Low complexity multiple antenna transmission solutions for next generation wireless communication systems

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by

Muhammad Hanif

B.Eng., National University of Sciences and Technology, Islamabad, Pakistan, 2009 M.S., National University of Sciences and Technology, Islamabad, Pakistan, 2012

A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of

DOCTOR OF PHILOSOPHY

in the Department of Electrical and Computer Engineering

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Low Complexity Multiple Antenna Transmission Solutions for Next Generation Wireless Communication Systems

by

Muhammad Hanif

B.Eng., National University of Sciences and Technology, Islamabad, Pakistan, 2009 M.S., National University of Sciences and Technology, Islamabad, Pakistan, 2012

Supervisory Committee

Dr. Hong-Chuan Yang, Supervisor

(Department of Electrical & Computer Engineering)

Dr. T. Aaron Gulliver, Departmental Member (Department of Electrical & Computer Engineering)

Dr. Xiaodai Dong, Departmental Member

Dr. Julie Zhou, Outside Member

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Supervisory Committee

Dr. Hong-Chuan Yang, Supervisor

Dr. T. Aaron Gulliver, Departmental Member (Department of Electrical & Computer Engineering)

Dr. Xiaodai Dong, Departmental Member

Dr. Julie Zhou, Outside Member

(Department of Mathematics & Statistics)

ABSTRACT

Two of the most prominent techniques to meet the next generation wireless com-munication system’s demands are cognitive radio and massive MIMO systems. Cog-nitive radio systems improve radio spectrum utilization either by spectrum sharing or by opportunistically utilizing the spectrum of the licensed users. Employing multiple antennas at the transmitter and/or the receiver of the radio can further improve the overall performance of the wireless systems. Massive MIMO systems, on the other hand, improve the spectral and energy efficiencies of currently deployed systems by reaping all the benefits of the multi-antenna systems at a very large scale. The price paid for employing a large number of antennas either at the transmitter or receiver is the high hardware cost. Judicious transmit or receive antenna selection can reduce this cost, while retaining most of the benefits offered by multiple antennas.

In my doctoral research, we have presented both upper and lower bounds on the capacity of a general selection diversity system. These novel bounds are simple to compute and can be used in a variety of different fading environments. We have

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also proposed and analyzed the performance of different antenna selection schemes for both an underlay cognitive radio and a massive MIMO system. Specifically, we have considered both receive and transmit antenna selection in an underlay cognitive radio based on the maximization of secondary link signal-to-interference plus noise ratio. Exact and asymptotic performance analyses of the secondary system with such selections are carried out, and numerical examples are presented to verify the correctness of the analytical results. Several sub-optimal antenna subset selection schemes for both a single-cell and a multi-cell multi-user massive MIMO system are also proposed. Numerical results on the sum rate of the system in different scenarios are presented to verify the superior performance of the proposed schemes over the existing sub-optimal antenna subset selection schemes. Lastly, we have also presented three novel hybrid analog/digital precoding schemes to reduce the hardware and software complexities of a sub-connected massive MIMO system.

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List of Figures

Figure 1.1 An example of a cognitive radio network. . . 2 Figure 1.2 Multiple antennas at the (a) ST (b) SR in an underlay cognitive

radio network. . . 3 Figure 2.1 Ergodic capacity and the bounds versus average SNR in a Weibull

fading environment for η = 2, k = 1, 4, 7, and M = 10. . . 15 Figure 2.2 OPRA capacity and its bounds versus the number of antennas, M ,

for µ = 1, σ = 3, and k = 1, 4, 7. . . 16 Figure 2.3 Ergodic and OPRA capacities with their bounds versus average

branch SNR, γ, for M = 4, L = 2 and k = 2. . . 17 Figure 3.1 An underlay cognitive radio setup with M antennas at the SR. . 25 Figure 3.2 Exact and asymptotic distribution of Z(M ) for PP = 2, λps = 1,

λss = 1 and m = 2. . . 30

Figure 3.3 Outage probability against the maximum secondary transmit power, Pmax, for M = 8, m = 3.5, λss = 10, PP = 1, N0 = 0.025, and

λps = λsp = 20. . . 31

Figure 3.4 Average BEPs of BFSK (C = α = 0.5) against Pmax for M = 16,

m = 3.5, λss = 10, PP = 1, λps = 20, λsp = 20, N0 = 0.025, and

J = 5. . . 33 Figure 3.5 Average BEPs of BFSK (C = α = 0.5) against the number of SR

antennas, M , for m = 3.5, λss = 10, PP = 1, Pmax = 1, λps = 20,

λsp = 20, J = 5, and T = 0.1. . . 35

Figure 4.1 An underlay cognitive radio with an ST equipped with M antennas serving an SR. . . 39

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Figure 4.2 Outage probability of the proposed antenna selection scheme against the normalized interference threshold level, T /PP, for PP = 0 dBm,

Pmax = −3 dBm, N0 = −20 dBm, J = 5, γT = 0.003PP and

M = 2, 4, 8 ST antennas. . . 47 Figure 4.3 Outage probability of the proposed TAS scheme against Pmax/N0

for PP = 0 dBm, N0 = −20 dBm, γT = 13 dBm, J = 5, T = −10

dBm and M = 2, 4, 8 ST antennas. . . 48 Figure 4.4 Performance comparison of the outage probabilities against T /PP

of the proposed scheme with optimal TAS employing discrete and continuous power adaptation for M = 2 ST antennas and J = 1, 5, 25 power levels with Pj = jPmax/J. . . 49

Figure 4.5 Ergodic capacities of the proposed and optimal TAS schemes against Pmax/N0 for M = 2, 4, 8 ST antennas, λps = λsp = 10, λss = 7,

T = −10 dBm, PP = 0 dBm, N0 = −20 dBm and J = 5 power

levels with Pj = jPmax/J. . . 51

Figure 4.6 Throughput of different schemes against the normalized peak ST transmit power for M = 4 ST antennas and Pj = Pmin+_J−1j−1(Pmax−

Pmin) with Pmin = Pmax/10. . . 53

Figure 4.7 Capacity of the proposed, MMI, MUC, MMSLIR, and MDS schemes for M = 4 ST antennas. . . 54 Figure 5.1 An example of a multi-cell MU-MIMO system. . . 61 Figure 5.2 Illustration of linear and circular arrays of antennas. . . 71 Figure 5.3 Comparison of Rsum for the proposed schemes against other

an-tenna subset selection schemes when K = 8 users are served by 8 out of M BS antennas. . . 73 Figure 5.4 Comparison of Rsum for the proposed schemes against other

an-tenna subset selection schemes for K = 8 and M = 16. . . 74 Figure 5.5 Comparison of Rsum for the trace-based AS scheme and others

against K for M = 1024 BS antennas in i.i.d. and equally cor-related Rayleigh fading environment with correlation coefficient ρ for different average SNR values. . . 75 Figure 5.6 Rsum of different antenna subset selection schemes for Kv = 6

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Figure 5.7 Comparison of Rsum for the proposed schemes for K = 1, Kv = 6,

and M = 12. . . 77 Figure 5.8 Comparison of Rsum for the low complexity schemes against K for

Kv = 6K and M = 512 in i.i.d. Rayleigh fading channel with

average SNR of 0 dB. . . 78 Figure 5.9 Sum rate of the low complexity AS schemes against the number of

BS antennas, M , arranged in a circle in an exponentially correlated Rayleigh fading environment with an average SNR of 10 dB. . . . 79 Figure 5.10 Rsumof the low-complexity schemes against the maximum

correla-tion coefficient, ρmax, in a correlated Rayleigh fading environment

with an average SNR of 10 dB. . . 80 Figure 5.11 Comparison of Rsumwith optimal transmit power allocation for the

low complexity schemes against K for Kv = 6K and M = 512 in

i.i.d. Rayleigh fading channel with average SNR of 0 dB. . . 82 Figure 6.1 An illustration of a fully-connected architecture for hybrid precoding. 84 Figure 6.2 An MU-MIMO system where the BS is equipped with R

phased-arrays. . . 86 Figure 6.3 Comparison of the proposed sequential-EGT and the interference

minimizer schemes. . . 91 Figure 6.4 An illustration of assignment of phased-arrays for the

sequential-EGT scheme. . . 93 Figure 6.5 An illustration of assignment of phased-arrays for the best-user

EGT scheme. . . 94 Figure 6.6 An illustration of assignment of phased-arrays for the

sequential-EGT and best-user sequential-EGT schemes when K = 2 < R = 4. . . 95 Figure 6.7 Rsumagainst the number of phased-arrays, R, in an i.i.d. Rayleigh

fading environment with an average SNR of 10 dB. The number of scheduled users is K = 8, and each phased-array has L phase shifters. . . 97 Figure 6.8 Rsumof the proposed schemes in an i.i.d. Rayleigh fading

environ-ment against the average SNR of individual channel. The number of phased-arrays is R = 8, and each phased-array has L = 16 phase shifters. . . 98

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Figure 6.9 Rsum of the proposed schemes against the number of scheduled

users, K, in an i.i.d. Rayleigh fading environment with different average SNRs. The number of phased-arrays is R = 16, and each phased-array has L = 32 phase shifters. . . 99 Figure 6.10 Comparison of the EGT-based schemes proposed for a fully-connected

architecture and a sub-connected architecture. . . 100 Figure 6.11 Effect of quantization on the sum rate for different number of

phase-shifters, L. . . 101 Figure 6.12 Sum rate of the min-trace-based AS and SVD-based EGT scheme

against average SNR in an i.i.d. Rayleigh fading environment for M = 32 and K = 8. . . 103 Figure 6.13 Sum rate of the min-trace-based AS and SVD-based EGT scheme

against average SNR in an i.i.d. Rayleigh fading environment for M = 128 and K = 8. . . 104 Figure 6.14 Sum-rate of the min-trace-based AS and SVD-based EGT schemes

against the average SNR in an i.i.d. Rayleigh fading environment for R = K = 8 and different values of L. . . 105 Figure 6.15 Performance comparison of the SVD-based EGT and

min-trace-based AS schemes in an i.i.d. Rayleigh fading environment with different average SNRs for K = 8 and L = 4. . . 106 Figure 6.16 Performance comparison of the SVD-based EGT and

min-trace-based AS schemes in an i.i.d. Rayleigh fading environment with different average SNRs for K = 8 and L = 16. . . 107 Figure 6.17 Sum-rates of the low-complexity schemes against R when K = R

and L = 4. . . 108 Figure 6.18 Sum-rates of the low-complexity schemes against R when K = R

and L = 16. . . 109 Figure 6.19 Sum-rate comparison of the min-trace-based AS scheme with the

SVD-based EGT scheme against individual link SNR for different levels of phase quantization. . . 110 Figure 6.20 Sum-rate comparison of the min-trace-based AS scheme with the

SVD-based EGT scheme against the number of RF chains for dif-ferent levels of phase quantization. . . 111

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List of Abbreviations

BEP Bit error probability

bps Bits per second

BS Base station

CDF Cumulative distribution function CoMP Coordinated multiple point

CoMP-CSB CoMP-Coordinated scheduling/beamforming CoMP-JPT CoMP-Joint processing/transmission

CR Cognitive radio

CSI Channel state information DPC Dirty paper coding

EGC Equal gain combining EGT Equal gain transmission FDD Frequency division duplex GSC Generalized selection combining

GSMuD Generalized selection multi-user diversity i.i.d. Independent and identically distributed ICI Inter-cell interference

LSAS Large scale antenna system LTE Long term evolution

MGF Moment generating function MIMO Multiple input multiple output MISO Multiple input single output MMSE Minimum mean square error MRC Maximum ratio combining MRT Maximum ratio transmission MU-MIMO Multi-user MIMO

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ORA Optimal rate allocation PDF Probability density function PMI Precoding matrix indicator

PR Primary receiver

PT Primary transmitter

RV Random variable

SC Selection combining

SIMO Single input multiple output

SINR Signal-to-interference and noise ratio SIR Signal-to-interference ratio

SNR Signal-to-noise ratio SR Secondary receiver ST Secondary transmitter TAS Transmit antenna selection TDD Time division duplex

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Mathematical Notations

AH _{Hermitian transpose of matrix A}

A 0 A is positive semidefinite argmax Argument of the maximum argmin Argument of the minimum B(·, ·) Beta function

CN (·, ·) Complex Gaussian distribution δ(·) Dirac delta function

δij Kronecker delta function

E[·] Mathematical expectation

2F1(·, ·) Gauss hypergeometric function

Γ(·) Euler Gamma function Γ(·, ·) Incomplete gamma function Ik Identity matrix of size k × k

Kν(·) Modified Bessel function of the second kind

max Maximum value of a function min Minimum value of a function

Q(·) Right tail probability function of a standard normal random variable tr(·) Trace of a matrix

u(·) Unit step function

U (·, ·, ·) Confluent hypergeometric function of the second kind (x)+ _{Equals max {x, 0}}

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ACKNOWLEDGEMENTS

I would like to take this opportunity to thank all those who contributed to the success of this work. I am deeply indebted to my parents for their help and prayers for my success throughout my life. I am also thankful to my supervisor, Dr. Hong-Chuan Yang, for supporting me both financially and academically during my graduate studies. He has been a constant source of knowledge, inspiration and encouragement for me. I am also very grateful to Dr. Mohamed-Slim Alouini, without whom I would not be able to accomplish this demanding yet rewarding task. Acting as my mentor, he helped me throughout my doctoral program.

I am also deeply grateful to Dr. T. Aaron Gulliver and Dr. Xiaodai Dong for serving as the department members in my supervisory committee. Their comments and suggestions have greatly improved my dissertation. I also really appreciate the time and effort spent by Dr. Julie Zhou and Dr. Witold Krzymien in reading my dissertation and providing some valuable comments. I am also appreciative of Dr. Wu-Sheng Lu who taught me a course on optimization. The knowledge I gained from him helped me derive some of the results given in Chapter 2. He also contributed to an algorithm that I used to compare the performance of a proposed hybrid precoding scheme with the interference minimization scheme explained in Chapter 6. Lastly, I would also like to thank Gary Boudreau, Edward Sich and Hossein Seyedmehdi from Ericsson Canada Inc. for their time and valuable comments on the proposed schemes for massive MIMO systems. Their contributions to the material presented in Chapter 5 and Chapter 6 are highly appreciated.

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Introduction

Researchers are looking for novel ways to improve the resource utilization of cur-rently deployed wireless communication systems to meet the ever-growing demands of higher throughput and better quality of service. The next generation wireless communication systems aim at improving the spectrum utilization to cope with the unprecedented demands of high quality communication. Cognitive radio networks and massive MIMO systems have emerged as two prominent techniques to improve the spectrum utilization [1–5]. Below is a brief description of these two key enabling technologies of the next generation communication systems.

1.1 Cognitive Radio Network

The idea of cognitive radio was first introduced by Mitola in 1999 [6]. Since then, a tremendous amount of research is being carried out to analyze the performance of different cognitive radio setups. Cognitive radios improve spectrum utilization by allowing the secondary cognitive users to use the spectrum licensed to the primary users, which has been reported to be highly under-utilized [7], while guaranteeing the performance of the primary users is not seriously affected. Figure 1.1 shows an example of a cognitive radio network where a secondary network coexists with the primary network.

Cognitive radio implementation can be classified broadly into two paradigms: Interweave and underlay implementation [2]. In the interweave implementation (also referred to as opportunistic spectrum access), the cognitive user first tries to detect the availability of spectrum holes (unoccupied band) in the licensed band through

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PR PT SR ST Primary Network Secondary Network

Figure 1.1: An example of a cognitive radio network.

spectrum sensing. Upon spectral hole detection, the cognitive users can communicate with one another. The main challenging aspect of this mode of communication is the spectrum sensing by the cognitive users, which is not only very challenging but also consumes a lot of power.

In the underlay implementation, the secondary users can access the spectrum simultaneously with the licensed (or primary) users. The secondary transmission, however, should be designed to ensure that the interference to the primary receivers is below a predetermined threshold. Meanwhile, the performance of the secondary network is doubly affected by the primary network. The first deterioration is caused by the interference arising from the primary transmitter, and the second is due to the strict interference constraint at the primary receiver [2].

Researchers have been looking for different ways to improve the secondary link performance while ensuring the interference caused by the secondary transmitter (ST) to the primary receiver (PR) is below a certain level. For example, adapting the transmit power at the ST can effectively limit the interference caused to the PR [8]. Although these systems reduce the probability of outage, they require variable power amplifiers, which are both expensive and inefficient as compared with fixed power amplifiers. Interference to the PR can also be effectively controlled by employing

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multiple antennas at the ST [9–12]. With beamforming transmission or antenna selection at the ST, the interference at the PR can either be eliminated or effectively contained within a limit. Multiple antennas at the secondary receiver (SR), if properly utilized, also improve the secondary link performance in the presence of interference from the primary users. Figure 1.2 shows two scenarios in which secondary users are equipped with multiple antennas. Sub-figure (a) shows the case where the ST is equipped with M antennas, while sub-figure (b) depicts the case where the SR employs multiple antennas to improve secondary link performance.

PR PT SR ST g1 gM hM h1 h0 _PR PT SR ST g1 gM h1 hM h0 (a) (b)

Figure 1.2: Multiple antennas at the (a) ST (b) SR in an underlay cognitive radio network.

Unfortunately, multiple antennas result in a high hardware cost as each RF-chain contains various components (like DAC/ADC and amplifiers) that are not only expen-sive but also consume a lot of circuit power. This problem becomes more serious for very large-scale antenna systems. Equipping the transmitter/receiver with a limited number of RF chains and selecting a proper antenna subset can significantly reduce the hardware cost and software complexity. Judicious antenna selection reduces the hardware cost without incurring a dramatic performance loss [13–16]. In cognitive radios, unlike in traditional wireless communication systems, antenna selection does not solely depend on the secondary link channel conditions. Rather, it also depends on the interference caused by the ST to the PR and the primary transmitter (PT) to the SR. Specifically, for the transmit antenna selection, the ST has to consider the channel gains from each antenna to both the PR and the SR to get the best performance. While, for multiple antennas at the SR, the interference caused by the ST to the PR cannot be controlled by selecting antennas at the SR. However, they

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can be used to improve the secondary link performance. Since the secondary link performance depends on the interference caused by the PT to the SR, the antenna selection at the SR must consider the channel gains both from the PT and the ST to the individual receive antenna to achieve the best performance. Chapter 3 and Chap-ter 4 respectively contain a detailed discussion on the receive and transmit antenna selection strategies in an underlay cognitive radio.

1.2 Massive MIMO Systems

The concept of massive MIMO systems, another key enabling technology for the next generation wireless communication systems, was introduced by Marzetta [17, 18]. A massive MIMO system (also referred to as very large MIMO and large-scale antenna system (LSAS)) is a multi-user MIMO system where the base station (BS) in a cell is equipped with a large number of antennas, and it serves a relatively small number of users simultaneously. Such systems support significant improvements simultaneously in both spectral and power efficiencies [17–19].

The idea of using multiple antennas for improving the spectral utilization of con-ventional communication systems has been extensively studied [13–16, 20, 21]. In particular, the capacity of a point-to-point MIMO system scales linearly with the minimum number of transmit and receive antennas [13–15]. In a cellular setting, where the mobile user equipment has a limited number of antennas, multi-user MIMO (MU-MIMO) system was proposed to achieve high throughput [22]. MU-MIMO sys-tems reap the benefits of a MIMO system by serving multiple users simultaneously with a multi-antenna BS. The initial work related to MU-MIMO did not consider the interference caused by the neighbouring cell transmission. Since the throughput of cell-edge users is severely affected by the co-channel inter-cell interference, the idea of network MIMO emerged [23–26] to mitigate the inter-cell interference.

In contrast with point-to-point MIMO systems, MU-MIMO systems require chan-nel state information at the BS to mitigate multiuser interference as the users cannot eliminate it due to the lack of inter-user cooperation [22]. Moreover, the dirty paper coding (DPC) based precoding, which achieves the best performance, is highly non-linear and computationally intensive [27–30]. On the other hand, non-linear beamform-ing techniques, such as zero-forcbeamform-ing (ZF) and minimum mean square error (MMSE) beamforming, perform much worse than the DPC-like precoding schemes, especially when the BS has only a few antennas [22, 31]. Recently, researchers have shown that

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the BS can use linear precoding schemes to achieve near-optimal performance in a massive MIMO setting [5]. In particular, [5] showed that ZF beamforming achieves almost the same sum-rate as the one achieved by ideal interference-free and dirty paper precodings in a massive MIMO setting.

While achieving huge improvements in system throughput, massive MIMO sys-tems face several challenges. For example, in order to precode the data, the BS requires knowledge of the downlink channel state information. For conventional fre-quency division duplex (FDD) based communication, this would require the trans-mission of a large number of pilot signals and a feedback of a large amount of data from every user to the BS. As a result, a lot of useful and expensive resources will be utilized just for the channel estimation and feedback. This problem, however, can be effectively resolved by operating in a time division duplex (TDD) fashion and estimating the downlink channel response through uplink pilot transmission by the users [5, 17, 18, 32, 33]. Another benefit of TDD is that the system becomes fully scal-able with respect to the number of service antennas as the uplink channel estimation overhead is independent of the number of BS antennas [18].

High hardware complexity is a much more serious challenge that massive MIMO systems face. Conventional implementations require an independent RF chain for each antenna. When the number of antennas becomes large, as in massive MIMO systems, the hardware complexity becomes prohibitively high. At the same time, operating a large number of RF chains will necessarily consume a lot of power dur-ing their operation. Furthermore, the baseband power consumption also increases significantly due to the large scale computationally intensive signal processing at the transmitter and/or the receiver. This problem becomes more prominent for high data rate communication, such as for mmWave communication systems [5, 32, 34].

Researchers have proposed different solutions to alleviate the problem of high hardware cost and power consumption for massive MIMO systems. One category of solutions deals with this problem by designing hardware friendly precoding schemes [35–37]. These schemes reduce the power consumption of the linear power amplifier by ensuring that the peak to average ratio of the transmitted signal is near or equal to unity. These solutions, however, result in degraded system performance and high computational complexity in the baseband. To deal with the high computational complexity of massive MIMO systems, some researchers have proposed using low-complexity precoding schemes [5,38,39]. These schemes, like matched-filter precoding, avoid the inversion of a large scale matrix, which is used in ZF beamforming or MMSE

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precoding, at the expense of system performance.

Another way of coping with the high hardware complexity and power consump-tion is to reduce the number of active RF chains. Since massive MIMO systems serve a small number of users simultaneously, the number of active RF chains can be re-duced significantly to control the hardware complexity and RF power consumption. Note that the minimum number of RF chains required to serve K users simultane-ously without inter-user interference is K [40]. Reducing the number of transmitting chains reduces not only the power consumption by the power hungry RF components but also the size of matrices used in the precoding design. This in turn reduces the baseband computational complexity and power consumption. Two prominent techniques amongst this category of solutions are given below.

Antenna subset selection In this scheme, the BS selects a subset of antennas to serve the scheduled users simultaneously. In order to remove the inter-user interference, the number of active RF chains should be at least equal to the number of the scheduled users [13–16, 41].

Hybrid analog/digital precoding Hybrid precoding is another method to reduce the hardware and software complexities of massive MIMO systems. Instead of using a complete RF chain for each antenna, the BS uses only a limited number of RF chains, and each RF chain feeds its signal to a phased-array. The output of the phased-arrays is then fed to the antennas [34].

Unfortunately, the optimal antenna subset selection requires an exhaustive search over all the antenna subsets, which makes it too computationally intensive to be implemented. Also, finding the optimal hybrid precoding scheme solution is still an open problem. Therefore, researchers have proposed different sub-optimal schemes for both antenna subset selection and hybrid precoding. In this dissertation, we propose several low-complexity schemes for both antenna subset selection and hybrid precoding and compare their performance with existing schemes in different scenarios. These two solutions will be discussed in a greater detail in the upcoming chapters.

1.3 Dissertation Outline

The main focus of the dissertation is on the low-complexity multiple antenna trans-mission solutions for the next generation wireless communication systems. In partic-ular, we propose to use only a limited number of RF chains to reduce the hardware

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cost, power consumption and baseband signal processing complexity. Since we have proposed to benefit from selection diversity systems, we dedicate Chapter 2 to the performance analysis of selection diversity system. Due to practical constraints, such as the amount of interference caused by the secondary transmitter to the primary receiver, the transmitter might not select the antenna that results in the highest re-ceived signal-to-noise ratio. Therefore, we consider the kth best selection diversity system, where the antenna that results in the kth highest signal strength at the re-ceiver side is selected for communication. In particular, we provide some fundamental results related to the capacity of the kth best path selection in a generalized fading environment in Chapter 2.

The following two chapters, Chapter 3 and Chapter 4, consider an underlay cognitive radio system. Chapter 3 investigates the performance of an optimal power adaptive receive antenna selection scheme for underlay cognitive radio networks and presents some analytical results regarding the outage probability and bit error rates of the secondary network. Chapter 4 considers two transmit antenna selection schemes, the optimal antenna selection with continuous power adaptation and a suboptimal antenna selection scheme with discrete power adaptation. Closed-form expressions for different performance metrics, such as the outage probability and ergodic capacity of the secondary network, are also presented and confirmed by Monte-Carlo simulations. After discussing the performance of selection diversity systems in an underlay cognitive radio, we focus on a key-enabling technology candidate for the next gen-eration wireless communication systems: Massive MIMO. In particular, we consider the problem of antenna subset selection in a massive MIMO system in Chapter 5. We consider both a single-cell and a multi-cell massive MIMO system and present three sub-optimal antenna subset selection schemes. In Chapter 6, we discuss the hybrid analog/digital precoding for a massive MIMO system. We also present three different hybrid precoding schemes for a multi-user massive MIMO system based on equal gain transmission.

Lastly, Chapter 7 provides the concluding remarks and points out some future research directions. Most of the technical content presented in this dissertation is either already published in or submitted to different journal and conference papers. These papers are listed at the end of the dissertation before the references.

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Chapter 2 Capacity Bounds for the kth Best

Path Selection over Generalized

Fading Channels

2.1 Introduction

Diversity combining techniques can mitigate the performance degradation of wireless communication systems due to multi-path fading. Selection combining (SC) is one of the low complexity diversity combining schemes where the diversity path with the highest signal-to-noise ratio (SNR) is selected for signal transmission or reception [13, 14]. In a practical implementation environment, however, the kth best path, instead of the best path, might be selected. For example, in a cognitive radio environment with underlay implementation, the secondary transmitter may transmit only when the amount of interference introduced to the primary receiver is below a certain level. When the antenna selection is employed at the secondary transmitter, it may not select the best path due to excessive interference to the primary receiver [42]. Selection of the kth best path may also occur in relay networks when the best path is not available due to scheduling or load-balancing conditions [43, 44]. Moreover, in the generalized selection multiuser diversity (GSMuD) scheme, the first k best users, instead of only the best one, are selected based on their SNRs for simultaneous channel access [45]. Therefore, it is of great practical interest to study the impact of selecting the kth best path on the system performance.

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communication systems [13, 14]. Unfortunately, the ergodic capacity, being the ex-pectation of instantaneous capacity over the distribution of received SNR, is often difficult to be expressed in a simple closed form for a generalized fading environment, especially when diversity combining is used. A recent attempt based on the moment generating function (MGF) results in single integration of Meijer’s G and Fox’s H functions but typically does not lead to a closed form expressions for the ergodic capacity [46, 47]. Therefore, to avoid time consuming numerical integration and sub-sequent analysis, researchers typically resort to tight bounds [48–51]. For example, the authors in [48] presented bounds on the ergodic capacity for a dual hop relay system, whereas [49] presents capacity bounds for MIMO channels under Nakagami-m fading. These bounds, however, cannot be directly extended to analyze selection diversity schemes. For selection combining systems, [50] presented some bounds on the ergodic capacity of a dual branch SC in a correlated Nakagami-m environment by truncating an infinite sum representation of the capacity. However, these bounds can-not be generalized to the kth best path selection in a generalized fading environment easily. On the other hand, [51] presented some simple bounds on the ergodic capac-ity, but these bounds are applicable for only the best path selection in a Rayleigh fading environment. To the best of authors’ knowledge, no bounds are available for the ergodic capacity of the kth best path selection over general fading channels.

Ergodic capacity of a communication system is closely related to its optimal rate allocation (ORA) capacity. The terminology of ORA capacity is used when both the transmitter and the receiver know the channel state information, and the transmitter adapts its transmission rate depending on the channel conditions [13, 14]. Mathe-matically, the ergodic and ORA capacities are similarly computed as the expectation of instantaneous channel capacity. If the transmitter also adapts its transmit power along with its transmission rate, it achieves a higher throughput. And the resultant capacity is referred to as optimal power and rate allocation (OPRA) capacity.

The closed-form expressions for OPRA capacity is very hard to obtain even for simple wireless communication systems. Therefore, we have only a limited number of results available for the OPRA capacity. Amongst them, [52] presented a closed-form expression of OPRA capacity for a Rayleigh fading environment. Recently, [53] presented the closed-form expression of OPRA capacity in terms of complementary error function for log-normal fading environment. The closed-form expressions for the capacity become even more challenging to obtain for selection diversity system. Consequently, there are very few results available for such systems. Amongst them,

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[52] presented a closed-form expression of OPRA capacity for the best path selection in a Rayleigh fading environment. The authors in [54] also presented a closed-form expression of OPRA capacity for dual-branch selection combining in a generalized Gamma fading environment. However, the approaches used in [52, 54] cannot be easily extended to the kth best path selection. The moment generating function (MGF) based approach also requires numerical integration to calculate the OPRA capacity, especially for the kth best path selection [55].

Numerical integration methods can be very time consuming and do not give any insight to the behaviour of the capacity. As such, researchers typically resort to the approximations or bounds on the capacity to characterize the performance of a diversity system in a generalized fading environment. Therefore, in this chapter, we provide upper and lower bounds on both the ergodic capacity and OPRA capacity for the kth best path in a generalized fading environment. Moreover, our bounds have very simple analytic expressions which allow their fast evaluation.

The chapter is organized as follows. First of all, the system model is described in Section 2.2. The lower and upper bounds on the ergodic and OPRA capacities are presented in Section 2.3. Different applications highlighting the usefulness of the bounds are illustrated in Section 2.4, which is followed by some concluding remarks. Lastly, appendices at the end of this chapter contain the proofs of the theorems and proposition introduced in Section 2.3.

2.2 System Model

Consider a system where the transmitter chooses the kth best path out of M inde-pendent and identically distributed (i.i.d.) paths for transmitting information to the receiver. Let hi denote the channel gain of the ith path, where i = 1, 2, · · · , M . The

received signal can be represented as

y = h(M −k+1)s + w, (2.1)

where s is the complex valued unity power transmit symbol, w represents the zero mean complex white Gaussian noise with variance E[wwH_{] = σ}2_{, and h}

(M −k+1) is

the flat fading channel gain of the kth best channel, where k = 1, 2, · · · , M . We use X(M −k+1) to represent the instantaneous received signal-to-noise ratio (SNR).

Mathematically, X(M −k+1) =

|h(M −k+1)|2

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It is well known that the probability density function (PDF) of X(M −k+1) can be

written in terms of PDF, f (γ), and cumulative distribution function (CDF), F (γ), of Xi as [56] fX(M −k+1)(γ) = k M k f (γ)F (γ)M −k(1 − F (γ))k−1. (2.2)

The (normalized) ergodic and OPRA capacities of the chosen path, respectively de-noted by Ck and eCk, are given by [14]

Ck= Z ∞ 0 log2(1 + γ) fX(M −k+1)(γ)dγ, (2.3) and e Ck = Z ∞ γ0 log2 γ γ0 fX(M −k+1)(γ)dγ, (2.4)

where γ0 is the SNR threshold below which no data is transmitted. This threshold

can be computed by the following constraint. Z ∞ γ0 1 γ0 − 1 γ fX(M −k+1)(γ)dγ = 1. (2.5)

In general, it is very difficult to obtain closed form expressions for Ck and eCk over

generalized fading channels. Simple yet tight bounds on the capacities can help a designer to quickly predict the system performance.

For a variety of fading channels, Xi can be modeled as a random variable (RV)

with generalized Gamma distribution. The PDF, f (γ), and CDF, F (γ), of generalized Gamma distributed random variable are given by [57]

f (γ) = λγ λc−1 βc_Γ(c)e −γλ_β u(γ), (2.6) and F (γ) = 1 −Γ(c, γ λ_/β) Γ(c) u(γ), (2.7)

respectively, where the parameters λ, c and β are all positive reals, Γ(c) =R₀∞tc−1_e−t_dt

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and u(x) is the unit step function. The generalized Gamma distribution specializes to channel gain distribution for Rayleigh, Weibull, and Nakagami-m fading channels for different values of c and λ [57].

Log-normal distribution is another important fading model used to describe the large scale fading [14] and small scale fading effects in ultra-wide band communication systems [58]. Log-normal fading model is also used to model scintillation effect in free space optical communication systems subjected to weak tubulence [59]. The PDF, f (γ), and CDF, F (γ), of a log-normal distributed random variable are [13]

f (γ) = √ ξ 2πσγ exp " −(10 log10γ − µ) 2 2σ2 # u(γ), (2.8) and F (γ) = 1 − Q 10 log₁₀γ − µ σ u(γ), (2.9)

respectively, where ξ = 10/ ln 10, µ is the mean, σ is the standard deviation of 10 log10γ, and Q(x) =

R∞ x

e−y2/2

√

2π dy is the right tail probability function of a standard

normal random variable.

2.3 Bounds on the Capacities

The following theorems define the bounds on ergodic and OPRA capacities of the kth best channel when the statistics of Xi have certain properties.

Theorem 1. If the survival function of Xi, F (γ) = 1 − F (γ), is log-concave then Ck

can be upper bounded as

Ck ≤ uk = log2 1 + F−1 1 − eH(k−1)−H(M )

, (2.10)

where H(k) is the kth Harmonic number defined as

H(k) = (Pk

j=11j, if k ∈ Z+;

0, otherwise, (2.11)

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And if 1/F (2γ_{− 1) is convex then C}

k can be lower bounded as

Ck≥ log2 1 + F−1 1 − k M = lk. (2.12)

Proof is given in Appendix A.

Theorem 2. If f (2γ_{) is log-concave then e}_C

k can be bounded as elk(γ0) ≤ eCk≤ euk(γ0), (2.13) where elk(γ0) = (1 − FX(M −k+1)(γ0)) log2     F−1 (1 − k/M )1−FX(M−k+1)_g(M,k,γ (γ0) 0) γ0     , (2.14) euk(γ0) = (1 − FX(M −k+1)(γ0)) log2 F−1 _{1 − e}h(M,k,γ0) γ0 ! , (2.15) h(M, k, γ0) = H(k − 1) − H(M ) +PM −k_r=0 (−1)r M −k r 1+F (γ0)r+k((k+r) ln[F (γ0)]−1) (k+r)2 1 − FX(M −k+1)(γ0) , (2.16) g(M, k, γ0) = 1 − k M − 1 k BF (γ0)(M − k, k), (2.17) and Bz(a, b) = Rz 0 t

a−1_{(1 − t)}b−1_{dt is the incomplete beta function.}

Proof is given in Appendix A.

Proposition 1. The bounds in (2.10) and (2.12) hold for the channels with general-ized Gamma distributed SNR with parameters λ ≥ 1 and c ≥ 1/λ.

Proof is given in Appendix B.

Proposition 2. The bounds in (2.13) hold for the channels with generalized Gamma or log-normal distributed SNR.

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Proof: For the generalized Gamma distribution, we have d2

dx2 (ln f (2

x_{)) = −}λ2ln(2)22λx

β < 0, (2.18)

and, for the log-normal distribution, we can easily show that d2

dx2 (ln f (2

x_{)) = −}100 log10(2)2

σ2 < 0. (2.19)

Remark

For the generalized Gamma and log-normal distributions, F−1_{(p) can easily be}

cal-culated using MATLAB’s in-built functions gaminv() and erfinv(), respectively.

2.4 Examples and Applications

2.4.1 Selection Combining over Weibull Fading Channels

Weibull distribution is used to characterize both the channel amplitude and the SNR for the mobile radio systems operating in 800/900 MHz band [13]. The PDF of the received SNR for Weibull fading with the shape parameter, η, and scale parameter, α, can be obtained by setting c = 1, λ = η, and β = αη _{in (2.6) [13, 57]. The inverse}

CDF, F−1_{(p), of Weibull RV can be calculated as F}−1_{(p) = α (− ln(1 − p))}1/η_{. If the}

Weibull fading parameter, η, is greater than or equal to 1 then the bounds in (2.12) hold.

Figure 2.1 shows the ergodic capacity for the kth best channel out of 10 i.i.d. Weibull fading channels along with its upper and lower bounds. Here the average SNR of each channel, γ, is computed as γ = αΓ(1 + 1/η). Also, the exact ergodic capacity is calculated by numerical integration of (2.3). It is clear from the figure that the bounds are tight for a wide range of system parameters.

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-50 0 5 10 15 20 25 1 2 3 4 5 6 7 8 9 10

Average SNR per Branch, γ (dB) Ck , lk and uk (bps/Hz) Upper bound, uk Exact, Ck Lower bound, lk -4 -2 0 0.5 1 1.5 k = 7 k = 1 k = 4 -5 -3 -1

Figure 2.1: Ergodic capacity and the bounds versus average SNR in a Weibull fading environment for η = 2, k = 1, 4, 7, and M = 10.

2.4.2 Selection Combining over Log-Normal Fading

Chan-nels

Now consider a log-normal fading environment in which the transmitter chooses the kth best antenna based on the channel gain. Figure 2.2 shows the exact OPRA capac-ity with its upper and lower bounds for different values of k. Here the exact OPRA capacity is computed using numerical integration. It is clear from the figure that the bounds become tighter with increasing k. Also, the upper bound, _euk(γ0), happens

to be tighter than the lower bound, elk(γ0), for the simulated scenario. However, the

tightness of the bounds euk(γ0) and elk(γ0) depends on the simulation parameters as it

will be clear from Figure 2.3, where the lower bound, elk(γ0), turns out to be tighter

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0 5 10 15 20 25 30 35 40 45 50 0.5 1 1.5 2 2.5 3

No. of Transmit Antennas, M eCk , el(γk 0 ) and euk (γ0 ) (bps/Hz) Upper bound,euk(γ0) Exact, eCk Lower bound, elk(γ0) k = 1 k = 4 k = 7

Figure 2.2: OPRA capacity and its bounds versus the number of antennas, M , for µ = 1, σ = 3, and k = 1, 4, 7.

2.4.3 Transmit Antenna Selection with MRC in Cognitive

Radio

Lastly, consider an underlay cognitive radio environment in which both the secondary transmitter and the receiver are equipped with multiple antennas. We presume that the secondary receiver employs maximal-ratio combining (MRC) over L branches resulting in the combined SNR to be a chi-square RV with 2L degrees of freedom. The PDF of the combined SNR can be written as

f (γ) = e−γ/γ γ

L−1

γL_{(L − 1)!}u(γ), (2.20)

where γ is the average SNR of the received signal on each branch.

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an-tennas for the transmission. Since the secondary network has to ensure that their communication is not detrimental to primary network’s performance, the secondary transmitter communicates with the secondary receiver over the best channel that does not cause excessive interference to the primary network. Consequently, the secondary transmitter would be choosing the kth best path for signal transmission. Since chi-square distribution is a special case of the generalized Gamma distribution and L ≥ 1, the bounds on both the ergodic and OPRA capacities hold for such a scenario.

Figure 2.3 shows the ergodic and OPRA capacities along with their bounds for the second best antenna selection out of 4 antennas. Observe that, at high enough

-5 0 5 10 15 -8 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Average Received SNR per Branch, ¯γ (dB)

Ergo dic and OPRA Capacities with Their Bounds (bps/Hz)

Upper bound OPRA,_euk(γ0) Exact OPRA, eCk

Lower bound OPRA, elk(γ0) Upper bound Ergodic, uk Exact Ergodic, Ck Lower bound Ergodic, lk

-14 -12 -10

0.1 0.2 0.3 0.4

Figure 2.3: Ergodic and OPRA capacities with their bounds versus average branch SNR, γ, for M = 4, L = 2 and k = 2.

branch average SNR, γ, the OPRA and ergodic capacities bounds approach each other. This is because of the fact that the optimal cutoff threshold, γ0, becomes

closer to 1, F (γ0) and FX(M −k+1)(γ0) also approach 0, and F

−1 _{1 −} k M

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γ. Therefore, limγ0→1elk(γ0) = log2 F−1 1 −Mk

≈ lk, and limγ0→1euk(γ0) = uk.

Also, for moderate to high average SNR, the lower and upper bounds on ergodic capacity can also be used as bounds on the OPRA capacity as the their values are very close to each other. Lastly, at very low average SNR, the best approximation of the ergodic and OPRA capacities are given by their upper and lower bounds, respectively.

2.5 Concluding Remarks

Simple yet tight bounds on the ergodic and OPRA capacities for the kth best channel selection were presented in this chapter. Numerical results for a variety of applica-tion scenarios were also presented to demonstrate the tightness of the bounds to the actual capacities. These bounds are quite general and can be applied to a variety of different scenarios provided that the received SNR satisfies the constraints presented in Theorem 1 and Theorem 2.

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Appendix A: Proofs of Theorems 1 and 2

The following lemma is used to derive the bounds on the capacities.

Lemma 1. Let Vi’s and Wi’s be two sets of M i.i.d. RVs with CDF FV(v) and

FW(w), respectively. If F_W−1(FV(v)) is convex then

FV Rv1 v0 vfV(i)(v)dv FV(i)(v1) − FV(i)(v0) ! ≤ FW Rw1 w0 wfW(i)(w)dw FW(i)(w1) − FW(i)(w0) ! , (2.21)

where V(i) and W(i) denote the (M − i + 1)st maximums amongst Vi’s and Wi’s,

respectively, FV(v0) = FW(w0), and FV(v1) = FW(w1). Inequalities are reversed if

F_W−1(FV(v)) is a concave function.

Proof: Inequality (2.21) can be proved by considering a new variate U = F−1

W (FV(V(i))).

It is easy to observe that

fU(u) = fW(u) fV(i)(F −1 V (FW(u))) fV(FV−1(FW(u))) . (2.22)

By using (2.2), we can prove that fU(u) = fW(i)(u). Result of (2.21) immediately

follows from the following Jensen’s inequality [60].

F_W−1 FV Rv1 v0 vfV(i)(v)dv FV(i)(v1) − FV(i)(v0) !! ≤ Rv1 v0 F −1 W (FV(v)) fV(i)(v)dv FV(i)(v1) − FV(i)(v0) = Rw1 w0 wfW(i)(w)dw FW(i)(w1) − FW(i)(w0) . (2.23)

It can similarly be shown that the inequality sign in (2.21) is reversed if F_W−1(FV(v))

is concave. Remark

Setting v0 = w0 = −∞ and v1 = w1 = ∞ in (2.21) results in

FV E[V(i)]

≤ FW E[W(i)]

. (2.24)

For proving the capacity bounds, we follow a similar approach of [61, Ch. 4]. In particular, we consider two sets of i.i.d. random variables, Yi’s and Zi’s, for i =

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1, 2, · · · , M . The CDF of Yi and Zi, respectively denoted by FY(y) and FZ(z), are

FY(y) = (1−e−y)u(y), and FZ(z) = −u(−z −1)/z +u(z +1). These random variables

will be used for the proofs of the capacity bounds in the following sections.

Ergodic Capacity Bounds

Since log2(x) is a concave function, we can write

Ck ≤ log2(1 + E[X(M −k+1)]). (2.25)

Moreover, since F_Y−1(F (x)) = − log(1 − F (x)) is convex, (2.24) implies

E[X(M −k+1)] ≤ F−1 1 − e− E[Y(M −k+1)]

. (2.26)

Using the fact that E[Y(M −k+1)] = PM_j=k1/j = H(M ) − H(k − 1), and log2(x) is a

monotonically increasing function, we can write

Ck ≤ log2(1 + E[X(M −k+1)]) ≤ uk. (2.27)

For the lower bound, consider M i.i.d. random variables Ri = log2(1 + Xi), where

i = 1, 2, · · · , M . Since FR(r) = F (2r− 1), and F_Z−1(FR(r)) = −1/FR(r) is concave,

FR E[R(M −k+1)]

≥ FZ E[Z(M −k+1)]

. (2.28)

Knowing that FZ E[Z(M −k+1)]

= 1 − k/M , (2.28) immediately results in Ck = E[R(M −k+1)] ≥ log2 1 + F−1(1 − k/M ) = lk. (2.29)

OPRA Capacity Bounds

For proving OPRA capacity bounds, we set Vi = log2

Xi

γ0

. It can easily be shown that F_V−1(p) = log2 F−1_(p) γ0 , (2.30) and fV(v) = γ0ln(2)2vf (γ02v) . (2.31)

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Since f (2v_{) is log-concave, (2.31) implies that f}

V(v) is also log-concave. If fV(v)

is log-concave then so are FV(v) and 1 − FV(v) [62]. The log-concavity of FV(v) also

implies that 1/FV(v) is a convex function [60].

The upper bound is proved by considering that fact that F_Y−1(FV(v)) = − ln(1 −

FV(v)) is convex. Based on (2.23) given in Lemma 1, we have

R∞ γ0 log2 γ γ0 fX(r)(γ)dγ 1 − FX(r)(γ0) ≤ log₂     F−1 1 − exp − R∞ y0yfY(r)(y)dy 1−F_X(r)(γ0) γ0     , (2.32)

where y0 = − ln (1 − F (γ0)). Lastly, the upper bound, euk(γ0), can be derived using

the following identity. Z ∞

y0

yfY(M −k+1)(y)dy = −(1 − FX(M −k+1)(γ0)) · h(M, k, γ0), (2.33)

where h(M, k, γ0) is defined in (2.16).

For the lower bound, since F_Z−1(FV(v)) = −1/FV(v) is concave,

R∞ γ0 log2 γ γ0 fX(M −k+1)(γ)dγ 1 − FX(M −k+1)(γ0) ≥ log2     F−1 −R−11−FX(M−k+1)(γ0) z0 zfZ(M−k+1)(z)dz γ0     , (2.34) where z0 = −1/F (γ0). However, Z −1 z0 zfZ(M −k+1)dz = M k − M + k M k BF (γ0)(M − k, k). (2.35) As such, we arrive at eCk ≥ elk(γ0).

Appendix B: Proof of Proposition 1

For the generalized Gamma distribution,

d2 dx2 (log f (x)) = − λc − 1 x2 − λ(λ − 1)xλ−2 β . (2.36)

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For λ ≥ 1 and c ≥ 1/λ, _dxd22 (log f (x)) ≤ 0. Therefore, the survival function, F (x), is

log-concave [62]. Hence, the upper bound on the capacity holds. Furthermore, the log-concavity of f (x) implies log-concavity of F (x) [62]. As such, we have [60]

f (x)2− f0(x)F (x) ≥ 0. (2.37) For the lower bound, we need to show that 1/F (2x_{− 1) is convex. Simple}

manipu-lations show that _dxd22

1 F (2x₋₁₎

≥ 0 is equivalent to

(y + 1) 2f (y)2− f0(y)F (y)≥ f (y)F (y), ∀ y ≥ 0. (2.38) Using (2.37), it is easy to see that

(y + 1) 2f (y)2− f0(y)F (y)≥ y 2f (y)2− f0(y)F (y). (2.39) As such, if

y 2f (y)2− f0(y)F (y)≥ f (y)F (y), ∀ y ≥ 0, (2.40) holds then so does (2.38). Inequality (2.40) obviously holds for y = 0. For y > 0, (2.40) can be shown to be equivalent to

−c + y λ β + 2β−c_e−yλ_/β yλc Γ(c) − Γ(c, yλ_/β) ≥ 0. (2.41)

By letting z = y_βλ and using the fact that c(Γ(c)−Γ(c, z)) = zc_e−z_{+Γ(c+1)−Γ(c+1, z),}

(2.41) simplifies to e−z ≥ Z z 0 t z c − t z c−1! e−tdt. (2.42)

Since for 0 ≤ t ≤ z, _ztc− _ztc−1 ≤ 0, we have indeed Z z 0 t z c − t z c−1! e−tdt ≤ 0 ≤ e−z, (2.43)

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Chapter 3 Receive Antenna Selection for an

Underlay Cognitive Radio

3.1 Introduction

Cognitive radio technology can help improve the spectrum utilization efficiency of wireless communication systems. In the underlay spectrum sharing mode, the sec-ondary network coexists with the primary network provided that the interference caused by the secondary transmitter to the primary network is below a certain thresh-old level [2]. Unfortunately, the interference caused by the primary transmission to the secondary receiver degrades the performance of the secondary network. The detri-mental effects of the interference can be mitigated by a diversity combining technique implemented at the secondary receiver. The most popular diversity combining tech-niques are the maximum-ratio combining (MRC), equal gain combining (EGC) and selection combining (SC) [13, 14]. Out of these combining schemes, selection combin-ing offers a reasonable performance with the lowest hardware complexity. As such, we consider a receive antenna selection scheme for an underlay cognitive radio and analyze its performance in this chapter.

Antenna selection can be performed based on different criteria in the presence of interference. For example, the authors in [63, 64] considered and analyzed the per-formance of an antenna selection scheme based on the desired signal received power. Antenna selection can also be carried based on the desired signal plus interference power and the desired signal-to-interference ratio (SIR) at the secondary receiver (SR). Performances of such selection diversity schemes were analyzed in [64–66].

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Re-cently, [67, 68] proposed and analyzed the performance of selection carried out based on the received signal-to-interference plus noise ratio (SINR) in a Rayleigh fading environment. In general, the SINR based selection scheme outperforms others [67]. As such, we consider the SINR based antenna selection in an underlay cognitive environment but over a more general fading environment.

In this chapter, we will derive the exact distribution of the received SINR at the selected secondary receiver antenna when the secondary link experiences Nakagami-m fading. The case where the performance of the secondary network is limited mainly by the interference from the primary transmitter (PT) is also considered. The asymptotic statistics of the received SINR and SIR for the large-scale antenna system [4] are also obtained. To the best of our knowledge, these statistical results are not previously reported in literature. These results are then applied to the performance analysis of the secondary transmission. Exact expressions for the outage probability and an approximated expression for the bit-error probability (BEP) based on the asymptotic distribution are also obtained under the assumption that the secondary transmitter (ST) adapts its power in finite discrete levels to satisfy the interference constraint at the primary receiver. Numerical simulations are also carried out to verify the analytical results and to demonstrate the approximation accuracy.

3.2 System Model

Consider an underlay cognitive radio environment in which the ST is equipped with a single antenna, while the SR has M antennas. We denote the channel gain from the PT to the ith SR antenna by gi, where i = 1, 2, · · · , M . The channel gains from

the ST to the primary receiver (PR) and the ith SR antenna are respectively denoted by h0 and hi for i = 1, 2, · · · , M as shown in Figure 3.1.

The primary network is presumed to be far away from the secondary network. Therefore, the channel gains |h0| and |gi| can be presumed to follow independent

Rayleigh fading implying that the channel power gains, |h0|2 and |gi|2, have

prob-ability density functions (PDFs) f0(x) = λspe−λspxu(x), and g(x) = λpse−λpsxu(x),

respectively, where u(x) is the unit step function, and the parameters λsp and λps

de-pend on the propagation distance and environment. Furthermore, the channel power gains in the secondary network are presumed to be independent and identically dis-tributed Gamma RVs. The PDF and cumulative distribution function (CDF) of the

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PR

PT

SR

ST

g

1

g

M

h

1

h

M

h

0 1 2 M

Figure 3.1: An underlay cognitive radio setup with M antennas at the SR. channel power gain, |hi|2, are

f (x) = λ m ssxm−1 Γ(m) e −λssx_u(x), _(3.1) and F (x) = 1 − Γ(m, λssx) Γ(m) u(x), (3.2)

respectively, where the parameters m and λss are positive reals, Γ(m) is the Euler

Gamma function, and Γ(m, x) is the incomplete Gamma function [69].

We assume that the ST adaptively changes its transmit power to control its in-terference to the PR. Let PS be the instantaneous transmit power of the ST. With

knowledge of |h0|2, either through feedback or channel reciprocity, PS is set to satisfy

the constraint PS|h0|2 ≤ T , where T is the maximum tolerable interference level at

the PR. For continuous power adaptation, PS is set as PS = min

Pmax,_|hT₀_|2

, where Pmax is the maximum instantaneous power available at the ST. In practice, however,

transmit power cannot be changed in a continuous manner. Therefore, we assume that the ST can adjust its power in J + 1 discrete power levels, denoted by Pj for

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designed based on the hardware limitations of the ST. Based on the channel power gain |h0|2, the transmit power, PS, is chosen according to the following rule:

PS =          Pj, for _P_j+1T < |h0|2 ≤ _PT_j, and j = 1, · · · , J − 1; PJ, for |h0|2 ≤ _PT_J; 0, for |h0|2 > _PT₁. (3.3)

To explore the diversity benefits, the SR selects the antenna that leads to the highest SINR for any chosen transmit power level. Mathematically speaking, the index of the chosen antenna, i∗_{, is given by}

i∗ = argmax

i=1,2,··· ,M

|hi|2

N0+ PP|gi|2

, (3.4)

where N0 is the common noise variance at each SR antenna, PP is the transmit power

of the PT, and PP|gi|2 is the PT interference power at the ith SR antenna. In what

follows, we derive the exact and asymptotic distribution of the received SINR at the chosen SR antenna, which will then be applied to the performance analysis of the secondary system in Section 3.4.

3.3 Statistics of the Received SINR

3.3.1 General Case

We first consider a general case where the noise variance, N0, cannot be neglected.

Exact distribution

Define a new RV, Yi = |hi |2

N0+PP|gi|2, which is related to the received SINR at the ith

SR antenna, denoted by γi, as γi = PSYi. The PDF of Yi, fY(y), can be found by

conditioning on N0+ PP|gi|2 as fY(y) = λpse λpsN0 PP PP Z ∞ N0 f (zy)ze−λpszPP _dz. _(3.5)

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Using (3.1), fY(y) can be shown to be fY(y) = λpsλmssN0m+1e λpsN0 PP PPΓ(m) ym−1E−m N0λssy + N0λps PP u(y), (3.6)

where En(z) is the exponential integral, defined as En(z) =

R∞ 1 t

−n_e−zt_{dt [70]. Using}

the fact that _dxdEn(x) = −En−1(x) and the following recurrence relationship of En(z)

[70] En(z) = e−z n − 1 − z n − 1En−1(z), (3.7)

we can show that

ym−1E−m(αy + β) = d dy[y m_E 1−m(αy + β)] + ym−1e−(αy+β) β . (3.8)

Using (3.8), the CDF of Yi, FY(y), can be shown to be

FY(y) =  1 − Γ(m, N0λssy) Γ(m) + (N0λssy) m eλpsN0PP E1−m N0λssy +N_P0λ_Pps Γ(m)   u(y).(3.9)

Lastly, the CDF of received SINR at the selected antenna, γ(M ), for PS = Pj is

Fγ(M )(γ) = FY γ Pj M [56]. Asymptotic distribution

To facilitate the asymptotic analysis, we present the asymptotic distribution of Y(M ) =

max {Yi} in the following proposition.

Proposition 1. The distribution of Y(M ) approaches the Gumbel distribution, exp

h − exp−x−bM

aM

i

, as the number of ST antennas, M , is increased. Here the constants bM and aM are given as

bM = FY−1(1 − 1/M ) , (3.10)

and

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Proof: First, observe that d dy 1 − FY(y) fY(y) = −1 − f 0 Y(y) fY(y) · 1 − FY(y) fY(y) . (3.12)

Furthermore, it can be shown that

f0 Y(y) fY(y) = m − 1 y − N0λss E−m−1 N0λps PP + N0λssy E−m N0λps PP + N0λssy . (3.13)

By using the recurrence relationship of En(z) as given in (3.7) and the asymptotic

series expansion of zez_E n+1(z) [70], we arrive at lim y→∞ f0 Y(y) fY(y) = −N0λss lim y→∞ e− N0λps PP +N0λssy N0λps PP + N0λssy E−m N0λps PP + N0λssy = −N0λss 1 2F0(1, −m; 0) = −N0λss6= 0. (3.14)

Here 2F0(a, b; x) is the generalized hypergeometric function [70]. We can also show

that lim y→∞ 1 − FY(y) fY(y) = − lim y→∞ fY(y) f0 Y(y) = 1 N0λss . (3.15)

Substituting (3.14) and (3.15) into (3.12) results in

lim y→∞ d dy 1 − FY(y) fY(y) = 0, (3.16)

Therefore, the asymptotic distribution of Y(M ) is Gumbel with the constants bM and

aM as defined in (3.10) and (3.11), respectively [56, Th. 10.5.2].

3.3.2 Interference Limited Scenario

We now consider a case where the noise at the chosen antenna is negligible as com-pared to the interference from the PT to the SR. The SINR at the SR can be approx-imated by the signal-to-interference ratio (SIR), i.e.

Yi ≈ Zi =

|hi|2

PP|gi|2

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Exact distribution

The exact statistics of Zi can easily be computed from (3.6) and (3.9) by setting

N0 = 0 and using the fact Eν(z) = zν−1Γ(1 − ν, z) [70]. For example, the CDF of Zi,

denoted by FZ(z), is given as FZ(z) = PPz λps/λss+ PPz m u(z). (3.18)

We can easily derive the exact distribution of SIR of the selected antenna, γ_{(M )}0 , using the relation γ0

(M ) = PSZ(M ), where Z(M ) denotes the largest Zi amongst M ones.

Asymptotic distribution

The asymptotic distribution of Z(M ) is given in the following proposition.

Proposition 2. The asymptotic distribution of the maximum of Zi’s, Z(M ), is the

Fr´echet distribution, i.e. for large M

FZ(M )(z) ≈ exp " λps λssPP 1 − 1 − _M1 −m1_z # u(z). (3.19)

Proof: It can be shown that

lim

z→∞

1 − FZ(z)

1 − FZ(rz)

= r, (3.20)

which implies that FZ(z) lies in the domain of maximal attraction of the Fr´echet

distribution [56, Th. 10.5.2], i.e. for large values of M

FZ(M )(z) ≈ exp " − z F_Z−1(1 − 1/M ) −1# , z ≥ 0. (3.21)

Eq. (3.19) follows directly by observing that F_Z−1(p) = λps

λssPP(p−1/m−1).

Figure 3.2 shows the exact and asymptotic CDFs of Z(M ) for different values of

M . Observe that the asymptotic distribution in (3.19) is a good approximation even for small values of M , and the approximation becomes more accurate by increasing the value of M .

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0 10 20 30 40 50 60 70 80 90 100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 z FZ (M ) (z ) and F r´ec het Distr ibution Exact Asymptotic M = 5 M = 10 M = 30

Figure 3.2: Exact and asymptotic distribution of Z(M ) for PP = 2, λps = 1, λss = 1

and m = 2.

3.4 Secondary System Performance Analysis

3.4.1 Outage Probability

Outage occurs whenever the SINR at the receiver is below a certain threshold level, γT. The probability of outage for the secondary system, Pout, can be computed

mathematically as Pout= p0+ J X j=1 pjPr PjY(M ) ≤ γT = p0+ J X j=1 pj FY γT Pj M , (3.22)

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where FY(·) is given in (3.9), and pj = Pr{PS = Pj} for j = 0, 1, · · · , J. For Rayleigh

fading case, pj can be computed as

pj =          e−λspTP1 _, _{for j = 0;} e− λspT Pj+1 _{− e}− λspT Pj _{, for j = 1, · · · , J − 1;} 1 − e−λspTPJ _, _{for j = J.} (3.23)

When the noise variance is negligibly small, the outage probability can be shown to simplify to Pout ≈ p0+ J X j=1 pj λssPPγT λpsPj + λssPPγT mM . (3.24) -10 -8 -6 -4 -2 0 2 4 6 8 10 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 Pmax(dB) Outage Probabilities, Pout Simulation Analytical T = 0.05 T = 0.1 T = 0.2

Figure 3.3: Outage probability against the maximum secondary transmit power, Pmax,

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Figure 3.3 shows the outage probability of the secondary system against the max-imum secondary transmit power, Pmax, for different values of the interference

thresh-old, T . The power levels used for simulation are Pj = jPmax/J, where J = 5, and

j = 0, 1, · · · , J. The outage threshold level, γT, is normalized to 1. We can see that

the outage probability first decreases with increasing Pmax due to the increase in the

received SINR at the SR. Also, the outage probability is higher for small values of T as expected. Furthermore, increasing Pmax too much results in an increase in Pout.

This is due to the fact that with a fixed number of discrete power levels, it becomes increasingly difficult to satisfy the interference constraint at the PR. Monte-Carlo simulation for 109 _{trials are also shown, which matches the analytical result very}

well.

3.4.2 Asymptotic Bit Error Probability

We consider a general class of modulation schemes whose conditional BEP, Pb(E|γ),

is given by

Pb(E|γ) = C exp(−αγ). (3.25)

Applying the moment generating function (MGF) based approach, the average BEP can be computed as [13]

Pb = CMγ(M )(−α), (3.26)

where Mγ(M )(s) = E [e

sγ(M )] is the MGF of the SINR at the chosen SR antenna, γ

(M ).

In the following, we present closed-form expressions for the BEP for both the general and interference-limited scenarios.

General Case

Using the asymptotic distribution of Y(M ), MY(M )(s) can be approximated as

MY(M )(s) ≈ −s Z ∞ 0 e−e −x−bM aM +sx_{dx − e}−eaMbM . (3.27)

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Solving the integral in (3.27), we get MY(M )(s) ≈ −aMse bMs h Γ − aMs − Γ− aMs, e bM aM i − e−e bM aM . (3.28)

Finally, noting that EesPSY(M ) _{= E}

PS

h EY(M )

esPSY(M )i_{, we can show that}

Pb = C J X j=1 pjMY(M )(−αPj). (3.29) where pj is defined in (3.23). -10 -8 -6 -4 -2 0 2 4 6 8 10 10-6 10-5 10-4 10-3 10-2 10-1 Pmax(dB) Exact and Asymptoti c Bit Error Pr obabilities Exact Asymptotic T = 0.025 T = 0.05 T = 0.1

Figure 3.4: Average BEPs of BFSK (C = α = 0.5) against Pmaxfor M = 16, m = 3.5,

λss = 10, PP = 1, λps = 20, λsp = 20, N0 = 0.025, and J = 5.

Figure 3.4 shows the average BEP of the secondary system against the maximum ST transmit power, Pmax, for different values of the interference threshold, T . The

Low complexity multiple antenna transmission solutions for next generation wireless communication systems

Contents

List of Figures

List of Abbreviations

Mathematical Notations

Introduction

1.1

Cognitive Radio Network

1.2

Massive MIMO Systems

1.3

Dissertation Outline

Chapter 2

Capacity Bounds for the kth Best

Path Selection over Generalized

Fading Channels

2.1

Introduction

2.2

System Model

2.3

Bounds on the Capacities

2.4

Examples and Applications

2.4.1

Selection Combining over Weibull Fading Channels

2.4.2

Selection Combining over Log-Normal Fading

Chan-nels

2.4.3

Transmit Antenna Selection with MRC in Cognitive

Radio

2.5

Concluding Remarks

Appendix A: Proofs of Theorems 1 and 2

Ergodic Capacity Bounds

OPRA Capacity Bounds

Appendix B: Proof of Proposition 1

Chapter 3

Receive Antenna Selection for an

Underlay Cognitive Radio

3.1

Introduction

3.2

System Model

PR

PT

SR

ST

g

g

h

h

h

3.3

Statistics of the Received SINR

3.3.1

General Case

3.3.2

Interference Limited Scenario

3.4

Secondary System Performance Analysis

3.4.1

Outage Probability

3.4.2

Asymptotic Bit Error Probability