Advanced modulation techniques in massive mimo

Orthogonal frequency division multiplexing with cyclic prex (CP-OFDM) is the primary modulation format in the massive multiple-input multiple-output (MIMO) literature as well as most of the current wireless communication standards. This waveform, however, suers from a number of drawbacks. In particular, the high spectral leakage of the OFDM subcarriers, the overhead due to the CP duration, and the high peak-to-average power ratio (PAPR) are the three most important shortcomings of this waveform. In this dissertation, we study some alternative modulation formats to address the limitations of CP-OFDM, particularly in massive MIMO systems. Three dierent designs are considered to target the above drawbacks of OFDM individually. In the rst one, lter bank multicarrier (FBMC) is considered to relieve the spectral leakage problem of OFDM in massive MIMO systems. We show that in FBMC-based massive MIMO, when linear equalization methods such as MRC, ZF, and MMSE, are applied, the signal-to-interference-plus-noise ratio (SINR) does not necessarily improve as the number of BS antennas grows large and saturates at a certain deterministic level. This phenomenon results from the law of large numbers and is due to the correlation between the combiner taps and the channel coecients. In order to resolve the saturation problem, we develop an ecient equalization method to remove this correlation. In the second design, we study the possibility of removing the CP in OFDM to increase its spectral eciency. Again, we show that a similar saturation problem exists in OFDM without CP systems. In this part, we develop another equalization structure based on the time-reversal technique. We show that by utilizing the proposed method, higher spectral eciency can be achieved by removing the CP overhead in massive MIMO systems. Finally, we consider single-carrier (SC) transmission in massive MIMO, to reduce PAPR. A novel receiver design based on time-reversal combining and frequency-domain equalization is proposed for this system. We show that through the proposed receiver structure, the performance of SC transmission can be enhanced in moderate to high signal-to-noise ratio (SNR) regimes.

ADVANCED MODULATION TECHNIQUES IN MASSIVE MIMO by Amir Aminjavaheri A dissertation submitted to the faculty of The University of Utah in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Electrical and Computer Engineering The University of Utah December 2018 Copyright c Amir Aminjavaheri 2018 All Rights Reserved The University of Utah Graduate School STATEMENT OF DISSERTATION APPROVAL The dissertation of Amir Aminjavaheri has been approved by the following supervisory committee members: Behrouz Farhang , Chair(s) 8/1/2018 Date Approved Rong-Rong Chen , Member 8/1/2018 Date Approved Neal Patwari , Member 8/1/2018 Date Approved Sneha Kasera , Member 8/1/2018 Date Approved Mingyue Ji , Member 8/1/2018 Date Approved by Florian Solzbacher , Chair/Dean of the Department/College/School of Electrical and Computer Engineering and by David B. Kieda , Dean of The Graduate School. ABSTRACT Orthogonal frequency division multiplexing with cyclic prefix (CP-OFDM) is the primary modulation format in the massive multiple-input multiple-output (MIMO) literature as well as most of the current wireless communication standards. This waveform, however, suffers from a number of drawbacks. In particular, the high spectral leakage of the OFDM subcarriers, the overhead due to the CP duration, and the high peak-to-average power ratio (PAPR) are the three most important shortcomings of this waveform. In this dissertation, we study some alternative modulation formats to address the limitations of CP-OFDM, particularly in massive MIMO systems. Three different designs are considered to target the above drawbacks of OFDM individually. In the first one, filter bank multicarrier (FBMC) is considered to relieve the spectral leakage problem of OFDM in massive MIMO systems. We show that in FBMC-based massive MIMO, when linear equalization methods such as MRC, ZF, and MMSE, are applied, the signal-to-interference-plus-noise ratio (SINR) does not necessarily improve as the number of BS antennas grows large and saturates at a certain deterministic level. This phenomenon results from the law of large numbers and is due to the correlation between the combiner taps and the channel coefficients. In order to resolve the saturation problem, we develop an efficient equalization method to remove this correlation. In the second design, we study the possibility of removing the CP in OFDM to increase its spectral efficiency. Again, we show that a similar saturation problem exists in OFDM without CP systems. In this part, we develop another equalization structure based on the time-reversal technique. We show that by utilizing the proposed method, higher spectral efficiency can be achieved by removing the CP overhead in massive MIMO systems. Finally, we consider single-carrier (SC) transmission in massive MIMO, to reduce PAPR. A novel receiver design based on time-reversal combining and frequency-domain equalization is proposed for this system. We show that through the proposed receiver structure, the performance of SC transmission can be enhanced in moderate to high signal-to-noise ratio (SNR) regimes. To my parents, Mahnaz and Mojtaba CONTENTS ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix NOTATION AND SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi CHAPTERS 1. 2. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 1.2 1.3 1.4 1.5 2 3 5 7 8 Next Generation Wireless Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . OFDM and Its Drawbacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Overview of Alternative Waveforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dissertation Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structure of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FUNDAMENTALS OF MASSIVE MIMO . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1 2.2 2.3 2.4 2.5 2.6 From MIMO to MU-MIMO to Massive MIMO . . . . . . . . . . . . . . . . . . . . . . . . Review of the Law of Large Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Asymptotic Optimality of Linear Processing . . . . . . . . . . . . . . . . . . . . . . . . . Obtaining the Channel State Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . Challenges and Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deployment in 3GPP Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 11 12 14 14 16 3. SYSTEM MODEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4. FILTER BANK MULTICARRIER IN MASSIVE MIMO . . . . . . . . . . . . 21 4.1 4.2 4.3 4.4 4.5 4.6 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FBMC Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Massive MIMO FBMC: Asymptotic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . Equalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frequency-Domain Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SINR Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 MRC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 ZF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Conclusion and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Proof of (4.28) and (4.35) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9.1 MRC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9.2 ZF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 23 26 29 34 36 39 41 41 45 46 46 48 5. OFDM WITHOUT CP IN MASSIVE MIMO . . . . . . . . . . . . . . . . . . . . . . 50 5.1 5.2 5.3 5.4 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . OFDM Without CP Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frequency-Domain Combining Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time-Reversal and Equalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 TR-MRC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 TR-FDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Efficient Implementation and Complexity Analysis . . . . . . . . . . . . . . . . . . . . 5.6 Analysis of SINR and Achievable Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 TR-MRC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.2 TR-FDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9 Proof of (5.8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. SINGLE CARRIER MODULATION WITH FREQUENCY DOMAIN EQUALIZATION IN MASSIVE MIMO . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 6.1 6.2 6.3 6.4 6.5 7. 50 54 55 59 59 61 63 66 68 71 73 78 79 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time-Reversal Combining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frequency Domain Equalization (FDE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 82 83 87 88 CONCLUSION AND FUTURE WORK . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 7.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Optimum CP Length Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Cell-Free Massive MIMO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Application to Underwater Acoustic Channels . . . . . . . . . . . . . . . . . . . . 90 91 91 91 91 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 vi LIST OF FIGURES 1.1 NR use cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 4.1 Block diagram of the FBMC transceiver in discrete time. . . . . . . . . . . . . . . . . . 25 4.2 The equivalent channel between the transmitted data symbol at time-frequency point (m0 , i0 ) and the demodulated symbol at time-frequency point (m, i). . . . 26 4.3 Block diagram of the proposed receiver structure to resolve the saturation issue. Here, only the portion of the receiver corresponding to subcarrier m and terminal k is shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.4 Block diagram of the simplified receiver. Utilizing multirate signal processing techniques, the additional equalization block can be moved to after the analysis filter bank and combiner to minimize the computational cost. . . . . . . . . . . . . . 32 4.5 Two equivalent systems considered in the proof of Proposition 3. . . . . . . . . . . 33 4.6 Illustration of the equivalent channel response. Here, we assume M = 512, and consider an exponentially decaying channel PDP with the decaying factor k,k (ω), for of 0.06 and the length of L = 50. (a) The equivalent channel, Cm subcarrier m = 0, without the proposed equalizer. (b) The equivalent channel, k,k (ω), for subcarrier m = 0, with the proposed equalizer. As the number C̃m of BS antennas increases, the equivalent channel becomes flat only when the proposed equalizer is in place. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.7 SINR performance comparison for the case that the proposed equalizer is not utilized. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.8 SINR performance comparison for the case that the proposed equalizer is utilized. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.9 SINR performance comparison as a function of different SNR values. In the case of FBMC, the proposed equalizer is incorporated at the BS. Here N = 100 BS antennas is considered. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.10 SINR performance comparison for different values of the FBMC subcarrier spacing ∆F , 1/M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.11 BER performance comparison. Here, N = 100 BS antennas and the ZF combiner are considered. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.1 Baseband system implementation of the proposed technique with a TR-FDE receiver. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 5.2 Computational complexity comparison of OFDM without CP with TR-MRC and TR-FDE techniques against CP-OFDM with MRC and ZF equalizations. Here, the following parameters are considered. M = 512, M̃ = 256, L = 40, and Q = 10. In (a), K = 10 is fixed and the value of N is varied, whereas in (b), N = 200 is fixed and the value of K is varied. . . . . . . . . . . . . . . . . . . . . . . 67 5.3 SINR saturation in the case of conventional frequency-domain combiners. Here, K = 10 terminals are considered and the number of BS antennas is varied. The SNR level is chosen to be 10 dB. The saturation level is calculated using (5.9). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 5.4 SINR performance comparison for time reversal methods. Here, K = 10 terminals are considered and the number of BS antennas is varied. The SNR level is chosen to be 10 dB. Asymptotic theoretical SINR values are calculated according to (5.30) and (5.35) for the cases of TR-MRC and TR-FDE, respectively. Using time reversal, arbitrarily large SINR values can be achieved by increasing the number of BS antennas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.5 Per user achievable information rate with and without the CP overhead. Here, the ratio L/M is approximately 7%, and the SNR level is chosen to be −10 dB. (a) K = 10, (b) K = 20 user terminals. . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.6 Per user achievable information rate as a function of the SNR level. Here, the ratio L/M is approximately 7%, and N = 200 BS antennas and K = 10 terminals are considered. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.7 Per user achievable information rate with and without the CP overhead. Here, K = 10 terminals are considered and the SNR level is chosen to be −10 dB. Moreover, the TDL-A channel with the RMS delay spread of 1100 ns is assumed. In this channel model, the ratio L/M is approximately 15%. . . . . . . 78 6.1 Block diagram of the proposed TR-FDE system. . . . . . . . . . . . . . . . . . . . . . . . 84 6.2 Simulated sum rate versus different number of BS antennas. 6.3 Simulated sum rate versus the input SNR level. . . . . . . . . . . . . . . . . . . . . . . . 89 6.4 Simulated sum rate versus the block size, M . . . . . . . . . . . . . . . . . . . . . . . . . . 89 viii . . . . . . . . . . . . . 88 LIST OF TABLES 5.1 Computational complexity of the conventional MRC and ZF detectors utilized in CP-OFDM systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 5.2 Computational complexity of different parts of the receivers proposed for OFDM without CP systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 NOTATION AND SYMBOLS a, a, A <{·} ={·} (·)∗ ? AT AH A−1 [A]mn tr{·} IM 0M ×N diag{a} |a|, |A| kak E{·} Var{·} CN (µ, σ 2 ) δij N n K k M m i σν2 pk [l] hn,k [l] L xk [l] rn [l] yk [l] Scalar, vector, and matrix quantities Real part of a complex number Imaginary part a complex number Conjugate operation Linear convolution Matrix transpose Matrix Hermitian Matrix inverse Element in the mth row and nth column of A Matrix trace operation Identity matrix of size M × M Zero matrix of size M × N Square diagonal matrix with the elements of a on the main diagonal Absolute value of a number, and matrix determinant Euclidean norm of vector a Expectation of a random variable Variance of a random variable Circularly symmetric normal distribution Kronecker delta function Number of BS antennas BS antenna index, n ∈ {0, . . . , N − 1} Number of user terminals User terminal index, k ∈ {0, . . . , K − 1} Number of subcarriers Subcarrier index, m ∈ {0, . . . , M − 1} Symbol time index Noise variance at the input of BS antennas Power delay profile corresponding to user k Channel impulse response between BS antenna n and user k Channel length Transmit signal of terminal k Received signal at BS antenna n Output of the time-reversal filter corresponding to terminal k ACKNOWLEDGEMENTS This work would have not been possible without the support of many people, whom I would like to thank here. I have been truly fortunate in the past few years to have the guidance and support of Professor Behrouz Farhang as my advisor. I am honored and privileged to have the opportunity to work with him and I cannot thank him enough. His knowledge and hardworking attitude inspired not only my vision toward research, but towards other aspects of life as well. Behrouz has been exceptionally supportive and has given me freedom to pursue various interesting research projects without any objection. I have learned invaluable lessons from him in so many dimensions, including management, research, and personality. I am also thankful to his family for their support and great hospitality during my studies. I would like to thank my committee members, Professor Rong-Rong Chen, Professor Neal Patwari, Professor Sneha Kasera, and Professor Mingyue Ji for showing interest in my work and for their guidance and support. I have received invaluable comments and feedback from them which resulted in great improvement of the quality and the presentation of my research. I would also like to extend my gratitude to Professor Joel Harley whose Advanced DSP course has been a great source of inspiration to me. During the course of this work, I was also fortunate to collaborate with my great friend and colleague, Professor Arman Farhang to whom my extensive gratitude is due. I have truly enjoyed our endless discussions throughout these years and have benefited a lot from his extensive knowledge. Collaborating with him has always been a source of inspiration and enthusiasm to hard work and achieving more. Successful completion of this work would have not been possible for me without his support. Besides my academic mentors, I was very fortunate to work with Pierre-Xavier Thomas and Dr. Rong Chen during my research internship at Cadence Design Systems in Summer 2017. Words cannot express my gratitude to them for their inspiration and support. I am also grateful to my officemates, Andy Laraway, Jonathan Hedstrom, George Yuen, Ahmad RezazadehReyhani, Daryl Wasden, Arslan Majid, Mohamed Abu Baker, Taylor Sibbett, Jon Driggs, and Tarek Haddadin. Accompanying them made research memorable and fun. Last but not least, I would like to thank my family. I am deeply indebted to my parents, Mahnaz and Mojtaba, my sister, Elham, and my brother, Iman, for their constant encouragement and support. xii CHAPTER 1 INTRODUCTION Massive multiple-input multiple-output (MIMO) is one of the primary technologies currently considered for the next generation of wireless networks [1]. In a massive MIMO system, the base station (BS) is equipped with a large number of antenna elements, in the order of hundreds or more, and is simultaneously serving tens of user terminals. By coherent processing of the signals over the BS antennas, the effects of uncorrelated noise and multiuser interference can be made arbitrarily small as the BS array size increases [2]. Hence, unprecedented network capacities can be achieved. Due to its simplicity and robustness against multipath channels, orthogonal frequency division multiplexing (OFDM) with cyclic prefix (CP) is the dominant modulation format that is considered in the massive MIMO literature (see for example [2–5]) as well as most of the current wireless standards such as the 3GPP long term evolution (LTE) standard. However, despite its many advantages, OFDM suffers from a number of drawbacks. In particular, due to the high side-lobe levels of the subcarriers, OFDM suffers from a large spectral leakage leading to high out-of-band emissions. Accordingly, stringent synchronization procedures are required in the uplink of multiuser networks. The users may experience different Doppler shifts, carrier frequency offsets, timing offsets, etc., and maintaining the orthogonality between the subcarriers may not be possible without energy-consuming and resource-demanding synchronization procedures. Furthermore, utilization of noncontiguous spectrum chunks through carrier aggregation for the future high data rate applications is very difficult in the uplink with OFDM as a result of high side-lobe levels of its subcarriers [6]. Additionally, to avoid interference, large guard bands are required between adjacent frequency channels, which in turn, lowers the spectral efficiency of OFDM. The CP duration also introduces an overhead to the network, leading to a further reduction in spectral efficiency. Being a multicarrier waveform, OFDM also suffers from a high peak-to-average 2 power ratio (PAPR). The goal of this dissertation is to study some alternative modulation formats in the context of massive MIMO. As a result, in the first chapter of this dissertation, we aim at motivating the problem by investigating the main drawbacks of OFDM in detail. We also review the characteristics of the next generation wireless networks in which addressing the shortcomings of OFDM becomes more critical. 1.1 Next Generation Wireless Networks In the recent past, we have observed an explosive increase in the amount of mobile data traffic and the number of mobile devices. It is anticipated that this trend also continues in the coming years. For example, the visual networking index reported by Cisco shows that the mobile data traffic globally will increase sevenfold in five years between 2016 and 2021 [7]. Motivated by this and to address the demands of the future mobile networks, the standardization bodies have recently laid down the foundation of the fifth generation of wireless networks (5G) by defining its primary objectives [8]. It is envisioned that the system capabilities in 5G are enhanced 10 to 100 times compared to the ones in the 4G LTE standard. For instance, peak data rates of 20 Gbps and peak spectral efficiency of 30 bps/Hz are required for the mobile broadband traffic in 5G [8]. The next generations of wireless networks are also expected to support a vast variety of requirements and applications. This is in contrary to the previous generation where the main objective was to provide a broadband connectivity to the mobile users. For instance, the following three different classes of use cases are anticipated for 5G: enhanced mobile broadband (eMBB), massive machine-type communications (mMTC), and ultra-reliable low-latency communications (URLLC). Each use cases requires different set of services and network capabilities. Note that the mobile broadband network of previous generations are enhanced through the eMBB framework. Thus, two entirely new use cases are added to the picture of cellular networks. Figure 1.1 illustrates some of the applications envisioned for the three use cases of 5G networks. The new applications and use cases defined in 5G demand more strict requirements in terms of data rate, number of connected devices, spectrum efficiency, battery life, and end-to-end latency as compared to the previous generations. As a result, a great deal of 3 eMBB Gigabytes in a second 3D video, UHD screens Work and play in the cloud Smart home Augmented reality Industry automation Voice Mission critical applications, e.g., e-health Smart city Self driving cars mMTC URLLC Figure 1.1. NR use cases. interest has emerged among the research and industrial communities motivating introduction of new technology enablers. Some of the enablers include massive MIMO, millimeter wave (mmWave), small cell deployment, advanced waveform design, flexible numerology, advanced channel coding, and device-centric architecture [1, 9, 10]. It is worth mentioning that from the above set of enablers, massive MIMO plays a key role in 5G. By using a large array of antennas at the BS, one can improve the capacity and coverage of the network. To compensate the free space path loss, it is also crucial to incorporate massive MIMO in mmWave, which itself is considered as an important 5G enabler. 1.2 OFDM and Its Drawbacks In recent years, OFDM has been among the most popular and accepted technologies in wireless broadband communications due to its many attractive features. In particular, the robustness of the OFDM waveform in wideband frequency-selective channels and the simplicity of the transceiver structure make OFDM an attractive solution in many applications. Another important feature of OFDM is its rather straightforward coupling with 4 the MIMO technology. OFDM also allows for dynamic bandwidth allocation to different users in multiple access scenarios. Thanks to the above features, the LTE standard, which has OFDM as its underlying physical layer modulation technology, offers great data rates and capacities, particularly in downlink. OFDM is also the dominant modulation format considered in the massive MIMO literature [2]. Despite its many advantages, OFDM suffers from a number of drawbacks, especially in the uplink. We note that OFDM waveform is built based on a rectangular pulse shape/prototype filter, and uses a CP to simplify the channel equalization. Although the rectangular pulse shape is well-localized in time, it is poorly localized in frequency due to the abrupt transitions in symbol boundaries. This fundamental structure of OFDM leads to a number of drawbacks as listed below: • To avoid interference, null guard tones are required at the spectrum edges, which in turn lowers the spectral efficiency of OFDM. For instance, in LTE systems operating at 10 MHz bandwidth, only 9 MHz of the band is used. • The CP duration adds an extra overhead to the network and further reduces the spectral efficiency. For example, the CP overhead in LTE is around 7%. Accordingly, the total loss due to CP and guard tones is about 16%. • The orthogonality of OFDM is based on strict synchronization between the users, and as soon as the synchronization is lost (for example due to multiple access, multicell operation or Doppler effects) multiple access interference prevails [11]. Accordingly, the uplink of LTE is based on resource demanding closed-loop procedures to establish the required synchronization. This clearly lowers the spectral/energy efficiency of the network and increases the end-to-end latency. • Utilization of noncontiguous spectrum chunks through carrier aggregation for future high data rate applications is very difficult in the uplink with OFDM as a result of high side-lobe levels of its subcarriers [6]. • OFDM suffers from a high peak-to-average power ratio (PAPR), requiring the power amplifiers (PAs) to work with a large power backoff. This leads to a lower energy 5 efficiency of the PAs and decreases the maximum communication range possible. To mitigate this problem, a linear precoding is utilized in the uplink of LTE [12]. • The abrupt transitions in symbol boundaries and the high spectrum leakage of OFDM increases its sensitivity to carrier frequency and timing offsets in the uplink of multiuser networks. A performance comparison between various waveforms with respect to multiuser synchronization errors has been presented in [13]. It is worth mentioning that the above challenges become more serious when considering the technological requirements of the next generations of wireless networks [9]. For example, some emerging applications in cellular networks such as the smart city and Internet of Things (IoT), by definition, need to support many machine-type communication (MTC) nodes with the design criteria of low implementation cost, long battery life, and extremely low latency message delivery [11]. This drives the idea of asynchronous communication in order to avoid the problems of LTE such as its high spectral loss and latency due to the sophisticated synchronization procedures[11, 14, 15]. 1.3 Overview of Alternative Waveforms As mentioned before, the new applications and use cases defined in the next generation of wireless networks demand more strict requirements in terms of data rate, number of connected devices, spectrum efficiency, battery life, and end-to-end latency as compared to the previous generations. Therefore, considering the drawbacks of OFDM and the new requirements of wireless networks, a great deal of interest in the area of waveform design has emerged among the research and industrial communities motivating introduction of alternative waveforms capable of keeping the advantages of OFDM while addressing its drawbacks; see for example [16–20]. In this section, we review some of the alternative waveforms that have been proposed in the literature. Filter bank multicarrier (FBMC) is a waveform that can achieve time-frequency localization by utilizing a well-designed pulse shape/prototype filter [21, 22]. Furthermore, to maintain the orthogonality in such systems, real and imaginary symbols are staggered in time and frequency [22]. Thanks to the spectrum confinement of the subcarriers in FBMC, the uplink synchronization requirements can be substantially relaxed [13], and carrier aggregation becomes a trivial task [23]. Due to the above advantages, FBMC is 6 currently being considered as an enabling technology in various research and industrial projects; see [23] and the references therein. Despite good time-frequency localization, FBMC has its own drawbacks. Specifically, application of FBMC to MIMO channels is limited [22] and also the ramp-up and ramp-down of the FBMC signal at the beginning and the end of each packet reduces its bandwidth efficiency in applications that demand communication of short bursts, for example in MTC. In order to overcome the above problems of FBMC, circularly pulse shaped waveforms such as generalized frequency division multiplexing (GFDM) and circular FBMC (C-FBMC) have been recently proposed [24, 25]. In GFDM, complex quadrature amplitude modulation (QAM) symbols are modulated using time and frequency localized pulses based on the Gabor system [26]. However, as a consequence of the Balian-Low theorem [27], orthogonality cannot be achieved in GFDM, which makes GFDM a nonorthogonal waveform. C-FBMC combines the ideas of real/imaginary staggering and circular pulse shaping in order to maintain the orthogonality as well as all the advantages of GFDM. Although circularly pulse shaped waveforms enhance the bandwidth efficiency of FBMC, they cannot achieve the same insensitivity to synchronization errors compared to the linear FBMC [13, 28]. Another waveform that has been recently proposed is the universal filtered multicarrier (UFMC) [15, 29]. UFMC modifies OFDM by removing the CP and applying a filter on each group of subcarriers, for example, a physical resource block in the context of LTE systems. This improves the robustness to synchronization errors by limiting the out of band emissions of the subcarriers. Single-carrier (SC) modulation is an alternative to the above multicarrier waveforms, offering a significantly lower PAPR. By utilization of this modulation format, one can maximize the efficiency of the PA and extend the battery life. This is, in particular, useful in mMTC and IoT applications. By maximizing the PA power output, one can also increase the range of communications. This becomes crucial in mmWave spectrum where coverage is limited due to path loss. Another limiting factor in mmWave systems is the high-power consumption of the mixed-signal components such as the analog-to-digital converter (ADC) [30]. Fortunately, the low PAPR advantage of SC modulation allows for lower resolution ADCs to be used at the receiver. This results in a substantially lower power consumption in a mmWave based massive MIMO system [31]. SC modulation have been widely used in 7 standards such as GSM and more recently in IEEE 802.11 ad. 1.4 Dissertation Contributions As mentioned before, CP-OFDM is the main modulation format considered in most of the massive MIMO literature. On the other hand, there has been a great deal of interest in the area of waveform design in the recent past. As a result, it is natural to investigate the performance of some alternative waveforms in the context of massive MIMO. This constitutes the main goal of the study in this dissertation. The contributions of this dissertation can be summarized as follows. • We consider the FBMC modulation in the context of massive MIMO and perform an in-depth analysis of this system. We mathematically show that the signal-tointerference-plus-noise ratio (SINR) saturates at a certain deterministic level. We derive an analytical expression for the SINR saturation level using three main multiantenna combining methods, namely, maximum ratio combining (MRC), zero forcing (ZF), and minimum mean square error (MMSE). We propose an effective equalization method to resolve the saturation problem. With the proposed equalizer in place, SINR grows without a bound by increasing the BS array size, and arbitrarily large SINR values are achievable. An efficient implementation of the proposed equalization method through using some concepts from multirate signal processing is also presented. We perform a thorough analysis of the proposed system, and find the analytical expressions for the SINR in the cases of MRC and ZF detectors. The publications related to this contribution include [32–36]. • We study the possibility of removing the CP in OFDM-based massive MIMO systems. To increase the spectral efficiency of massive MIMO systems, we show that the CP overhead can be successfully eliminated. This is a result of the coherent combining of the received signals at the BS antennas that yields the channel distortions to disappear as the BS array size increases. We show that in the absence of the CP, the SINR performance of the conventional frequency-domain combining methods, namely, MRC, ZF, and MMSE, saturates at a certain deterministic level. Hence, arbitrarily large SINR values cannot be achieved by increasing the array size at the BS. We propose to use the time reversal (TR) technique to resolve the above saturation 8 problem. Although the conventional TR technique can achieve reasonable SINR values and is a viable option in many scenarios, it suffers from a high level of multiuser interference in multiuser cases. We propose a novel frequency-domain equalization (FDE) technique to be applied after the TR operation to reduce the MUI level. We refer to the conventional TR technique as TR-MRC, while the proposed TR-based method with additional frequency-domain equalization is referred to as TR-FDE. We introduce efficient methods to minimize the computational cost of both TR-MRC and TR-FDE receivers, and compare the complexity of the proposed receiver structures with that of the conventional CP-OFDM with MRC and ZF detectors. We perform a thorough analysis and obtain closed-form expressions for the SINR and achievable rate performance of both TR-MRC and TR-FDE receivers. Corresponding publications include [37], and [38]. • Finally, we consider SC modulation in the context of massive MIMO. Previous studies have shown that the time-reversal technique is a simple yet powerful method to be used for SC transmission in massive MIMO systems under low SNR conditions. We propose to use the FDE in addition to the time-reversal operation, and show that this can enhance the performance in moderate to high SNR conditions. Furthermore, we show that the added computational complexity depends on the number of user terminals and does not grow with the number of BS antennas. This is an important feature when the number of BS antennas is large. The publication corresponding to this contribution is [39]. It should be noted that other than the above contributions, which are the main focus of the dissertation, other publications have emanated from our research; see [13, 28, 40, 41]. 1.5 Structure of the Dissertation With the background presented in this chapter, this dissertation proceeds as follows. To pave the way for our analyses and discussions throughout the dissertation, we review the fundamentals of massive MIMO in Chapter 2. This includes an overview of the attractive features as well as the practical challenges of massive MIMO. The trend that has been taken in the standardization community to commercialize this technology is also given in this chapter. 9 In Chapter 3, we introduce the system model and the general assumptions that are used in the subsequent chapters. Chapters 4, 5, and 6 constitute the main body of the dissertation corresponding to the three main contributions. In Chapter 4, we conduct an in-depth study of the FBMC waveform in massive MIMO. In this chapter, we formally introduce the FBMC modulation format and give an asymptotic analysis of its performance in the context of massive MIMO. This analysis reveals that the SINR is upper bounded by a certain deterministic value, and arbitrarily large SINR values cannot be achieved by increasing the BS array size. We propose an efficient equalization technique to resolve this problem and mathematically analyze the SINR performance when the proposed equalizer is in place. In Chapter 5, we study the possibility of removing the CP overhead from the OFDM waveform. By having an excess number of BS antennas, we show that the channel distortions can be compensated through coherent combining in time domain and an effective equalization. We also analyze the spectral efficiency of the proposed system in this chapter and show that a higher spectral efficiency is indeed achieved as compared to the conventional CP-OFDM systems. In Chapter 6, we consider SC transmission in massive MIMO. A novel receiver design based on time-reversal combining and frequency-domain equalization is proposed in this chapter. We show that through the proposed system, the performance of SC transmission can be enhanced in moderate to high SNR regime as compared to the conventional algorithms. Finally, concluding remarks are given in Chapter 7 along with possible future research activities. CHAPTER 2 FUNDAMENTALS OF MASSIVE MIMO In this chapter we review the fundamentals of massive MIMO in some detail. This will pave the way for our analyses and discussions in later chapters of this dissertation. 2.1 From MIMO to MU-MIMO to Massive MIMO MIMO technology has emerged in the recent past as a unique solution for increasing the throughput and reliability of wireless communication systems. Point-to-point MIMO systems incorporate multiple antennas at both sides of the communication link and thus can benefit from the additional dimension of space. This, however, requires expensive multiantenna terminals. Moreover, multiplexing gains may not be satisfactory in the low signal-to-noise (SNR) regime or in line-of-sight (LOS) conditions where the signals from different antennas cannot be resolved [3]. As an alternative to point-to-point MIMO, multiuser-MIMO (MU-MIMO) has been suggested to increase the network capacity [42]. In this case, the mobile terminals can be cheap single-antenna devices that are separated from each other by tens or hundreds of wavelengths, and the multiplexing gains are shared between them. A BS with multiple antennas is considered to be at the other end of this communication scenario. In this setup, although some user terminals may experience poor propagation conditions (for example low SNR or LOS channel), promising multiplexing gains of MIMO are expected for the network as a whole. Built upon the MIMO technology, massive MIMO has been recently suggested to further improve the throughput and reliability of wireless communication systems [2]. Massive MIMO, in essence, considers a MU-MIMO system which incorporates a BS with a large number of antennas; an order of magnitude larger than the number of users that it serves. By coherent processing of the users’ signals over the BS antennas, the effects of uncorrelated noise and multiuser interference can be made arbitrarily small as the BS array size increases 11 [2]. This also allows for focusing of the BS’s emitted energy into specific regions in space, where the terminals reside. As a result, the interference between users is reduced and the energy efficiency of the network is significantly improved. These benefits of massive MIMO have initiated a broad range of research studies that seriously consider Massive MIMO as a strong technology enablement for the fifth generation of wireless cellular systems, 5G [1]. On the other hand, the availability of a vast usable spectrum in the millimeter wave band (30-300 GHz) has driven a significant interest in millimeter wave (mmWave) communications in the recent past. This paradigm can address the explosive demand for mobile traffic envisioned in the future 5G networks. Although the mmWave band has been previously utilized in radar and sensing applications, its use in communication systems was limited mainly due the poor propagation properties. Fortunately, the emergence of massive MIMO has unleashed the capabilities of mmWave communication. In particular, using a large number of antenna elements at the BS, one can direct the signal energy to specific points in space and overcome the propagation pathloss limitations of mmWave frequencies. Interestingly, the short wavelengths at mmWave frequencies allow for large antennas to be packed into small form factors, making mmWave and massive MIMO two paradigms that are usually considered together. The necessity of utilizing massive MIMO in mmWave frequencies constitutes another reason for adopting large array sizes in the next generation of wireless communication systems. 2.2 Review of the Law of Large Numbers The law of large numbers plays a pivotal role for understanding the theoretical aspects of massive MIMO networks. In fact, most of the asymptotic results in the massive MIMO literature are derived using this theorem; see for example [2, 4]. Naturally, it is also extensively used throughout this dissertation. Hence, in this section, we provide a brief overview of this fundamental theorem of statistics. The reader is encouraged to refer to the classical probability and statistics text books, for example [43], for a more in-depth review. In essence, the law of large numbers states that the sample average obtained from a large number of random variables converges to the statistical mean as the number of samples increases. To formally express the theorem, let a = [a1 , . . . , an ]T and b = [b1 , . . . , bn ]T be two random vectors each containing i.i.d. elements with E{|ai |2 } = σa2 and E{|bi |2 } = σb2 12 for i = 1, . . . , n. Furthermore, assume that ith elements of a and b are correlated according to E a∗i bi = Cab , i = 1, . . . , n. Then, according to the law of large numbers, the sample P mean n1 kak2 = n1 ni=1 |ai |2 converges almost surely to the statistical mean σa2 as n tends to infinity, that is, 1 kak2 → σa2 as n → ∞, n with almost sure convergence. Similarly, the sample correlation (2.1) 1 H na b = 1 n Pn i=1 ai bi converges almost surely to the statistical correlation Cab as n tends to infinity, that is, 1 H a b → Cab as n → ∞, n (2.2) with almost sure convergence. 2.3 Asymptotic Optimality of Linear Processing In general, in a MU-MIMO system, the optimum performance can be achieved using complex processing methods such as dirty paper coding in downlink [44], and spheredecoding in uplink [45]. Fortunately, the availability of a large number of BS antennas can greatly simplify the processing. In particular, it can be proved mathematically that if the number of BS antennas is much larger than the number user terminals, then linear methods such as maximum-ratio and zero-forcing combining/processing are optimal [46]. This provides a great flexibility in algorithm implementation. The following example gives an intuition why linear processing is asymptotically optimum in massive MIMO. Consider the uplink of a narrow band massive MIMO system with K users and a BS with an array of N K antenna elements. Let x be a K × 1 vector representing the transmit symbols of terminals, and y be a N × 1 vector of received symbols at the BS. Hence, we have y = Hx + ν, (2.3) where H is a N × K matrix of channel coefficients and ν is the vector of additive noise. Assume that the elements of H are i.i.d. CN (0, 1) random variables. Similarly, we assume that the elements of the noise vector are i.i.d. CN (0, σν2 ). Assuming that the channel matrix H is available at BS, the transmit vector x can be estimated using the simple matched filter (MF) combining matrix W = 1 N H. The result is 13 x̂ = W H y = 1 H 1 H Hx + HH ν. N N (2.4) According to the law of large numbers, as the number of BS antennas N grows large, converges almost surely to IK . Similarly, the noise contribution 1 H NH ν 1 H NH H tends to zero as N increases. As a result, the effect of multiuser interference and uncorrelated noise becomes negligible in massive MIMO by using simple linear processing. This phenomenon is referred to as favorable propagation condition in the massive MIMO literature [3, 47]. According to this phenomenon, the vector corresponding to the channel between a user and the BS antennas becomes asymptotically orthogonal to the one corresponding to other users. The above result was derived based on the Rayleigh fading channel assumption. The validity of favorable propagation condition in real massive MIMO implementations has been studies in [48, 49]. As the above example illustrates, a simple linear processing can eliminate noise and interference completely provided that N K. Hence, optimum performance can be achieved asymptotically without necessity of utilizing complex processing algorithms. We now obtain the information theoretic achievable rate performance of a massive MIMO system in the uplink. This analysis can help us better understand some of the advantages of massive MIMO. Also, we may utilize some of these relationships in later chapters where we analyze the achievable rate performance of alternative waveforms in a massive MIMO system. The sum-capacity (total throughput) of a narrow band MU-MIMO system as represented by (2.3) can be calculated as [50], C = log2 IK + 1 H H H σν2 (bits/sec/Hz). (2.5) Assuming N K and using the favorable propagation condition discussed above, we have HH H ≈ N IK . As a result, the asymptotic sum-rate can be expressed as N CN K ≈ K · log2 1 + 2 σν (bits/sec/Hz). (2.6) According to the above result, using a large array of BS antennas, we can essentially decompose the channel into K parallel links each with the effective SNR of N . σν2 Notice that 14 the above asymptotic capacity can be achieved using the simple MF combining method given in (2.4). Hence, such linear processing becomes optimum asymptotically. 2.4 Obtaining the Channel State Information One of the main concerns in a any MIMO system is the availability of channel state information (CSI) needed for detection/precoding. This concern clearly becomes more pronounced in a massive MIMO network where the number of channel coefficients to be estimated is very large. Hence, it is intuitive that this issue can impose some limitations to the massive MIMO systems. For example, as discussed in [2], the number of users that can be supported by a massive MIMO BS is limited by the coherence time of the channel. This constraint arises from the limitations of CSI acquisition. To better understand this, we note that in a typical wireless communication system, a time period with duration equal to the coherence time of the channel is divided into two intervals: (i) training period and (ii) data transmission period. The CSI is acquired during the training period and is used in the second interval for data transmission. Since the overall time interval is less than the coherence time of the channel, the CSI estimated in the first period is also valid during the second one. However, this also means that the time allocated for training is naturally upper limited by a fraction of the coherence time of the channel. The implication of this constraint in a massive MIMO system is that the time needed for training does not scale with the number of BS antennas. Another point to be noted is that time division duplex (TDD) operation with reciprocal uplink/downlink channels are often assumed in the massive MIMO literature. Under this condition, the channel estimates can be obtained using uplink pilots, providing us with the CSI needed for downlink precoding and uplink combining. In this scheme, orthogonal uplink pilots are assigned to different users. Hence, the duration of the training period will scale by the number of users and is independent of the number of BS antennas. 2.5 Challenges and Limitations In this section, we discuss some of the main challenges and limitations that have emerged in massive MIMO. Multicellular massive MIMO networks based on TDD operation suffer from a pilot 15 contamination problem which was first reported in [51]. This is a major factor in limiting the capacity of those networks. Following the CSI acquisition method described in the previous section, the channel estimates are obtained at the BS during the uplink transmission. Since the channel coherence time is usually not long enough to allow for utilization of orthogonal pilot sequences in different cells, nonorthogonal pilots of neighboring cells will contaminate the pilots of each other [2, 51, 52]. Thus, the channel estimates at each BS will contain the channel information of the user terminals located in the other cells as well as its own users. As a result, when the BS linearly combines the received signal in order to decode the transmitted symbols of its users, it also combines the data symbols of the users of other cells which results in inter-cell interference. The corresponding inter-cell interference does not vanish even when the number of BS antenna elements tends to infinity [2]. To tackle the pilot contamination problem, a number of solutions have been proposed in the literature. In references [53–56], the effect of pilot contamination is mitigated by optimization of the pilot assignment between the neighboring cells. In [34, 57–59], nonlinear channel estimation algorithms are incorporated to reduce the channel estimation errors caused by the pilot contamination. The authors in [60, 61] considered a cooperative multicell network and developed some algorithms to decode the users’ data by joint processing among different BS units. Mobility of the user terminals is another limiting factor in massive MIMO. Mobility reduces the coherence time of the channel, and hence, a shorter time period will be available for channel training and obtaining the CSI. This has a direct impact on the number of users that can be served in the network. In particular, as we mentioned in the previous section, in the massive MIMO literature, the de facto strategy for obtaining the CSI is to operate in the TDD mode and rely on the uplink pilots and channel reciprocity to obtain the channel estimates. With this technique, the duration of the training interval is proportional to the number of user terminals. As the duration of this interval is reduced due to the mobility, fewer users can be accommodated in a network. Incorporating massive MIMO also affects the choice of analog and digital hardware architecture used. This, in particular, is more pronounced in mmWave frequencies. To understand this better, we note that a full digital implementation of such a communication system requires one radio frequency (RF) chain per antenna element. Consequently, the cost 16 and power consumption of mixed-signal hardware components become a critical issue, making the full-digital architecture less practical. On the other hand, the hardware architecture based on hybrid analog/digital beamforming has been shown to be a promising approach providing a decent balance between performance and hardware constraints [62–65]. In the hybrid architecture, part of the beamforming occurs in analog RF domain using a network of phase shifters, and further precoding/combining in the digital domain ensures having performances close to the full-digital solutions. Accordingly, the number of RF chains can be significantly reduced while the performance remains almost identical. Acquiring the CSI and selecting the beamforming coefficients in the hybrid architecture is a challenging task and is currently the subject of active research. 2.6 Deployment in 3GPP Standards As mentioned in the previous chapter, massive MIMO is a principal technology enabler in 5G new radio (NR). The 3GPP release 15 contains the specifications corresponding to 5G NR. It is also worth mentioning that to facilitate the deployment of massive MIMO in 5G, some preliminary version of this technology was also developed in the preceding releases. Hence, in this section, we overview the path that the 3GPP community has taken to bring massive MIMO into real life. The first industrial version of massive MIMO was defined in LTE-advanced Pro (3GPP release 13). Before this, up to 8 BS antenna ports was supported by the standard. Moreover, the only antenna configuration assumed was a uniform linear array (ULA). Therefore, the focus was primarily on one-dimensional beamforming in the azimuth domain. With the advent of massive MIMO and advancement of RF technologies, deployment of a larger number of antenna elements was feasible in release 13. Accordingly, the concept of fulldimension MIMO (FD-MIMO) was introduced in this release [66]. In FD-MIMO, the BS is equipped with a two-dimensional rectangular antenna array. Thus, beamforming can be done in both azimuth and elevation directions in the space. In release 13, up to 16 antenna ports is supported and the CSI feedback mechanism is improved to take the larger number of antennas into account. In release 14, the maximum number of antenna ports supported is extended to 32. In release 15, the concept of beam management is introduced to facilitate the CSI 17 acquisition and tracking. Beam management consists of four different procedures: (i) beam sweeping, (ii) beam measurement, (iii) beam determination, and (iv) beam reporting [67]. The objective is to provide an efficient means for aligning of the transmitter and the receiver beams with high resolution. Moreover, to lower the cost of implementation, the idea of multipanel MIMO is introduced. In multipanel MIMO, the antenna elements are configured in multiple panels [68]. Each panel consists of an antenna array integrated in a silicon package using advanced chip design techniques. The above developments pave the way for supporting 64+ antenna elements at the BS in release 15. CHAPTER 3 SYSTEM MODEL The system model that is used throughout this dissertation is introduced in this chapter. We consider a single-cell massive MIMO setup [2], with K single-antenna user terminals that are simultaneously communicating with a BS equipped with an array of N antenna elements. Throughout this dissertation, we consider the uplink transmission while the results and our proposed techniques are applicable to the downlink transmission as well. We consider a discrete-time model for our analysis. Let xk [l] represent the transmit signal of terminal k in discrete time. The received signal at the nth BS antenna can be expressed as rn [l] = K−1 X xk [l] ? hn,k [l] + νn [l], (3.1) k=0 where hn,k [l] is the channel impulse response between the k th terminal and the nth BS antenna, and νn [l] is the additive noise at the input of the nth BS antenna. We assume that the samples of the noise signal νn [l] are a set of i.i.d. CN (0, σν2 ) random variables both in time and across the BS antennas. The transmit signal xk [l] is synthesized using different modulation formats as discussed in the following chapters. Also, for a given terminal k, we model the corresponding channel responses using the channel power delay profile (PDP) pk [l], l = 0, . . . , L − 1. In particular, we assume that the channel tap hn,k [l], l ∈ {0, . . . , L − 1}, follows a CN (0, pk [l]) distribution, and different taps are assumed to be independent. Additionally, we assume that the channels corresponding to different terminals and different BS antennas are independent. The above assumption implies that the BS antenna array is sufficiently compact so that the channel responses corresponding to a particular user and different BS antennas are subject to the same channel PDP. Throughout this dissertation, we consider normalized channel PDPs for each terminal such that 19 L−1 X pk [l] = 1, l=0 k ∈ {0, . . . , K − 1}. (3.2) This implies a perfect power control at the BS. Moreover, we assume that for each terminal, the average transmitted power is equal to one, that is, E{|xk [l]|2 } = 1. As a result, considering the above channel model, the average SNR at the input of the BS can be calculated as SNR = 1 . σν2 (3.3) Throughout this dissertation, we assume that perfect CSI is available at the BS. This simplifies the analysis and helps us to understand the underlying phenomena when considering novel waveform design methods in massive MIMO systems. Extending the results to include the respective channel estimation techniques and the corresponding performance measurements are left for a future study. When considering multicarrier waveforms, detection is usually performed on a persubcarrier basis. Let M denote the number of subcarriers, and ym be the N × 1 vector containing demodulated symbols corresponding to subcarrier m ∈ {0, . . . , M − 1} for different BS antennas. Hence, linear multiantenna combining can be done according to WH m ym , where W m is an N × K combining matrix corresponding to subcarrier m. This results in the detected symbols of different terminals. We consider three conventional linear combiners, namely, maximum ratio combining (MRC), zero forcing (ZF) and minimum mean square error (MMSE). For these combiners, we have  −1  for MRC, Hm Dm , −1 H W m = Hm Hm Hm , for ZF,  −1  H 2 Hm Hm Hm + σν IK , for MMSE, (3.4) where Hm is the N × K matrix of frequency-domain channel coefficients for the mth P −j 2πlm M . In the case of MRC, Dm is a K × K subcarrier, that is, [Hm ]nk = L−1 l=0 hn,k [l]e diagonal matrix whose diagonal elements are formed by the diagonal elements of HH m Hm . The role of Dm is just to normalize the amplitude of the MRC output. Without this term, the amplitude grows linearly without a bound as the number of BS antennas increases. We note that for large number of BS antennas N and using the law of large numbers, 1 H N Hm Hm tends to IK [4]. Similarly, the matrix 1 N Dm tends to IK as the number of BS antennas increases. Hence, all of the above combiners tend to 1 N Hm , namely, MF combining, 20 as the number of BS antennas increases [4]. In light of this observation, throughout this dissertation, to perform an analysis in the asymptotic regime, that is, as the number of BS antennas N approaches infinity, we consider MF multiantenna combining according to Wm = 1 N Hm . CHAPTER 4 FILTER BANK MULTICARRIER IN MASSIVE MIMO In this chapter, we perform an asymptotic study of the performance of FBMC in the context of massive MIMO. We show that the effects of channel distortions, that is, intersymbol interference and intercarrier interference, do not vanish as the BS array size increases. As a result, the SINR cannot grow unboundedly by increasing the number of BS antennas, and is upper bounded by a certain deterministic value. We show that this phenomenon is a result of the correlation between the multiantenna combining tap values and the channel impulse responses between the mobile terminals and the BS antennas. To resolve this problem, we introduce an efficient equalization method that removes this correlation, enabling us to achieve arbitrarily large SINR values by increasing the number of BS antennas. We perform a thorough analysis of the proposed system and find analytical expressions for both equalizer coefficients and the respective SINR. 4.1 Introduction FBMC is a waveform that has gained an increased attention in the recent years due to its improved spectral properties compared to OFDM [13, 22, 23]. The application of FBMC to massive MIMO channels has been recently studied in [69], where its so-called self-equalization property leading to a channel flattening effect was reported through simulations. According to this property, the effects of channel distortions (intersymbol interference and intercarrier interference) will diminish by increasing the number of BS antennas. The authors in [70] obtain the asymptotic mean squared error (MSE) performance of FBMC in massive MIMO channels. Their analysis shows that the MSE becomes uniform across different subcarriers as a result of the channel hardening effect. In [36], multitap equalization per subcarrier is proposed for FBMC-based massive MIMO systems to improve the equalization accuracy as compared to the single-tap equalization at the expense of a higher computational 22 complexity. The authors in [34] show that the pilot contamination problem in multicellular massive MIMO networks can be resolved in a straightforward manner with FBMC signaling due to its special structure. These studies prove that FBMC is an appropriate match for massive MIMO and vice versa as they can both bring pivotal properties into the picture of the next generations of wireless systems. Specifically, this combination is of a great importance as not only the same spectrum is being simultaneously utilized by all the users but it is also used in a more efficient manner compared to OFDM. Since the literature on FBMC-based massive MIMO is not mature yet, these systems need to go through meticulous analysis and investigation. In particular, in this chapter, we perform an in-depth analysis on the performance of FBMC in massive MIMO channels. We consider single-tap equalization per subcarrier, and investigate the performance of three most prominent linear combiners, namely, MRC, ZF, and MMSE. We show that the self-equalization property shown through simulations and claimed in [69] and [36] is not very accurate. More specifically, by increasing the number of BS antennas, the channel distortions average out only up to a certain extent, but not completely. Thus, the SINR saturates at a certain deterministic level. This determines an upper bound for the SINR performance of the system. Our main contributions in this chapter are the following; (i) We derive an analytical expression for the SINR saturation level using MRC, ZF, and MMSE combiners. (ii) We propose an effective equalization method to resolve the saturation problem. With the proposed equalizer in place, SINR grows without a bound by increasing the BS array size, and arbitrarily large SINR values are achievable. (iii) An efficient implementation of the proposed equalization method through using some concepts from multirate signal processing is also presented. (iv) Finally, we perform a thorough analysis of the proposed system, and find the analytical expressions for the SINR in the cases of MRC and ZF detectors. All the above analyses are evaluated and confirmed through numerical simulations. It is worth mentioning that although the theories developed in this chapter are applicable to all types of FBMC systems, the formulations are based on the most common type in the literature that was developed by Saltzberg [71], and is known by different names including OFDM with offset quadrature amplitude modulation (OFDM/OQAM), FBMC/OQAM, and staggered multitone (SMT) [22]. Throughout this chapter, we refer to it as FBMC for 23 simplicity. The rest of the chapter is organized as follows. To pave the way for the derivations presented in the chapter, we review the FBMC principles in Section 4.2. In Section 4.3, we present the asymptotic equivalent channel model between the mobile terminals and the BS in an FBMC massive MIMO setup. This analysis will lead to an upper bound for the SINR performance of the system. Our proposed equalization method is introduced in Section 4.4. In Section 4.5, we study the FBMC in massive MIMO from a frequency-domain perspective, leading to some insightful remarks regarding these systems. In Section 4.6, we find the SINR performance of the FBMC system incorporating the proposed equalization method. The mathematical analysis of the chapter as well as the efficacy of the proposed filter design technique are numerically evaluated in Section 4.7. Finally, we conclude the chapter in Section 4.8. 4.2 FBMC Principles We present the theory of FBMC in discrete time. Let dm,i denote the real-valued data symbol transmitted over the mth subcarrier and the ith symbol time index. The total number of subcarriers is assumed to be M . In order to avoid interference between the symbols and, thus, maintain the orthogonality, the data symbol dm,i is phase adjusted using the phase term ejθm,i , where θm,i = π 2 (m + i). Accordingly, each symbol has a ± π2 phase difference with its adjacent neighbors in both time and frequency. The symbols are then pulse-shaped using a prototype filter f [l], which has been designed such that q[l] = f [l]?f ∗ [−l] is a Nyquist pulse with zero crossings at M sample intervals. The length of the prototype filter, f [l], is usually expressed as Lf = κM , where κ is called the overlapping factor1 . To express the above procedure in a mathematical form, the discrete-time FBMC waveform can be written as [21] x[l] = +∞ M −1 X X dm,i am,i [l], (4.1) i=−∞ m=0 where am,i [l] = fm [l − iM/2]ejθm,i . 1 (4.2) The overlapping factor indicates the number of adjacent FBMC symbols overlapping in the time domain. 24 Here, fm [l] , f [l]ej 2πml M is the prototype filter modulated to the center frequency of the mth subcarrier, and the functions am,i [l], for m ∈ {0, . . . , M − 1} and i ∈ {−∞, . . . , +∞}, can be thought as a set of basis functions that are used to modulate the data symbols. Note that the spacing between successive symbols in the time domain is M/2 samples. In the frequency domain, the spacing between successive subcarriers is 1/M in normalized frequency scale. It can be shown that the basis functions am,i [l] are orthogonal in the real domain [21], that is, ham,i [l], am0 ,i0 [l]i< = < X +∞ am,i [l]a∗m0 ,i0 [l] l=−∞ = δmm0 δii0 . (4.3) As a result, the data symbols can be extracted from the synthesized signal, x[l], according to dm,i = hx[l], am,i [l]i< . (4.4) Figure 4.1 shows the block diagram of the FBMC transceiver. Note that considering the transmitter prototype filter f [l], and the receiver prototype filter f ∗ [−l], the overall effective pulse shape q[l] = f [l] ? f ∗ [−l] is a Nyquist pulse by design. Also, in practice, in order to efficiently implement the synthesis (transmitter side) and analysis (receiver side) filter banks, one can incorporate the polyphase implementation to reduce the computational complexity [22]. The presence of a frequency-selective channel leads to some distortion in the received signal. Thus, one may adopt some sort of equalization to retrieve the transmitted symbols at the receiver side. Here, we limit our study to a case where the channel impulse response remains time-invariant over the interval of interest. Accordingly, the received signal at the receiver can be expressed as y[l] = h[l] ? x[l] + ν[l], (4.5) where h[l] represents the channel impulse response, and ν[l] is the additive white Gaussian noise (AWGN). 25 ejθ0,i M 2 d0,i Synthesis filter bank Analysis filter bank f0 [l] f0∗ [−l] e−jθ0,i M 2 f1 [l] Σ x[l] y[l] Ideal Channel y1,i M 2 f1∗ [−l] ejθM −1,i dM −1,i ℜ{·} dˆ0,i ℜ{·} dˆ1,i e−jθ1,i ejθ1,i d1,i y0,i M 2 e−jθM −1,i M 2 M 2 ∗ fM −1 [−l] fM −1 [l] yM−1,i ℜ{·} dˆM −1,i Figure 4.1. Block diagram of the FBMC transceiver in discrete time. At the receiver, after matched filtering and phase compensation, and before taking the real part (see Figure 4.1), the demodulated signal ym,i can be expressed as +∞ M −1 X X ym,i = Hmm0 ,ii0 dm0 ,i0 + νm,i , (4.6) i0 =−∞ m0 =0 where νm,i is the noise contribution, and the interference coefficient Hmm0 ,ii0 can be calculated according to Hmm0 ,ii0 = hmm0 [i − i0 ]ej(θm0 ,i0 −θm,i ) , ∗ hmm0 [i] = fm0 [l] ? h[l] ? fm [−l] M . ↓ The symbol ↓ M 2 denotes M 2 -fold (4.7a) (4.7b) 2 decimation. In (4.7), hmm0 [i] is the equivalent channel impulse response between the transmitted symbols at subcarrier m0 and the received ones at subcarrier m. This includes the effects of the transmitter filtering, the multipath channel, and the receiver filtering; see Figure 4.2. According to (4.6), the demodulated symbol ym,i suffers from interference originating from other time-frequency symbols. In practice, the prototype filter f [l] is designed to be well localized in time and frequency. As a result, the interference is limited to a small neighborhood of time-frequency points around the desired point (m, i). In order to devise a simple equalizer to combat the frequency-selective effect of the channel, it is usually assumed that the symbol period M/2 is much larger than the channel length L, or equivalently, the channel frequency response is approximately flat over each subcarrier band. With this assumption, the demodulated signal ym,i can be expressed as [72] ym,i ≈ Hm dm,i + um,i + νm,i , (4.8) 26 e−jθm,i ejθm′ ,i′ M 2 dm′ ,i ′ fm′ [l] h[l] ∗ fm [−l] M 2 ym,i Figure 4.2. The equivalent channel between the transmitted data symbol at time-frequency point (m0 , i0 ) and the demodulated symbol at time-frequency point (m, i). where Hm , PL−1 l=0 h[l]e−j 2πml M is the channel frequency response at the center of the mth subcarrier. The term um,i is called the intrinsic interference and is purely imaginary. This term represents the contribution of the intersymbol interference (ISI) and intercarrier interference (ICI) from the adjacent time-frequency symbols around the desired point (m, i). Based on (4.8), the effect of channel distortions can be compensated using a single-tap equalizer per subcarrier. After equalization, what remains is the real-valued data symbol dm,i , the imaginary term um,i , and the noise contribution. By taking the real part from the equalized symbol, one can remove the intrinsic interference and obtain an estimate of dm,i . It should be noted that the performance of the above single-tap equalization primarily depends on the validity of the assumption that the symbol duration is much larger than the channel length, or equivalently, the frequency response of the channel is approximately flat over the pass-band of each subcarrier. On the other hand, in highly frequency-selective channels, where the above assumption is not accurate any more, more advanced multitap equalization methods (see [23, 73]) should be deployed to counteract the multipath channel distortions. 4.3 Massive MIMO FBMC: Asymptotic Analysis In this section, we first extend the formulation of the previous section to massive MIMO channels. Then, we show that linear combining of the signals received at the BS antennas using the channel frequency coefficients leads to a residual interference that does not fade away even with an infinite number of BS antennas. Hence, we conclude, the SINR is upper bounded by a certain deterministic value, and arbitrarily large SINR performances cannot be achieved as the number of BS antennas grows. Following (3.1) and the system model introduces in Chapter 3, we can extend (4.6) to the MIMO case according to 27 ym,i = +∞ M −1 X X i0 =−∞ Hmm0 ,ii0 dm0 ,i0 + ν m,i , (4.9) m0 =0 where ym,i is an N ×1 vector containing the demodulated symbols corresponding to different BS antennas, dm,i is a K × 1 vector containing the real-valued data symbols of all the K terminals transmitted at the mth subcarrier and the ith time instant, ν m,i is the noise contribution across different BS antennas, and Hmm0 ,ii0 is an N × K channel matrix. The element (n, k) of Hmm0 ,ii0 can be calculated according to n,k n,k 0 j(θm0 ,i0 −θm,i ) Hmm , 0 ,ii0 = hmm0 [i − i ]e ∗ 0 hn,k mm0 [i] = fm [l] ? hn,k [l] ? fm [−l] (4.10a) ↓M 2 . (4.10b) We assume that the BS uses a single-tap equalizer per antenna per subcarrier. Accordingly, combining the elements of ym,i using an N × K matrix W m , and taking the real part from the resulting signal, the estimate of the transmitted data symbols for all the terminals can be obtained as d̂m,i = < W H m ym,i +∞ −1 n X M o X H =< WH m Hmm0 ,ii0 dm0 ,i0 + W m ν m,i i0 =−∞ m0 =0 =< +∞ M −1 n X X o Gmm0 ,ii0 dm0 ,i0 + ν 0m,i , (4.11) i0 =−∞ m0 =0 H 0 where Gmm0 ,ii0 , W H m Hmm0 ,ii0 , and ν m,i , W m ν m,i . Here, we examine MRC, ZF, and MMSE linear combiners as introduced in (3.4). Moreover, as discussed in Chapter 3, all of the these combiner matrices tend to 1 N Hm , namely, MF combiner, as the number of BS antennas increases. Therefore, in the following, to find the various interference terms in the asymptotic regime, that is, as the number of BS antennas N approaches infinity, we consider MF multiantenna combining according to W m = 1 N Hm . In the asymptotic regime, that is, as N tends to infinity, the elements of Gmm0 ,ii0 = WH m Hmm0 ,ii0 can be calculated using the law of large numbers. In particular, as N grows large, the element (k, k 0 ) of Gmm0 ,ii0 converges almost surely to o n ∗ 0 n,k0 n,k → E H H Gk,k m mm0 ,ii0 , mm0 ,ii0 n,k where Hm = PL−1 l=0 hn,k [l]e−j 2πlm M (4.12) is the element (n, k) of the channel matrix Hm . To calculate the right hand side of (4.12), we first find the equivalent time-domain channel 28 impulse response after multiantenna combining. 0 k,k In particular, let gmm 0 [i] denote the equivalent channel impulse response between the transmitted symbols at subcarrier m0 of terminal k 0 and the received ones at subcarrier m of BS output corresponding to terminal k after combining2 . Following (4.10), we have k,k0 gmm 0 [i] N −1 1 X n,k ∗ ∗ Hm fm0 [l] ? hn,k0 [l] ? fm [−l] ↓ M . = 2 N (4.13) n=0 Hence, as the number of BS antennas grows large, the asymptotic equivalent channel response can be obtained using the law of large numbers according to n ∗ o k,k0 ∗ n,k 0 0 f [l] ? h [l] ? f [−l] gmm [i] → E H 0 m n,k m m ↓M n ∗ o 2 n,k ∗ = fm0 [l] ? E Hm hn,k0 [l] ? fm [−l] M . ↓ (4.14) 2 n,k The above expression includes a correlation between the channel frequency coefficient Hm and the channel impulse response hn,k0 [l]. This correlation can be calculated as n ∗ o L−1 X 2π`m n,k E Hm hn,k0 [l] = E h∗n,k [`]hn,k0 [l] ej M `=0 = pk [l]ej where pk,m [l] , pk [l]ej 2πlm M 2πlm M δkk0 = pk,m [l]δkk0 , (4.15) is the channel PDP of terminal k modulated to the center frequency of the mth subcarrier. The result in (4.15) shows the correlation between the combiner taps at the receiver and the channel impulse responses between the terminals and the BS antennas. The following proposition states the impact of this correlation on the SINR at the receiver outputs. Proposition 1. In an FBMC massive MIMO system, as the number of BS antennas tends to infinity, the effects of multiuser interference and noise vanish. However, some residual ISI and ICI from the same user remain even with infinite number of BS antennas. In particular, 2 Note that we have used the letters g and G, respectively, to denote the equivalent time and frequency channel coefficients after combining. On the other hand, letters h and H have been used in (4.10) to refer to the respective channel coefficients before combining. 29 for a given user k, the equivalent channel impulse response between the transmitted data symbols at subcarrier m0 and the received ones at subcarrier m tends to k,k ∗ 0 , gmm [i] → f [l] ? p [l] ? f [−l] 0 k,m m m M ↓ (4.16) 2 which is dependent on the channel PDP. As a result, the SINR converges almost surely to <2 Gk,k mm,ii k SINRm,i → +∞ M −1 , (4.17) P P 2 k,k < Gmm0 ,ii0 i0 =−∞ m0 =0 (m0 ,i0 )6=(m,i) k,k 0 j(θm0 ,i0 −θm,i ) . The above value constitutes an upperbound where Gk,k mm0 ,ii0 = gmm0 [i − i ] e for the SINR performance of the system. Hence, arbitrarily large SINR values cannot be achieved by increasing the BS array size. Proof. As suggested by (4.15), when k 0 6= k, the channel response tends to zero. Thus, multiuser interference tends to zero. A similar argument can be made for the additive noise. This results from the law of large numbers and the fact that the combiner coefficients are uncorrelated with the filtered noise samples. When k 0 = k, which implies the interference from the same user on itself, the channel response tends to (4.16). Notice that due to the presence of pk,m [l], the orthogonality condition of (4.3) does not hold anymore even with an infinite number of BS antennas. Consequently, some residual ISI and ICI remain and cause the SINR to saturate at a deterministic level given in (4.17). We note that according to (4.12), the asymptotic SINR saturation results from the statistical correlation between the multiantenna combiner taps and the interference coefficients. This correlation is an inherent property of FBMC-based massive MIMO systems and is due to the transients of the channel impulse response since no CP is used. In particular, when the multiantenna combining is performed in the frequency domain according to (3.4), such correlation appears as a result of the leakage due to the absence of CP. This result is general as a similar phenomenon also emerges in massive MIMO systems based on OFDM without CP; see Chapter 5. 4.4 Equalization As discussed in the previous section, even with an infinite number of BS antennas, some residual ICI and ISI remain due to the correlation between the combiner taps and the 30 channel impulse responses between the user terminals and the BS antennas. As a solution to this problem, in this section, we propose an efficient equalization method to remove the above correlation. In (4.16), the problematic term that leads to the saturation issue is the modulated k,k channel PDP, pk,m [l]. In the absence of this term, the channel response gmm 0 [i] = fm0 [l] ? ∗ [−l] does not incur any interference provided that q[l] = f [l] ? f ∗ [−l] is a Nyquist fm ↓M 2 pulse. This observation suggests that we can resolve the saturation issue by equalizing the effect of pk,m [l]. Let Pk (ω) denote the discrete-time Fourier transform (DTFT) of pk [l]. Similarly, we define Pk,m (ω) = Pk (ω − 2πm/M ) as the DTFT of pk,m [l]. This observation implies that one can equalize the effect of pk,m [l] by introducing a filter φk,m [l] with transfer function Φk,m (ω) = 1 , Pk,m (ω) (4.18) ∗ [−l] to achieve the desired equivalent channel response g k,k [i] → f 0 [l]? in cascade with fm m mm0 ∗ [−l] in the asymptotic regime. This modifies the receiver structure as illustrated in fm ↓M 2 Figure 4.3. Proposition 2. In an FBMC massive MIMO system, as the number of BS antennas tends to infinity and by using the proposed equalization method, the channel distortions, that is, ICI and ISI, as well as MUI and noise effects will disappear, and arbitrarily large SINR performances can be achieved. Proof. Using the equalizer in (4.18), the distortion due to the channel PDP pk,m [l] in the equivalent channel impulse response in (4.16) is removed. Hence, the equivalent channel impulse response tends to that of an ideal channel. As a result, the effects of ICI and ISI will vanish asymptotically. Note that in the presence of the proposed equalizer, multiuser interference still tends to zero. This is due to the fact that the asymptotic values of the multiuser interference coefficients are given by (4.12) for k 6= k 0 . Since the channels of different users are independent, the effect of multiuser interference tends to zero whether or not the proposed equalizer is in place. This argument also holds for the noise contribution since the combining coefficients and the filtered noise samples are independent. 31 y0 [l] yN −1[l] 0,k Wm ∗ fm [−l] φk,m [l] M 2 e−jθm,i ℜ{·} N −1,k Wm ∗ [−l] fm φk,m [l] dˆkm,i M 2 Figure 4.3. Block diagram of the proposed receiver structure to resolve the saturation issue. Here, only the portion of the receiver corresponding to subcarrier m and terminal k is shown. It is worth mentioning that in the above analysis, we did not make any assumption about the flatness of the channel response over the bandwidth of each subcarrier. Thus, the result obtained in Proposition 2 is valid for any frequency-selective channel. It is worth mentioning that according to (4.18), the proposed filter response depends on the channel PDPs. Hence, the BS needs to estimate the channel PDP for each terminal to be able to avoid the saturation issue. Fortunately, in massive MIMO systems, the channel PDP can be estimated in a relatively easy and feasible manner. In particular, the channel PDP for each terminal can be determined by calculating the mean power of each tap of the respective channel impulse responses across different BS antennas. As the number of BS antennas increases, according to the law of large numbers, this estimate becomes closer to the exact channel PDP. Although the above method resolves the saturation problem, it may not be of practical interest as it may lead to a very complex receiver. The source of the complexity lies in the requirement of a separate filter φk,m [l] per user per antenna. Hence, the receiver front-end processing has to be repeated for each terminal separately. Next, we utilize multirate signal processing techniques and propose the following steps to resolve the complexity issue. Proposition 3. In an FBMC massive MIMO system, the channel PDP equalization can be performed after analysis filter bank and combiner as in Figure 4.4. Here, φ̃k [i] , φk [l] ? sinc(2l/M ) where φk [l] , φk,0 [l] and sinc(t) , low-pass filter with bandwidth 2π M. sin(πt) πt . ↓M 2 , (4.19) Note that the term sinc(2l/M ) acts as an ideal 32 0,k Wm y0 [l] ∗ fm [−l] yN −1[l] M 2 e−jθm,i N −1,k Wm ∗ [−l] fm φ̃k [n]ejπmi ℜ{·} dˆkm,i M 2 Figure 4.4. Block diagram of the simplified receiver. Utilizing multirate signal processing techniques, the additional equalization block can be moved to after the analysis filter bank and combiner to minimize the computational cost. Proof. The FBMC prototype filter is normally designed such that its frequency response 2π is almost perfectly confined to the interval [− 2π M , M ]. Hence, in Figure 4.3, after filtering ∗ [−l], the frequency response of the result is almost perfectly the incoming signal ym [l] by fm , 2π(m+1) . This implies that the input to φk,m [l] confined to the frequency interval 2π(m−1) M M is band-limited. It is intuitive that since the input to the equalizer is band-limited, the equalization processing can take place in the low rate (after decimation). Subsequently, the filtering can be moved to after the combining due to the linearity. This leads to the structure in Figure 4.4. Note that the equalizer used for any particular subcarrier can be obtained from the one used for subcarrier 0. In the following, we rigorously prove that the equalization can be performed after the decimation. For simplicity, consider the two systems given in Figure 4.5. Here, x[l] is an arbitrary 2π band-limited signal whose spectrum is confined to the frequency interval [− 2π M , M ], and h[l] is an arbitrary filter impulse response. Moreover, let xm [l] , x[l]ej h[l]ej 2πml M and hm [l] , 2πml M represent the modulated versions of x[l] and h[l], respectively, and h̃[i] , h[l] ? sinc(2l/M ) ↓ M denote the band-limited and decimated version of h[l]. We prove that the 2 two systems shown in Figure 4.5 are equivalent. 2 First consider the top system in Figure 4.5, and let ĥ[l] , h[l] ? M sinc(2l/M ) which has the transfer function Ĥ(ω) = 2π H(ω), ω ∈ [− 2π M , M ], . 0, else. Note that since the input signal does not have any frequency component outside of the 2πml frequency interval 2π(m−1) , 2π(m+1) , it is possible to use the filter ĥm [l] , ĥ[l]ej M instead M M of hm [l] in the top system in Figure 4.5. Subsequently, after the decimation operation, the 33 xm [l] xm [l] M 2 y[n] h̃[n]ejπmi y[n] hm [l] M 2 Figure 4.5. Two equivalent systems considered in the proof of Proposition 3. DTFT of the output signal y[i] can be expressed as, [74], M −1 2 2 X 2ω − 2π(2u + m) 2ω − 2π(2u + m) Y (ω) = X Ĥ . M M M u=0 2π Using the fact that both X(ω) and Ĥ(ω) are band-limited to [− 2π M , M ], we find that in the summation above, only one of the terms is nonzero. In particular, for even m we have Y (ω) = 2 2ω 2ω X Ĥ , M M M − π ≤ ω ≤ +π, and for odd m we have Y (ω) = 2 2ω − 2π 2ω − 2π X Ĥ , M M M Here, it is worth to mention that when m is even, 0 ≤ ω ≤ 2π. 2 2ω MX M and 2 2ω M Ĥ M represent the DTFT of the decimated versions of xm [l] and ĥm [l], respectively. Similarly, when m is 2 2 odd, M X 2ω−2π and M Ĥ 2ω−2π express the DTFT of the decimated versions of xm [l] M M and ĥm [l], respectively. Consequently, instead of passing xm [l] through the filter ĥm [l] and decimating the result, one can decimate both xm [l] and ĥm [l] separately, and then convolve them together in the low rate. Before we finish the proof, we just aim to derive the decimated version of ĥm [l] in terms of h[l]. We have 2πml M M 2 ĥm [l] M = h[l] ? sinc(2l/M ) ej M 2 2 M ↓2 ↓M 2 = h̃[i]ejπmi . This results in the system given in Figure 4.5. This completes the proof. As suggested by the above proposition, one can incorporate the receiver structure shown in Figure 4.4 to resolve the saturation issue in an efficient manner. In particular, after the analysis filter bank and multiantenna combining, the filter φ̃k [i]ejπmi can be incorporated to equalize the effect of the problematic term pk,m [l] in (4.16). Note that in this approach, the 34 main parts of the receiver front-end including the analysis filter bank and the multiantenna combiner will remain unchanged. The advantages of this simplified structure as compared to the previous one include: (i) The analysis filter bank is common for all terminals and can be performed once. (ii) The additional equalizer has a very short length since it is performed at the low rate after decimation, and (iii) the equalizer is performed after the multiantenna combining, hence, its computational cost is independent of the number of BS antennas. Before we end this section, we note that according to (4.16), a frequency shifted version of the power delay profile pk [l] distorts the equivalent channel. As a result, only the frequency 2π response of pk [l] limited to the interval ω ∈ − 2π M , + M affects the respective equivalent channel response. This interval corresponds to the width of a single subcarrier. 4.5 Frequency-Domain Perspective In this section, we aim at studying the results of the previous sections from the frequencydomain point of view. As we show, this study leads to a deeper understanding of FBMC in massive MIMO channels. In OFDM-based systems, presence of the CP greatly simplifies the equalization procedure. In particular, as long as the length of the CP is larger than the duration of channel impulse response, one can utilize a single-tap equalizer per subcarrier to undo the effect of the channel and retrieve the transmitted data symbols. On the other hand, in FBMC-based systems, since no CP is adopted, single-tap equalization does not fully compensate the channel frequency-selectivity across subcarrier bands. However, assuming that the number of subcarriers is sufficiently large so that the channel frequency response is approximately flat over each subcarrier band, then the model described by (4.8) is going to be valid. Therefore, the task of equalization can be simplified by using single-tap equalization per subcarrier. In this section, we aim at discussing the fact that in massive MIMO systems, by using the equalization method developed in Section 4.4, it is not necessary to have a flat channel response over the band of each subcarrier in order to use single-tap equalizer. In particular, by using the simple single-tap per subcarrier equalization even in strong frequency selective channels and by incorporating a large number of antennas at the BS, the effective channel 35 response becomes flat. It is clear that this property has a number of advantages from the system implementation point of view. In particular, since there is no need for flat-fading assumption over the band of each subcarrier, one can widen the subcarrier widths (or equivalently decrease the symbol duration). Consequently, the following advantages can be achieved [69]. 1. The sensitivity to carrier frequency offset (CFO) in the uplink of multiple access networks is decreased by widening the subcarrier bands. 2. The PAPR is lowered, which leads to larger coverage and higher battery efficiency in mobile terminals. This is a direct consequence of reducing the number of subcarriers in a synthesized signal. 3. The sensitivity to channel time variations within the FBMC symbol duration is reduced. This advantage arises from the reduction of the symbol duration. As a result, a higher quality of service is expected in highly time-varying channels such as in high speed trains. 4. The latency between the terminals and the BS is decreased, as a result of shorter symbol durations. This is crucial for addressing the low-latency requirements of the 5G networks. 5. The inefficiency due to the ramp-up and ramp-down of the prototype filter at the beginning and the end of each packet, especially in bursty communications, is decreased. This results from the shortening of the symbol duration which in turn leads to a shorter prototype filter in the time domain [28]. Following (4.13), we can obtain the frequency response of the equivalent channel after combining. To this end, consider a given terminal k, and let Gk,k mm0 (ω) denote the frequency response of the high-rate (without decimation) equivalent channel between the transmitted symbols at subcarrier m0 and the received ones at subcarrier m. We have Gk,k mm0 (ω) = = N −1 1 X n,k ∗ ∗ Hm Fm0 (ω)Hn,k (ω)Fm (ω) N n=0 k,k ∗ Cm (ω)Fm0 (ω)Fm (ω), (4.20) 36 where k,k Cm (ω) N −1 1 X n,k ∗ = Hm Hn,k (ω). N (4.21) n=0 ∗ (ω) are two modulated square-root Nyquist filters, that is, Q(ω) = In (4.20), Fm0 (ω) and Fm k,k k,k |F (ω)|2 is a Nyquist pulse, and Cm (ω) is due to the multipath channel. Ideally, Cm (ω) should be flat over the pass band of the subcarrier m so that the symbols of subcarrier m can be perfectly reconstructed without any interference. However, when there exists a k,k frequency-selective channel, the term Cm (ω) may incur some distortion over the pass band of subcarrier m and, accordingly, lead to some interference in the detected symbols. As the number of BS antennas grows large, using the law of large numbers and according to (4.15), k,k Cm (ω) tends to Pk,m (ω). Therefore, the flat-fading condition may not be achieved by just increasing the BS array size. On the other hand, when the equalizer in (4.18) is utilized, the equivalent channel in the frequency domain can be expressed as k,k ∗ 0 G̃k,k mm0 (ω) = C̃m (ω)Fm (ω)Fm (ω), (4.22) where k,k C̃m (ω) 1 N = NP −1 n,k Hm ∗ Hn,k (ω) n=0 Pk,m (ω) = k,k Cm (ω) . Pk,m (ω) (4.23) k,k k,k Therefore, since Cm (ω) asymptotically tends to Pk,m (ω), C̃m (ω) will in turn tend to a frequency flat channel. Thus, no interference is expected in large antenna regime. This channel flattening effect of FBMC-based massive MIMO systems is illustrated in Figure 4.6. 4.6 SINR Analysis In this section, we analyze the SINR performance of an FBMC-based massive MIMO system in the uplink incorporating the proposed equalization method. We limit our study to the two most prominent linear combiners namely, MRC and ZF. As mentioned in Chapter 3, in the large antenna regime, all the combiners in (3.4) tend to 1 N Hm , and hence, the same asymptotic SINR performance as in MRC and ZF is expected for the MMSE combiner. As mentioned earlier, the equalization approaches given in Figures 4.3 and 4.4 are equivalent. Although the method given in Figure 4.4 is preferred for implementation, here, for the purpose of analysis, we consider the approach given in Figure 4.3. 37 2 2 2 2 1 1 1 1 0 -0.02 0 0.02 0 -0.02 0 0.02 0 -0.02 0 0.02 0 -0.02 0 0.02 0 0.02 (a) 2 2 2 2 1 1 1 1 0 -0.02 0 0.02 0 -0.02 0 0.02 0 -0.02 0 0.02 0 -0.02 (b) Figure 4.6. Illustration of the equivalent channel response. Here, we assume M = 512, and consider an exponentially decaying channel PDP with the decaying factor of 0.06 and k,k the length of L = 50. (a) The equivalent channel, Cm (ω), for subcarrier m = 0, without k,k (ω), for subcarrier m = 0, with the proposed equalizer. (b) The equivalent channel, C̃m the proposed equalizer. As the number of BS antennas increases, the equivalent channel becomes flat only when the proposed equalizer is in place. ∗ [−l] and the equalizer φ In Figure 4.3, the receiver filter fm k,m [l] can be combined together ∗ [−l] , f ∗ [−l] ? φ as a single filtering block with impulse response f˜k,m k,m [l]. Therefore, we m ∗ [−l] in place, and use (4.11) to obtain the can consider having the new receiver filter f˜k,m estimated data symbols. To this end, let H̃kmm0 ,ii0 be an N × K matrix with elements given ∗ [−l] in place instead of f ∗ [−l]. Moreover, we form by (4.10) but with the new filter f˜k,m m the K × K matrix G̃mm0 ,ii0 similar to Gmm0 ,ii0 . In particular, the k th row of G̃mm0 ,ii0 can H H̃k th column of the combiner matrix W . be calculated as wm,k m mm0 ,ii0 , where wm,k is the k Following the above definitions, the interference coefficients are determined by the real part of the elements of G̃mm0 ,ii0 . In order to pave the way for our SINR analysis, we desire to find the elements of G̃mm0 ,ii0 in a matrix form. Towards this end and based on (4.10) and (4.11), the convolution, downsampling, multiantenna combining, and phase compensation operations can all be expressed compactly as 38 0 k G̃k,k mm0 ,ii0 = ψ mm0 ,ii0 where k,k0 gm = N −1 X n=0 n,k Wm H ∗ 0 k,k gm , (4.24) hn,k0 , (4.25) and ψ kmm0 ,ii0 H = ej(θm0 ,i0 −θm,i ) eT ii0 F̃k,m Fm0 . (4.26) 0 k,k The vector gm is the effective multipath channel impulse response between terminals k n,k and k 0 at subcarrier m, after the combining operation. Wm is the element (n, k) of the T combining matrix W m , and hn,k , hn,k [0], . . . , hn,k [L − 1] is the vector of channel impulse response between nth BS antenna and k th terminal. Fm0 and F̃k,m are two Toeplitz matrices that are defined in (4.27a) and (4.27b), respectively, and signify the synthesis filter at subcarrier m0 and the new analysis filter at subcarrier m, respectively. Note that the size of the matrix Fm0 is (Lf + L − 1) × L. To determine the size of F̃k,m , we follow (4.18) to note that fm [l] = f˜k,m [l] ? p∗k,m [−l]. Hence, the length of the new filter f˜k,m [l] can be obtained as Lf̃ = Lf − L + 1. As a result, the size of F̃k,m can be calculated as (2Lf − 1) × (Lf + L − 1). The (2Lf −1)×1 vector eii0 is accounted for the downsampling operation and contains zeros th entry which is equal to one. Finally, ej(θm0 ,i0 −θm,i ) is due to except on its (Lf + (i − i0 ) M 2 ) the phase compensation.  Fm0 F̃k,m fm0 [0] 0 fm0 [1] fm0 [0]   .. =  ... .   0 0 0 0 ··· ··· .. . ··· ··· 0 0 .. .  0 0 .. .    ,  fm0 [Lf − 1] fm0 [Lf − 2] 0 fm0 [Lf − 1]  ∗ f˜k,m [Lf̃ − 1] 0 f˜∗ [L − 2] f˜∗ [L − 1]  k,m f̃ k,m f̃  .. .. = . .   0 0  0 0 ··· ··· .. . ··· ··· 0 0 .. . 0 0 .. . (4.27a)     .   ∗ ∗ ˜ ˜ fk,m [0] fk,m [1] ∗ [0] 0 f˜k,m (4.27b) 0 k,k Note that in (4.24), the term ψ kmm0 ,ii0 is completely deterministic, whereas gm is a random vector. Therefore, in this equation, we have decomposed the interference coefficients into random and deterministic components. Moreover, while ψ kmm0 ,ii0 does not depend on 0 k,k the type of combining, gm is directly related to the combining method and should be evaluated for each combiner separately. 39 4.6.1 MRC In MRC, as the number of BS antennas grows large, Dm in (3.4) tends to N IK . P −1 n,k ∗ k,k0 Therefore, we can write gm = N1 N hn,k0 . In Section 4.9, we have calculated n=0 Hm 0 k,k the first and second order statistics of the complex random vector gm . The result is where Dpk T 1] . k,k0 0 µk,k = δkk0 pk,m , (4.28a) m ,E gm n o k,k0 1 0 0 H k,k0 k,k0 (4.28b) Γk,k gm −µk,k = Dpk0 , m ,E gm −µm m N n o 1 0 0 T k,k0 k,k0 k,k0 Km ,E gm −µk,k gm −µk,k = δkk0 pk,m pT (4.28c) m m k,m , N T , diag pk [0], pk [1], . . . , pk [L − 1] , and pk,m , pk,m [0], pk,m [1], . . . , pk,m [L − 0 0 0 0 k,k k,k k,k Let γ k,k m be a zero-mean random vector defined as γ m , gm − µm . Thus, from (4.24) and (4.28a) we have 0 0 k G̃k,k mm0 ,ii0 = ψ mm0 ,ii0 H k γ k,k m + δkk0 ψ mm0 ,ii0 = ψ kmm0 ,ii0 H 0 γ k,k m + δkk0 δmm0 δii0 + jAmm0 ,ii0 , where Amm0 ,ii0 , = P+∞ ∗ l=−∞ am0 ,i0 [l]am,i [l] H pk,m (4.29) . The second line of (4.29) follows from the real-orthogonality property of FBMC given in (4.3). We recall that by incorporating the equalizer φk,m [l], the effect of the modulated channel PDP pk,m [l] is removed and the real H orthogonality condition is satisfied. Hence, the term ψ kmm0 ,ii0 pk,m = ej(θm0 ,i0 −θm,i ) eT ii0 F̃k,m Fm0 pk,m is equal to δmm0 δii0 + jAmm0 ,ii0 since the matrix F̃k,m compensates the effect of pk,m . As mentioned above, the interference coefficients are given by the real part of the elements of G̃mm0 ,ii0 . Let Rmm0 ,ii0 , <{G̃mm0 ,ii0 }, and ν 00m,i , <{ν 0m,i }. Accordingly, (4.11) can be reformulated as d̂m,i = +∞ M −1 X X Rmm0 ,ii0 dm0 ,i0 + ν 00m,i . (4.30) i0 =−∞ m0 =0 By stacking the real and imaginary parts of the matrices and vectors that constitute the elements of G̃mm0 ,ii0 , it is possible to find an expression for the elements of Rmm0 ,ii0 . In T particular, for an arbitrary complex matrix or vector a, we define ǎ , <{aT }, ={aT } . Thus, following (4.29) we can find the elements of Rmm0 ,ii0 as 0 k k,k Rmm 0 ,ii0 = ψ̌ mm0 ,ii0 T 0 γ̌ k,k m + δkk0 δmm0 δii0 . (4.31) 40 0 We note that the real-valued random vector γ̌ kk m is zero-mean and its covariance matrix can be determined using (4.28) as where D pk0 n k,k0 T o k,k0 k,k0 0 k,k0 γ̌ m γ̌ − E γ̌ γ̌ − E Ck,k , E m m m m # " 0 k,k0 k,k0 k,k0 1 <{Γk,k m + Km } ={−Γm + Km } = 0 0 k,k0 k,k0 2 ={Γk,k <{Γk,k m + Km } m − Km } 1 = D pk0 + δkk0 P k,m , N T ={pk,m pT 0 1 <{pk,m pk,m } 1 Dpk 0 k,m } and P k,m , 2 . ,2 T 0 Dpk0 ={pk,m pT k,m } −<{pk,m pk,m } (4.32) 0 k,k Following (4.31), the instantaneous power corresponding to Rmm 0 ,ii0 can be calculated as 2 k,k0 k,k0 = R Pmm 0 ,ii0 mm0 ,ii0 T k,k0 0 T 0 k = γ̌ k,k Ψkmm0 ,ii0 γ̌ k,k γ̌ m , m m + δkk0 δmm0 δii0 + 2δkk0 δmm0 δii0 ψ̌ mm0 ,ii0 k k where Ψkmm0 ,ii0 , ψ̌ mm0 ,ii0 ψ̌ mm0 ,ii0 T . From the above equation, the average power, with averaging over different channel realizations, can be calculated according to [75, p. 53], o n k,k0 k,k0 k + δkk0 δmm0 δii0 P̄mm 0 ,ii0 = tr Cm Ψmm0 ,ii0 n o 1 = tr (D pk0 + δkk0 P k,m )Ψkmm0 ,ii0 + δkk0 δmm0 δii0 . N (4.33) Thus, the SINR can be calculated as given in the following proposition. Proposition 4. In the uplink of an FBMC massive MIMO system with MRC combiner and the proposed PDP equalizer, the effective SINR can be calculated according to N + tr D pk + P k,m Ψkmm,ii k SINRm,i = K−1 +∞ M −1 . +∞ −1 P P P P MP tr D pk0 Ψkmm0 ,ii0 + tr D pk + P k,m Ψkmm0 ,ii0 + σν2 k0 =0 i0 =−∞ m0 =0 k0 6=k i0 =−∞ m0 =0 (m0 ,i0 )6=(m,i) (4.34) Proof. This follows from (4.33), and noting that SINRkm,i , k,k P̄mm,ii +∞ P M −1 P i0 =−∞ m0 =0 (m0 ,i0 )6=(m,i) k,k0 P̄mm 0 ,ii0 + +∞ P M −1 K−1 P P i0 =−∞ m0 =0 k0 =0 k0 6=k . k,k0 P̄mm 0 ,ii0 + σν2 41 4.6.2 ZF In Section 4.9, it is shown that for the ZF combiner, provided that N ≥ K + 1, the first 0 k,k and second order statistics of the random vector gm can be calculated according to 0 µk,k m = δkk0 pk,m , (4.35a) 1 Dpk0 − pk0 ,m pH k0 ,m , N −K 0 Γk,k m = 0 k,k Km = 0. (4.35b) (4.35c) 0 Hence, the covariance matrix of γ̌ k,k m is determined by 1 D pk0 − P̃ k0 ,m , N −K H H 1 <{pk,m pk,m } −={pk,m pk,m } . ,2 <{pk,m pH ={pk,m pH k,m } k,m } 0 Ck,k m = where P̃ k,m (4.36) Proposition 5. In the uplink of an FBMC massive MIMO system with ZF combiner and the proposed PDP equalizer, and provided that N ≥ K + 1, the effective SINR can be calculated according to SINRkm,i = K−1 P +∞ P M −1 P k0 =0 i0 =−∞ m0 =0 (m0 ,i0 )6=(m,i) N −K (4.37) tr D pk0 − P̃ k0 ,m Ψkmm0 ,ii0 + σν2 Proof. This follows from the covariance matrix given in (4.36) and similar analysis as in the MRC case. 4.7 Numerical Results In this section, we deploy computer simulations to evaluate the efficacy of the proposed equalization method as well as the analysis of the previous sections. For all the simulations in this section, we let M = 512 and assume K = 10 terminals in the network. We consider the PHYDYAS prototype filter [76], with the overlapping factor κ = 4. Normalized PL−1 −α ` k exponentially decaying channel PDPs pk [l] = e−αk l / , l = 0 . . . , L − 1 for `=0 e k = 0, · · · , K − 1 with different decaying factors αk = (k + 1)/20 for different terminals and length L = 50 are assumed3 . We present the SINR performance corresponding to terminal k = 0. 3 A similar approach has been taken in [77] to choose the channel PDPs for different terminals. 42 First, we show the SINR for the case where the proposed equalization is not incorporated at the BS. Figure 4.7 shows the average SINR performance (with averaging over different channel realizations) of MRC, ZF, and MMSE combiners as a function of different number of BS antennas. The noise level is selected such that the SNR at the input of the BS antennas is equal to 10 dB. From Figure 4.7, we can see that without the proposed equalization, the SINR performance of all three linear detectors, namely, MRC, ZF, and MMSE, tend to the saturation level predicted by (4.17) as N grows large. Accordingly, arbitrarily large SINR values cannot be achieved by increasing the BS array size. Also, the SINR performance of ZF and MMSE combiners converges faster to the saturation level as compared to the one in MRC. In practice, when considering a finite number of BS antennas, the impact of SINR saturation depends on the combining method used as well as the channel PDP and noise level. In the next set of simulations, we evaluate the performance of FBMC with the proposed equalizer in place. Figure 4.8 shows the SINR performance of MRC, ZF, and MMSE combiners as a function of different number of BS antennas. The noise level is selected such that the SNR at the input of the BS antennas is equal to 10 dB. As it is shown, using the proposed equalization method, the saturation problem of the conventional FBMC systems in massive MIMO channels is avoided and arbitrarily large SINR values can be achieved by increasing N . In Figure 4.8, we have also shown the theoretical SINR values for MRC and ZF combiners, as calculated in (4.34) and (4.37), respectively. This figure confirms that the theoretical SINR values match the simulated ones. This verifies the accuracy of the analysis of Section 4.6. Figure 4.9 shows the theoretical SINR performance of the MRC and ZF combiners and with the proposed equalization as a function of different input SNR values. Moreover, the SINR performance of OFDM with MRC and ZF combiners is shown as a benchmark; see [4] for the SINR expressions of OFDM. In this figure, we consider N = 100 BS antennas. As the figure shows, OFDM and FBMC have almost identical SINR performance when MRC is utilized. On the other hand, in the case of ZF combiner, although the performance of OFDM and FBMC are very close in the low SNR regime, a better SINR is expected for OFDM in the high SNR region. The reason for this phenomenon is that in OFDM, the interference is entirely removed using the CP. Hence, by increasing the input SNR, a 43 30 25 20 15 10 5 0 10 1 10 2 10 3 Figure 4.7. SINR performance comparison for the case that the proposed equalizer is not utilized. better SINR at the output is also expected. In contrast, the FBMC waveform is designed to increase the bandwidth efficiency by excluding the CP overhead and providing much lower out-of-band emission than OFDM. Hence, due to the absence of CP, some residual interference remains after the ZF combining. This residual interference becomes noticeable only in the very high SNR regime. As discussed in Section 4.5, by incorporating a large number of BS antennas, one can widen the subcarrier bands in an FBMC system. This, in turn, brings a number of advantages such as robustness to CFO and channel time variations, lower PAPR, lower latency, higher bandwidth efficiency. These benefits are crucial for the next generation of wireless systems. In the next experiment, we aim at evaluating the SINR performance as we widen the subcarrier bands. Figure 4.10 shows the SINR for different values of FBMC subcarrier spacings, ∆F , 1/M . In this experiment, the input SNR of 0 dB is considered. To use the simple single-tap equalizer per subcarrier, the design norm is to choose the symbol spacing to be about an order of magnitude larger than the channel length. In this case, with L = 50, this leads to the symbol spacing of around M/2 = 500, which in turn yields the 44 40 35 30 25 20 15 10 5 0 10 1 10 2 10 3 Figure 4.8. SINR performance comparison for the case that the proposed equalizer is utilized. subcarrier spacing of ∆F = 0.001. However, as the figure shows, by incorporating a large number of BS antennas as well as the proposed equalizer, one can considerably increase the subcarrier spacing while the SINR performance has a slight degradation. In particular, increasing the subcarrier spacing by an order of magnitude leads to about 0.7 dB SINR degradation when using ZF combiner. In MRC, the degradation is less than 0.3 dB and is negligible. Figure 4.11 presents the uncoded bit error rate (BER) performance comparison. In this experiment, N = 100 BS antennas is considered. Moreover, the transmitted symbols belong to a 64-QAM constellation. We compare the performance of FBMC with and without our proposed channel PDP equalizer. We also show the performance of OFDM as a benchmark. For all cases, ZF combiner is utilized. As the figure shows, the BER performance is improved significantly when the proposed channel PDP equalizer is in place. Furthermore, we achieve the same performance as in OFDM, where the channel frequency response is completely flat over each individual subcarrier band. 45 50 40 30 20 10 0 -20 -10 0 10 20 30 Figure 4.9. SINR performance comparison as a function of different SNR values. In the case of FBMC, the proposed equalizer is incorporated at the BS. Here N = 100 BS antennas is considered. 4.8 Conclusion and Discussion In this chapter, we studied the performance of FBMC transmission in the context of massive MIMO. We considered single-tap-per-subcarrier equalization using the conventional linear combiners, namely, MRC, ZF, and MMSE. It was shown that the correlation between the multiantenna combining tap weights and the channel impulse responses leads to an interference which does not fade away even with an infinite number of BS antennas. Hence, arbitrarily large SINR values cannot be achieved, and the SINR is upper-bounded by a certain deterministic value. We derived a closed-form expression for this upper bound, identified the source of SINR saturation, and proposed an efficient equalization method to remove the above correlation and resolve the problem. We mathematically analyzed the performance of the FBMC system incorporating the proposed equalization method and derived closed-form expressions for the SINR in the cases of MRC and ZF. The analyses in this chapter was based on single-tap per subcarrier equalization. However, we note that as mentioned in Section 4.3, the asymptotic SINR saturation is an 46 30 28 26 24 22 20 18 16 14 12 10 8 1 2 3 4 5 6 7 8 9 10 10 -3 Figure 4.10. SINR performance comparison for different values of the FBMC subcarrier spacing ∆F , 1/M . inherent property of FBMC-based massive MIMO systems due to the absence of CP. As a result, one may expect the SINR saturation issue to appear also in FBMC systems incorporating multitap per subcarrier equalization methods such as those in [36] and [73] if we do not equalize the channel PDP. Using multitap equalizers, however, can increase the saturation level in expense of a higher computational cost. We can also realize this point from the results of [36], where the performance of multitap and single-tap equalizers are compared with each other for different number of BS antennas. Therefore, our proposed channel PDP equalizer can also be adopted in multitap systems to further improve the performance. 4.9 Proof of (4.28) and (4.35) 4.9.1 MRC In MRC, as the number of BS antennas grows large, Dm in (3.4) tends to N IK . P −1 k,k0 n,k ∗ Therefore, we can write gm = N1 N hn,k0 . Hence, the mean of the `th element n=0 Hm 0 k,k of gm , for ` ∈ {0, . . . , L − 1}, can be calculated as 47 10 0 10 -1 10 -2 10 -3 10 -4 10 -5 10 -6 -10 -5 0 5 10 Figure 4.11. BER performance comparison. Here, N = 100 BS antennas and the ZF combiner are considered. 0 k,k E{gm [`]} = N −1 L−1 2πlm 1 XX E{h∗n,k [l]hn,k0 [`]}ej M N n=0 l=0 = δkk0 pk,m [`]. 0 0 k,k k,k 0 This leads to (4.28a). We now calculate the correlation between gm [`] and gm [` ], for `, `0 ∈ {0, . . . , L − 1}. We consider the case that k 6= k 0 . Hence, N −1 N −1 k,k0 1 X X n,k ∗ n0 ,k k,k0 0 ∗ E gm [`] gm [` ] = 2 E Hm Hm hn,k0 [`]h∗n0 ,k0 [`0 ] N 0 = = 1 N2 n=0 n =0 N −1 N −1 L−1 X X X L−1 X E{h∗n,k [l]hn0 ,k [l0 ]hn,k0 [`]h∗n0 ,k0 [`0 ]}ej 2π(l−l0 )m M n=0 n0 =0 l=0 l0 =0 1 δ``0 pk0 [`], N for k 6= k 0 . The above correlation for the case of k = k 0 can be determined using a similar line of derivations. The result is k,k 1 k,k 0 ∗ [`] gm [` ] = δ``0 pk [`] + pk,m [`]p∗k,m [`0 ]. E gm N 48 0 k,k This leads to (4.28b). Moreover, the pseudo-covariance matrix Km in (4.28c) can be derived using the same line of derivations as above. 4.9.2 ZF 0 k,k H h 0 [`], where w Here, we use similar techniques as in [78]. We have gm [`] = wm,k m,k k is the k th column of the combiner matrix W m , and hk0 [`] is an N × 1 vector with its nth −1 element equal to hn,k0 [`]. In the case of ZF equalizer, we have W m = Hm (HH m Hm ) . Also, 0 k,k let hm,k denote the k th column of Hm . Hence, the mean of gm [`] can be determined as follows. 0 k,k H hk0 [`]} E{gm [`]} = E{wm,k M −1 2πm0 ` 1 X H hm0 ,k0 }ej M E{wm,k = M 0 (a) = (b) = 1 M 1 M m =0 M −1 L−1 X X m0 =0 l=0 M −1 L−1 X X H hm,k0 }pk0 ,m [l]ej E{wm,k δkk0 pk,m [l]ej 2πm0 (`−l) M 2πm0 (`−l) M m0 =0 l=0 = δkk0 pk,m [`]. This results in (4.35a). In the above equation, (a) follows from the fact the channel frequency response hm0 ,k0 can be expressed as a combination of a term that is correlated with hm,k0 and a term that is independent of hm,k0 , that is, hm0 ,k0 = αmm0 ,k0 hm,k0 + hindep mm0 ,k0 , (4.38) 0 0 0 where hindep mm0 ,k0 is independent of hm,k and the correlation coefficient αmm ,k can be calcu- lated as n,k0 n,k0 ∗ αmm0 ,k0 = E Hm Hm = Pk0 [m0 − m], 0 where Pk [m] , PL−1 l=0 pk [l]e−j 2πml M is the mth coefficient of the M -point discrete Fourier transform of the channel PDP pk [l]. The step (b) above follows from the fact that in the H H h case of ZF equalization, we have wm,k m,k0 = δkk0 , which results from W m Hm = Ik . 49 0 In order to calculate the covariance matrix Γk,k m in (4.35b), we now find the correlation 0 0 k,k k,k 0 between gm [`] and gm [` ], for `, `0 ∈ {0, . . . , L − 1}. We have, k,k0 k,k0 0 ∗ E gm [`] gm [` ] H 0 = E{wm,k hk0 [`]hH k0 [` ]wm,k } (a) = δkk0 pk,m [`]p∗k,m [`0 ] + (b) = δkk0 pk,m [`]p∗k,m [`0 ] + M −1 M −1 2πm00 `0 2πm0 ` 1 X X indep H H E{wm,k hindep wm,k }ej M e−j M 0 ,k 0 hmm00 ,k 0 mm 2 M 0 00 m =0 m =0 M −1 M −1 X X Pk0 [m0 m0 =0 m00 =0 (c) = δkk0 pk,m [`]p∗k,m [`0 ] + − m00 ] − Pk0 [m0 − m]Pk0 [m − m00 ] j 2π(m0 `−m00 `0 ) M e M 2 (N − K) 1 δ``0 pk0 [`] − pk0 ,m [`]p∗k0 ,m [`0 ] . N −K This results in (4.35b). In the above equation, equality (a) follows from (4.38). Then, indep equality (b) follows from the independence of wm,k from hindep mm0 ,k0 and hmm00 ,k0 , the correlation n o indep H 0 00 0 00 0 0 0 E hindep h = P [m − m ] − P [m − m]P [m − m ] IN , k k k mm0 ,k0 mm00 ,k0 and the identity h h −1 i −1 i H = = E tr H H E tr W H W m m m m K , N −K for N ≥ K + 1. The latter identity is based on the fact that HH m Hm is a K × K complex central Wishart matrix with N degrees of freedom and covariance IK [79]. Finally, the equality (c) above follows using some straightforward algebraic manipulations. We note that using a similar line of derivations as above, one can find the pseudo-covariance matrix given in (4.35c). CHAPTER 5 OFDM WITHOUT CP IN MASSIVE MIMO In this chapter, we study the possibility of removing the CP overhead from OFDM in massive MIMO systems. The absence of CP increases the spectral efficiency in expense of intersymbol interference (ISI) and intercarrier interference (ICI). It is known that in massive MIMO, the effects of uncorrelated noise and multiuser interference vanish as the number of BS antennas tends to infinity. To investigate if the channel distortions in the absence of CP fade away, we study the performance of the standard MRC receiver. Our analysis reveals that in this receiver, there always remains some residual interference leading to saturation of SINR. To resolve this problem, we propose to use the TR technique. Moreover, in order to further reduce the multiuser interference, we propose a frequency-domain equalization to be deployed after the TR combining. We compare the achievable rate of the proposed system with that of the conventional CP-OFDM. We show that in realistic channels, a higher spectral efficiency is achieved by removing the CP from OFDM, while reducing the computational complexity. 5.1 Introduction In the massive MIMO context, OFDM with CP is particularly attractive because it enables the conversion of the frequency-selective channels between each mobile terminal antenna and the BS antennas into a set of flat-fading channels over each subcarrier band. Therefore, the users’ data streams can be distinguished from each other through the respective channel responses. Hence, most of the literature deals with massive MIMO while utilizing OFDM with CP (CP-OFDM) [2–5]. However, the CP duration adds an extra overhead to the network and reduces the spectral efficiency. Therefore, in order to increase the transmission rate, it is desirable to eliminate the CP duration from OFDM. This, however, comes at the expense of ISI and ICI, imposed by the multipath channel. Here, it is worth mentioning that in massive MIMO, the effects of uncorrelated noise as well as 51 various types of interference/imperfections such as multiuser interference (MUI), imperfect channel state information, hardware imperfections, phase noise, etc., will vanish as the number of BS antennas grows large [3, 4, 80, 81]. Therefore, the core question at the heart of this chapter is: “Can massive MIMO average out the ISI and ICI introduced by the multipath channel in OFDM without CP?” There are a number of methods in the literature tackling the ISI and ICI problem of OFDM with insufficient CP [82–86]. References [82] and [83] suggest to remove the effect of ICI and ISI by utilizing the previously detected symbols and using successive interference cancellation (SIC). In [84] and [85], a MIMO-OFDM scenario is considered, and iterative interference cancellation using turbo equalization is proposed. The authors in [86] propose an interference cancellation algorithm based on some structural properties obtained from shifting the received OFDM blocks. We note that the above methods are designed for the conventional OFDM (or MIMO-OFDM) scenarios, and do not take advantage of the excessive number of BS antennas in a massive MIMO setup. In [87], the authors consider the conventional frequency-domain combining methods and deploy computer simulations to show that the CP duration can be shortened to achieve a higher spectral efficiency in a massive MIMO system. However, no detailed mathematical analysis of the proposed approach is presented. In this chapter, to investigate if the channel distortions (ISI and ICI) in the absence of CP fade away as the number of BS antennas grows large, we first study the performance of the conventional frequency-domain combining methods, such as MRC, ZF, and MMSE detectors. We mathematically analyze the SINR performance of the above detectors when the CP is removed from the OFDM signal. Our SINR analysis reveals that when the above combining methods are applied, the channel distortions arising from the absence of CP do not average out as the number of BS antennas tends to infinity. Thus, SINR saturates at a certain deterministic level and arbitrarily large SINR values cannot be achieved by increasing the BS array size. To resolve the saturation issue, we propose to use a technique known as time reversal (TR) to combine/precode the signals of different BS antennas in the time domain instead of the frequency domain. This technique is based on a pivotal phenomenon in physics 52 that harnesses the principle of channel reciprocity and multipath effects to concentrate the signal energy at a certain point in space (spatial focusing) and compress the channel impulse response in the time domain (temporal focusing). This spatial-temporal focusing effect mitigates the ISI, ICI, and MUI [88]. Time reversal has been extensively studied and utilized in underwater acoustic channels; see for example [89–92]. In [93] and [94], the authors utilize the temporal focusing property of TR and propose a CP length design method to satisfy specific performance requirements in underwater acoustic channels. The authors, consequently, balance the trade-off between the CP length and the resulting interference due to the residual ISI and ICI imposed by the insufficient length of CP. The scope of [93] and [94] is limited to small-scale underwater acoustic networks without any consideration of multiuser scenarios. Recently, there has been an emerging interest in the application of TR for the future generation of wireless networks [95]. Moreover, the application of TR to massive MIMO in the context of single-carrier transmission has been studied extensively [77, 81, 96]. Application of TR to CP-OFDM has been studied in [97–99], where the authors consider a single-user massive MIMO scenario and show that TR can be applied to a CPOFDM system either in the time or in the frequency domain. Moreover, the authors show that TR allows the CP length to be reduced thanks to its spatial-temporal focusing property. As it is shown in [77] for the case of single-carrier transmission, with the TR technique, the channel distortions tend to zero as the number of BS antennas goes to infinity. We show that this result is also applicable to the case of OFDM without CP transmission. Thus, arbitrarily large SINR values can be achieved by increasing the BS array size. However, as we show in this chapter, the performance of the conventional TR is limited due to the excessive amount of MUI when the number of BS antennas is finite. We show that OFDM allows for a straightforward frequency-domain equalization (FDE) to be utilized after the TR combining. With this approach, the MUI level is significantly reduced and larger SINR values can be achieved compared to the conventional TR method, while the SINR saturation problem is also avoided. Throughout the chapter, we refer to the conventional TR technique as TR-MRC, while the proposed TR-based method with additional FDE is referred to as TR-FDE. It is worth mentioning that in a typical communication system, a time period with duration equal to the coherence time of the channel is divided into two intervals: (i) 53 training period, and (ii) data transmission period. In this study, we only focus on the data transmission period and consider removing the CP overhead during this period. Throughout the chapter, we consider perfect knowledge of the CSI at the BS and assume that the CP is included in the course of training to establish the carrier frequency and timing synchronization and obtain an accurate CSI. Studying the problem of CP removal/shortening during the training interval in the context of massive MIMO remains for the future study. Also, in this chapter, we focus on the uplink transmission, but the results and algorithms are trivially applicable to the downlink as well. We analytically derive the SINR performance of the TR-MRC receiver as well as our proposed TR-FDE technique. Based on our SINR derivations, we obtain a lower bound on the achievable information rate for both the TRMRC and TR-FDE receivers. We show that higher spectral efficiency can be achieved using OFDM without CP as compared to CP-OFDM. More specifically, we show that using TR-MRC and TR-FDE techniques in OFDM without CP, higher information rate is achievable as compared to the case of CP-OFDM with the conventional MRC and ZF detection methods, respectively. Furthermore, we analyze the computational complexity of both TR-MRC and TR-FDE methods and introduce computationally efficient ways to implement them. We show that while the complexity of TR-MRC is almost similar to the frequency-domain MRC, a significantly lower complexity is obtained when utilizing the TR-FDE equalizer as compared to the conventional ZF detector. To summarize, we list the contributions of this chapter as follows: • To increase the spectral efficiency of massive MIMO systems, we show that the CP overhead can be successfully eliminated. This is a result of the coherent combining of the received signals at the BS antennas that yields the channel distortions to disappear as the BS array size increases. • We show that in the absence of the CP, the SINR performance of the conventional frequency-domain combining methods, namely, MRC, ZF, and MMSE, saturates at a certain deterministic level. Hence, arbitrarily large SINR values cannot be achieved by increasing the array size at the BS. • We propose to use the TR technique to resolve the above saturation problem. 54 • Although the conventional TR technique can achieve reasonable SINR values and is a viable option in many scenarios, it suffers from a high level of MUI in multiuser cases. We propose a novel frequency-domain equalization technique to be applied after the TR operation to reduce the MUI level. • We introduce efficient methods to minimize the computational cost of both TR-MRC and TR-FDE receivers. We also compare the complexity of the proposed receiver structures with that of the conventional CP-OFDM with MRC and ZF detectors. • We perform a thorough analysis and obtain closed-form expressions for the SINR and achievable rate performance of both TR-MRC and TR-FDE receivers. The rest of the chapter is organized as follows. After presenting the transmission framework in Section 5.2, we discuss the saturation problem of the conventional frequency-domain combiners that arises in the absence of CP in Section 5.3. The TR technique is introduced as a remedy to this problem in Section 5.4, where we also propose a novel FDE to further reduce the MUI. In Section 5.5, we present a complexity analysis of the receiver structures that are introduced in this chapter, and compare them with that of the conventional CP-OFDM. The asymptotic performance, in terms of SINR and achievable rate, of the TR-MRC and the proposed TR-FDE receivers is analyzed in Section 5.6. Our discussions in this chapter are numerically evaluated in Section 5.7. Finally, we conclude the chapter in Section 5.8. 5.2 OFDM Without CP Transmission We assume OFDM modulation is used for data transmission with the total number of M subcarriers. To increase the bandwidth efficiency, we do not insert CP/guard interval between the successive OFDM symbols. Therefore, the ith OFDM symbol of terminal k (i) (i) can be obtained as xk = FH M dk , where FM is the M -point normalized DFT matrix, and (i) (i) (i) dk = [dk,0 , . . . , dk,M −1 ]T is the transmit data vector of terminal k on symbol time index (i) i. The elements of dk are i.i.d. zero-mean complex random variables with the variance of unity. The baseband transmit signal of terminal k can be obtained by concatenation (i) of vectors xk , i ∈ {· · · , −1, 0, +1, · · · }. The received signal at the BS can be obtained according to (3.1). 55 5.3 Frequency-Domain Combining Approach Conventionally, in CP-OFDM systems, MRC, ZF and MMSE combiners are applied in the frequency domain. With such a setup and in a large-scale multiuser MIMO scenario, the multiuser interference and noise effects average out as the number of BS antennas tends to infinity [2]. Hence, SINR increases without any bound as the number of BS antennas increases. In the case of interest to this chapter, that is, in the absence of CP, SINR saturation occurs, and thus, arbitrary large information rates cannot be achieved by increasing the BS antennas. In this section, we dig into the mathematical details that explain this limitation of the conventional frequency-domain combiners when applied to the OFDM without CP signal. In the next section, we introduce the TR combining as a remedy to this problem. Let us consider the equalization of the ith OFDM symbol. To this end, we form the (i) M × 1 vector rn = [rn [iM ], . . . , rn [iM + M − 1]]T by considering the ith segment of the signal rn [l], and follow [82–85] to express (3.1) in the matrix form as rn(i) = i K−1 X X (i,i0 ) (i0 ) Hn,k xk + ν n(i) , (5.1) i0 =i−1 k=0 where,  (i,i−1) Hn,k 0 0   .. .  = 0 0   .. . ··· ··· .. . ··· ··· .. . 0 ···  (i,i) Hn,k hn,k [L − 1] hn,k [L − 2] · · · 0 hn,k [L − 1] · · · .. .. .. . . . ··· ··· ··· ··· ··· ··· .. .. .. . . . ··· hn,k [0] hn,k [1] .. . ··· 0 0 0 .. . (i,i−1) (i,i)       hn,k [L − 1] ,  0   ..  . 0 ··· ··· .. .  hn,k [0]   ..  .   = hn,k [L − 1] hn,k [L − 2] hn,k [L − 3] · · ·  0 hn,k [L − 1] hn,k [L − 2] · · ·   .. .. .. ..  . . . . 0 0 0 ··· The M × M convolution matrices Hn,k (i) ··· hn,k [1] hn,k [2] .. . 0 0 .. . (5.2a)       0  . 0   ..  .  hn,k [0] (5.2b) (i−1) and Hn,k , when multiplied to the vectors xk and xk , create the tail of the symbol i − 1 overlapping with L − 1 samples in the beginning 56 (i) of the symbol i and the channel affected symbol i, respectively. The vector ν n includes M samples of the AWGN signal νn [l] at the position of symbol i. Next, the received signals at different BS antennas are passed through OFDM demodulators (DFT blocks), and then, the outputs of the DFT blocks across different BS antennas are combined using the frequency-domain channel coefficients between the terminals and BS antennas. To cast this procedure into a mathematical formulation and pave the way for our analysis, we obtain the output of the OFDM demodulator at BS antenna n as i K−1 X X r̃n(i) = (i,i0 ) (i0 ) FM Hn,k xk + FM ν (i) n i0 =i−1 k=0 i K−1 X X = (i,i0 ) (i0 ) (i) FM Hn,k FH M dk + FM ν n i0 =i−1 k=0 i K−1 X X = (i,i0 ) (i0 ) H̃n,k dk + ν̃ (i) n , (5.3) i0 =i−1 k=0 (i) where the tilde symbol in r̃n (i) and ν̃ n is to indicate that they are in the frequency (i,i−1) domain. Similarly, the matrices H̃n,k (i,i−1) , FM Hn,k (i,i) (i,i) H FH M and H̃n,k , FM Hn,k FM are the frequency-domain intersymbol and intercarrier interference matrices, respectively. Note (i,i−1) that in the case of CP-OFDM transmission, H̃n,k (i,i) = 0M ×M and H̃n,k is a diagonal matrix with the diagonal entries given by the frequency-domain channel coefficients, that (i,i) P −j 2πlm M . is, H̃n,k mm = L−1 l=0 hn,k [l]e Let W m be the N ×K combining matrix corresponding to subcarrier m ∈ {0, . . . , M −1}, (i) and the N ×1 vector tm contain the mth outputs of the DFT blocks at different BS antennas. Accordingly, the output of the combiner can be obtained as (i) (i) ŝm = WH m tm , (5.4) (i) where the K × 1 vector ŝm contains the detected symbols of all terminals at subcarrier m and time index i. We consider three conventional linear combiners, namely, MRC, ZF and MMSE, as introduced in (3.4). Moreover, as discussed in Chapter 3, all of these combiner matrices tend to 1 N Hm , that is, MF combiner, as the number of BS antennas increases. Therefore, in the following, to find the various interference terms in the asymptotic regime, that is, as the number of BS antennas N approaches infinity, we consider MF multiantenna combining according to W m = 1 N Hm . 57 Following (5.3) and (5.4), the data symbols undergo an equivalent channel that arises (i) from the multiantenna combining and the multipath channel. Let dˆk,m denote the mth (i) element of ŝm . Accordingly, (5.4) can be expanded as (i) (i,i) (i) dˆk,m = Hkk,mm dk,m + | {z } + k0 =0 m0 =0 k0 6=k m0 =0 m0 6=m (i,i) (i) Hkk,mm0 dk,m0 {z | Desired Signal K−1 −1 XM X M −1 X + m0 =0 } ICI M −1 X (i,i−1) (i−1) Hkk,mm0 dk,m0 {z | ISI (i) (i,i−1) (i−1) (i,i) (i) Hkk0 ,mm0 dk0 ,m0 + Hkk0 ,mm0 dk0 ,m0 + ν̃k,m , {z | } MUI } (5.5) |{z} Noise (i,i) (i,i−1) where the equivalent channel coefficients Hkk0 ,mm0 and Hkk0 ,mm0 determine the amount (i) (i−1) of the interference from symbols dk0 ,m0 and dk0 ,m0 , respectively, on the detected symbol (i) (i) dˆk,m , and ν̃k,m denotes the effective noise after combining. These interference coefficients capture the effects of the combiner gains together with the ICI and ISI coefficients in (5.3). (i,i) (i,i−1) Mathematically, we can calculate Hkk0 ,mm0 and Hkk0 ,mm0 according to (i,i) Hkk0 ,mm0 = (i,i−1) Hkk0 ,mm0 = n,k respectively, where Hm , PL−1 l=0 N −1 h (i,i) i 1 X n,k ∗ Hm , H̃n,k0 N mm0 1 N n=0 N −1 X n,k Hm n=0 hn,k [l]e−j 2πlm M ∗ h (i,i−1) H̃n,k0 i mm0 (5.6a) , (5.6b) is the element (n, k) of the matrix Hm . Using the law of large numbers, the interference coefficients given in (5.6) converge almost surely to the following values as the number of BS antennas N tends to infinity. (i,i) Hkk0 ,mm0 (i,i−1) Hkk0 ,mm0 n o h (i,i) i n,k ∗ → E Hm H̃n,k0 , mm0 n h i o (i,i−1) n,k ∗ → E Hm H̃n,k0 . 0 mm (5.7a) (5.7b) Note that the asymptotic values in (5.7a) and (5.7b) are the statistical correlation of the (i,i) (i,i−1) n,k combiner tap value Hm with the interference components H̃n,k0 mm0 and H̃m,k0 mm0 , respectively. In Section 5.9, we have simplified the expressions in (5.7). The result is that as N grows (i,i) (i,i−1) large, the coefficients Hkk0 ,mm0 and Hkk0 ,mm0 for k 6= k 0 tend to zero. Accordingly, the MUI 58 term in (5.5) fades away asymptotically. On the other hand, the ICI and ISI terms remain as specified according to the following coefficients: τ̄k , M 1 − Pk [m0 − m] (i,i) Hkk,mm → 1 − (i,i) Hkk,mm0 → (i,i−1) Hkk,mm (i,i−1) Hkk,mm0 j 2π(m0 −m) M (5.8a) for m 6= m0 , , M (1 − e ) τ̄k → , M Pk [m0 − m] − 1 → , 2π(m0 −m) M (1 − ej M ) (5.8b) (5.8c) for m 6= m0 , (5.8d) PL−1 l=0 lpk [l], is the average delay spread of the channel corresponding to user k, PL−1 2πlm and Pk [m] , l=0 pk [l]e−j M . where τ̄k , Proposition 6. In the absence of CP and with the conventional MRC, ZF, or MMSE combiners, as the number of BS antennas tends to infinity, SINR for user terminal k converges almost surely to 2 1 − τ̄k /M . −1 2 MP |1−Pk [u]|2 τ̄k /M + 2M 2 sin2 (πu/M ) SINRk → (5.9) u=1 Hence, SINR saturation occurs and arbitrary large SINR values cannot be achieved by increasing the number of BS antennas. (i,i) Proof. As the number of BS antennas N tends to infinity, the coefficients Hkk0 ,mm0 and (i,i−1) Hkk0 ,mm0 for k 6= k 0 tend to zero; see Section 5.9. Hence, the contribution of multiuser interference becomes negligible. A similar argument can be developed for the noise contribution. Thus, the SINR of terminal k at subcarrier m is determined based on the ICI and ISI terms and can be calculated as (i,i) Hkk,mm (i) SINRk,m = M −1 P m0 =0 m0 6=m 2 (i,i) Hkk,mm0 + 2 M −1 P m0 =0 (i,i−1) 2 Hkk,mm0 . (5.10) This reduces to (5.9), following (5.8a) through (5.8d) and noting that the asymptotic SINR value is equal for all subcarriers. 59 We note that although the analysis in this section was based on the OFDM without CP, one can follow a similar line of derivations to show that, in general, when insufficient CP lengths are utilized, the SINR saturation problem occurs. 5.4 Time-Reversal and Equalization As it was shown in the previous section, when CP is removed from the OFDM signal, the conventional frequency-domain combining methods lead to some residual ICI and ISI components that will not fade away even with infinite number of BS antennas. Consequently, SINR saturates at a certain deterministic level. In order to resolve this problem, in this section, we propose to use TR to combine the signals of different BS antennas in the time domain instead of the frequency domain. As it is shown in [77] for the case of single-carrier transmission, with TR combining, intersymbol interference and multiuser interference tend to zero as the number of BS antennas goes to infinity. Thus, arbitrarily large SINR values can be achieved by increasing the BS array size. However, as we show in this chapter, performance of the conventional TR combining is rather limited due to the excessive amount of multiuser interference when the number of BS antennas is finite. We show that OFDM allows for a straightforward frequency-domain equalization to be utilized after the TR combining. With this approach, the MUI level is significantly reduced and larger SINR values can be achieved compared to the conventional TR method, while the saturation problem is also resolved. A more detailed discussion on the TR-MRC and TR-FDE receivers is presented in the following subsections. 5.4.1 TR-MRC In TR-MRC, for a given terminal k, the received signals at the BS antennas are first prefiltered with the time-reversed and conjugated versions of the CIRs between that terminal and the corresponding BS antennas. Then, the resulting signals are combined with each other. Mathematically, the output of the TR filter corresponding to user k can be expressed P −1 ∗ as yk [l] = √1N N n=0 rn [l] ? hn,k [−l]. Moreover, following (3.1), we have yk [l] = K−1 X k0 =0 where xk0 [l] ? gkk0 [l] + nk [l], (5.11) 60 g kk0 N −1 1 X √ [l] , hn,k0 [l] ? h∗n,k [−l], N n=0 (5.12) is the equivalent CIR after the TR operation. In particular, gkk0 [l], for k 0 6= k, is the cross-talk CIR between the terminals k and k 0 , and gkk [l] is the time-reversal equivalent P −1 ∗ CIR of terminal k. Also, nk [l] , √1N N n=0 νn [l] ? hn,k [−l] is the noise contribution after the TR operation. (i) Here, we focus on the equalization of the ith OFDM symbol. Hence, let the M × 1 vector yk = [yk [iM ], . . . , yk [iM +M −1]]T contain the ith segment of the signal yk [l]. Accordingly, (5.11) can be expressed in a matrix form as (i) yk = i+1 K−1 X X (i,i0 ) (i0 ) (i) Gkk0 xk0 + nk , (5.13) i0 =i−1 k0 =0 where  (i,i−1) Gkk0  (i,i) Gkk0 0 ··· 0 · · ·  . . ..  ..   .. . = 0  . 0 . .   .. . . . . 0 ··· gkk0 [L − 1] gkk0 [L − 2] · · · 0 gkk0 [L − 1] · · · .. .. .. . . . .. .. .. . . . .. .. .. . . . .. .. .. . . . ··· ··· ··· ···  ···   ..  .   . .. = gkk0 [1 − L]  .  0 gkk0 [1 − L] . .   .. .. ..  . . . 0 0 ··· gkk0 [0] gkk0 [−1] .. .  (i,i+1) Gkk0 gkk0 [1] gkk0 [0] .. . .. . ···  ..  .   . ..  0   .. = 0 . gkk [1 − L] ..  ..  . .   gkk0 [−2] · · · gkk0 [−1] · · · 0 .. . gkk0 [1] gkk0 [2] .. .        gkk0 [L − 1] ,   0   ..  . 0 (5.14a)  gkk0 [L − 1] 0 ··· 0  ··· gkk0 [L − 1] ··· 0   . .. .. .. .  . . . .   .. .. .. . . . gkk0 [L − 1] ,  .. .. .. ..  . . . .   .. .. . . gkk0 [0] gkk0 [1]  gkk0 [1 − L] ··· gkk0 [−1] gkk0 [0] (5.14b) ··· .. . .. . .. . .. . ··· .. . .. . .. . .. . ··· .. . .. . .. . .. . gkk0 [1 − L] 0 ··· gkk0 [2 − L] gkk0 [1 − L] · · ·  0 ..  .   0  , 0  ..  .  0 0 (5.14c) 61 (i) and the vector nk includes M samples of the AWGN signal nk [l] at the position of symbol (i,i−1) i. The matrices Gkk0 (i,i) (i,i+1) , Gkk0 and Gkk0 are M × M convolution matrices comprising the ISI components due to the tail of the symbol i − 1, the ICI components within the symbol i and the ISI components originating from the beginning of the symbol i + 1, respectively. (i) Applying an M -point DFT block to yk , we obtain the following frequency-domain signal. (i) ỹk = i+1 K−1 X X i0 =i−1 (i,i−1) where G̃kk0 (i) (i,i−1) , FM Gkk0 (i,i0 ) (i0 ) (i) G̃kk0 dk0 + ñk , (5.15) k0 =0 (i,i) (i,i) (i,i+1) H FH M , G̃kk0 , FM Gkk0 FM , G̃kk0 (i,i+1) , FM Gkk0 FH M and (i) ñk , FM nk . In Section 5.6, we analyze the SINR performance of TR-MRC and show that in this case, the SINR will grow without a bound as N grows large. Consequently, the SINR saturation problem is resolved through deployment of TR-MRC. To further compensate the frequency-selectivity of the channel, we propose to use an FDE after the time-reversal operation. The proposed FDE jointly processes the combined signals from all the users for per-user equalization and separation of signals from different users. 5.4.2 TR-FDE As mentioned in Section 5.4.1, the SINR saturation problem is resolved through deployment of TR-MRC. Hence, as the number of BS antennas grows large, the power of different interference terms tends to zero and arbitrarily large SINR values can be achieved. However, for finite number of BS antennas, this receiver suffers from a significant amount of interference in multiuser networks. This is mainly due to the interference originating from the symbols of different terminals transmitted on the same time and frequency slots. To gain a better intuition, we note that the TR-MRC receiver can be analogous to the MRC receiver used in CP-OFDM systems. The MRC receiver is simple and allows for arbitrarily large SINR values in CP-OFDM systems by increasing the number of BS antennas. However, multiuser interference is an important issue in MRC. Therefore, to tackle the multiuser interference and improve the SINR, ZF or MMSE detectors are utilized. In light of this discussion, in the following, we consider the time-reversal technique and aim at designing an additional equalization step in the frequency domain to reduce the residual interference 62 in TR-MRC. We utilize the structure of OFDM to design a multiuser equalizer after time-reversal. In particular, we consider each subcarrier individually, and apply an equalizer to eliminate the interference coming from different user terminals. Collecting the samples of all the user terminals on a given DFT frequency bin m, where m ∈ {0, . . . , M − 1}, into a K × 1 vector, one can perform frequency domain equalization on the individual bins. To this end, we (i) reorder the signal samples given in (5.15) as follows. Let the K × 1 vector sm contain the (i) (i) mth elements of the vectors dk , k = 0, . . . , K − 1. Similarly, we form the K × 1 vectors tm (i) (i) (i) and η m using the corresponding elements of ỹk and ñk , respectively. Accordingly, the (i) (i) relation between sm and tm can be expressed as (i,i) (i) t(i) m = Rmm sm + | {z } Desired Signal where M −1 X ξ (i) m = (i,i) (i) Rmm0 sm0 | , (5.16) Noise + Interference i+1 N −1 X X + m0 =0 m0 6=m (i) ξm |{z} (i,i0 ) (i0 ) (i) . Rmm0 sm0 + η m (5.17) i0 =i−1 m0 =0 i0 6=i {z ICI } | {z } ISI |{z} Noise (i,i0 ) In the above model, the element kk 0 of the K × K matrix Rmm0 is given by the element (i,i0 ) mm0 of the matrix G̃kk0 . (i) Based on the channel model in (5.16), the MMSE estimate of sm can be obtained as (i,i) ŝ(i) m = Rmm (i) where Cov ξ m H (i,i) (i,i) Rmm Rmm H + Cov ξ (i) m −1 t(i) m, (5.18) (i) represents the covariance matrix of ξ m and can be computed as i+1 M −1 M −1 X X X (i,i0 ) (i,i0 ) H (i,i) (i,i) H Rmm0 Rmm0 +σν2 IK . Cov ξ (i) = R R + m mm0 mm0 i0 =i−1 m0 =0 i0 6=i m0 =0 m0 6=m As a result, we can simplify (5.18) according to (i,i) ŝ(i) m = Rmm where Λm = Pi+1 i0 =i−1 (i,i0 ) m0 =0 Rmm0 PM −1 H Λm + σν2 IK (i,i0 ) H Rmm0 −1 t(i) m, (5.19) . (i) One major problem associated with the calculation of the MMSE estimate ŝm according to (5.19) is the high computational cost due to matrix multiplications involved in computing 63 Λm . In order to lower this computational burden, we note that when the number of BS antennas is much larger than the number users, that is, N K, the effects of intra-block and inter-block interference become small [77]. This is due to the coherent processing using the time-reversal combining. Consequently, we can approximate the matrix Λm by (i,i) (i,i) H Λm ≈ Rmm Rmm . Based on our simulations, the performance loss resulting from this approximation is negligible when N K. As a result, we can express the proposed MMSE estimator with approximation as (i,i) ŝ(i) m ≈ Rmm H (i,i) (i,i) Rmm Rmm H + σν2 IK −1 (i) tm . (5.20) We refer to this equalizer as TR-FDE. This additional equalization step leads to a substantial SINR performance improvement compared to the conventional TR-MRC. This is theoretically and numerically evaluated in Sections 5.6 and 5.7, respectively. Figure 5.1 illustrates the baseband system implementation of the TR combining with the proposed FDE. 5.5 Efficient Implementation and Complexity Analysis In this section, we study the computational complexity of the TR-MRC and TR-FDE receivers and compare the results with those of the conventional MRC and ZF methods utilized in CP-OFDM. The proposed receiver structures can be divided into two parts: (i) the TR part, and (ii) the FDE part. We discuss actions that should be taken to minimize the complexity of each part. According to (5.11), the TR part consists of a set of FIR filters whose complexity depends on the channel impulse responses between the BS antennas and the terminals. In particular, if the respective CIRs are sparse, that is, are characterized by a small number of multipath components, one can directly implement the TR part in the time domain. However, in general, the direct implementation of (5.11) may be computationally intensive in a wide-band OFDM transmission scenario, as the number of channel taps can be large. Fortunately, the above issue can be resolved by utilizing the fast-convolution techniques such as overlap-add and overlap-save [100]. Thus, the TR convolutions in (5.11) are implemented efficiently in the frequency domain using the fast Fourier transform (FFT) algorithm. In the overlap-add and overlap-save methods, the processing is performed on a block-by-block basis, where each block is of length M̃ and is constructed from the samples 64 Base Station Wireless Channel Mobile Terminals OFDM Modulator OFDM Modulator OFDM Demodulator ZF Equalizer OFDM Demodulator One demodulator per terminal Time Reversal Combining Figure 5.1. Baseband system implementation of the proposed technique with a TR-FDE receiver. of the input signal rn [l]. Here, M̃ is a design parameter and is usually selected from the range 4L ≤ M̃ ≤ 8L to minimize the computational cost. Accordingly, an M̃ -point FFT is applied to each block to obtain the frequency-domain samples of the input signal. Then, these samples are multiplied with the respective frequency-domain channel coefficients. At this point, in order to minimize the number of required inverse FFT (IFFT) blocks, we can combine the signals corresponding to different BS antennas directly in the frequency domain, and then, apply a single IFFT block to the resulting signal to obtain the samples of yk [l]. The above procedure significantly reduces the computational cost of the TR operation. We now focus on the implementation of the second part, that is, FDE. Direct calculation of the matrices involved in the FDE introduced in Section 5.4.2 imposes a substantial amount of computational burden to the system. In particular, considering a given subcarrier (i,i) m, the matrix Rmm should be computed to perform the equalization according to (5.20). (i,i) (i,i) (i,i) The element Rmm kk0 is equal to the the mth diagonal element of G̃kk0 = FM Gkk0 FH M. Therefore, the direct approach requires the computation of the mth diagonal elements of the (i,i) matrices G̃kk0 , for k, k 0 ∈ {0, . . . , K − 1}, to form the FDE matrix. This involves a great number of calculations especially when the number of subcarriers is large. In particular, the number of complex multiplications using the direct method for all the subcarriers has a complexity that is of order K 2 M 3 . We denote this complexity by O(K 2 M 3 ). Clearly, the direct method becomes computationally very expensive when M is large. Fortunately, this issue can be resolved through the method that we introduce in the following. 65 (i,i) (i,i) (i,i) (i,i) According to the expression G̃kk0 = FM Gk0 FH M , we have G̃kk0 mm = Rmm kk0 = (i,i) TG ∗ fm kk0 fm , where fm is the column m of the DFT matrix FM . Therefore, one can obtain (i,i) the element Rmm kk0 as a linear combination of the samples of the time-reversal channel (i,i) PL−1 impulse response gkk0 [l], that is, Rmm kk0 = l=1−L βm [l] gkk0 [l], for some coefficients βm [l]. After some algebraic manipulations, the coefficients βm [l] can be found as βm [l] = M −|l| −j 2πml M . M e Hence, we have L−1 X M − |l| (i,i) 2πml Rmm kk0 = gkk0 [l]e−j M . M (5.21) l=1−L (i,i) Therefore, Rmm can be efficiently computed from gkk0 [l] using the FFT algorithm. Also, (i,i) when the channel length L is much smaller than the block length M , we have Rmm kk0 ≈ PL−1 −j 2πml M . Following (5.21), the number of complex multiplications needed for l=−L+1 gkk0 [l]e deriving the matrices involved in the FDE is reduced to O K 2 M log2 M . As a result, a substantial computational complexity reduction is achieved. We now compare the computational cost of TR-MRC and TR-FDE for OFDM without CP with those of the conventional MRC and ZF in CP-OFDM. In Table 5.1, we have presented the number of complex multiplications needed to perform the MRC and ZF methods in CP-OFDM. Here, Q represents the number of OFDM symbols. In Table 5.1, for both cases of MRC and ZF, the first and second terms represent the complexity due to the time-to-frequency conversion using M -point FFT blocks and frequency-domain combining, respectively. In the case of ZF, the third and fourth terms are due to the calculation of −1 the ZF combining matrices W m = Hm (HH m Hm ) , m ∈ {0, . . . , M − 1}. This needs to be calculated once for the transmitted packet consisting of Q symbols. Table 5.2 shows the number of complex multiplications needed to perform the TR combining and the FDE using the procedures discussed in this section. More specifically, the first three terms in the case of TR combining are due to the implementation of (5.11) using fast-convolution as discussed above. Moreover, the fourth is arising from the calculation (i) of ỹk (i) from yk using M -point FFT blocks. In the case of FDE, the first two terms given in Table 5.2 account for the calculation of the equivalent channel responses gkk0 [l] given in (5.12) using fast-convolution1 . The third term, 21 K 2 M log2 M , is arising from the 1 ∗ Here, FFT size of M̃ is considered. Moreover, we have used the fact that gk0 k [l] = gkk 0 [−l] to reduce 66 Table 5.1. Computational complexity of the conventional MRC and ZF detectors utilized in CP-OFDM systems. Technique Number of Complex Multiplications 1 MRC 2 QN M log2 M + QN M K 1 3 1 2 3 ZF QN M log 2 M + QN M K + 2 N M K + 3 M K 2 Table 5.2. Computational complexity of different parts of the receivers proposed for OFDM without CP systems. Technique Number of Complex Multiplications QN M K M̃ 1 QN M 1 QM K 1 2 M̃ −L+1 M̃ log2 M̃ + M̃ −L+1 + 2 M̃ −L+1 M̃ log2 M̃ + 2 QKM log2 M K(K+1) N M̃ + K(K+1) M̃ log2 M̃ + 12 K 2 M log2 M + 34 M K 3 + QM K 2 2 4 TR FDE (i,i) calculation of the coefficients Rmm kk0 according to (5.21). The fourth term is due to the −1 (i,i) H (i,i) (i,i) H matrix operations involved in Rmm Rmm Rmm + σν2 IK . Finally, the last term accounts for the multiplication of the FDE matrix to the input vector as in (5.20). Figure 5.2 compares the computational complexity of OFDM without CP with TRMRC and TR-FDE techniques against CP-OFDM with MRC and ZF methods. Here, the following parameters are considered. M = 512, M̃ = 256, L = 40, and Q = 10. In Figure 5.2(a), we have fixed K = 10 and varied the value of N , whereas in Figure 5.2(b), N = 200 is fixed and the value of K is varied. As the figures show, while the complexity of MRC and TR-MRC are approximately the same, the TR-FDE receiver has a lower computational cost compared to the ZF receiver. The reason for this is that the proposed frequency-domain equalization takes place after multiantenna combining; see Figure 5.1. Hence, the number of input signals to the FDE is significantly reduced as compared to the case of conventional ZF equalizer. 5.6 Analysis of SINR and Achievable Rate In this section, we analyze the SINR performance of both TR-MRC and TR-FDE receivers. This SINR analysis will ultimately lead us to find a lower-bound for the achievable information rate of each equalization technique. the number of computations. 67 8 10 7 7 6 5 4 3 2 1 0 100 200 300 400 500 30 40 50 (a) 5 10 8 4 3 2 1 0 10 20 (b) Figure 5.2. Computational complexity comparison of OFDM without CP with TR-MRC and TR-FDE techniques against CP-OFDM with MRC and ZF equalizations. Here, the following parameters are considered. M = 512, M̃ = 256, L = 40, and Q = 10. In (a), K = 10 is fixed and the value of N is varied, whereas in (b), N = 200 is fixed and the value of K is varied. 68 5.6.1 Let (i,i0 ) Gkk0 ,mm0 TR-MRC (i,i0 ) (i) denote the the element (m, m0 ) of the matrix G̃kk0 , and nk,m be the element (i) m of the vector ñk . Then, according to (5.15), the SINR of the TR-MRC receiver can be calculated as Sig SINRTR-MRC k,m = Pk,m ICI + P ISI + P MUI + P Noise Pk,m k,m k,m k,m , (5.22) where n o 2 (i,i) Sig Pk,m = E Gkk,mm , −1 nM X (i,i) ICI =E Gkk,mm0 Pk,m 2 , m0 =0 m0 6=m −1 nM X ISI Pk,m =E (i,i−1) 2 Gkk,mm0 m0 =0 −1 n K−1 XM X MUI Pk,m =E k0 =0 m0 =0 k0 6=k (i,i+1) + Gkk0 ,mm0 (i,i−1) Gkk0 ,mm0 2 2 (i,i) o , + Gkk0 ,mm0 2 (i,i+1) + Gkk0 ,mm0 2 o , n o (i) 2 Noise Pk,m = E nk,m . (5.23) Using the channel model introduced in Chapter 3 and after some straightforward calculaNoise = σ 2 . In the following, in order to tions, the average noise power can be obtained as Pk,m ν (i,i−1) simplify the above SINR expression, we aim to analyze the interference coefficients Gkk0 ,mm0 , (i,i) (i,i+1) (i,i−1) (i,i) Gkk0 ,mm0 , and Gkk0 ,mm0 . By utilizing the Toeplitz structure of the matrices Gmm0 , Gmm0 (i,i+1) and Gmm0 , one can obtain these coefficients through the following expressions. (i,i−1) (i,i−1) ∗ fm0 = aH mm0 gkk0 , (i,i) (i,i) = bH mm0 gkk0 , (i,i+1) (i,i+1) ∗ fm0 = cH mm0 gkk0 , T Gkk0 ,mm0 = fm Gkk0 T ∗ Gkk0 ,mm0 = fm Gkk0 fm 0 T Gkk0 ,mm0 = fm Gkk0 (5.24) where fm is the mth column of the M -point DFT matrix FM , and the vector gkk0 , [gkk0 (1 − L), . . . , gkk0 (L − 1)]T contains the samples of the TR channel impulse response gkk0 [l]. Also, the vectors amm0 , bmm0 , and cmm0 are determined by 69 1 amm0 = √ TL0 ×M M 1 bmm0 = √ TL0 ×M M 1 cmm0 = √ TL0 ×M M respectively, where ωm , e−j 2πm M M −1 M −L+1 01×M +L−1 , ωm , . . . , ωm 0 0 T ∗ fm , M −1 0 01×L−1 , ωm , . . . , ωm 0 , 01×L−1 0 T ∗ fm , L−2 0 ωm 0 , . . . , ωm0 , 01×M +L−1 T ∗ fm , (5.25) and L0 , 2L − 1 is the length of the vector gkk0 . The notation A = TM ×N (a) for an (N + M − 1) × 1 vector a, represents an M × N Toeplitz matrix, in which [A]mn = [a]m−n+N . Accordingly, the vector a is formed by starting from the top right element of A, going along the first row to the top left element and then going along the first column to the bottom left element. We note that while the TR channel impulse response gkk0 has a random nature, the vectors amm0 , bmm0 , and cmm0 are deterministic. Accordingly, in (5.24), we have separated (i,i−1) (i,i) (i,i+1) the random and deterministic parts of the coefficients Gkk0 ,mm0 , Gkk0 ,mm0 , and Gkk0 ,mm0 . This will help us to find their statistics. Following the definition of the time-reversal equivalent channel response gkk0 [l] given in (5.12), the mean of the complex random vector gkk0 can be obtained as √ E gkk0 = N δkk0 δ L0 , (5.26) T where δ L0 , 01×(L−1) , 1, 01×(L−1) , and δkk0 is the Kronecker delta function. Moreover, the covariance matrix of gkk0 is calculated according to n H o Γkk0 , E gkk0 − E gkk0 gkk0 − E gkk0 = diag{pkk0 }, (5.27) where the elements in the vector pkk0 = [pkk0 [1 − L], . . . , pkk0 [L − 1]]T are obtained by P convolving pk0 [l] by pk [−l], that is, pkk0 [l] = L−1 `=0 pk0 [`]pk [` − l]. According to (5.24), the SINR expression given in (5.22) can be written as SINRTR-MRC k,m H where QIntf k,m , gkk Φm gkk + PK−1 k0 =0 k0 6=k Sig E Qk,m , = Intf E Qk,m + σν2 (5.28) H Ψ g 0 includes the interference power due to the gkk 0 m kk Sig H B g ICI, ISI, and MUI components, and Qk,m , gkk m kk is the desired signal power. Here, Bm = bmm bH mm and the matrices Ψm , and Φm are defined according to 70 Ψm = Φm = M −1 X m0 =0 M −1 X H H amm0 aH mm0 + bmm0 bmm0 + cmm0 cmm0 , bmm0 bH mm0 + m0 =0 m0 6=m M −1 X H amm0 aH mm0 + cmm0 cmm0 (5.29a) m0 =0 = Ψ m − Bm , (5.29b) respectively. We note that QIntf k,m is a summations of K quadratic terms in the complex Sig random vectors gkk0 , k 0 ∈ {0, . . . , K − 1}. Similarly, Qk,m is quadratic in the complex random vector gkk . Proposition 7. In the absence of CP and with TR-MRC equalization, the SINR can be calculated as SINRTR-MRC = k,m where λk , PL−1 l=1−L 1− |l| 2 M pkk [l]. N + λk , K − λk + σν2 (5.30) Sig Proof. According to (5.26) and (5.27), the mean value of the quadratic term Qk,m can Sig be obtained as, E Qk,m = N + tr {Γkk Bm }, [75, p. 53], where we have used the fact that [bmm ]L = 1. Similarly, by noting that [amm0 ]L = [cmm0 ]L = 0 for any m0 and m, and [bmm0 ]L = 0 for m0 6= m, we can find the mean of the quadratic expression QIntf k,m as Intf PK−1 E Qk,m = tr {Γkk Φm } + k0 =0 tr {Γkk0 Ψm }. To simplify this, we note that the diagonal k0 6=k P elements of Ψm are all equal to one. Accordingly, tr{Γkk0 Ψm } = tr{Γkk0 } = l pkk0 [l] = Intf P P = K −tr {Γkk Bm }. The value of tr{Γkk Bm } can l ` pk0 [`]pk [`−l] = 1. Hence, E Qk,m be obtained as follows. From (5.25) we can find the elements of the vector bmm according to h M − L + 1 2π M − L + 2 j 2π (L−2)m ej M (L−1)m , e M , . . . , 1, M M M − L + 2 j 2π (2−L)m M − L + 1 j 2π (1−L)m iT ..., e M , e M . M M P |l| 2 Hence, tr{Γkk Bm } = λk = L−1 l=1−L 1 − M pkk [l]. This completes the proof. bmm = (5.31) Remark 1. The SINR gain of O(N ) is achievable with TR-MRC and the SINR saturation problem is resolved. It is worth mentioning that the parameter λk is a positive constant that depends on P the channel PDP. Moreover, using l pkk [l] = 1, we can find that λk is always less than or 71 equal to one, that is, λk ≤ 1. When the channel length is much smaller than the symbol |l| 2 duration, that is, L M , we have 1 − M ≈ 1 for l ∈ {1 − L, · · · , L − 1}. This leads to λk ≈ 1. For a fixed channel PDP, as the symbol duration M becomes smaller, the value of λk decreases. Using the result of the Proposition 7, a lower bound on the achievable information rate at the output of the TR-MRC equalizer can be obtained by considering the worst case uncorrelated additive noise. Assuming that terminals transmit Gaussian data symbols, it is proven in [101] that the worst case uncorrelated noise is circularly symmetric Gaussian with the same variance as the effective additive noise. Accordingly, a lower bound on the achievable rate in the case of TR-MRC can be obtained as N + λk TR-MRC . = log2 1 + Rk K − λk + σν2 (5.32) On the other hand, a lower bound on the achievable information rate of CP-OFDM transmission with MRC equalizer is given by [4, 46] CP-OFDM Rk MRC where the term M M +L M N −1 log2 1 + , = M +L K − 1 + σν2 (5.33) represents the rate loss due to the CP overhead. In Section 5.7, we numerically evaluate the rate given in (5.32) and compare it against (5.33) as a benchmark. Before we end our discussion in this section, we note that for large values of N and K, N we have RkTR-MRC ≈ log2 1 + K+σ 2 . This matches the achievable rate reported in [77] for ν the case of single-carrier transmission when TR-MRC is applied. This implies that when TR-MRC is utilized, and for large values of N and K, the same information rate can be achieved either by the OFDM without CP or the single-carrier transmission. 5.6.2 TR-FDE In the case of TR-FDE, the additional FDE equalization step removes a significant portion of the remaining interference after the TR operation. Here, we mathematically analyze the SINR and achievable rate performance of this scheme. In order to find the SINR performance of the TR-FDE receiver, we focus on the FDE −1 (i,i) H (i,i) (i,i) H (i,i) matrix Rmm Rmm Rmm + σν2 IK . We note that Gkk0 ,mm is the element kk 0 of (i,i) (i,i) the matrix Rmm . Moreover, according to (5.24) and (5.26), Gkk0 ,mm can be expressed √ (i,i) as Gkk0 ,mm = N δkk0 [bmm ]L + bH mm g̃kk0 , where g̃kk0 , gkk0 − E{gkk0 }. Furthermore, as 72 calculated in (5.31), the Lth entry of the vector bmm is equal to [bmm ]L = 1. Based on the (i,i) above analysis, we can express the matrix Rmm as (i,i) Rmm = √ N IK + ∆m , (5.34) where the elements of the matrix ∆m can be obtained according to [∆m ]kk0 = bH mm g̃kk0 . (i,i) According to (5.34), as the number of BS antennas N grows large, the matrix Rmm tends √ −1 (i,i) H (i,i) (i,i) H Rmm Rmm + σν2 IK tends to √1N IK as to N IK . Hence, the FDE matrix Rmm N grows large. Using this, the following proposition finds the asymptotic (N → ∞) SINR in the case of TR-FDE. Proposition 8. In the absence of CP and with TR-FDE equalization, the SINR tends to = SINRTR-FDE k,m as N grows large. We recall that λk , N , K(1 − λk ) + σν2 PL−1 l=1−L 1− |l| 2 M pkk [l] (i,i) H Proof. According to (5.34), the FDE matrix Rmm √1 IK N (5.35) (i,i) ≤ 1. (i,i) H Rmm Rmm + σν2 IK −1 tends to as N grows large. Moreover, the interference coming from the first term in (5.16) becomes negligible by performing the equalization. Therefore, the interference originating from the second term in (5.16) is dominant and tends to √1 N (i) ξ m asymptotically. We note that this term constitutes the residual interference after the TR-FDE. Using the same line (i) of derivation as in Proposition 7, we can find the variance of the element k in ξ m as PK−1 2 2 k0 =0 tr{Γkk0 Φm } + σν = K(1 − λk ) + σν . This leads to the SINR expression given in (5.35). Remark 2. Similar to the case of TR-MRC, the SINR gain of O(N ) is achievable using TR-FDE receiver and the SINR saturation is avoided. The above result suggests that SINR saturation can be avoided through utilization of TR. The additional FDE step further improves the SINR level in multiuser systems. According to (5.35), a lower bound on the asymptotic achievable information rate at the output of the TR-FDE equalizer can be obtained as R̃kTR-FDE = log2 1 + N , K(1 − λk ) + σν2 (5.36) 73 where the tilde sign in R̃ signifies that it is an asymptotic information rate, that is, it tends to the actual information rate as the number of BS antennas N increases. On the other hand, the achievable information rate of CP-OFDM transmission with ZF equalizer is given by [4, 46] CP-OFDM Rk ZF where the term M M +L M N −K , = log2 1 + M +L σν2 (5.37) represents the rate loss due to the CP overhead. We note that comparing (5.36) and (5.37) may not be fair as the former is derived using asymptotic analysis, and the latter is valid for finite values of N as well. Hence, for the purpose of comparison, we also consider the asymptotic version of (5.37) given by [4] CP-OFDM R̃k ZF N M log2 1 + 2 . = M +L σν (5.38) In Section 5.7, we numerically evaluate the rate given in (5.36) and compare it against (5.38) as a benchmark. 5.7 Numerical Results In this section, we evaluate the analyses and discussions of the previous sections through numerical simulations. We consider the Extended Typical Urban (ETU) channel model as defined in the LTE standard [102]. We adopt the LTE air interface parameters to OFDM without CP. Specifically, the OFDM useful symbol duration of T = 66.7 µs, which translates to the subcarrier spacing of ∆f = 15 kHz is considered. Note that when considering OFDM without CP transmission, the useful symbol duration is equal to the total symbol duration, and delay spread of the ETU model covers about 7% of the OFDM symbol duration. We choose the DFT size of M = 512, and 300 active subcarriers. This corresponds to the 5 MHz bandwidth scenario defined in the LTE standard. We first evaluate the SINR performance of various methods discussed in this chapter. In Figure 5.3, we have demonstrated the SINR saturation of the conventional frequency-domain combining methods, namely MRC, ZF, and MMSE. In this experiment, K = 10 active terminals are considered, and the noise level is chosen such that the average SNR at the input of the BS antennas is 10 dB. We show the average SINR values over different channel realizations with the power delay profile of the ETU channel model. The saturation level is calculated using (5.9) and is compared with the simulated SINR values. As we expect, in 74 25 20 15 10 5 0 10 1 10 2 10 3 Figure 5.3. SINR saturation in the case of conventional frequency-domain combiners. Here, K = 10 terminals are considered and the number of BS antennas is varied. The SNR level is chosen to be 10 dB. The saturation level is calculated using (5.9). all three frequency-domain combining methods, SINR does not improve beyond a certain deterministic level. As mentioned in Section 5.4, this problem can be resolved by using the TR technique. Figure 5.4 shows the SINR performance of TR-MRC and TR-FDE methods. Again, as expected, for both cases of TR-MRC and TR-FDE, SINR will grow unboundedly as the number of BS antennas grows. Moreover, since the proposed TR-FDE method significantly reduces the MUI level compared to the conventional TR-MRC technique, it yields to an improved SINR performance. In Figure 5.4, the SNR at the input of the BS antennas is 10 dB. Moreover, we have also shown the theoretical SINR values calculated according to (5.30) and (5.35) for the cases of TR-MRC and TR-FDE, respectively. As the number of BS antennas N grows large, the simulated SINR values coincide with the values derived using asymptotic analysis in Section 5.6. We next conduct an experiment to evaluate the achievable information rate with and without including the CP overhead. Figure 5.5(a) shows the theoretical achievable rate of OFDM without CP with TR-MRC and TR-FDE equalizers as well as that of CP-OFDM 75 50 40 30 20 10 0 10 1 10 2 10 3 Figure 5.4. SINR performance comparison for time reversal methods. Here, K = 10 terminals are considered and the number of BS antennas is varied. The SNR level is chosen to be 10 dB. Asymptotic theoretical SINR values are calculated according to (5.30) and (5.35) for the cases of TR-MRC and TR-FDE, respectively. Using time reversal, arbitrarily large SINR values can be achieved by increasing the number of BS antennas. with MRC and ZF detectors. In the cases of OFDM without CP with TR-FDE and CPOFDM with ZF equalizer, asymptotic rates given by (5.36) and (5.38), respectively, are considered. In this experiment, K = 10 terminals are considered and the noise level is chosen such that SNR at the input of the BS antennas is −10 dB. Figure 5.5(b) shows the results for the case where K = 20 terminals are active. As shown in Figures 5.5(a) and 5.5(b), with OFDM without CP and TR-MRC equalization, we can achieve a higher spectral efficiency as compared to in CP-OFDM with MRC equalizer. A similar argument applies for OFDM without CP with TR-FDE and CP-OFDM with ZF detector. Hence, as expected, by eliminating the CP overhead we can achieve a higher spectral efficiency compared with the conventional CP-OFDM systems. It should be noted that according to Figures 5.5(a) and 5.5(b), for a fixed achievable rate performance, one can decrease the number of BS antennas (and hence the implementation cost) by removing the CP overhead. In Figure 5.6, we compare the achievable rate performance of OFDM without CP and 76 6 5 4 3 2 100 200 300 400 500 600 400 500 600 (a) 6 5 4 3 2 1 100 200 300 (b) Figure 5.5. Per user achievable information rate with and without the CP overhead. Here, the ratio L/M is approximately 7%, and the SNR level is chosen to be −10 dB. (a) K = 10, (b) K = 20 user terminals. 77 10 9 8 7 6 5 4 3 2 -20 -15 -10 -5 0 5 10 Figure 5.6. Per user achievable information rate as a function of the SNR level. Here, the ratio L/M is approximately 7%, and N = 200 BS antennas and K = 10 terminals are considered. CP-OFDM for various levels of SNR. In this experiment, N = 100 BS antennas and K = 10 terminals are considered. As shown, for typical SNR levels, higher spectral efficiency can be achieved using OFDM without CP. On the other hand, in very low SNR regime, the noise level dominates the overall interference plus noise, and hence, similar rates can be achieved using OFDM with/without CP deploying various equalization methods. On the other hand, when the SNR level is high, the residual interference dominates the noise, hence the performance of OFDM without CP with TR-MRC/TR-FDE and CP-OFDM with MRC becomes saturated and does not improve with increasing the transmission power. So far in this section, we considered the ETU channel model, which covers about 7% of the OFDM symbol duration of T = 66.7 µs. In the next experiment, we aim to show the advantage of the elimination of CP in channels with larger delay spreads. Accordingly, we consider the TDL-A channel PDP with the RMS (root mean square) delay spread of 1100 ns. This channel model has been recently proposed for the frequency spectrum above 6 GHz [103], and covers about 15% of the OFDM symbol duration. Figure 5.7 shows the achievable 78 6 5 4 3 2 100 200 300 400 500 600 Figure 5.7. Per user achievable information rate with and without the CP overhead. Here, K = 10 terminals are considered and the SNR level is chosen to be −10 dB. Moreover, the TDL-A channel with the RMS delay spread of 1100 ns is assumed. In this channel model, the ratio L/M is approximately 15%. rate comparison of OFDM without CP and CP-OFDM considering the above channel model. Here, K = 10 terminals and the SNR level of SNR = −10 dB are considered. As shown, here due to a larger CP duration, the spectral efficiency is improved more considerably by eliminating the CP overhead. 5.8 Conclusion It is known that in massive MIMO channels uncorrelated noise and multiuser interference vanish as the number of BS antennas grows large. Motivated by this, in this chapter, we studied OFDM without CP under such channels to investigate if the channel distortions (ISI and ICI) average out in the large antenna regime. To this end, we mathematically analyzed the asymptotic SINR performance of the conventional frequency-domain combining methods, namely, MRC, ZF, and MMSE. Our analysis revealed that in these cases, there always exists some residual interference even for an infinite number of BS antennas leading to the saturation of the SINR performance. To solve this saturation issue, we proposed to use the 79 TR technique. Moreover, we introduced a frequency-domain equalizer to be incorporated after the TR combining to further reduce the multiuser interference. We mathematically analyzed the asymptotic achievable information rate of the proposed receiver design. We showed that by removing the CP overhead and using the proposed technique, a higher spectral efficiency is achievable as compared to the conventional CP-OFDM systems, while the computational complexity is also reduced. 5.9 The elements of (i,i−1) H̃n,k h i (i,i−1) H̃n,k mm0 (i,i) H̃n,k and = Proof of (5.8) can be expanded as [82] M −1 L−1 2π 1 XX 0 0 hn,k [l]ej M (um −lm −um) w[u − l + M ], M u=0 l=0 and h i (i,i) H̃n,k = mm0 M −1 L−1 2π 1 XX 0 0 hn,k [l]ej M (um −lm −um) w[u − l], M u=0 l=0 where w[l] is the windowing function, which is considered to be a rectangular window, that 1, 0 6 l 6 M − 1, . Accordingly, following (5.7), we have is, w[l] = 0, otherwise. (i,i) Hkk,mm h (i,i) i n,k ∗ → E Hm H̃n,k mm M −1 L−1 X X 1 lm n,k ∗ −j 2π M = E w[u − l] Hm hn,k [l]e M u=0 l=0 M −1 L−1 X X L−1 X 2π 0 1 E h∗n,k [l0 ]hn,k [l]ej M (l −l)m w[u − l] = M 0 u=0 l=0 l =0 = 1 M L−1 X =1− where τ̄k , PL−1 l=0 l=0 (M − l)pk [l] τ̄k , M lpk [l], is the average delay spread of the channel corresponding to user k. For m 6= m0 we have, 80 h (i,i) i (i,i) n,k ∗ Hkk,mm0 → E Hm H̃n,k mm0 M −1 L−1 X X L−1 X 1 0 −lm0 +l0 m−um) (um ∗ 0 j 2π E w[u − l] = hn,k [l ]hn,k [l]e M M 0 u=0 l=0 l =0 = = 1 M 1 M M −1 L−1 X X u=0 l=0 L−1 X pk [l]e−j 2πl(m0 −m) M (i,i−1) Hkk,mm0 PL−1 l=0 M −1 X ej 2πu(m0 −m) M u=l L−1 X pk [l]e −j 2πl(m0 −m) M l=0 L−1 X −1 2π(m0 −m) M (1 − ej M ) 1 − Pk [m0 − m] , = 2π(m0 −m) M 1 − ej M where Pk [m] , 0 l=0 1 =− M = 2π pk [l]e−j M (l−u)(m −m) w[u − l] pk [l]e−j can be calculated as 2πlm M 1 − ej 1 − ej pk [l]e 2πl(m0 −m) M 2π(m0 −m) M −j 2πl(m0 −m) M l=0 − L−1 X pk [l] l=0 . Similarly, the asymptotic value of the ISI coefficient (i,i−1) Hkk,mm → τ̄k M (i,i−1) and Hkk,mm0 → Pk [m0 −m]−1 M 1−ej 2π(m0 −m) M , when m 6= m0 . n,k Moreover, with similar derivations it is possible to show that Hm is uncorrelated with h i h i (i,i) (i,i) (i,i−1) and H̃n,k0 , when k 6= k 0 . Accordingly, the MUI coefficients Hkk0 ,mm0 H̃n,k0 0 0 and mm (i,i−1) Hkk0 ,mm0 mm tend to be zero as N grows large. CHAPTER 6 SINGLE CARRIER MODULATION WITH FREQUENCY DOMAIN EQUALIZATION IN MASSIVE MIMO Single-carrier (SC) modulation has been proposed as an effective waveform for massive MIMO systems. Previous studies have shown that the time-reversal technique is a simple yet powerful method to be used for SC transmission in massive MIMO systems under low SNR conditions. In this chapter, we propose to use the frequency domain equalization (FDE) in addition to the time-reversal operation. We show that this can enhance the performance in moderate to high SNR conditions. Furthermore, the added computational complexity depends on the number of user terminals and does not grow with the number of BS antennas. 6.1 Introduction OFDM is a modulation format that has been adopted in many recent standards such as IEEE 802.11ac and 3GPP LTE. However, it is well known that OFDM suffers from a high peak-to-average power ratio (PAPR), requiring the power amplifiers (PAs) to work with a large power backoff. This leads to a lower energy efficiency of the PAs and decreases the maximum communication range possible. This is a critical concern in mmWave systems, in particular, since the range is also limited by the path loss [30]. SC modulation is an alternative to OFDM, offering a significantly lower PAPR. By utilization of this modulation format, one can maximize the PA output power and in turn achieve a higher coverage. Another limiting factor in mmWave systems is the high-power consumption of the mixed-signal components such as the analog-to-digital converter (ADC) [30]. Fortunately, the low PAPR advantage of SC modulation allows for lower resolution ADCs to be used at the receiver, which results in a substantially lower power consumption in a mmWave based massive MIMO system [31]. 82 The optimality of SC transmission in downlink massive MIMO systems has been studied in [77], where the authors propose a low-complexity transmission scheme based on the time-reversal technique. It is shown that the proposed scheme can achieve a near-optimal sum-rate performance in the low SNR regime. Moreover, the transmission method in [77] can be performed in a decentralized manner at each BS antenna, and does not require any receiver equalization. This simplifies the processing and considerably decreases the computational complexity of the system. The effect of oscillator phase noise in the uplink of this system is also studied in [81]. Application of frequency domain equalization (FDE) in SC systems has been studied extensively in the literature; see [104, 105] and the references therein. In SC-FDE, the channel distortions are equalized in the frequency domain similar to OFDM. This allows for an effective equalization using the FFT algorithm. In this chapter, we propose FDE in addition to time-reversal for SC transmission in massive MIMO. In particular, we propose to use FDE after the time-reversal operation to equalize the effective multiuser channel that arises from the combination of multipath channel and time-reversal coefficients. We show that the combination of FDE and time reversal can enhance the performance significantly while the added complexity (compared to time reversal alone) depends on the number of user terminals and does not grow with the number of BS antennas. It should be mentioned that since the time-reversal scheme proposed in [77] is near-optimum in low SNR regime, the advantage of FDE becomes significant in moderate to high SNR conditions. In this chapter, we focus on the uplink direction only. However, we note that extension of the concept to the downlink is straightforward. The rest of the chapter is organized as follows. In Section 6.2, we present the timereversal technique for combining the received signals across BS antennas. In Section 6.3, we introduce our proposed FDE method is detail. Our discussions in this chapter are numerically evaluated in Section 6.4. Finally, we conclude the chapter in Section 6.5. 6.2 Time-Reversal Combining We follow the system model introduced in Chapter 3. In addition, similar to [77], we assume that the BS incorporates the TR technique to combine the received signals across different BS antennas. Accordingly, the output of the time-reversal combining operation 83 corresponding to terminal k can be expressed as yk [l] = √1 N PN −1 n=0 rn [l] ? h∗n,k [−l]. Following (3.1), this can be written as yk [l] = K−1 X gkk0 [l] ? xk0 [l] + νk0 [l], (6.1) k0 =0 where νk0 [l] , √1 N PN −1 n=0 νn [l] ? h∗n,k [−l] represents the noise contribution after the time- reversal operation, and g kk0 N −1 1 X √ [l] , hn,k0 [l] ? h∗n,k [−l], N n=0 (6.2) is the equivalent channel impulse response after time-reversal combining. Assuming that the channel impulse responses hn,k [l] are zero outside of the interval 0 ≤ l ≤ L − 1, then the time-reversal channel gkk0 [l] will be zero outside of the interval −L + 1 ≤ l ≤ L − 1. 6.3 Frequency Domain Equalization (FDE) To further compensate the frequency-selectivity of the channel, we propose to use an FDE after the time-reversal operation. The proposed FDE jointly processes the combined signals from all the users for per-user equalization and separation of signals from different users. To implement the FDE, we consider block-processing with block length of M . In practice, M is chosen to be much larger than the channel length L so that the interference due to multipath channel is limited. Furthermore, to minimize the channel time-variation within each block, the block length should be selected to be much smaller than the coherence time of the channel. To obtain the frequency domain symbols, we apply an M -point DFT to each block. After equalization, the time domain symbols are obtained using an inverse DFT (IDFT) of size M . Figure 6.1 shows the block diagram of the proposed system. To pave the way for our development, we express (6.1) in a vector form. Let the M × 1 T (i) vector yk = yk [iM ], yk [iM + 1], . . . , yk [iM + M − 1] contain the samples of the signal (i) yk [l] corresponding to block i. Similarly, we define the vector xk = xk [iM ], xk [iM + T 1], . . . , xk [iM + M − 1] . Noting that the convolution in (6.1) leads to interference among the samples of the data sequences xk [l] both along time, and among different users data, (6.1) may be expressed as (i) yk = i+1 K−1 X X i0 =i−1 k0 =0 (i,i0 ) (i0 ) (i) Gkk0 xk0 + ν k , (6.3) 84 S/P DFT time-reversal combining IDFT P/S IDFT P/S FDE S/P BS antennas DFT branches branches Figure 6.1. Block diagram of the proposed TR-FDE system. (i) (i) where xk and ν k contain the samples of the signal contribution xk [l] and noise contribution (i,i−1) νk0 [l], respectively, corresponding to block i. The matrices Gkk0 (i,i) (i,i+1) , Gkk0 , and Gkk0 , respectively, model the distortion due to samples of blocks i − 1, i, and i + 1 on the current processing block i. The mathematical expression for these matrices follow the Toeplitz structure given as (i,i−1) Gkk0  (i,i) Gkk0  0 ··· 0 · · ·  . . ..  ..  = 0 . . .   .. . . . . 0 ··· gkk0 [L − 1] gkk0 [L − 2] · · · 0 gkk0 [L − 1] · · · .. .. .. . . . .. .. .. . . . .. .. .. . . . ··· ··· ··· ···  ···   ..  .   . .. = gkk0 [1 − L]  .  0 gkk0 [1 − L] . .   .. .. ..  . . . 0 0 ··· gkk0 [0] gkk0 [−1] .. . gkk0 [1] gkk0 [0] .. . .. .  (i,i+1) Gkk0 ···  ..  .   . ..  = gkk0 [1 − L]  .. ..  . .   gkk0 [−2] · · · gkk0 [−1] · · · 0 .. . gkk0 [1] gkk0 [2] .. .       , gkk0 [L − 1]   ..  . 0 (6.4a)  gkk0 [L − 1] 0 ··· 0  ··· gkk0 [L − 1] ··· 0   . .. .. .. .  . . . .   .. .. .. . . . gkk0 [L − 1] ,  .. .. .. ..  . . . .   .. .. . . gkk0 [0] gkk0 [1]  gkk0 [1 − L] ··· gkk0 [−1] gkk0 [0] (6.4b) ··· ··· ··· .. .. .. . . . .. .. .. . . . .. .. .. . . . gkk0 [1 − L] 0 ··· gkk0 [2 − L] gkk0 [1 − L] · · ·  0 ..  .   0 . ..  .  0 0 (6.4c) 85 (i) (i) Let ỹk = FM yk , where FM is the normalized M -point DFT matrix with the elements [FM ]mm0 = 2πmm √1 e−j M M 0 . Multiplying (6.3) from left by FM , we get i+1 K−1 X X (i) ỹk = i0 =i−1 (i,i0 ) (i,i0 ) (i,i0 ) (i0 ) (i) G̃kk0 x̃k0 + ν̃ k , (6.5) k0 =0 (i) (i) (i) (i) where G̃kk0 , FM Gkk0 FH M , ν̃ k , FM ν k , and x̃k , FM xk . We refer to the elements (i) of x̃k as the synthetic subcarriers. Collecting the samples of all the user terminals on a given DFT frequency bin m, where m ∈ {0, . . . , M − 1}, into a K × 1 vector, one can perform frequency domain equalization on the individual bins. To this end, we reorder the signal samples given in (6.5) as follows. (i) (i) Let the K × 1 vector sm contain the mth elements of the vectors x̃k , k = 0, . . . , K − 1. (i) (i) (i) Similarly, we form the K × 1 vectors rm and η m using the corresponding elements of ỹk (i) (i) (i) and ν̃ k , respectively. Accordingly, the relation between sm and rm can be expressed as (i) (i,i) (i) + rm = Rmm s | {z m} Desired Signal where M −1 X ξ (i) m = (i,i) | i0 =i−1 i0 6=i {z } Intrablock Interference | , (6.6) Noise + Interference i+1 N −1 X X (i) Rmm0 sm0 + m0 =0 m0 6=m ξ (i) m |{z} (i,i0 ) (i0 ) Rmm0 sm0 + η (i) m . (6.7) m0 =0 {z Interblock Interference } |{z} Noise (i,i0 ) In the above model, the element kk 0 of the K × K matrix Rmm0 is given by the element (i,i0 ) mm0 of the matrix G̃kk0 . As signified in (6.7), the interference can be categorized into intrablock interference and interblock interference. While the former results from the symbols transmitted in the current block i, the latter originates from blocks i − 1 and i + 1. (i) Based on (6.6), the MMSE estimate of sm can be obtained as (i,i) ŝ(i) m = Rmm (i) where Cov ξ m H (i,i) (i,i) Rmm Rmm H + Cov ξ (i) m −1 (i) rm , (i) represents the covariance matrix of ξ m and can be computed as M −1 i+1 M −1 X X X (i,i0 ) (i,i0 ) H (i,i) (i,i) H Rmm0 Rmm0 +σν2 IK . Cov ξ (i) = R R + 0 0 m mm mm m0 =0 m0 6=m i0 =i−1 m0 =0 i0 6=i (6.8) 86 As a result, we can simplify (6.8) according to (i,i) ŝ(i) m = Rmm where Λm = Pi+1 i0 =i−1 (i,i0 ) m0 =0 Rmm0 PM −1 H Λm + σν2 IK (i,i0 ) H Rmm0 −1 (i) rm , (6.9) . (i) One major problem associated with the calculation of the MMSE estimate ŝm according to (6.9) is the high computational cost due to matrix multiplications involved in computing Λm . In order to lower this computational burden, we note that when the number of BS antennas is much larger than the number users, that is, N K, the effects of intrablock and interblock interference become small [77]. This is due to the coherent processing using the time-reversal combining. Consequently, we can approximate the matrix Λm by (i,i) (i,i) H Λm ≈ Rmm Rmm . Based on our simulations, the performance loss resulting from this approximation is negligible when N K. As a result, we can express the proposed MMSE estimator with approximation as (i,i) ŝ(i) m ≈ Rmm H (i,i) (i,i) Rmm Rmm H + σν2 IK −1 (i) rm . (6.10) We refer to this equalizer as TR-FDE. In order to further simplify the processing, we calculate the elements of the matrix (i,i) (i,i) (i,i) (i,i) (i,i) Rmm . According to the expression G̃kk0 = FM Gk0 FH M , we have [G̃kk0 ]mm = [Rmm ]kk0 = (i,i) TG ∗ fm kk0 fm , where fm is the column m of the DFT matrix FM . Therefore, one can obtain (i,i) the element [Rmm ]kk0 as a linear combination of the samples of the time-reversal channel PL−1 (i,i) impulse response gkk0 [l], that is, [Rmm ]kk0 = l=1−L βl gkk0 [l], for some coefficients βl . 2πml −|l| −j M . e After some algebraic manipulations, the coefficients βl can be found as βl = MM P (i,i) (i,i) M −|l| −j 2πml M . Therefore, Rmm can be efficiently Hence, we have [Rmm ]kk0 = L−1 l=1−L M gkk0 [l]e computed from gkk0 [l] using the FFT algorithm. Also, when the channel length L is much (i,i) P −j 2πml M . smaller than the block length M , we have Rmm kk0 ≈ L−1 l=−L+1 gkk0 [l]e Before we end the discussion in this section, we note that the proposed FDE is performed after the time-reversal combining. Hence, its computational cost is independent of the number of BS antennas. This, clearly, is an important factor in massive MIMO applications, where N K. 87 6.4 Numerical Results In this section, we numerically evaluate the performance of our proposed TR-FDE scheme. We consider a single-cell scenario with K = 10 user terminals. We follow [77], and PL−1 −α ` k consider an exponential PDP pk [l] = e−αk l / , l = 0, . . . , L − 1 with decaying `=0 e factor αk = k/5 and channel length L = 4 is considered for terminal k ∈ {0, . . . , K − 1}. We compare the simulated sum rate (total throughput of the network in bits/sec/Hz) performance of the proposed approach with that of time-reversal maximum ratio combining (TR-MRC), [77], as a benchmark. The sum rate is obtained using simulations according to Rsum = K−1 X log2 1 + Pksig /Pkintf+noise , (6.11) k=0 where Pksig and Pkintf+noise are the average signal, and average interference-plus-noise power, respectively, corresponding to user k. The averaging is over 103 channel realizations. We also show the sum capacity of the cooperative frequency-selective MIMO channel as an upper bound on the sum rate [77]. Figure 6.2 shows the sum rate for different number of BS antennas, N . The noise level is chosen such that the input SNR is 0 dB. We have also shown the performance for various block lengths M = 32, 64, 128, 256, and 512. As the figure shows, the performance is significantly improved by adding FDE. As the block length M increases, the sum rate first improves and then almost saturates. This is due to the fact that as we increase the block length, the equalization resolution in the frequency domain increases. Assuming that the channel variation within a block is negligible, this can lead to a more effective compensation of the frequency selectivity of the channel. Moreover, as the subcarrier widths become smaller than the channel coherence bandwidth, higher resolution of the equalizer will not have any significant advantage. Figure 6.3 compares the sum rate as a function of input SNR level. Again, the effect of varying the block size M is shown. Here, the BS array size is N = 50. As the figure shows, the advantage of FDE becomes evident in moderate to high SNR conditions. This is due to the fact that TR-MRC is nearly optimum in low SNR conditions [77], where the effect of multiuser interference becomes negligible compared to additive noise. Another observation is that as the SNR decreases, the sum rate performance of TR-FDE becomes less dependent on the selection of block size. 88 100 90 80 70 60 50 40 30 20 10 0 200 400 600 800 1000 Figure 6.2. Simulated sum rate versus different number of BS antennas. Figure 6.4 presents the sum rate as a function of block length M . Here, the BS array size is N = 50. In this figure, we also have shown the performance of the full MMSE implementation given in (6.8). As the figure shows, the loss due to approximation in (6.10) is negligible. 6.5 Conclusion In this chapter, we considered the uplink of a SC-based massive MIMO system and proposed a simple FDE technique to enhance the sum rate performance of the network. The proposed method incorporates the time-reversal operation to coherently combine the received signals across BS antennas. Subsequently, the effective multiuser channel is efficiently compensated in the frequency domain. We referred to this structure as TR-FDE, and showed that it leads to substantial performance improvement in mid to high SNR conditions. 89 10 9 8 7 6 5 4 3 2 -20 -15 -10 -5 0 5 10 Figure 6.3. Simulated sum rate versus the input SNR level. 80 60 40 20 100 200 300 ‘ Figure 6.4. Simulated sum rate versus the block size, M . 400 500 CHAPTER 7 CONCLUSION AND FUTURE WORK 7.1 Conclusion Three advanced waveforms were studies in this dissertation in the context of massive MIMO. Each waveform can address a particular limitation of CP-OFDM. First, FBMC was considered to relieve the large spectral-leakage problem of OFDM. We showed that in FBMC, linear combining of the signals received at the BS antennas using the channel frequency coefficients leads to a residual interference that does not fade away even with an infinite number of BS antennas. As a result, the SINR is upper bounded by a certain deterministic value, and arbitrarily large SINR performances cannot be achieved as the number of BS antennas grows. To resolve this issue, we proposed an efficient equalization method to remove the above residual interference. As the second design, we considered removing the CP overhead from the OFDM waveform. It is desirable to eliminate the CP duration to achieve a higher energy and spectral efficiency. This, however, comes at the expense of ISI and ICI. Fortunately, as we showed, massive MIMO can average out the ISI and ICI introduced as a result of the CP removal. This is contingent upon using an appropriate receiver design. The approach proposed in this dissertation is based on time-reversal combining and a frequency-domain equalization. Finally, SC transmission was considered to relieve the high PAPR issue of multicarrier waveforms. Previous studies have shown that time-reversal can be incorporated in SC-based massive MIMO systems to achieve a near-optimum performance in low SNR regime. We showed that the performance can be significantly improved in mid to high SNR conditions through using a FDE technique. The proposed FDE is performed after the time-reversal combining and its complexity does not grow with the number of BS antennas. 91 7.2 Future Work In this section, we briefly highlight some possible future research ideas. 7.2.1 Optimum CP Length Design The design in Chapter 5 was based on a complete removal of the CP duration from the OFDM waveform. We showed that a higher spectral efficiency can be achieved as compared to the case where a full CP is included. It should be noted, however, that a complete removal of CP does not necessarily maximize the spectral efficiency. Instead, one can obtain the optimum CP length depending on the channel conditions and the system parameters, for example, the number of BS antennas. This optimization problem can be formulated as Lopt = arg max ` n M o log2 1 + SINReff (`) , M +` (7.1) where SINReff (`) denotes the effective SINR achieved using a given CP length `, and Lopt is the optimum CP length. While complete removal of CP maximizes the first term in (7.1), the logarithmic term can be maximized by including a full CP duration. A balance between these two extreme designs can be reached according to the above optimization framework. Study of this problem is suggested as a future work. 7.2.2 Cell-Free Massive MIMO Throughout this dissertation, we assumed a co-located BS antenna array that is sufficiently compact so that the channel responses corresponding to a particular user and different BS antennas are subject to the same channel PDP. It is worth mentioning that there exist another type of massive MIMO setup in which the elements of the BS array are distributed in a large area. This setup is called distributed or cell-free massive MIMO [106]. In this scenario, for a given user, channel responses corresponding to different BS antennas undergo different PDPs. This is a completely different problem than what we are considering in this dissertation, and remains as a future study. 7.2.3 Application to Underwater Acoustic Channels All underwater acoustic (UWA) channels share the following features [107]: (i) fast variation in time; equivalently, wide dispersion in frequency, and (ii) wide dispersion in time. Because of these two properties, UWA channels are said to be doubly dispersive. 92 Obviously, any design of a communication system for UWA channels should take the above features into account. For instance, if OFDM is adopted, a long symbol guard interval (cyclic prefix or zero padding) is required to take care of the extended impulse response of the UWA channels. But, with a long symbol guard interval, the duration of each OFDM symbol should also be made long (a few times longer than the guard interval length) to keep a good bandwidth efficiency. But, such a choice may contradict the fast variation of UWA channels in time which dictates the use of shorter OFDM symbols. To address the above problem, the FBMC waveform together with a large array of BS antennas can be used. Specifically, as suggested in Chapter 4, by increasing the number of BS antennas in a MIMO-FBMC system, the symbol duration may be decreased by almost an order of magnitude. 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