Algorithms based on PDA can be employed for low complexity MIMO detection. PDA was originally developed for target tracking in remote sensing applications like radar, sonar, etc. [1]–[3]. In these applications, tracking a target (potentially a non-cooperative hostile target) or multiple targets is of interest. Signals from such targets of interest can be weak. To detect such weak signals, the detection threshold has to be lowered, which may lead to the detection of other background signals (e.g., signals from other unwanted targets) and sensor noise, yielding spurious measurements which are clutter or false-alarm originated. Target tracking loss may result when such spurious measurements are used in a tracking filter (e.g., a Kalman filter). The selection of which measurements to use to update the estimate of the target state for tracking is known as data association. Data association uncertainty arises from ambiguity as to which measurements are appropriate for use in the tracking filter. Tracking of targets of interest in the presence of data association uncertainty has been studied extensively. The goal is to obtain accurate estimates of the target state and the associated uncertainty.

Optimal estimation of the target state involves recursive computation of the conditional probability density function (pdf) of the state, given all the information available, namely, (i) the prior information about the initial state, (ii) the intervening known inputs, and (iii) the set of measurements up to that time.

In the previous chapters, large MIMO detection algorithms were presented under the assumption of perfect knowledge of channel gains at the receiver. However, in practice, these gains are estimated at the receiver, either blindly/semi-blindly or through pilot transmissions (training). In FDD systems, channel gains estimated at the receiver are fed back to the transmitter (e.g., for precoding purposes). In TDD systems, where channel reciprocity holds, the transmitter can estimate the channel and use it for precoding. Due to noise and the finite number of pilot symbols used for channel estimation, the channel estimates are not perfect, i.e., there are estimation errors. This has an influence on the achieved capacity of the MIMO channel and the error performance of detection and precoding algorithms. This chapter addresses the effect of imperfect CSI on MIMO capacity, how much training is needed for MIMO channel estimation and channel estimation algorithms and their performance on the uplink in large-scale multiuser TDD MIMO systems.

MIMO capacity with imperfect CSI

The capacity of MIMO channels can be degraded if the CSI is not perfect. Gaussian input distribution, which is the capacity achieving distribution in the perfect CSI case, is suboptimal when CSI is imperfect [1],[2]. Lower and upper bounds on the mutual information for iid frequency-flat Rayleigh fading point-to-point MIMO channels have been derived for the imperfect CSI case in [3] assuming Gaussian input, where the MMSE channel estimate is assumed at the receiver and the same channel estimate is assumed to be available at the transmitter.

Channel models play a crucial role in the design and analysis of wireless communication systems. They enable the system designers to analyze the performance of wireless systems and optimize design parameters even before the systems are actually built. They are key ingredients in such design and performance evaluation exercises, which are often carried out through mathematical analysis or computer simulation or a combination of both. A good channel model that accurately captures the real channel behavior is a very valuable tool that can accelerate the development of practical wireless systems. The need for good channel models to aid the design, analysis, and development of MIMO systems in general, and large MIMO systems in particular, is immense. A lot of effort has been directed towards MIMO channel sounding campaigns and MIMO channel modeling. Measurements from these campaigns have aided the formulation of MIMO channel models in wireless standards [1]–[4]. Channel sounding campaigns with large numbers of antennas, in both outdoor and indoor settings, have also appeared, though sparsely, in the literature. Now, with the increasing interest in large MIMO system implementations, there is renewed interest and activity in large MIMO channel sounding. While these channel measurements are expected to yield accurate and realistic models for large MIMO channels, the traditional analytical MIMO channel models which are widely known in the literature, are expected to find continued use.

Detection of MIMO encoded signals, be it for spatial multiplexing or space-time coding or SM, is one of the crucial receiver functions in MIMO wireless communication [1]. Compared to detection in SISO or SIMO communication in fading channels, detection in MIMO communication is more involved. This is because, in addition to fading, the receive antennas encounter spatial interference due to simultaneous transmission from multiple transmit antennas. Efficient detection of signals in the presence of this spatial interference is therefore a demanding task, and sophisticated signal processing algorithms are needed for this purpose. Consequently, design, analysis, and implementation of efficient algorithms for MIMO detection continues to attract the attention of researchers and system developers.

Often, the roots of several MIMO detection algorithms in the literature can be traced to algorithms for multiuser detection (MUD) in CDMA which have been studied since the mid-1980s [2]. This is because CDMA systems and MIMO systems are both described by a linear vector channel model with the same structural format. In the case of a CDMA system the channel matrix is defined by the normalized cross-correlations between the signature sequences of the active users, whereas the channel matrix in a MIMO system is defined by the spatial signatures between the transmit and receive antennas.

Large MIMO systems are systems which use tens to hundreds of antennas in communication terminals. Depending on the application scenario, different MIMO system configurations are possible. These include point-to-point MIMO and multiuser MIMO configurations. In multiuser MIMO, point-to-multipoint (e.g., downlink in cellular systems), and multipoint-to-point (e.g., uplink in cellular systems) configurations are common. In a point-to-point communication scenario (Fig. 1.1), the number of transmit antennas nt at the transmitter and the number of receive antennas nr at the receiver can be large. A typical application scenario for a point-to-point large MIMO configuration is providing high-speed wireless backhaul connectivity between BSs using multiple antennas at each BS. Since space constraint need not be a major concern at the BSs, a large number of antennas can be used at both the transmit and receive BSs in this application scenario.

In multiuser MIMO (Fig. 1.2), the communication is between a BS and multiple user terminals. These user terminals can be small devices like mobile/smart phones or medium sized terminals like laptops, set-top boxes, TVs, etc. In mobile applications where mobile/smart phones and personal digital assistants are the user terminals, only a limited number of antennas can be mounted on them because of space constraints. However, in applications involving user terminals like TVs, set-top boxes, and laptops a larger number of antennas can be used on the user terminal side as well.

The job of MIMO encoding is to map the input symbols, say, from a modulation alphabet, to symbols to be transmitted over multiple transmit antennas. Spatial multiplexing and space-time coding are two well-known MIMO encoding techniques [1],[2]. Spatial modulation (SM) is a more recently proposed scheme for multiantenna communications [3]. These MIMO encoding schemes do not require any knowledge of the CSI at the transmitter, and hence are essentially “open-loop” schemes. MIMO encoding using CSI at the transmitter is referred to as MIMO precoding, which is treated in Chapter 10. Spatial multiplexing is an attractive architecture for achieving high rates. Space-time coding is attractive for achieving increased reliability through transmit diversity. SM serves a different purpose. It allows fewer transmit RF chains than the number of transmit antennas to be used without compromising much on the rate. This reduces the RF hardware complexity, size, and cost. In spatial multiplexing and spacetime coding, information is carried on the modulation symbols. In SM, on the other hand, in addition to modulation symbols, the indices of the antennas on which transmission takes place also convey information. This is why SM does not compromise much on the rate. Among the three MIMO encoding schemes, spatial multiplexing is the simplest, and its complexity rests more at the receiver in detecting the transmitted symbol vector. SM, though simple conceptually, needs additional memory to construct the encoding table at the transmitter for selecting the antennas for transmission.

Probability theory and graph theory are two branches of mathematics that are widely applicable in many different domains. Graphical models combine concepts from both these branches to provide a structured framework that supports representation, inference, and learning for a broad spectrum of problems [1]. Graphical models are graphs that indicate inter-dependencies between random variables [2],[3]. Distributions that exhibit some structure can generally be represented naturally and compactly using a graphical model, even when the explicit representation of the joint distribution is very large. The structure often allows the distribution to be used effectively for inference, i.e., answering certain queries of interest using the distribution. The framework also facilitates construction of these models by learning from data.

In this chapter we consider the use of graphical models in: (1) the representation of distributions of interest in MIMO systems, (2) formulation of the MIMO detection problem as an inference problem on such models (e.g., computation of posterior probability of variables of interest), and (3) efficient algorithms for inference (e.g., low complexity algorithms for computing the posterior probabilities). Three basic graphical models that are widely used to represent distributions include Bayesian belief networks [4], Markov random fields (MRFs) [5], and factor graphs [6]. Message passing algorithms like the BP algorithm [7] are efficient tools for inference on graphical models. In this chapter, a brief survey of various graphical models and BP techniques is presented.

In large systems where the physical laws governing system behavior are inherently probabilistic and complicated, traditional methods of obtaining closed-form analytic solutions may not be adequate for the level of detailed study needed. In such situations, one could simulate the behavior of the system in order to estimate the desired solution. This approach of using randomized simulations on computers, which came to be called the Monte Carlo methods, is powerful, elegant, flexible, and easy to implement. From the early days of their application in simulating neutron diffusion evolution in fissionable material in the late 1940s, Monte Carlo methods have found application in almost every discipline of science and engineering. Central to the Monte Carlo approach is the generation of a series of random numbers, often a sequence of numbers between 0 and 1 sampled from a uniform distribution, or, in several other instances, a sequence of random numbers sampled from other standard distributions (e.g., the normal distribution) or from more general probability distributions that arise in physical models. More sophisticated techniques are needed for sampling from more general distributions, one such technique being acceptance–rejection sampling. These techniques, however, are not well suited for sampling large-dimensional probability distributions. This situation can be alleviated through the use of Markov chains, in which case the approach is referred to as the Markov chain Monte Carlo (MCMC) method [1]. Typically, MCMC methods refer to a collection of related algorithms, namely, the Metropolis–Hastings algorithm, simulated annealing, and Gibbs sampling.

As in any new or emerging technology, demonstrators, testbeds, and prototypes play an important role in the development of large MIMO systems. The terms demonstrators, testbeds, and prototypes are often used loosely and interchangeably to refer to practical proof-of-concept-like implementations. The following broad definitions from [1],[2] bring out some key differences between them.

1. • A demonstrator is meant primarily to showcase and demonstrate technology to customers. Generally, it involves implementation of a new idea, concept, or standard that has been already established and has been finalized to some extent. Therefore, the requirements on functionality and design time are more important than scalability.

2. • A testbed is meant for research in general. It is a platform that allows the testing or verification of new algorithms or ideas under real-world conditions. Therefore, testbeds are expected to be more modular, scalable, and extendable.

3. • A prototype is meant to be serve as the initial realization of a research idea or a standard in real time, as a reference, a proof-of-concept, or a platform for future developments and improvements. It is often intended to evolve a prototype into a product.

From these definitions, one can see that testbeds and prototypes can play crucial roles in the research and development phase. While prototypes need necessarily to operate in real time, a testbed can be a real-time testbed or a non-real-time (offline) testbed depending on the available resources in comparison with the real-time computation need.

Channel state information at the transmitter (CSIT) can be exploited to improve performance in MIMO wireless systems through a precoding operation at the transmitter. Precoding techniques use CSIT to encode the information symbols into transmit vectors. Typically, an information symbol vector u is encoded into a transmit vector x using the transformation x = Tu, where T is referred to as the precoding matrix. The precoding matrix T is chosen based on the available CSIT. Precoding on point-to-point MIMO links and multiuser MIMO links is common. In addition, multiuser MIMO precoding in multicell scenarios is of interest. In this chapter, precoding schemes for large MIMO systems are considered.

Precoding in point-to-point MIMO

In point-to-point MIMO links, precoding techniques can achieve improved performance in terms of enhanced communication reliability, which is typically quantified in terms of the diversity gain/order achieved by the precoding scheme. In addition to performance, precoding/decoding complexities are also of interest. Often, one encounters a tradeoff between performance (diversity gain) and precoding/decoding complexity. Well-known precoders for point-to-point MIMO are presented in the following subsections.

System model

Consider an nt × nr point-to-point MIMO system (nr ≤ nt), where nt and nr denote the number of transmit and receive antennas, respectively. Assume CSI to be known perfectly at both the transmitter and the receiver. Let x = (x1,…,xnt)T be the vector of symbols transmitted by the nt transmit antennas in one channel use.

The practical demonstration of the vertical Bell laboratories layered space-time architecture (V-BLAST) multiantenna wireless system by Bell Labs [1], and the theoretical prediction of very high wireless channel capacities in rich scattering environments by Telatar in [2] and Foschini and Gans in [3] in the late 1990s opened up immense possibilities and created wide interest in multiantenna wireless communications. Since then, multiantenna wireless systems, more commonly referred to as multiple-input multiple-output (MIMO) systems, have become increasingly popular. The basic premise of the popularity of MIMO is its theoretically predicted capacity gains over single-input single-output (SISO) channel capacities. In addition, MIMO systems promise other advantages like increased link reliability and power efficiency. Realizing these advantages in practice requires careful exploitation of large spatial dimensions.

Significant advances in the field of MIMO theory and practice have been made as a result of the extensive research and development efforts carried out in both academia and industry [4]–[7]. A vast body of knowledge on MIMO techniques including space-time coding, detection, channel estimation, precoding, MIMO orthogonal frequency division multiplexing (MIMO-OFDM), and MIMO channel sounding/modeling has emerged and enriched the field. It can be safely argued that MIMO systems using 2 to 4 antennas constitute a fairly mature area now. Technological issues in such small systems are fairly well understood and practical implementations of these systems have become quite common.

The physical layer capabilities in wireless transmissions are growing. In particular, the growth trajectory of the achieved data transmission rates on wireless channels has followed Moore's law in the past decade and a half. Over a span of 15 years starting mid-1990s, the achieved wireless data transmission rates in several operational scenarios have increased over 1000 times. The data transmission rate in WiFi which was a mere 1 Mbps in 1996 (IEEE 802.11b) had reached 1 Gbps by 2011 (IEEE 802.11ac). During the same span of time, the data rate in cellular communication increased from about 10 kbps in 2G to more than 10 Mbps in 4G (LTE). One of the promising technologies behind such a sustained rate increase is multiantenna technology – more popularly referred to as the multiple-input multiple-output (MIMO) technology, whose beginnings date back to the late 1990s.

The interest shown in the study and implementation of MIMO systems stems from the promise of achieving high data rates as a result of exploiting independent spatial dimensions, without compromising on the bandwidth. Theory has predicted that the greater the number of antennas, the greater the rate increase without increasing bandwidth (in rich scattering environments). This is particularly attractive given that the wireless spectrum is a limited and expensive resource.

More than a decade of sustained research, implementation, and deployment efforts has given MIMO technology the much needed maturity to become commercially viable.