilizes all of the time-frequency
bins. However, multi-user MIMO, as originally envisioned with roughly equal
numbers of service-antennas and terminals and frequency division duplex
operation, is not a scalable technology. Massive MIMO (also known as
"Large-Scale Antenna Systems", "Very Large MIMO", "Hyper MIMO", "Full-Dimension
MIMO" & "ARGOS") makes a clean break with current practice through the use of a
large excess of service-antennas over active terminals and time division duplex
operation. Extra antennas help by focusing energy into ever-smaller regions of
space to bring huge improvements in throughput and radiated energy efficiency.
Other benefits of massive MIMO include the extensive use of inexpensive
low-power components, reduced latency, simplification of the media access
control (MAC) layer, and robustness to intentional jamming. The anticipated
throughput depend on the propagation environment providing asymptotically
orthogonal channels to the terminals, but so far experiments have not disclosed
any limitations in this regard. While massive MIMO renders many traditional
research problems irrelevant, it uncovers entirely new problems that urgently
need attention: the challenge of making many low-cost low-precision components
that work effectively together, acquisition and synchronization for
newly-joined terminals, the exploitation of extra degrees of freedom provided
by the excess of service-antennas, reducing internal power consumption to
achieve total energy efficiency reductions, and finding new deployment
scenarios. This paper presents an overview of the massive MIMO concept and
contemporary research.
Wireless networks are fundamentally limited by the intensity of the received signals and by their interference. Since both of these quantities depend on the spatial location of the nodes, mathematical techniques have been developed in the last decade to provide communication-theoretic results accounting for the networks geometrical configuration. Often, the location of the nodes in the network can be modeled as random, following for example a Poisson point process. In this case, different techniques based on stochastic geometry and the theory of random geometric graphs -including point process theory, percolation theory, and probabilistic combinatorics-have led to results on the connectivity, the capacity, the outage probability, and other fundamental limits of wireless networks. This tutorial article surveys some of these techniques, discusses their application to model wireless networks, and presents some of the main results that have appeared in the literature. It also serves as an introduction to the field for the other papers in this special issue.
We present a simple algorithm for numerically inverting Laplace transforms. The algorithm is designed especially for probability cumulative distribution functions, but it applies to other functions as well. Since it does not seem possible to provide effective methods with simple general error bounds, we simultaneously use two different methods to confirm the accuracy. Both methods are variants of the Fourier-series method. The first, building on Dubner and Abate (Dubner, H., J. Abate. 1968. Numerical inversion of Laplace transforms by relating them to the finite Fourier cosine transform. JACM 15 115–123.) and Simon, Stroot, and Weiss (Simon, R. M., M. T. Stroot, G. H. Weiss. 1972. Numerical inversion of Laplace transforms with application to percentage labeled experiments. Comput. Biomed. Res. 6 596–607.), uses the Bromwich integral, the Poisson summation formula and Euler summation; the second, building on Jagerman (Jagerman, D. L. 1978. An inversion technique for the Laplace transform with applications. Bell System Tech. J. 57 669–710 and Jagerman, D. L. 1982. An inversion technique for the Laplace transform. Bell System Tech. J. 61 1995–2002.), uses the Post-Widder formula, the Poisson summation formula, and the Stehfest (Stehfest, H. 1970. Algorithm 368. Numerical inversion of Laplace transforms. Comm. ACM 13 479–490 (erratum: 13 624).) enhancement. The resulting program is short and the computational experience is encouraging.
INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.
This paper surveys recent advances in the area of very large MIMO systems.
With very large MIMO, we think of systems that use antenna arrays with an
order of magnitude more elements than in systems being built today, say a
hundred antennas or more. Very large MIMO entails an unprecedented number of
antennas simultaneously serving a much smaller number of terminals. The
disparity in number emerges as a desirable operating condition and a practical
one as well. The number of terminals that can be simultaneously served is
limited, not by the number of antennas, but rather by our inability to acquire
channel-state information for an unlimited number of terminals. Larger numbers
of terminals can always be accommodated by combining very large MIMO technology
with conventional time- and frequency-division multiplexing via OFDM. Very
large MIMO arrays is a new research field both in communication theory,
propagation, and electronics and represents a paradigm shift in the way of
thinking both with regards to theory, systems and implementation. The ultimate
vision of very large MIMO systems is that the antenna array would consist of
small active antenna units, plugged into an (optical) fieldbus.
Cambridge Core - Communications and Signal Processing - Foundations of MIMO Communication - by Robert W. Heath Jr
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.
In this paper, we investigate the downlink throughput performance of a massive multiple-input multiple-output (MIMO) system that employs superimposed pilots for channel estimation. The component of downlink (DL) interference that results from transmitting data alongside pilots in the uplink (UL) is shown to decrease at a rate proportional to the square root of the number of antennas at the BS when the least-squares (LS) channel estimate is employed in a matched-filter (MF) precoder. The normalized mean-squared error (NMSE) of the channel estimate is compared with the Bayesian Cramér-Rao lower bound that is derived for the system, and the former is also shown to diminish with increasing number of antennas at the base station (BS). Furthermore, we show that staggered pilots are a particular case of superimposed pilots and offer the downlink throughput of superimposed pilots while retaining the UL spectral and energy efficiency of regular pilots. We also extend the framework for designing a hybrid system, consisting of users that transmit either regular or superimposed pilots, to minimize both the UL and DL interference. The improved NMSE and DL rates of the channel estimator based on superimposed pilots are demonstrated by means of simulations.
This paper characterizes, through a stochastic geometry analysis, the increase in spectral efficiency that full-duplex transmission brings about in wireless networks. While, on isolated links, full-duplex promises a doubling of the spectral efficiency, in the context of a network this is weighted down by the corresponding rise in interference, and our characterization captures the balance of these effects.
The analysis encompasses both the forward link (FL) and the reverse link (RL) with single-user and multiuser transmissions. And, as a complement to the analysis, Monte-Carlo simulations on a Vodafone LTE field test network are also presented. In the FL, the rise in interference is found to have minor impact and a doubling in spectral efficiency can indeed be approached, especially in microcellular networks. In the RL, however, a major difficulty arises in the form of exceedingly strong interference among base stations. This renders full-duplex transmission all but unfeasible in macrocellular networks (unless major countermeasures could be implemented) and undesirable in dense microcellular networks. Only in microcells with sufficient spacing among base stations does RL full-duplex pay off. Thus, full-duplex is seen not to blend easily with densification.
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.
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.
We consider the Massive Multiple-Input Multiple-Output (MIMO) downlink with maximum-ratio and zero-forcing processing and time-division duplex (TDD) operation. To decode, the terminals must know their instantaneous effective channel gain. Conventionally, it is assumed that by virtue of channel hardening, this instantaneous gain is close to its average and hence that terminals can rely on knowledge of that average (also known as statistical channel information). However, in some propagation environments, such as keyhole channels, channel hardening does not hold. We propose a blind algorithm to estimate the effective channel gain at each user, that does not require any downlink pilots. We derive a capacity lower bound of each user for our proposed scheme, applicable to any propagation channel. Compared to the case of no downlink pilots (relying on channel hardening), and compared to training-based estimation using downlink pilots, our blind algorithm performs significantly better. The difference is especially pronounced in environments that do not offer channel hardening.
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.
Massive multiple-input multiple-output (MIMO) communication promises the full potential of MIMO communication realized through the deployment of hundreds of antennas at each base station. Accurate channel state information (CSI) is crucial for making the most of massive MIMO. In time-varying channels, however, if not periodically updated, CSI obtained at the base station becomes progressively outdated. In this paper, we formulate a training optimization problem for maximizing uplink sum-rates. The optimal training period and update interval are characterized as solutions to fixed-point equations, which depend on the normalized Doppler shift and large-scale channel coefficients. It is found that, for a sufficiently large number of antennas at each base station, the optimal training period is equal to the number of users per cell.
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.
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.
Massive multiple-input multiple-output (MIMO) wireless communications refers to the idea equipping cellular base stations (BSs) with a very large number of antennas, and has been shown to potentially allow for orders of magnitude improvement in spectral and energy efficiency using relatively simple (linear) processing. In this paper, we present a comprehensive overview of state-of-the-art research on the topic, which has recently attracted considerable attention. We begin with an information theoretic analysis to illustrate the conjectured advantages of massive MIMO, and then we address implementation issues related to channel estimation, detection and precoding schemes. We particularly focus on the potential impact of pilot contamination caused by the use of non-orthogonal pilot sequences by users in adjacent cells. We also analyze the energy efficiency achieved by massive MIMO systems, and demonstrate how the degrees of freedom provided by massive MIMO systems enable efficient single-carrier transmission. Finally, the challenges and opportunities associated with implementing massive MIMO in future wireless communications systems are discussed.
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.
We consider the uplink (UL) and downlink (DL) of non-cooperative multi-cellular time-division duplexing (TDD) systems, assuming that the number N of antennas per base station (BS) and the number K of user terminals (UTs) per cell are large. Our system model accounts for channel estimation, pilot contamination, and an arbitrary path loss and antenna correlation for each link. We derive approximations of achievable rates with several linear precoders and detectors which are proven to be asymptotically tight, but accurate for realistic system dimensions, as shown by simulations. It is known from previous work assuming uncorrelated channels, that as N→∞ while K is fixed, the system performance is limited by pilot contamination, the simplest precoders/detectors, i.e., eigenbeamforming (BF) and matched filter (MF), are optimal, and the transmit power can be made arbitrarily small. We analyze to which extent these conclusions hold in the more realistic setting where N is not extremely large compared to K. In particular, we derive how many antennas per UT are needed to achieve η% of the ultimate performance limit with infinitely many antennas and how many more antennas are needed with MF and BF to achieve the performance of minimum mean-square error (MMSE) detection and regularized zero-forcing (RZF), respectively.
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.
Euler summation is a convergence-acceleration technique which has proved very effective in Fourier-series methods for Laplace transform inversion. We present an analysis of the effect of Euler summation that explains its excellent performance. The central result is a bound on the truncation error when Euler summation is used. Our analysis supports some of the parameter-selection strategies proposed in J. Abate and W. Whitt’s recent comprehensive treatment of Fourier series methods [Queueing Syst. 10, No. 1/2, 5-87 (1992; Zbl 0749.60013)]. Specifically, we show that if our goal is to minimize the required number of terms of the Fourier series subject to the truncation error bound being no more than a specified target accuracy, then the “degree of averaging parameter” m should depend only on the desired accuracy, and not on properties of the function in question (so long as that function is sufficiently smooth). We also present a framework, here called “product smoothing”, for constructing related summation methods with desirable properties.
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.
The Signal to Interference Plus Noise Ratio (SINR) on a wireless link is an important basis for consideration of outage, capacity, and throughput in a cellular network. It is therefore important to understand the SINR distribution within such networks, and in particular heterogeneous cellular networks, since these are expected to dominate future network deployments [1]. Until recently the distribution of SINR in hetero-geneous networks was studied almost exclusively via simulation, for selected scenarios representing pre-defined arrangements [2] of users and the elements of the heterogeneous network such as macro-cells, femto-cells, etc. However, the dynamic nature of heterogeneous networks makes it difficult to design a few rep-resentative simulation scenarios from which general inferences can be drawn that apply to all deployments. In this paper, we examine the downlink of a heterogeneous cellular network made up of multiple tiers of transmitters (e.g., macro-, micro-, pico-, and femto-cells) and provide a general theoretical analysis of the distribution of the SINR at an arbitrarily-located user. Using physically realistic stochastic models for the locations of the base stations (BSs) in the tiers, we can compute the general SINR distribution in closed form. We illustrate a use of this approach for a three-tier network by calculating the probability of the user being able to camp on a macro-cell or an open-access (OA) femto-cell in the presence of Closed Subscriber Group (CSG) femto-cells. We show that this probability depends only on the relative densities and transmit powers of the macro-and femto-cells, the fraction of femto-cells operating in OA vs. Closed Subscriber Group (CSG) mode, and on the parameters of the wireless channel model. For an operator considering a femto overlay on a macro network, the parameters of the femto deployment can be selected from a set of universal curves.
A cellular base station serves a multiplicity of single-antenna terminals over the same time-frequency interval. Time-division duplex operation combined with reverse-link pilots enables the base station to estimate the reciprocal forward- and reverse-link channels. The conjugate-transpose of the channel estimates are used as a linear precoder and combiner respectively on the forward and reverse links. Propagation, unknown to both terminals and base station, comprises fast fading, log-normal shadow fading, and geometric attenuation. In the limit of an infinite number of antennas a complete multi-cellular analysis, which accounts for inter-cellular interference and the overhead and errors associated with channel-state information, yields a number of mathematically exact conclusions and points to a desirable direction towards which cellular wireless could evolve. In particular the effects of uncorrelated noise and fast fading vanish, throughput and the number of terminals are independent of the size of the cells, spectral efficiency is independent of bandwidth, and the required transmitted energy per bit vanishes. The only remaining impairment is inter-cellular interference caused by re-use of the pilot sequences in other cells (pilot contamination) which does not vanish with unlimited number of antennas.
This paper presents a simple new expression for the exact evaluation of averages of the form E [ln (1+x1+...xN/y1+...+yM+1)], where x1,..., xN, y1..., yM are arbitrary non-negative random variables, in terms of the joint moment generating functions of these random variables. Application examples are given for the ergodic capacity evaluation of some multiuser wireless communication systems which are difficult to solve by the known classical methods.
Fundamentals Massive MIMO
Stochastic geometry modeling and performance evaluation of MIMO cellular networks using the equivalent-in-distribution (EiD)-based approach
- Di Renzo
- W Lu