We provide a comprehensive overview of mathematical models and analytical techniques for millimeter wave (mmWave) cellular systems. The two fundamental physical differences from conventional sub-6-GHz cellular systems are: 1) vulnerability to blocking and 2) the need for significant directionality at the transmitter and/or receiver, which is achieved through the use of large antenna arrays of small individual elements. We overview and compare models for both of these factors, and present a baseline analytical approach based on stochastic geometry that allows the computation of the statistical distributions of the downlink signal-to-interference-plus-noise ratio (SINR) and also the per link data rate, which depends on the SINR as well as the average load. There are many implications of the models and analysis: 1) mmWave systems are significantly more noise-limited than at sub-6 GHz for most parameter configurations; 2) initial access is much more difficult in mmWave; 3) self-backhauling is more viable than in sub-6-GHz systems, which makes ultra-dense deployments more viable, but this leads to increasingly interference-limited behavior; and 4) in sharp contrast to sub-6-GHz systems cellular operators can mutually benefit by sharing their spectrum licenses despite the uncontrolled interference that results from doing so. We conclude by outlining several important extensions of the baseline model, many of which are promising avenues for future research.

Modeling and Analyzing Millimeter Wave Cellular Systems

In a device-to-device (D2D) underlaid cellular network, the uplink spectrum is reused by the D2D transmissions, causing mutual interference with the ongoing cellular transmissions. Massive MIMO is appealing in such a context as the base station's (BS's) large antenna array can nearly null the D2D-to-BS interference. The multi-user transmission in massive MIMO, however, may lead to increased cellular-to-D2D interference. This paper studies the interplay between massive MIMO and underlaid D2D networking in a multi-cell setting. We investigate cellular and D2D spectral efficiencies under both perfect and imperfect channel state information (CSI) at the receivers that employ partial zero-forcing. Compared to the case without D2D, there is a loss in cellular spectral efficiency due to D2D underlay. With perfect CSI, the loss can be completely overcome if the number of canceled D2D interfering signals is scaled with the number of BS antennas at an arbitrarily slow rate. With imperfect CSI, in addition to pilot contamination, a new asymptotic effect termed underlay contamination arises. In the non-asymptotic regime, simple analytical lower bounds are derived for both the cellular and D2D spectral efficiencies.

The Interplay Between Massive MIMO and Underlaid D2D Networking

Although the Poisson point process (PPP) has been widely used to model base station (BS) locations in cellular networks, it is an idealized model that neglects the spatial correlation among BSs. This paper proposes the use of the determinantal point process (DPP) to take into account these correlations, in particular the repulsiveness among macro BS locations. DPPs are demonstrated to be analytically tractable by leveraging several unique computational properties. Specifically, we show that the empty space function, the nearest neighbor function, the mean interference, and the signal-to-interference ratio (SIR) distribution have explicit analytical representations and can be numerically evaluated for cellular networks with DPP-configured BSs. In addition, the modeling accuracy of DPPs is investigated by fitting three DPP models to real BS location data sets from two major U.S. cities. Using hypothesis testing for various performance metrics of interest, we show that these fitted DPPs are significantly more accurate than popular choices such as the PPP and the perturbed hexagonal grid model.

Statistical Modeling and Probabilistic Analysis of Cellular Networks With Determinantal Point Processes

This paper proposes a stochastic geometry framework to analyze the signal-to-noise-and-interference ratio (SINR) and rate performance in a large-scale uplink massive multiple-input and multiple-output (MIMO) network. Based on the model, expressions are derived for spatial average SINR distributions over user and base station distributions with maximum ratio combining (MRC) and zero-forcing (ZF) receivers. We show that, using massive MIMO, the uplink SINR in certain urban marcocell scenarios is limited by interference. In the interference-limited regime, the results reveal that for MRC receivers, a superlinear (polynomial) scaling law between the number of base station antennas and scheduled users per cell preserves the uplink signal-to-interference ratio (SIR) distribution, while a linear scaling applies to ZF receivers. ZF receivers are shown to outperform MRC receivers in the SIR coverage, and the performance gap is quantified in terms of the difference in the number of antennas to achieve the same SIR distribution. Numerical results verify the analysis. It is found that the optimal compensation fraction in fractional power control to optimize rate is generally different for MRC and ZF receivers. Besides, simulations show that the scaling results derived from the proposed framework apply to the networks, where base stations are distributed according to a hexagonal lattice.

Analyzing Uplink SINR and Rate in Massive MIMO Systems Using Stochastic Geometry

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Software Defined Radio Architectures Systems And Functions

We study a multiple input multiple output (MIMO) cellular system where each base-station (BS) is equipped with a large antenna array and serves some single antenna mobile stations (MSs). With the same setup as in [1], the influence of orthogonal and non-orthogonal pilot sequences on the system performance is analytically characterized when each BS has infinitely many antennas. Using stochastic geometric modeling of the BS and MS locations, closed-form expressions are derived for the distribution of signal-to-interference-ratio (SIR) for both uplink and downlink. Moreover, they are shown to be equivalent for the orthogonal pilots case. Further, it is shown that the downlink SIR is greatly influenced by the correlations between the pilot sequences in the non-orthogonal pilots case. Finally, the mathematical tools can be used to study system performances with other general channel estimation methods and transmission-reception schemes.

https://opendl.ifip-tc6.org/db/conf/wiopt/wiopt2013/MadhusudhananLLB13.pdf

Stochastic geometric modeling and interference analysis for massive MIMO systems

Systems and methods are described for jointly optimizing communications in a multi-user network to maximize the weighted sum-rate of its data links. For example, embodiments seek to maximize overall link transmit rates, while minimizing link interference and/or noise in general multi-in multi-out (MIMO) interference networks having multiple interfering data links, each with one or more antennas at the transmitter and receiver. Novel algorithmic approaches, including an “iterative minimax” approach and a “dual link” approach, are described for solving the weighted sum-rate maximization in a manner that converges quickly and reliably on an optimal point. Some embodiments implement these approaches in a centralized transmission controller, while others implement these approaches in a distributed fashion.

Maximizing efficiency of multi-user communications networks

MIMO interference network optimization is important for increasingly crowded wireless communication networks. This paper presents a new algorithm, named Dual Link Algorithm, for weighted sum-rate maximization where the interference is efficiently managed. We consider general MIMO interference channels with Gaussian input and a total power constraint. Two of the previous state-of-the-art algorithms are the WMMSE algorithm and the polite water-filling (PWF) algorithm. The WMMSE algorithm is provably convergent, while the PWF algorithm takes the advantage of the optimal transmit signal structure and converges the fastest in most situations but is not guaranteed to converge in all situations. Therefore, it is highly desirable to design an algorithm that has the advantages of both algorithms. The proposed dual link algorithm is such an algorithm. Its fast and guaranteed convergence is important to distributed implementation and time varying channels. In addition, the technique and a scaling invariance property used in the convergence proof may find applications in other non-convex problems in communication networks.

A new algorithm for the weighted sum rate maximization in MIMO interference networks

This paper proposes a three dimensional channel model for the study of distributed MIMO communication systems, where the transmit antennas may be at different directions when viewed from the receiver. Examples of such systems include cooperative satellites and air based base stations. Both line of sight and multipath signal components are considered. A common difficulty of such channel models is that there are too many geometric parameters to produce analytical correlation functions of the antennas in order to gain insight on how the channel depends on those parameters. The difficulty is solved here by identifying a single key parameter, the projected distance, to summarize the effect of all the geometric parameters, and thus, the model is simple enough to be useful for system designers. Analytical correlation functions are provided, which is used to compare the rate of the MIMO and SIMO systems in order to demonstrate that with high probability, significant rate gain can be obtained by the distributed MIMO systems.

A 3-D Channel Model for Distributed MIMO Satellite Systems

In this paper, we propose a data-aided channel estimator, which can improve the performance of the linear minimum-mean-squared-error (LMMSE) by clustering and reinforcement-learning for multiple-input multiple-output (MI-MO) systems. For clustering-based data detection, we develop a system constrained Gaussian mixture model (SCGMM), in which the a posteriori probabilities (APPs) can be calculated by the expectation-maximization (EM) algorithm. The initial centroids of the SCGMM are sensitive to the channel estimation. To obtain robust channel estimation, we design initial pilots that can reduce the estimated error of the LMMSE. To further improve the quality of channel estimation, we propose a data-aided channel estimation algorithm, which exploits the techniques of coding and reinforcement-learning to obtain soft symbol decisions. Numerical results show that the proposed method can approach the bit-error-rate (BER) performance with perfect channel state information (CSI) in the high signal-to-noise (SNR) region.

Xing Li

Papers

Stochastic geometric modeling and interference analysis for massive MIMO systems

Maximizing efficiency of multi-user communications networks

A new algorithm for the weighted sum rate maximization in MIMO interference networks

A 3-D Channel Model for Distributed MIMO Satellite Systems

Data-Aided MIMO Channel Estimation by Clustering and Reinforcement-Learning