Universit`
a di Pisa
Dipartimento di Ingegneria dell’Informazione
Tesi di Laurea Magistrale in
Ingegneria delle Telecomunicazioni
Resource Allocation Algorithms for Modern and
Future Wireless Systems
Candidato: Relatori:
Gabriele Antonio Innocenti Marco Moretti Michele Morelli YingJun Angela Zhang
Chance and chance alone has a message for us. Everything that occurs out of necessity, everything expected, repeated day in and day out, is mute. Only chance can speak to us. We read its message much as gypsies read the images made by coffee grounds at the bottom of a cup. M.Kundera, The Unbearable Lightness of Being
Contents
1 The Problem of Managing and Allocating Resources 4
1.1 Introduction . . . 4
1.2 Resource Management and Allocation . . . 7
1.3 Goals and Metrics . . . 12
1.4 Models and Problem Formulations . . . 15
2 Multiuser MIMO-OFDM Systems 21 2.1 Introduction . . . 21
2.2 The Wireless Channel . . . 23
2.3 Why MIMO-OFDM Systems? . . . 28
2.3.1 OFDM Technique . . . 30
2.3.2 MIMO Technology . . . 33
3 Resource Allocation in Multiuser MIMO-OFDMA Systems 39 3.1 Introduction . . . 39
3.2 Fairness Based Resource Allocation Scheme for Downlink . . 41
3.3 A Resource Allocation Scheme for Uplink . . . 44
3.4 Efficient Margin Adaptive Scheduling for MIMO-OFDMA Systems . . . 56
4 Deployments in Next Generation Systems 64 4.1 Introduction . . . 64
4.2 Scenarios for 5G Mobile and Wireless Communications . . . 67
4.3 Cloud-RAN for Mobile Networks . . . 68
4.4 Clustering and Radio Resource Management . . . 74
Chapter 1
The Problem of Managing and
Allocating Resources
1.1
Introduction
Wireless Communications are revolutionizing the way modern people com-municate, live and think. As such, they have captured the attention and the imagination of media and researchers. The vision of wireless communi-cations supporting information exchange between people (users) or devices in an untethered way has recently been one amongst the most important in-ventions in globalizing the world, canceling distances; it has allowed people to work directly from home, create remote classrooms, monitor fire hazards and send e-mails from everywhere: all this has already become reality to-day and it is clear that there will be many more wireless applications in the foreseeable future. Receiving telephone calls at a fixed location or waiting to get home to download one video are, for example, just old story.
So what exactly is a wireless communication system? A wireless commu-nication system actually enables commucommu-nication between two or more users (or people, or devices); it offers services that may be impractical in a wired network, making communication possible even in case of moving users (or in case of time-varying, frequency selective and random communication chan-nel). Wireless communications has resulted in a considerable fragmentation in the industry as evidenced by many different products, standards and services being designed and implemented: personal communication (PCS), satellite and commercial broadcast, aid-to-navigation, environment-sensing systems; given that, it is clear that a comprehensive definition of such a wide
topic is hard to find. The main reason for such a division is that different wireless applications have different requirements and specified functions. For example, voice systems require short latencies but can tolerate higher Bit-Error-Rate (BER); data systems have very low BER requirements; and on the other hand, short messaging services have no strict delay constraints. This variety also makes it difficult to build one universal wireless system that can simultaneously satisfy all of them, and as a consequence, provide a gen-eral definition. In the following, wireless systems will be gengen-erally referred to as systems, in which unguided electromagnetic-wave propagation is used to transfer information (or more generally, to perform one or more specified functions).
Although the very first radiocommunications date back to the end of 1800, the past two decades have been crucial to the development of re-search activities in this field. Today, wireless technology is still considered as the enabling technique of future consumer products and wireless devices have increasingly become more pervasive and common, as shown in the GSM-Association forecast (2014):
Figure 1.1: The growth of Machine-to-Machine (M2M) communications Considered the actual and expected future demands, it is clear that communication systems still need to rapidly evolve and many technical chal-lenges remain in designing wireless networks that could improve the actual performances to support emerging applications. Systems will be required to support a variety of highspeed data communication services, and at the same time, serve a growing number of users in a fair manner, even with
flexible Quality-of-Service (QoS). This enforces a robust and application-specific optimization of limited system Resources and focus has often been given towards increasing both the spectral and energy efficiency (specifically, SE and EE).
Figure 1.2: 5G Challenges: What is expected from new generation Systems? Next generation communication systems must address challenges of mul-timedia broadcast due to wide variations of the wireless channel in both the time and frequency domains, higher data-rate demands and increased mo-bility of users. Furthermore, since battery technology has not progressed as rapidly as Information and Communication Technology (ICT), power effi-ciency has become increasingly important; device power consumption is, in fact, one of the main bottlenecks of modern wireless systems. For example, Telecom Italia is the second largest energy consumer in Italy; and in com-parison, energy consumption of italian mobile networks is growing much faster than ICT on the whole [7]. It is stated that as silicon technology is progressing exponentially, processor power consumption is increasing by 150% every two years while the improvement in battery technology is much slower, leading to an evident gap between the demand for energy and the battery capacity offered.
The technical problems to meet the increasing demand extend across all levels of the system design stack, ranging from the silicon to the appli-cation layer. Cross-layer approaches exploit interactions between different layers and can definitely improve the EE. In the following, various Resource Allocation (RA) approaches for wireless communication systems will be dis-cussed. Chapter 2 will introduce Orthogonal Frequency Division
Modula-tion (OFDM) and Multiple Input Multiple Output (MIMO) as two advanced physical layer techniques, which can increase the system capacity, combat channel impairments, and reduce energy requirements. Chapter 3 deals with how these technologies could be exploited for an efficient allocation of resources, both in the uplink and the downlink segments of a wireless system. The Last Chapter will consider the actual and most modern ideas of access networks, with particular attention to the problem of the Radio Resource Management (RRM) and Allocation.
1.2
Resource Management and Allocation
Given the recent proliferation of wireless devices in our everyday lives (com-pare fig. 1.1), a rising problem is the scarcity and variability of resources, in terms of time (or processor speed), frequency (or available bandwidth) and power. This is the result of the too rapid growth of wireless technology, coupled with the explosive demand of data connections and the diffusion of mobile devices (laptops, palmtop computers, etc.), which has dramatically changed users’ services requirements and needs. As a result, available re-sources must be managed and allocated in an efficient way in order to fulfil and satisfy the present and future demands in wireless communications; un-fortunately, increasing available resources does not necessarily imply system performance enhancements. Higher system efficiencies can be achieved by managing and exploiting the available resources: this is a problem of Op-timization. The challenge is clear: despite the limited available frequency spectrum, the battery power and the wireless channel fluctuations (due to different phenomena, such as Shadowing, Multipath, etc.), an increasing number of users keep on asking for faster and more reliable transmissions: how to deal with that?
Let us distinguish between two different issues, Resource Management and Allocation. Specifically, Resource Management encompasses processes that determine timing, ordering procedures and the amount of system re-source to allocate. On the other hand, Rere-source Allocation refers to the decision of how a set of resources should be assigned to the user and im-pacts on user throughputs and power consumption. In particular, Adaptive Resource Allocation seems to be a promising technique to achieve better system efficiencies as it refers to the assignment of a specific set of resources
according to the contingent state of the system, which can include wireless channel conditions, the state of the queues, the number of users, QoS re-quirements etc. with reference to a specific optimality criterion. Resources can be allocated considering different degrees of freedom, such as frequency subchannels, power and rate. The performance of the peculiar allocation is measured either by the amount of power required for transmitting a certain data rate or the maximum rate achievable with a certain amount of power. The problem is even more challenging if we consider a multiuser environ-ment. As introduced above, larger frequency bandwidths would not always lead to faster and more reliable transmissions because of the dynamic nature of the wireless channel. At the same time, one may think that increasing the transmit power could lead to better system performances; unfortunately, in this case interference would determine a reduction in the overall system per-formance. The most desirable solution appears to be the one that optimally manages wireless resources, that improves a specified quality or performance metric, that adapts to the channel characteristics and QoS requirements.
In this work, two instances of system efficiency will be of interest: the SE and EE. Since the system or user Throughput (transmitted information over a channel per second) is naturally a primary requirement, much effort has been made to enhance these parameters and advanced technologies have also been exploited (such as MIMO, OFDMA) both in literature and practice. In particular, the SE refers to the information rate that can be transmitted over a given bandwidth (1Hz) in a specific communication system; SE, as defined in 1.1, is widely used as indicator for the design of wireless communication systems:
η = Rb
B(RF ), [bit/s/Hz] (1.1)
where Rb is the transmission rate and B(RF ) is the bandwidth at RF. SE
could be seen as a measure of how efficiently the bandwidth is exploited, from the lower layers perspective. Higher throughputs usually imply large energy consumption, which is sometimes unaffordable given the energy-limited devices; for this reason, also the EE must be taken into account:
ηEE = C P = 2R N0(22R− 1) , [bit/s/W] (1.2) where C is the channel capacity ([bit/s]), P the transmit power ([W]), N0
the noise power spectral density([W/Hz]) and R is the channel capacity expressed in bits per degrees of freedom, related to C by the sampling
rate 2B (C = 2BR bit/s). For an additive white gaussian noise channel (AWGN), the channel capacity R is:
R = 1 2log2 1 + P N0B , [bit] (1.3) Equation 1.2 shows the EE metrics considered as bits per Joule, or the system throughput per unit of energy consumption; it represents a theo-retic result, and as such might not be achieved in practical systems due to the performance losses of capacity-approaching channel codes, imperfect knowledge of channel state information (CSI), cost of synchronization and electronic circuit power consumption. It can be considered as an upper bound.
As power consumption is affected by all aspects of the system design, ranging from silicon to applications layers, recent efforts have been made to its reduction from several points of view. Possible solutions in such highly dynamic environments, as discussed above, may require optimization cross-layer adaptive algorithms and schemes for architectures. Moreover, as multiuser wireless systems are based on a shared medium, the system performance is not only affected by a layer-based optimization, but also by the interaction between links in the network. Among the layers, we will particularly focus on the Physical layer (PHY) and the Medium Access Control layer (MAC).
In particular, the PHY layer deals with the data transmission over wire-less channels and consists of the modulation choice, power control, channel codings. Traditional wireless systems are designed to support the high-est feasible PHY rate and therefore, they are supposed to transmit at the maximum power without adaption. This could not represent the optimal solution, for many reasons; hence, a set of PHY parameters should be set to adapt the user requirements (for example, in terms of throughput and delay) and wireless channel impairments to trade off SE and EE. PHY plays an important role in wireless communications and power consumption due to the time and frequency selectiveness of the communication medium. On the other hand, the MAC layer manages wireless resources for the PHY layer and ensures their efficient allocation in order to maximize system per-formance metrics while mantaining user QoS requirements. For the MAC layer, protocols have been designed to efficiently exploit resources (power, transmission formats, frequency bandwidths).
Optimization across the PHY and MAC layers is desired for network resources allocation and scheduling, as it attempts to dynamically match the requirements of data-link connections to the available physical layer re-sources. Advanced data-compression algorithms and channel-coding schemes have also been proposed for the PHY layer, as a only-layer approach; even though such an approach may seem appealing, the results could be lim-ited for different reasons: modern channel coding has already enhanced the transmission rate very close to the information-theoretic capacity in Gaussian channels; transmission technologies deal with specific objectives, generally not yielding to a universally optimal performance solution; future wireless systems, which aim at supporting different services, will need to manage transmission technologies, according to the QoS requirements and the channel conditions. Moreover in wireless systems, where the impor-tance of information exchanges across layers has been highlighted, capabil-ity is highly dependant on the proper utilization of its resources. This is the reason to incorporate the effect of PHY in upper layers and viceversa, collapsing the traditional layered architecture.
Figure 1.3: Framework of cross-layer optimized resource allocation From a mathematical point of view, allocating resources may generally be formulated as an optimization problem (minimizing or maximizing a spe-cific parameter or goal function, according to the context), under particular constraints of power, delay, outage probability, data rate, etc, depending on the specific application. But, in what do resource allocation and manage-ment practically consist?
• Allocating Time resources is the process of selecting which link(s) should be active during a specific time slot. A single time slot can be assigned to multiple links as long as they do not interfere; in Chapter 2, an algorithm will be discussed to this scope. From an energy-efficiency point of view, it has been noticed that buffers don’t always have data to transmit, and traffic load should be considered in order to save this resource (compare Chapter 4).
• In terms of Frequency, each active link needs a certain bandwidth to transmit and receive. Frequency resources require careful atten-tion as the bandwidth is rigidly limited (regulated by internaatten-tional frequency licences and agreements) and in wireless systems it often has to be shared among multiple users. Interference between inter-fering links can here arise. Dealing with frequencies is even more complicated, due to the properties of wireless channels: for frequency selective channels, for examples, modulation and power allocation de-pend on the subchannel assignment and the state of the subchannel, while the subchannel assignment itself reciprocally depends on the modulation and power allocation to each user.
• In the Spatial domain, allocation is related to the presence of a set of transmitting and receiving antennas. MIMO systems provides spatial diversity in terms of channels or degrees of freedom, which can be exploited by users, increasing the system capacity and enabling the system to multiplex more users or streams transmitted by the same user. Even if it comes with a higher power consumption and necessi-tates more space to include many antennas (and thus, not always fea-sible especially in the Mobile Equipment), MIMO performance made the spatial domain a promising domain to be exploited ever more in future wireless networks.
• Power allocation is mainly referred to as setting transmit power for each of the available channels; in order to maximize/minimize a spe-cific function, a certain amount of power will be distributed among the subchannels and hence, users. The transmitted power is related to the capacity of the wireless link and its availability. Typically in wireless devices an additional circuit power need during transmission
should be taken into account, which is independent from the transmis-sion rate: thus, a fixed cost must be accounted for optimizing energy consumption.
1.3
Goals and Metrics
As the goal is to exploit the limited radio frequency spectrum and the network infrastrucure as more efficiently as possible, techniques and algo-rithms for controlling parameters such as transmit power, beamforming, throughput, fairness, modulation scheme, utility, etc. have commonly been published, studied and compared. In literature, the above mentioned pa-rameters are considered as general quality and performance measures, and they are used to compare different schemes. Efficient dynamic and adaptive schemes may increase the overall system efficiency, often considerably more than what may be possible by introducing advanced channel coding and source coding schemes.
Managing and allocating resources is especially important in systems, where co-channel interference is more limiting than noise, for example in cellular systems, broadcast networks covering large areas, or wireless net-works consisting of adjacent access points that may reuse the same channel frequencies. The performance of these systems, from the radio resource point of view, can be optimized by three main mechanisms in OFDMA Systems: subcarrier assignment, bit loading and power loading. Optimal subcarrier assignment determines which subchannel must be allocated to the user for the temporal slot; the following power and bit loadings opti-mally decide the best allocation of power to each subchannel and the number of bits (in other words, the modulation type). This optimization process depends heavily on the CSI, which is used for the decision. Although a perfect CSI is never possible and its accuracy must be taken into account in the optimization process, recent efforts have been done for adaptive al-location mechanisms. It means that resources management and alal-location adapts to the users’ varying channel conditions on a temporal basis (or to available CSI). The condition to check is that the channel coherence Time, Tc, should be much longer than the time slot, TS, so that the channel
es-timation could be considered valid for the entire TS and the equalization
In recent years, researchers have tried to explore the idea of adaptively assigning radio resources to users. The problem of adaptive radio resource allocation has been tackled from various angles, resulting in a long list of approaches and ideas; hundred of different algorithms and schemes have been proposed. Since it may be hard to group them all and give a detailed description of every single quality metric that they aim to optimize, in the following, it has been summarized a list of the most focused performance metrics:
1. Fairness: In many applications, the main concern is to provide all the users the available resources in a fair fashion. Fairness implies that all users get an equitable amount of system resources, without starvation (users should fairly transmit, even if they experience worse channel conditions). Attention towards the fairness issue has particu-larly raised together with the increasing number of users bacause the traditional goal of maximising the summation of throughput on all links could result in an unbalanced use of network resources.
2. Link Utilization: The algorithm must lead to an efficient utilization of the channel capacity, in this case; this implies that an adaptive scheme should decide the most appropriate modulation type and power allo-cation, based on the CSI, to get closer to the channel capacity or to get the highest feasible data-rate on that channel.
3. Complexity and Scalability: The less the implementation complex-ity is, the faster the algorithm will run. It may be one of the most desirable features, especially for high-speed applications. Moreover, the algorithm should operate efficiently even if the number of users or data-rates increase, providing versatility and scalability in various applications.
4. Energy Efficiency: Recently, green communication topics have raised attention and efforts have been done in order to lower the power con-sumption (and thus, costs); algorithms have been proposed with the goal of prolonging the mobile terminal battery life or just to maximize the energy efficiency of data transmission (bit/Joule delivered to the users). Moreover, the network costs may be further reduced with an accurate design of the circuitry and so the need for more power.
5. Throughput : Some algorithms aim at maximing the throughput, so the information rate over a communication channel. It is different from the Link Utilization target, as it considers the metric from the users point of view. They should be able to provide short-term throughput guarantee for error-free sessions; sometimes, a specified throughput is required to be temporally achieved in an average sense. In order to maximize system throughput, the system will allocate more resources to the users, who experience better channel conditions. This may cause radio resource monopolized by a small number of users, leading to unfairness.
6. QoS : Quality-of-Service requirement is one of the most challenging issues for future wireless communication systems; the application pro-vides and defines different service classes, which have specific features in terms of rate constraints or delay bounds, for example; the algo-rithm should manage them in order to satisfy different sets of require-ments.
7. Utility: Here the focus is given on the ’user satisfaction’ to avoid such a throughput/fairness dilemma. The approach is to maximize the metric that quantifies users’ satisfaction, also known as Utility. The utility of certain resource can be perceived differently by each user. Users’ satisfaction who are given a certain amount of resource can be described by the utility function U(r), where r is the allocated resource. The more the resource is allocated to that user, the more the user will be satisfied; the exact expression of U(r) may depend on traffic types and can be obtained by studying the behaviour and satisfaction of users.
Future wireless networks challenge scholars to find solutions and further investigations in this field. There are still many unsolved problems that need additional study. Next generation systems will be required to deal with a greater amount of users, since the aggressive proliferation of mobile devices; the mobility issue will also be of primary importance, as users re-quire to communicate even when they are travelling and the conditions of the channel vary as fast as they move. Different QoS will also be demanded for the upcoming new services; at the same time, the general trend is to
guarantee forever higher and more reliable data-rates for faster communi-cations and lower or affordable power consumption, as the actual batteries don’t offer optimal performances. For this reason, developments in this field focus primarily on the improvement of the system efficiency based on the interaction between layers. For example, in order to support voice and data services, the MAC layer needs to obtain information such as QoS, priorities, utility, from the upper levels, which will be then mapped to the appropriate PHY layer parameters. The power consumption is considered however the main issue for the development of future wireless network.
Another issue, which is worth highlighting, is the Signaling overhead. Some applications assume slow varying wireless channels, that do not trigger adaptive resource allocation so often, so that the assumptions on accurate channel estimation and negligible signaling overhead are appropriate and reasonable. When it comes to fast varying channel conditions, channel estimation must be performed and fed back to the resource allocator very often in order to update with fast fading channels, resulting in an increase in signaling overhead (even if sometimes, the selectivity of the channel is too rapid to consider possible to have a valid channel estimation). In particular, this overhead in fast fading scenario induced by mobility is still open for investigation, since optimization is here complicated to be performed as the variations may be different in different applications and scenarios. Even if the majority of algorithms and proposed schemes assume perfect knowledge of the channel state, it does not come for free; perfect CSI is not always an appropriate assumption and this calls for new researches, in particular if networks will need to support a greater number of users.
1.4
Models and Problem Formulations
Solutions to the optimization problem in wireless communications often show a high degree of complexity; this consequently results in suboptimal reduced-complexity algorithms and heuristics being developed. Neverth-less, even with reduced-complexity algorithms, resource allocation is still a centralized task (executed at the BS) because of the computational burden. In this section, we briefly discuss different classical problems, together with their objective functions and constraints; different solving strategies have been proposed in literature. The goal is to take advantage of the multiuser,
frequency diversity and channel gains; generally, schemes are based on the fact that channels are statistically independent and a channel that is good for one user may be bad for another and viceversa.
Maximum Sum-Rate Algorithms. Given a fixed transmit power, PT, the objective of the Maximum Sum Rate (MSR) scheme is to maximize
the sum of users’ data rates (or, the overall throughput). It is also known as ’Raw Data Rate Maximization’ scheme. According to Shannon-Hartley theorem, the maximum data rate on any link is a nonlinear function of global variables, such as transmit power, channel gains and noise power; hence, it is unlikely that algorithms with a polynomial complexity could solve it.
As wireless scenarios present varying channel gains, depending on rela-tive user locations, scatterers, shadowing and multipath fading phenomena, etc, only estimations could be executed, as gains cannot be known in ad-vance. Here, resources are allocated to the users who can achieve the best system performance (or that maximizes system gains), i.e. the users with the actual best channel conditions are selected. It means that all the sys-tem resources will be allocated to only few users (probably the nearest ones to the base station, with highest channel gains); users with bad channel conditions may starve and never get the chance to access and exploit the resources. If the scheme is optimal to maximize users’ data rate, on the other hand, it lacks fairness. Given SIN Rk,n
SINRk,n =
PnHk,n
σ2 (1.4)
the Signal-to-Interference-and-Noise ratio of k -th user on the n-th subcar-rier, where Pn is the transmitted power on the subchannel n, Hk,nthe
chan-nel gain of user k on that subchanchan-nel and σ2 the noise power, the problem
can be formulated as follows:
max P,ρ K X k=1 N X n=1 log2(1 + SINRk,n)ρk,n s.t. K X k=1 N X n=1 Pk,n≤ PT Pk,n ≥ 0, ∀k, n ρk,n∈ {0, 1}, ∀k, n (1.5)
where Pk,n is the transmit power of k -th user on the n-th subcarrier, that
must be non-negative; and ρk,nis a binary allocation variable, which is equal
to 1 if subchannel n is allocated to user k. The scheme belongs to the general category of rate adaptive schemes and it is referred to as a system-centric approach because its main focus is to optimize a system-centric measure, the overall data rate, subject to a global power constraint. The sum-rate maximization problem could be then solved by performing waterfilling over all the allocated subcarriers.
Maximum Fairness Algorithm. Raw-Rate Maximization scheme may lack fairness; as channel condition varies with high dynamic range among users, some of them may be underserved and starve. Thus, the maximum fairness problem is proposed; it aims at resource allocation in such a way that the minimum data rate among users is maximized, and implicitly leads to global fairness. It is also known as Max-Min problem for obvious reasons. max P,ρ mink N X n=1 log2(1 + PnHk,n σ2 )ρk,n s.t. K X k=1 ρk,n ≤ 1, ∀n X n Pn ≤ PT ρk,n ∈ {0, 1}, ∀k, n (1.6)
The basis of such a scheme is that no user gets benefit by increasing its allocation power. Here, the optimal solution is considerably difficult to de-termine because the objective function is not convex and then computation-ally expensive; therefore suboptimal algorithms are necessary, where bits and power allocation are done separately, for example. Different heuristic implementations have been proposed, but a common approach is to assume that equal power is initially allocated to each subcarrier; then, iteratively each available subcarrier is assigned to the user with the best channel condi-tions (for example, a greedy assignment strategy that allocates to the user with the smallest rate). The scheme can however exhibit drawbacks. In fact, it does not lead to flexible rate distribution among users; moreover, the total throughput is largely limited by the user with the worst SINR, as
most of the resources would be allocated to that user.
Proportional Rate Constraint Scheduling As discussed above, in wireless networks data rate requirements for different services can vary sub-stantially: that is the reason for the introduction of the Proportional Rate Constraint Scheduling, where the goal is to maximize the sum data rate, with the additional constraint that each user data rate is proportional to a set of pre-determined system parameters {wk}Kk=1. It means that if different
users require different data rates, radio resources can be allocated propor-tionally to the users’ rate constraints.
Being ynk the k-th user data rate on the n-th subcarrier, Cn the capacity
of subchannel n, PT the total transmit power, pnk(ynk) is the power needed
to transmit on subcarrier n at data rate yn
k, and γk,n an indicator set to
1 if subcarrier n is assigned to user k, otherwise 0, the objective function becomes: max K X k=1 wk( N X n=1 ykn) s.t. K X k=1 N X n=1 pnk(ykn) ≤ PT 0 ≤ ynk ≤ γknCn K X k=1 γkn ≤ 1 γkn∈ {0, 1} (1.7)
Constraints in 1.7 could also be relaxed to find an easier solution to the prob-lem. In fact, the optimization problem stated above involves both binary variables γk,nand continuous variables pnk(ykn) and one inequality constraint.
A typical heuristic approach is the one that split the allocation in two dif-ferent phases: the subcarrier allocation phase, where assuming an uniform power distribution, subcarriers are allocated complying as much as possible with the proportional rate constraints; and the power allocation phase, in which the power is distributed so that the proportional rate constraints are exactly met.
A similar approach can be applied to OFDMA systems, where a map-ping between subcarriers and users is required, such that the total rate is maximized. Given K users and N subcarriers, finding the global
max-imum among KN possible subcarriers allocation may be computationally
very heavy. A possible approach is to separate the subcarrier and power al-location procedures, setting a suboptimal/near-optimal problem with more manageable complexity.
Minimum Transmit Power Scheme As mentioned above, transmit power is a limited and limiting resource in wireless application, as the power and energy issue is an actual important concern; in the uplink of cellular networks for example, user terminals are limited by total available power, which is here referred to as the scarcest resource. The related approach is to assign resources with the goal of minimizing the overall transmitted power under user data rate or BER constraints, for example. Also in downlink, power minimization can often be an important optimization problem. Different schemes and algorithms have been proposed recently for OFDMA MIMO systems, where the minimization problem finds the distribution of subcarriers that minimizes the total power required with the data rate con-straint and each user experiences a flat-fading channel. This can be modeled as: min K X k=1 N X n=1 Pk,n ≤ PT s.t. ωk ≥ γk, ∀k (1.8)
where PT is the maximum available transmit power, k and n the user and
subcarrier indices, Pk,n the allocated power to the k-th user on n-th
subcar-rier, ωk the k-th user data rate and γkthe minimum data rate guarantee for
that user. In literature, it is also referred to as the Margin Adaptive scheme, another case of system-centric approach; its objective is still to minimize the overall power subject to the different users’ rate constraints.
The stated problem is computationally intractable, and a suboptimal min-imization solution is possible, based on Lagrangian multipliers (a weighted sum of objective and constraint functions). The problem is dealt using linear programming methods by decoupling subcarrier allocation and bit loading optimization.
Proportional Fairness Scheduling The algorithms discussed so far aim at achieving a specific objective, but no one of them deals with the latency matter. In addition to throughput and fairness, a third element
enters the trade-off, which is the attempt of achieving objectives over time. Let Rk(t) be the k−th user achievable data rate at time t, and Tk(t) the
average throughput for user k, up to time slot t. The Proportional Fairness scheduler selects the k∗ user that maximizes the ratio Rk(t) over Tk(t); in
other words, this means to select the user with the highest instantaneous data rate over its mean rate. The average throughput Tk(t) is updated at
time t+1, according to:
Tk(t + 1) = (1 −t1 c)Tk(t) + 1 tcRk(t) , if k = k ∗ (1 − 1 tc)Tk(t) , otherwise (1.9)
where tc controls the latency; if tc is large, then the latency increases, with
the benefit of higher throughput. Otherwise the latency decreases, the aver-age throughout values change more quickly at the expense of sum through-put.
The scheme aims at balancing the fairness among users in terms of through-put; it does not guarantee fairness in a strict sense. For example, if a session experiences a prolonged period of poor channel condition, it may not get the desired allocation even though the channel improves.
Proportional Fairness can be adapted to an OFDMA system, considering each subcarrier indipendently; here another index, n, will be necessary, to determine the selected subcarrier. Multiple users can transmit on different subcarriers simultaneously. Let Rk(t; n) be the supportable data rate for
user k in subcarrier n at time slot t; then for each subcarrier, the user with the highest ratio is selected for transmission. In this case, the average user throughput is updated as:
Tk(t + 1) = (1 − 1 tc )Tk(t) + 1 tc X n∈Ωk Tk(t) (1.10)
Chapter 2
Multiuser MIMO-OFDM
Systems
2.1
Introduction
Wireless channel, or the physical medium over which a wireless communi-cation takes place, operates through unguided electromagnetic propagation from the transmitter to the receiver stage. Once the transmitted signal is radiated, it can experience several disturbing phenomena (see details in the following), which can result in signal distortions and a consequent re-duction of system performance. Wireless channel is generally time-varying and affected by the multipath propagation; this is a consequence of the time-selective and frequency-selective nature of the channel. In particular, in mobile wireless channels, where the transmitter and/or receiver are not always in a fixed location, a changing environment is expected and expe-rienced, consequently random fluctuations in the signal power strength are evident; moreover, the characteristics of the wireless channel also change randomly with time, making it even more difficult to design reliable com-munication systems. It is known that the obstructions caused by vehicles, buildings and ground in the vicinity of the point of impingement of the electromagnetic wave are not the only cause of this channel variability; the speed of the mobile equipment can also generate intereference, due to the Doppler effect that spreads the signal bandwidth according to the velocity, the carrier frequency and the angle of arrival of the wave. In this Chapter, a detailed description of channel impairments and a possible solution to combat them is summarized.
Figure 2.1: USA - Example of Bandwidths Allocation.
Since there already exists an extended infrastructure of wired networks, the design of protocols to interface wireless and wired networks is today still a challenging topic of research. However, wireless systems with mo-bile users could never compete with wired systems in terms of data rate and reliability: this lower performance along with connectivity and network topology changes over time defines another dynamic aspect of the wireless channel. Hence, the challenge: high-performance wireless networks must be optimized and adapted to the channel variability, requiring integrated and adaptive protocol stack across all layers of the OSI model. Thus, in order to implement the wireless vision, techniques to mitigate channel impairments and to improve the quality and system efficiency together with better means of sharing the limited resources still need investigation.
The scarcity of radio spectrum frequencies leads to the definition of poli-cies of bandwidth allocation. Nowadays, regulatory institutions and units define bandwidths, powers, and tolerated level of interference for wireless systems both over a regional and a global extension. The Federal Communi-cations Commission (FCC) is, for example, the USA responsible allocation resources group, while the European Telecommunications Unit (ETSI) coor-dinates the frequencies allocation in Europe; globally, spectrum is controlled by the International Communications Unit (ITU). These organizations allo-cate bandwidths to the specific operators, and control that systems respect their restrictions. Wireless applications mostly reside between 30MHz and 30GHz; such frequencies are natural for wireless systems since they are not
affected by the curvature of the Earth, require only moderately sized anten-nas (related to the frequency), and can penetrate the ionosphere. Spectrum is a very expensive resource, since licenses are auctioned to the highest bidder, and reasonable returns on investment through its exploitment are aimed.
In the following sections, the properties of the wireless channels and the resulting signal impairments will be discussed; in 2.3, two techniques that provide system performance improvement, deal with signal distortion over wireless channels and exploit various system diversities will be presented as feasible solutions. Finally, implementation examples of resource allocation algorithms in a MIMO-OFDMA wireless system will be given in the next Chapter.
2.2
The Wireless Channel
Wireless environments change over time in unpredictable ways due to the user mobility, posing limitations for reliable high-speed communications. The path between the transmitter and the receiver can vary (also, over time) from a simple line-of-sight to one that is severely obstructed by build-ings, mountains, cars, etc. Measurements from experimental field studies show that in presence of several obstacles between the transmitter and the receiver, power can be in part absorbed and/or scattered. In general, there are more mechanisms affecting radio propagation: refraction, diffraction, reflections. Propagation models suggest that while power decay like r2 in
the vicinity of the transmitter in conditions of free space, at large distance transmit power decays exponentially with distance (or in other words, with a higher exponent than 2). Many physical models can take into account the exact physics of the propagation environment and provide estimates of the power propagation behaviour (for example, ray tracing). However, the complexity and variability of the channel makes it often difficult to consider that channel as deterministic; hence, statistical models are frequently used. Even if less accurate, they can well represent the varying channel conditions from an average point of view, and by using the Vertical Bell Labs Space-Time (V-BLAST) architecture scheme, it could be proved that the system is however able to reach the capacity of the channel.
the dissipation of the radiated power over distance as well as the effects of the propagation channel. The path-loss modeling can predict the received signal mean power for a certain transmitter-receiver (T-R) separation dis-tance: PL(d)|dB = 10log10 PT x PRx ∝ 10nlog10 d d0 (2.1) where d0 is the reference distance determined from measurements close to
the transmitter, d is the T-R separation distance and n is the path loss exponent depending on the environment (typical values for an Urban area are 2-3). The simplified model in 2.1 describes how fast signal power de-grades with distance but does not consider that signal power measurements in different points, at the same distance from the transmitter, may be differ-ent (depending on the surrounding environmdiffer-ental clutter): this discrepancy can be taken into account introducing a log-normal random variable in 2.1, Xσ, which is zero-mean with standard deviation σ (in dB). Xσ describes
the random Shadowing effect. Xσ represents the received power at a fixed
distance from the transmitter as a log-normal random variable.
Figure 2.2: Path Loss, Shadowing and Multipath over Distance In fact, received power tipically presents random variations due to ob-jects that absorbs power in the propagation path, giving rise to a random variation with reference to the path loss at given distances. Large-scale fading, resulting from Shadowing, is specifically due to the three major propagation mechanisms:
objects (compared to the wavelength);
• Diffraction, when signal is obstructed by objects with sharp irregular-ities;
• Scattering, which happens in case of many rough surfaces.
Xσ has empirically confirmed to model the variation in received power levels
in both outdoor and indoor radio propagation environments, in an accurate way. Shadowing depends on the scenario that alterates the natural signal propagation and its effect results in a slow fluctuation of the signal power level (known in literature as Large-Scale Fading or Slow Fading, since vari-ations due to path loss and shadowing occur over relatively large distances between transmitter and receiver). Slow Fading arises when the coherence time of the channel is large relative to the delay requirement of the applica-tion; coherence time, Tc which is a measure of the minimum required time
for the magnitude or phase change of the channel to become uncorrelated from its previous value, is normally used to distinguish fast and slow fad-ing. Slow Fading does not represent a hard obstacle, as it may be estimated, given its slow variability.
On the other hand, Small-Scale fading describes the rapid fluctuations of the received power over a short period of time or short travel distance. Fading is here caused by the interference between two or more versions of the transmitted signal which arrives at the receiver at slightly different times, since each signal replica experiences a different path (and then a different phase shift) from source to receiver. The receiver collects a superposition of multiple copies of the transmitted signal, whose delays also depend on the relative speed between receiver and transmitter (not only on the relative locations of them) and whose gains are modeled as random processes. Small-Scale fading results from a constructive or destructive recombination of multiple copies originated from several paths between the transmitter and receiver and variability depends on the order of the carrier wavelength. Multipath propagation can create small-scale fading effects, due to the presence of reflecting objects and scatterers in the channel that make the environment changing and energy dissipating, in which signals can combine in different ways. The relative speed of the device and the signal bandwidth can influence this phenomenon.
An important feature in the definition of multipath channels is the delay spread, which characterizes the received signal. Delay spread is the time delay between the arrival of the first signal component (LOS or multipath) and the last received replica associated to a single transmitted pulse:
στ =p ¯τ2− ¯τ2 (2.2) being: ¯ τ2 = L−1 P l=0 α2 lτl2 L−1 P l=0 α2 l , and ¯τ = L−1 P l=0 α2 lτl L−1 P l=0 α2 l (2.3)
respectively the compound average delay (by amplitudes αl) and the mean
excess delay. If the delay spread is small compared to the inverse of the signal bandwidth (or the signaling time, then there is little time spreading in the received signal (which can be then neglected, in practical application), resulting in a frequency-flat fading behaviour. However when the delay spread is relatively large, there is significant time spreading of the received signal, leading to a substantial signal distortion. In other words, the channel itself introduces distortion, as it is frequency-selective.
Wireless channels can also be time-selective. From an analytical point of view, it means that the impulse response heavily depends on the instant of transmission of the signal. This time variation arises because either the transmitter or the receiver are moving (then caused by the Doppler spread), but also because clutter conditions can vary over time, determining different combinations of the transmitted signals at the receiver. Time selectivity can be fought by designing the coherence time greater than the symbol interval, in order to consider that the experienced channel gain is not changed during the symbol transmission and the consequent equalization can be relatively simple.
The following picture 2.3 illustrates small- and large-scale fading effects. The signal fades fastly as the receiver slightly moves, but the local average signal changes much more gradually with distance. They represent the effects of the mentioned phenomena.
Large and Small Scale Fadings, that describe fluctuations in the enve-lope of a transmitted signal over a long and a short observation interval (both modeled as random processes), are not the only phenomenona affect-ing signals over wireless channels. In classical multiuser scenarios, users can
Figure 2.3: Example of small and large-scale Fading
experience significant Interference among them for different reasons (among others, coupling effects and Doppler effects). In wireless applications two types of interference can be detected: self- and other users interference. The first is due to the time dispersion: signal can be dispersed over time due to the multipath effect. This causes interference among and with “neigh-boring” symbols and is referred to as Inter Symbol Interference (ISI). Self-Interference can also be caused by the Doppler frequency shift, when the relative velocity of the receiver to the transmitter leads to frequency vari-ations of the received signal (included a bandwidth spread). On the other hand, interference from other users can result both from Cochannel Inter-ference (CCI) and Adjacent Channel Interferce (ACI). CCI can arise, for example, in cellular mobile networks owing to the frequency reuse; it results in crosstalk between users or signal transmitted over the same frequency. Differently, ACI is caused by the transmitted power on adjacent channels, and can result from inadequate filtering, improper tuning or poor frequency control.
In practice, channel gains are modeled in different ways, according to the propagation environment. If every distinct path at the receiver is assumed as the sum of a very large number of independent paths, each arriving with approximately the same delay (and therefore, more or less carrying the same power), channel gains can be considered to follow a complex Gaussian dis-tribution; in particular, they will exhibit a phase, φ, uniformly distributed
in [0,2π] and an amplitude, α, which is Rayleigh distributed: p(α) = α σ2e −α2 2σ2 α ≥ 0 0 α < 0 (2.4)
Equation 2.4 represents a pessimistic case, as it considers all the received paths as equivalent. There could exist cases where one path is particularly strong to be considered as ’dominant’ and deterministic (because only de-pending on the geometry of the problem); let A be the strength of this direct path (and thus A2/2 its power) and K = A2/2σ the ratio of the direct path
and the other paths power or the Rice factor. Here, channel gains do not show zero mean and the resulting distribution is Rician:
p(α) = α σ2e −α2 2σ2e−kI0 √ 2Kα σ u(α) (2.5) where I0 is the first type Bessel function.
2.3
Why MIMO-OFDM Systems?
Figure 2.4: The block diagram of a MIMO-OFDM System
How to deal with Fading, Shadowing, Multipath and Interference is cen-tral to the design of wireless communication systems, if higher performance networks are aimed. Orthogonal Frequency Division Multiplexing (OFDM) and Multiple Input Multiple Output (MIMO) technologies have recently
attracted substantial interest, since they offer the opportunity to properly exploit channel impairments as a vantage, and not as a problem. Here, the channel diversity experienced by users can be adapted to assign subcarriers, choose the modulation and coding rate (known in literature as the ACM, Adaptive Coding and Modulation), and transmit power to users based on their CSI. Next generation wireless networks are supposed to be based on a combination of OFDM and MIMO schemes, together with dynamic resource allocation algorithms, which provide a simple way of exploiting resources minimizing or maximizing certain system parameters. These techniques have been widely adopted in modern cellular standards such as IEEE 802.16 (WiMAX) and the 3GPP Long Term Evolution (LTE) to accomodate high speed mobility and support higher data rates (around 100Mbit/s).
The combination of MIMO and OFDM is the dominant air interface for the most modern broadband wireless systems; as for the name, they combine multiple antennas technology, that increases channel capacity by transmitting through different antennas, and orthogonal frequency-division multiplexing, which divides a radio channel into several closely spaced sub-channels to provide more reliable communications, combating ISI and delay spread. Several precoded data streams can be here transmitted at the same time onto the allocated subchannels and sent over multiple paths: signal transmission exploits multiple domains (space, frequency and time). MIMO scheme can provide power gain by enabling receive antennas to construc-tively combine data streams arriving from different paths and at slightly different times; in other words, MIMO increases the overall efficiency for a given total transmit power by multiplexing parallel channels and taking advantage of antenna diversities. Moreover, as OFDM converts a high-speed data channel into a number of lower-high-speed, parallel and flat channels, MIMO processing can be fastly executed. Thus, these technologies are able to overcome the effects of wireless channels on the transmitted signals and are particularly powerful as further techniques to reduce channel effects are not needed. Infact, MIMO does not attempt to mitigate multipath and OFDM avoids complex equalization (each subcarrier represents a flat-fading channel), but they combat signal impairments from a different point of view. MIMO systems with OFDM signalling in multiuser applications accomo-date users in both frequency and spatial domain, and provide finer granu-larity of resource allocation than simple time or frequency division multiple
access (TDMA, FDMA) systems, typical of the first generations standards fot mobile systems. However, how spatial and frequency resources are allo-cated to active users remains an important problem, as discussed: systems have to cope with not only an increase in the number of users but also in higher data rate requirements with differentiated QoS. MIMO-OFDMA sys-tems also address to these two concerns. Recently, algorithms that jointly allocate spatial and frequency resource to improve EE have been researched; however, the complexity of the joint design may be prohibitive (simple algo-rithms need to be implemented to trade off complexity and performance).
Benefits don’t come for free, however: as these systems use many anten-nas at both the receiver and transmitter stage, more space in the devices and at the Base Station is needed, together with a overhead in circuit im-plementation; as a result, more active circuit components are required, that increase costs for energy consumption both for the transmit and the re-ceive and circuit power. Moreover, additional time or frequency resources are spent on the signalling overhead and computations are required. To this purpose, several projects and commissions, such as the Energy Aware Radio and Network Technologies (EARTH), have been created to develop new generation techniques and architectures to limit the power and energy consumption.
2.3.1
OFDM Technique
OFDM can be referred to as a mean of dealing with the problem of fre-quency selective wireless channels. In section 2.2, multipath was introduced as one of the main causes of signal distortion in wireless channels, for which OFDM represents an efficient approach to combat its effects, together with the mitigation of ISI (as a matter of fact, as data rates increase, multipath can heavily reduce the system performance in single carrier transmissions). OFDM is a frequency-division multiplexing (FDM) scheme used as a digi-tal multi-carrier modulation; specifically, a given frequency-selective broad-band channel is splitted into a large number of flat-fading subchannels, where each subcarrier can be modulated with a conventional modulation scheme at a lower symbol rate. In other words, high data rates require shorter duration symbols; OFDM enables longer duration symbols by di-viding high data rates into several low-rate data streams. The number of
Figure 2.5: Simplified OFDM Block Diagram
subchannels is chosen such that each subcarrier has a bandwidth less than the coherence bandwidth of the channel, so that subcarriers experience rel-atively frequency-flat fading. This enables OFDM to efficiently resist to the effects of frequency selectivity, since rates and power can be adjusted on each subcarrier individually. Moreover, by adapting transmission pa-rameters to the different channel gains (of each subchannel n), frequency diversity can be exploited. However, if the number of subcarriers increases, then the scheduling complexity also increases, since the allocation scheme is dependant on the number of subcarriers.
In Figure 2.5, the serial data stream is splitted into K low-rate paral-lel subchannels; data symbols on every subchannel are then applied to a different modulator, whose carrier frequencies are f0, f1, ...fK. Each
sub-channel has a bandwidth ∆f , resulting in a total bandwidth W equal to K∆f . Hence, the benefit of this approach: symbol duration is extended by a factor K and, as mentioned above, each subcarrier signal is likely to remain unaffected by multipath effect and ISI, as the channel’s delay spread becomes a shorter fraction of a symbol interval. Heavy channel equaliza-tion operaequaliza-tions are then unnecessary, being the signal no longer subject to frequency-selective fading on these low-rate subchannels. The K modulated carriers are then combined to generate the OFDM signal, which can now be sent over the channel. At the receiver stage, OFDM signal is demultiplexed into K frequency sub-bands, and then demodulated; baseband signals are
recombined using a parallel-to-serial convertor and it is proved that in a synchronized OFDM system, the signal received on subcarrier n is indepen-dent from all the others system subchannels (this is a consequence of the orthogonality of the subcarriers). The implementation complexity can be further reduced, by applying a Fast Fourier Transform (FFT)-based imple-mentation, where the subchannel signal can be directly modulated onto the subcarriers in a single step.
In order to mitigate ISI, a guard interval is typically adopted, whose duration is not shorter than the maximum delay spread introduced by the channel. This solution helps to keep the symbols orthogonal. During this in-terval, a prefix is transmitted, consisting in a cyclic extension of the channel IFFT symbols: at the receiver, it will be cancelled as it has no information content. The cyclix prefix makes the convolution between the transmitted signal and the channel pulse response equivalent to the circular convolution. The addiction of the cyclix prefix increases the necessary bandwidth and reduces the bitrate: in practical applications, the duration of the cyclic pre-fix is chosen so that the spread of the bandwidth is the 20% of the duration of the pulse response.
Apart from robustness to ISI and multipath, OFDM enables a flexible users subchannels allocation, that is known as Orthogonal Frequency Di-vision Multiple Access (OFDMA), an attractive way to divide users in the frequency domain by allocating orthogonal subcarriers to different users. In a multiuser scenario, each subcarrier at a particular instant may expe-rience different attenuations in terms of fading, due to the variable nature of the wireless channel. OFDMA provides the opportunity to efficiently exploit the given bandwidth by allocating the subchannels to the users who can utilize them best at that moment (for the duration of the time slot); in fact, adaptive resource allocation can assign to each user its best sub-channels according to different optimality criteria. Here, fading gains are considered independent on each subchannel from user to user. OFDMA can increase SE of the system, but also the power efficiency. Furthermore, in OFDM systems different modulation schemes can be employed and adapted on different subcarriers (or users). For example, users near to the BS may experience better channel quality than the furthest ones; thus, high-order modulation schemes to increase data rates could be used. On the other hand, if users are far from BS or are located in highly crowded areas,
low-order modulations may be preferrable.
In conclusion, Multiuser Diversity (MUD) can also be exploited by allo-cating different portions of bandwidth along with transmission power and modulation type to different users, and provides other Degrees Of Free-dom (DOF) for adaption. Hence, experiments confirm that in OFDMA systems transmit power can be drastically reduced and the system capacity increased. However, in wireless systems fairness is an important concern; OFDMA as described does not guarantee it. This is the reason why adap-tive resource allocation schemes considering fairness are often taken into account. Moreover, OFDM as a technique provides a very high Peak to Average Power Ratio (PAPR) consequently to the indipendence between streams on different subcarriers: High Power Amplifiers (HPA) could not be completely then exploited, as the fluctuation of the envelope of the trans-mitted signal must be considered, in order to avoid the saturation of the device.
2.3.2
MIMO Technology
In order to provide higher spectral efficiency, data-rates and to improve the channel capacity, MIMO has recently been indicated as a very promising scheme and has been employed in 4G wireless networks. It is worth men-tioning that ’MIMO’ does not refer only to the presence of multiple transmit and receive antennas, but also to the possibility to have simultaneous trans-missions of signals and streams onto the same channel (which is known as the spatial multiplexing gain). Since the signal propagates through different paths, MIMO systems can exploit the diversity introduced by the system, by simply using the several streams in order to reduce the deep fading prob-ability. The diversity gain can here be provided by considering that if all the channel gains are independent and complex-gaussian uniformly distributed, then their sum at the receiver stage could be modeled as a χ2 with 2N
degrees of freedom, being N the number of the independent paths. This distribution is known to assume near 0-values with a lower probability than the gaussian distribution: in other words, by combining different streams it is less likely that the received signal has been received in deep fading condi-tions (it would mean that all the independent channels offer simultaneously very low gains). As stated, diversity gain can be achieved by sending signals
that carry the same information through different paths between the trans-mit and receive antennas; on the other hand, multiplexing gain is provided by transmitting independent information streams on parallel spatial chan-nels. The technology achievements are here not provided at the expenses of power or bandwidth; the capacity of a wireless link increases linearly only with the minimum of the number of transmitter or receiver antennas (specifically, the number of users that a single channel can support is lim-ited by the minimum number of antennas at the receiver and transmitter). By the way, the multiplexing gain is also limited by the rank of the channel matrix, which can also be lower than the minimum number of antennas.
Figure 2.6: Multiple Input Multiple Output Scheme
The use of multiple antennas enables Space Division Multiple Access (SDMA), that may be considered as a particular case of MIMO, where the paths between transmit and receive antennas are independent. It is clear that the number of users that can be serviced using a single base transceiver station in a cellular system can be increased, and so the capacity; theoreti-cally, SDMA can be included into any multiple access standard at the cost of a limited increase in system complexity. For instance, by combining TDMA and SDMA two or more users can share the same time slots. But, with multiple antennas, the coverage area can be significantly larger than that of any single antenna system. Moreover, multiple receive antennas can provide power gain through beamforming: it means that the received signals can be combined in order to maximize the SNR, or in other words, combining them so that they could be summed in phase. This gain was already known, before the introduction of MIMO systems, and was achievable in SIMO or MISO architectures.
In Figure 2.6, a communication system employing n transmit and m receive antennas is shown. This system can be represented by the following
discrete time model: y1 .. . ym = h11 · · · h1n .. . . .. ... hm1 · · · hmn x1 .. . xn + n1 .. . nm (2.6) or equivalently, y = Hx + n (2.7) where x represents the n-dimensional transmitted symbols (prefiltered as will be clear in the following), n is the m-dimensional additive white Gaussian noise vector (AWGN), and H is the Cm×n channel matrix, whose
elements hij are the channel coefficients between the j-th transmit antenna
and the i-th receive antenna. These are typically assumed to be zero mean (Rayleigh fading), i.i.d. complex circular Gaussian random variables that model fading gain. It is worth noticing that this representation is valid only for narrowband point to point systems, in flat fading condition; moreover, although the dependence on time is here suppressed, H, x, y, n are in general all stochastic processes. In the model 2.7, fading has also been modeled flat in the time domain and in particular, we will consider matrix H as deterministic and known at the receiver. This hypothesis is not really difficult to meet, as in deterministic and time-invariant channel the CSI can be easily shared between transmitter and receiver though feedback.
As discussed, an important assumption in 2.6 and 2.7 is that the receiver is able to estimate the channel state perfectly. This can be obtained by transmitting pilot signals over the channel in an initial phase. Practically, this is a realistic hypothesis, but viceversa is rarely verified: such a channel knowledge at transmitter stage is rarely available, especially for random and time-varying channels. H can be then considered known at the receiver and in our applications. Although knowledge of the CSI is required, in the majority of literature energy consumption of CSI signalling information is ignored and supposed known; when the energy consumption of signaling information is considered, the CSI accuracy and the total energy efficiency should be traded off.
The above-mentioned model describes a Gaussian channel, whose ca-pacity can be easily computed by a decomposition in a set of parallel and independent subchannels, also known as MIMO - SVD architecture. As every linear transformation can be splitted into the composition of three
Figure 2.7: Parallel Decomposition of the MIMO Channel
operations (a rotation, a scale multiplication and another rotation) by ap-plying the Singular Value Decomposition (SVD) to the channel matrix H, it results:
H = UΛV∗ (2.8) where U ∈ Cm×m and V ∈ Cn×n are unitary matrices, and Λ ∈ Rm×n
is a diagonal matrix (even if it is not generally square), whose diagonal elements λ1 ≥ λ2 ≥ · · · ≥ λnmin are the real and non-negative eigenvalues of
H (being nmin is the min(m,n)). The nmin is the maximum number of data
streams in a n × m MIMO system. Generally, at the transmitter the data vector to be transmitted is multiplexed on the antennas by a specified linear precoder B (a filtering operation); and the received vector is multiplied by the spatial receive filter WH.
A typical approach, to combat the propagation channel impairments, is the Zero-Forcing (ZF) strategy, where the linear precoder B or the spatial receive filter WH are equals to H−1to invert the channel behaviour.
Unfor-tunately, this means that the channels for each user and for each frequency should be perfectly known, with a great burden of calculation and control signalling. By choosing different transmit precoders and spatial receiver filters (specifically, pre-multiplying by V∗ the data and by U∗ the received vector, that is a spatial filtering operation) and for the unitary matrices properties, it results that the MIMO channel can transformed into r (r ≤ min(m,n)) parallel non-interfering SISO gaussian channels:
z = U∗UΛV∗Vx + U∗n = Λs + w (2.9) where w is statistically equivalent to the noise vector n, since U is uni-tary; z is the received signal at the output of the spatial receive filter and
s is the vector of transmitted symbols. Equation 2.9 shows the parallel De-composition of the MIMO-SVD channel, with maximally nmin indipendent
and parallel channels over which simbols are transmitted; nmin is, in that
case, the number of non-zero eigenvalues and the number of significative components of y. In general, the channel gains are the r singular values of the matrix H. The same concept is expressed in Figure 2.7.
Equation 2.9 also explains how a MIMO system in a multi-user (MU) scenario can work, in case of complete knowledge of the channel realisation. As the base station is equipped with Nt antennas and each user with Nr
antennas (considering, for example, the downlink of a OFDMA-MIMO sys-tem), users signals are separated by the implementation of linear pre-coding and receiving spatial filters (specifically, U∗ and V). First, the transmitted signal is prefiltered at the BS:
xk,n = Vk,nsk,n (2.10)
being sk,n the transmitted symbols to user k on subchannel n before the
filtering operation; the received signal, y, is then filtered by the spatial receive filter, resulting in:
zk,n = U∗k,nyk,n = λnsk,n+ U∗k,n(Hk,n
X
j6=k
xj,n+ nk,n) (2.11)
being zk, n as defined above. Equation 2.11 highlights that after the spatial
filtering, the signal is composed of different parts: the information signal (λnsk,n), multiple access interference (generated by the j-th users assigned
to subchannel n) and noise (Uk,n∗ nk,n).
Let us finally consider the Channel Capacity of a MIMO System, or the maximum transmission rate at which information can be reliably transmit-ted over the MIMO channel. According to the model 2.9, there are nmin
available parallel channels, with deterministic and known gains, λ1, λ2, · · · , λnmin
(i.e., the main diagonal coefficients in Λ). It is known that the maximum achievable spectral efficiency on the i-th channel is:
Ci = log2 1 + Piλ 2 i N0 , bit/s/Hz (2.12) and given the statistical independence of the channels, the overall capacity results: Ci(P1, · · · , Pnmin) = nmin X i=1 Ci = nmin X i=1 log2 1 + Piλ 2 i N0 , bit/s/Hz (2.13)
where Pi are the allocated powers on each i-th channel and N0 is the noise
power density. The solution of the maximization problem, considering the constraint of the total transmit power, PT, is the well-known
Waterfill-ing strategy; the waterfillWaterfill-ing solution gives the optimal power distribution among all the nmin independent channels. Generally, the maximum
achiev-able spectral efficiency is expressed in a vectorial notation as:
C = log2det Im+ 1 N0 HRxHH (2.14) where Rx is the correlation matrix of the transmitted signals.
It could be proved that even in case of random channels, channel capacity could also be achieved by using a typical V-BLAST architecture with an accurate choice of the multiplexing matrix. In MIMO-SVD systems, it has been shown that the optimal architecture to reach the capacity of the channel is to use the waterfilling distribution together with representation of the transmit and receive signals on the coordinate system given by the columns of U and V. If the CSI is not available at the transmitter stage, then the multiplexing matrix could be chosen as the unitary identity matrix, leading to the same result as in the MIMO-SVD architecture for high SNR. The difference consists in a non-optimal distribution of power among the parallel channels (waterfilling could not be applied).
In conclusion, even if MIMO and OFDM architectures may seem very similar at first sight, as they both consist of a transformation (specifically, a rotation by an unitary matrix) that converts a non-diagonal channel into a set of parallel and independent subchannels, it should be kept in mind that these operations are very different as in OFDM these matrices are the direct and inverse Fourier transformation, whose elements don’t depend on the channel; in MIMO systems, of course, U and V are strictly dependent on the channel.
