DMT of Wireless Networks: An overview

The e�cient operation of single-source, single-sink wireless network is considered with the diversity-multiplexing gain tradeo� (DMT) as the measure of performance. Whereas in the case of a point-to-point MIMO channel the DMT is determined by the fading statistics, in the case of a network, the DMT is additionally, a function of the time schedule according to which the network is operated, as well as the protocol that dictates the mode of operation of the intermediate relays.In general, it is only possible at present, to provide upper bounds on the DMT of the network in terms of the DMT of the MIMO channel appearing across cuts in the network. This paper presents a tutorial overview on the DMT of half-duplex multi-hop wireless networks that also attempts to identify where possible, codes that achieve the DMT.For example, it is shown how one can construct codes that achieve the DMT of a network under a given schedule and either an amplify-and-forward or decode-and-forward protocol. Also contained in the paper,are discussions on the DMT of the multiple-access channel as well as the impact of feedback on the DMT of a MIMO channel.

On the Successive Refinability of MIMO channels: DMT and codes

Diversity embedded space time codes are high rate codes that are designed such that they have a high diversity code embedded within them. A recent work by Diggavi and Tse characterizes the performance limits that can be achieved by diversity embedded space-time codes in terms of the achievable Diversity Multiplexing Tradeoff (DMT). In particular, they have shown that the trade off is successively refinable for rayleigh fading channels with one degree of freedom using superposition coding and Successive Interference Cancellation (SIC). However, for Multiple-Input Multiple-Output (MIMO) channels, the questions of successive refinability remains open. We consider MIMO Channels under superposition coding and SIC. We derive an upper bound on the successive refinement characteristics of the DMT. We then construct explicit space time codes that achieve the derived upper bound. These codes, constructed from cyclic division algebras, have minimal delay. Our results establish that when the channel has more than one degree of freedom, the DMT is not successive refinable using superposition coding and SIC. The channels considered in this work can have arbitrary fading statistics.

Multi-hop Cooperative Wireless Networks: Diversity Multiplexing Tradeoff and Optimal Code Design

We consider single-source single-sink (ss-ss) multi-hop relay networks, with slow-fading links and single-antenna half-duplex relay nodes. While two-hop cooperative relay networks have been studied in great detail in terms of the diversity-multiplexing tradeoff (DMT), few results are available for more general networks. In this paper, we identify two families of networks that are multi-hop generalizations of the two-hop network: K-Parallel-Path (KPP)networks and layered networks.KPP networks, can be viewed as the union of K node-disjoint parallel relaying paths, each of length greater than one. KPP networks are then generalized to KPP(I) networks, which permit interference between paths and to KPP(D) networks, which possess a direct link from source to sink. We characterize the DMT of these families of networks completely for K > 3. Layered networks are networks comprising of layers of relays with edges existing only between adjacent layers, with more than one relay in each layer. We prove that a linear DMT between the maximum diversity dmax and the maximum multiplexing gain of 1 is achievable for single-antenna fully-connected layered networks. This is shown to be equal to the optimal DMT if the number of relaying layers is less than 4.For multiple-antenna KPP and layered networks, we provide an achievable DMT, which is significantly better than known lower bounds for half duplex networks.For arbitrary multi-terminal wireless networks with multiple source-sink pairs, the maximum achievable diversity is shown to be equal to the min-cut between the corresponding source and the sink, irrespective of whether the network has half-duplex or full-duplex relays. For arbitrary ss-ss single-antenna directed acyclic networks with full-duplex relays, we prove that a linear tradeoff between maximum diversity and maximum multiplexing gain is achievable.Along the way, we derive the optimal DMT of a generalized parallel channel and derive lower bounds for the DMT of triangular channel matrices, which are useful in DMT computation of various protocols. We also give alternative and often simpler proofs of several existing results and show that codes achieving full diversity on a MIMO Rayleigh fading channel achieve full diversity on arbitrary fading channels. All protocols in this paper are explicit and use only amplify-and-forward (AF) relaying. We also construct codes with short block-lengths based on cyclic division algebras that achieve the optimal DMT for all the proposed schemes.Two key implications of the results in the paper are that the half-duplex constraint does not entail any rate loss for a large class of cooperative networks and that simple AF protocols are often sufficient to attain the optimal DMT

A Distributed Approach to Interference Cancellation

In this paper, we propose and analyze a novel idea of performing interference cancellation (IC) in a distributed/cooperative manner, with a motivation to provide multiuser detection (MUD) benefit to nodes that have only a single user detection capability. In the proposed distributed interference cancellation (DIC) scheme, during phase-1 of transmission, an MUD capable cooperating relay node estimates all the sender nodes' bits through multistage interference cancellation. These estimated bits are then sent by the relay node on orthogonal tones in phase-2 of transmission. The destination nodes receive these bit estimates and use them for interference estimation/cancellation, thus achieving IC benefit in a distributed manner. For this DIC scheme, we analytically derive an exact expression for the bit error rate (BER) in a basic five-node network (two source-destination node pairs and a cooperating relay node) on AWGN channels. Analytical BER results are shown to match with simulation results. For more general system scenarios, including more than two source-destination pairs and fading channels without and with space-time coding, we present simulation results to establish the potential for improved performance in the proposed distributed approach to interference cancellation. We also present a linear version of the proposed DIC.

Spectral Efficiency of Channel-Aware Schedulers in Non-identical Composite Links with Interference

Accurate system planning and performance evaluation requires knowledge of the joint impact of scheduling, interference, and fading. However, current analyses either require costly numerical simulations or make simplifying assumptions that limit the applicability of the results. In this paper, we derive analytical expressions for the spectral efficiency of cellular systems that use either the channel-unaware but fair round robin scheduler or the greedy, channel-aware but unfair maximum signal to interference ratio scheduler. As is the case in real deployments, non-identical co-channel interference at each user, both Rayleigh fading and lognormal shadowing, and limited modulation constellation sizes are accounted for in the analysis. We show that using a simple moment generating function-based lognormal approximation technique and an accurate Gaussian-Q function approximation leads to results that match simulations well. These results are more accurate than erstwhile results that instead used the moment-matching Fenton-Wilkinson approximation method and bounds on the Q function. The spectral efficiency of cellular systems is strongly influenced by the channel scheduler and the small constellation size that is typically used in third generation cellular systems.

D-MG Tradeoff and Optimal Codes for a Class of AF and DF Cooperative Communication Protocols

Cooperative relay communication in a fading channel environment under the orthogonal amplify-and-forward (OAF), non-orthogonal and orthogonal selection decode-and-forward (NSDF and OSDF) protocols is considered here. The diversity-multiplexing gain tradeoff (DMT) of the three protocols is determined and DMT-optimal distributed space-time code constructions are provided. The codes constructed are sphere decodable and in some instances incur minimum possible delay. Included in our results is the perhaps surprising finding that the OAF and NAF protocols have identical DMT when the time durations of the broadcast and cooperative phases are optimally chosen to suit the respective protocol. Two variants of the NSDF protocol are considered: fixed-NSDF and variable-NSDF protocol. In the variable-NSDF protocol, the fraction of time occupied by the broadcast phase is allowed to vary with multiplexing gain. In the two-relay case, the variable-NSDF protocol is shown to improve on the DMT of the best previously-known static protocol for higher values of multiplexing gain. Our results also establish that the fixed-NSDF protocol has a better DMT than the NAF protocol for any number of relays.

Jointly Optimal Power Control and Hop Length for a Single Cell, Dense, Ad hoc Wireless Network

We consider a dense, ad hoc wireless network confined to a small region, such that direct communication is possible between any pair of nodes. The physical communication model is that a receiver decodes the signal from a single transmitter, while treating all other signals as interference. Data packets are sent between source-destination pairs by multihop relaying. We assume that nodes self-organise into a multihop network such that all hops are of length d meters, where d is a design parameter. There is a contention based multiaccess scheme, and it is assumed that every node always has data to send, either originated from it or a transit packet (saturation assumption). In this scenario, we seek to maximize a measure of the transport capacity of the network (measured in bit-meters per second) over power controls (in a fading environment) and over the hop distance d, subject to an average power constraint. We first argue that for a dense collection of nodes confined to a small region, single cell operation is efficient for single user decoding transceivers. Then, operating the dense ad hoc network (described above) as a single cell, we study the optimal hop length and power control that maximizes the transport capacity for a given network power constraint. More specifically, for a fading channel and for a fixed transmission time strategy (akin to the IEEE 802.11 TXOP), we find that there exists an intrinsic aggregate bit rate (Thetaopt bits per second, depending on the contention mechanism and the channel fading characteristics) carried by the network, when operating at the optimal hop length and power control. The optimal transport capacity is of the form dopt(Pmacrt) x Thetaopt with dopt scaling as Pmacrt 1 /eta, where Pmacrt is the available time average transmit power and eta is the path loss exponent. Under certain conditions on the fading distribution, we then pro- - vide a simple characterisation of the optimal operating point.

Tracking analysis of an LMS decision feedback equalizer for a wireless channel

We consider a time varying wireless fading channel, equalized by an LMS Decision Feedback equalizer (DFE). We study how well this equalizer tracks the optimal MMSEDFE (Wiener) equalizer. We model the channel by an Autoregressive (AR) process. Then the LMS equalizer and the AR process are jointly approximated by the solution of a system of ODEs (ordinary differential equations). Using these ODEs, we show via some examples that the LMS equalizer moves close to the instantaneous Wiener filter after initial transience. We also compare the LMS equalizer with the instantaneous optimal DFE (the commonly used Wiener filter) designed assuming perfect previous decisions and computed using perfect channel estimate (we will call it as IDFE). We show that the LMS equalizer outperforms the IDFE almost all the time after initial transience.

Capacity Optimizing Hop Distance in a Mobile Ad Hoc Network with Power Control

In a dense multi-hop network of mobile nodes capable of applying adaptive power control, we consider the problem of finding the optimal hop distance that maximizes a certain throughput measure in bit-metres/sec, subject to average network power constraints. The mobility of nodes is restricted to a circular periphery area centered at the nominal location of nodes. We incorporate only randomly varying path-loss characteristics of channel gain due to the random motion of nodes, excluding any multi-path fading or shadowing effects. Computation of the throughput metric in such a scenario leads us to compute the probability density function of random distance between points in two circles. Using numerical analysis we discover that choosing the nearest node as next hop is not always optimal. Optimal throughput performance is also attained at non-trivial hop distances depending on the available average network power.

An Accurate Model for EESM and its Application to Analysis of CQI Feedback Schemes and Scheduling in LTE

The Effective Exponential SNR Mapping (EESM) is an indispensable tool for analyzing and simulating next generation orthogonal frequency division multiplexing (OFDM) based wireless systems. It converts the different gains of multiple subchannels, over which a codeword is transmitted, into a single effective flat-fading gain with the same codeword error rate. It facilitates link adaptation by helping each user to compute an accurate channel quality indicator (CQI), which is fed back to the base station to enable downlink rate adaptation and scheduling. However, the highly non-linear nature of EESM makes a performance analysis of adaptation and scheduling difficult; even the probability distribution of EESM is not known in closed-form. This paper shows that EESM can be accurately modeled as a lognormal random variable when the subchannel gains are Rayleigh distributed. The model is also valid when the subchannel gains are correlated in frequency or space. With some simplifying assumptions, the paper then develops a novel analysis of the performance of LTE's two CQI feedback schemes that use EESM to generate CQI. The comprehensive model and analysis quantify the joint effect of several critical components such as scheduler, multiple antenna mode, CQI feedback scheme, and EESM-based feedback averaging on the overall system throughput.

Degrees of freedom of relay aided MIMO X network with orthogonal components

We consider the one-way relay aided MIMO X fading Channel where there are two transmitters and two receivers along with a relay with M antennas at every node. Every transmitter wants to transmit messages to every other receiver. The relay broadcasts to the receivers along a noisy link which is independent of the transmitters channel. In literature, this is referred to as a relay with orthogonal components. We derive an upper bound on the degrees of freedom of such a network. Next we show that the upper bound is tight by proposing an achievability scheme based on signal space alignment for the same for M = 2 antennas at every node.

Performance Analysis of a Cooperative System with Rateless Codes and Buffered Relays

In a cooperative relay-assisted communication system that uses rateless codes, packets get transmitted from a source to a destination at a rate that depends on instantaneous channel states of the wireless links between nodes. When multiple relays are present, the relay with the highest channel gain to the source is the first to successfully decode a packet from the source and forward it to the destination. Thus, the unique properties of rateless codes ensure that both rate adaptation and relay selection occur without the transmitting source or relays acquiring instantaneous channel knowledge. In this paper, we show that in such cooperative systems, buffering packets at relays significantly increases throughput. We develop a novel analysis of these systems that combines the communication-theoretic aspects of cooperation over fading channels with the queuing-theoretic aspects associated with buffering. Closed-form expressions are derived for the throughput and end-to-end delay for the general case in which the channels between various nodes are not statistically identical. Corresponding results are also derived for benchmark systems that either do not exploit spatial diversity or do not buffer packets. Altogether, our results show that buffering - a capability that will be commonly available in practical deployments of relays - amplifies the benefits of cooperation.

Algebraic Approaches to Space-Time Code Construction for Multiple-Antenna Communication

A major challenge in wireless communications is overcoming the deleterious effects of fading, a phenomenon largely responsible for the seemingly inevitable dropped call. Multiple-antennas communication systems, commonly referred to as MIMO systems, employ multiple antennas at both transmitter and receiver, thereby creating a multitude of signalling pathways between transmitter and receiver. These multiple pathways give the signal a diversity advantage with which to combat fading. Apart from helping overcome the effects of fading, MIMO systems can also be shown to provide a manyfold increase in the amount of information that can be transmitted from transmitter to receiver. Not surprisingly,MIMO has played, and continues to play, a key role in the advancement of wireless communication.Space-time codes are a reference to a signalling format in which information about the message is dispersed across both the spatial (or antenna) and time dimension. Algebraic techniques drawing from algebraic structures such as rings, fields and algebras, have been extensively employed in the construction of optimal space-time codes that enable the potential of MIMO communication to be realized, some of which have found their way into the IEEE wireless communication standards. In this tutorial article, reflecting the authors’interests in this area, we survey some of these techniques.

DMT of Multihop Networks: End Points and Computational Tools

In this paper, the diversity-multiplexing gain tradeoff (DMT) of single-source, single-sink (ss-ss), multihop relay networks having slow-fading links is studied. In particular, the two end-points of the DMT of ss-ss full-duplex networks are determined, by showing that the maximum achievable diversity gain is equal to the min-cut and that the maximum multiplexing gain is equal to the min-cut rank, the latter by using an operational connection to a deterministic network. Also included in the paper, are several results that aid in the computation of the DMT of networks operating under amplify-and-forward (AF) protocols. In particular, it is shown that the colored noise encountered in amplify-and-forward protocols can be treated as white for the purpose of DMT computation, lower bounds on the DMT of lower-triangular channel matrices are derived and the DMT of parallel MIMO channels is computed. All protocols appearing in the paper are explicit and rely only upon AF relaying. Half-duplex networks and explicit coding schemes are studied in a companion paper.

New Insights into Optimal Discrete Rate Adaptation for Average Power Constrained Single and Multi-Node Systems

The throughput-optimal discrete-rate adaptation policy, when nodes are subject to constraints on the average power and bit error rate, is governed by a power control parameter, for which a closed-form characterization has remained an open problem. The parameter is essential in determining the rate adaptation thresholds and the transmit rate and power at any time, and ensuring adherence to the power constraint. We derive novel insightful bounds and approximations that characterize the power control parameter and the throughput in closed-form. The results are comprehensive as they apply to the general class of Nakagami-m (m >= 1) fading channels, which includes Rayleigh fading, uncoded and coded modulation, and single and multi-node systems with selection. The results are appealing as they are provably tight in the asymptotic large average power regime, and are designed and verified to be accurate even for smaller average powers.