Analysis of advanced waveforms for 5G

The fifth generation of cellular networks is going to radically change the current wireless networks paradigm and provide novel services ranging from broadband data connection to machine to machine communications. Many different elements of the wireless networks have to be upgraded to fulfill the expectations and requirements of 5G. One of the challenging aspects of 5G networks design is finding a suitable waveform for its physical layer. The currently used waveform, Orthogonal Frequency Division Multiplexing (OFDM), has some serious limitations and fails to provide the expected performance of 5G. Therefore, there is active research to find a suitable waveform for 5G and beyond. In this dissertation, we study three major candidate waveforms of 5G; Generalized Frequency Division Multiplexing (GFDM), windowed circular filterbank multicarrier offset QAM (C-FBMC), and Orthogonal Time-Frequency Space (OTFS). We present GFDM and C-FBMC waveforms and describe novel efficient transceiver implementations where we compare their computational complexity and showthat the GFDMreceiver has a very high implementation complexity. Moreover, we compare Bit Error Rate (BER) performance of both waveforms and show that C-FBMC outperforms GFDM. Thus, we conclude C-FBMC outperforms GFDM by providing a lower complexity and superior BER performance. In addition, we derive equations that quantify out-of-band (OOB) emissions and multiuser interference (MUI) of the C-FBMC waveform when utilized in an asynchronous scenario. We show how different OOB suppression methods result in an improved MUI performance. We show that our analysis can also be applied to GFDM. We present a detailed information theoretic analysis of filterbank multicarrier offset QAM waveform (FBMC-OQAM) and show that the conventional FBMC-OQAM receivers incur a marginal performance loss when they discard imaginary interference of matched filter output. We attribute this marginal performance loss to the ramp-up and ramp-down portions of an FBMC-OQAM signal, noting that the circular version of FBMC-OQAM, that is, C-FBMC, does not suffer from this information loss. Furthermore, we present a capacity analysis of C-FBMC from a signal processing perspective and prove that regardless of statistics of wireless channel and additive noise, real and imaginary parts of the matched filter output of a C-FBMC receiver are related though an orthonormal transformation. Thus, they both carry the same information content and utilization of an imaginary part in the detection process cannot improve the receiver performance. Finally, we present a novel vectorized formulation for a MIMO OFDM-based OTFS setup that streamlines analysis of OTFS systems. We use this formulation to derive ergodic capacity of OFDM-based OTFS waveform and prove that both OFDM and OFDM-based OTFS have the same ergodic capacity under a time varying channel. iv

ANALYSIS OF ADVANCED WAVEFORMS FOR 5G by Ahmad RezazadehReyhani 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 Ahmad RezazadehReyhani 2018 All Rights Reserved The University of Utah Graduate School STATEMENT OF DISSERTATION APPROVAL The dissertation of Ahmad RezazadehReyhani has been approved by the following supervisory committee members: Behrouz Farhang-Boroujeny , Chair(s) 16 July 2018 Date Approved Tolga Tasdizen , Member 16 July 2018 Date Approved Rong Rong Chen , Member 16 July 2018 Date Approved John Belz , Member 16 July 2018 Date Approved Mingyue Ji , Member 16 July 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 The fifth generation of cellular networks is going to radically change the current wireless networks paradigm and provide novel services ranging from broadband data connection to machine to machine communications. Many different elements of the wireless networks have to be upgraded to fulfill the expectations and requirements of 5G. One of the challenging aspects of 5G networks design is finding a suitable waveform for its physical layer. The currently used waveform, Orthogonal Frequency Division Multiplexing (OFDM), has some serious limitations and fails to provide the expected performance of 5G. Therefore, there is active research to find a suitable waveform for 5G and beyond. In this dissertation, we study three major candidate waveforms of 5G; Generalized Frequency Division Multiplexing (GFDM), windowed circular filterbank multicarrier offset QAM (C-FBMC), and Orthogonal Time-Frequency Space (OTFS). We present GFDM and C-FBMC waveforms and describe novel efficient transceiver implementations where we compare their computational complexity and show that the GFDM receiver has a very high implementation complexity. Moreover, we compare Bit Error Rate (BER) performance of both waveforms and show that C-FBMC outperforms GFDM. Thus, we conclude C-FBMC outperforms GFDM by providing a lower complexity and superior BER performance. In addition, we derive equations that quantify out-of-band (OOB) emissions and multiuser interference (MUI) of the C-FBMC waveform when utilized in an asynchronous scenario. We show how different OOB suppression methods result in an improved MUI performance. We show that our analysis can also be applied to GFDM. We present a detailed information theoretic analysis of filterbank multicarrier offset QAM waveform (FBMC-OQAM) and show that the conventional FBMC-OQAM receivers incur a marginal performance loss when they discard imaginary interference of matched filter output. We attribute this marginal performance loss to the ramp-up and ramp-down portions of an FBMC-OQAM signal, noting that the circular version of FBMC-OQAM, that is, C-FBMC, does not suffer from this information loss. Furthermore, we present a capacity analysis of C-FBMC from a signal processing perspective and prove that regardless of statistics of wireless channel and additive noise, real and imaginary parts of the matched filter output of a C-FBMC receiver are related though an orthonormal transformation. Thus, they both carry the same information content and utilization of an imaginary part in the detection process cannot improve the receiver performance. Finally, we present a novel vectorized formulation for a MIMO OFDM-based OTFS setup that streamlines analysis of OTFS systems. We use this formulation to derive ergodic capacity of OFDM-based OTFS waveform and prove that both OFDM and OFDM-based OTFS have the same ergodic capacity under a time varying channel. iv To Bahar for her kindness, encouragement, endless support, and sacrifices during my study. CONTENTS ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi NOTATION AND SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii CHAPTERS 1. 2. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Candidate Waveforms for 5G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 OFDM-based waveforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1.1 Weighted Overlap and Add OFDM (WOLA-OFDM) . . . . . . . . . . . 1.1.1.2 Filtered-OFDM (F-OFDM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1.3 Universal Filtered MultiCarrier (UFMC) . . . . . . . . . . . . . . . . . . . . . . 1.1.1.4 N-Continuous OFDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 FBMC-based waveforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2.1 FBMC-OQAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2.2 FBMC-QAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2.3 Filtered Multi Tone (FMT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2.4 Lapped-OFDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Circularly pulse shaped waveforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3.1 Generalized Frequency Division Multiplexing (GFDM) . . . . . . . . . 1.1.3.2 Windowed CP-Circular OQAM (WCP-COQAM) . . . . . . . . . . . . . . 1.1.4 Orthogonal Time-Frequency Space (OTFS) . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Contributions and Organization of The Dissertation . . . . . . . . . . . . . . . . . . . . . 4 6 6 7 7 8 8 8 9 10 10 10 10 11 11 12 CIRCULARLY PULSE SHAPED WAVEFORMS . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.1 Generalized Frequency Division Multiplexing . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Circular Filterbank Multicarrier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Implementations and Complexity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 A new GFDM transmitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 C-FBMC transmitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 GFDM receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 C-FBMC receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 BER Performance Analysis of Circularly Waveforms . . . . . . . . . . . . . . . . . . . . 2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. 17 21 21 23 24 24 25 25 30 OUT-OF-BAND EMISSIONS AND MULTIUSER INTERFERENCE ANALYSIS 32 3.1 Alternate C-FBMC Transmitter Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Out-of-Band Emissions Analysis of C-FBMC . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Multiuser Interference Analysis of C-FBMC . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Fully synchronized . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Timing offset only . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Frequency offset only . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Extension to GFDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. CAPACITY ANALYSIS OF FILTERBANK MULTICARRIER . . . . . . . . . . . . . . . . 48 4.1 Filterbank Multicarrier-Offset QAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Synthesis filter bank and perfect reconstruction . . . . . . . . . . . . . . . . . . . . 4.1.2 Channel impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Channel equalization and data recovery . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4 Real-valued signal models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Capacity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Channel equalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Data recovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 C-FBMC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. C-FBMC Tranceiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Properties of A and G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relating zR and zI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proofs of Properties of A and G Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Properties of the prototype filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Proof of (5.13) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Proof of (5.15) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 61 63 64 64 66 67 67 CAPACITY ANALYSIS OF OTFS WAVEFORM . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 6.1 Orthogonal Time-Frequency Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 OFDM-based OTFS: Single Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Separable windows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Rectangular window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Frequency domain representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 OFDM-based OTFS: MIMO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Ergodic Capacity Analysis of MIMO OFDM-based OTFS . . . . . . . . . . . . . . . . 6.5 Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. 49 49 50 51 51 52 52 53 54 55 57 CAPACITY ANALYSIS OF CIRCULAR FILTERBANK MULTICARRIER . . . . . 58 5.1 5.2 5.3 5.4 6. 34 35 38 40 41 42 42 43 45 71 73 75 76 76 77 79 82 83 CONCLUSIONS AND FUTURE RESEARCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 7.1 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 7.1.1 Detector design for OTFS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 vii 7.1.2 OOB and MUI analysis of other waveforms . . . . . . . . . . . . . . . . . . . . . . . 86 7.1.3 OTFS waveform with different multicarrier . . . . . . . . . . . . . . . . . . . . . . . 87 APPENDICES A. KRONECKER PRODUCT PROPERTIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 B. ACRONYMS AND ABBREVIATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 viii LIST OF FIGURES 1.1 Time and frequency response of Rectangular window (red) and Raised Cosine window (blue). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1 Frequency Overlap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2 GFDM Transmitter block diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3 GFDM Receiver block diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.4 C-FBMC Transmitter block diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.5 C-FBMC Receiver block diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.6 BER performance for PHYDYAS prototype filter, 16-QAM . . . . . . . . . . . . . . . . . 27 2.7 Frequency response comparison of prototype filters . . . . . . . . . . . . . . . . . . . . . . 27 2.8 BER performance for M=5, and 4-QAM Modulation . . . . . . . . . . . . . . . . . . . . . . 28 2.9 BER performance for M=5, and 16-QAM Modulation . . . . . . . . . . . . . . . . . . . . . 28 2.10 BER performance for M=5, and 64-QAM Modulation . . . . . . . . . . . . . . . . . . . . . 29 2.11 BER performance for M=5, and 256-QAM Modulation . . . . . . . . . . . . . . . . . . . . 29 2.12 BER performance for M=6, and 16-QAM Modulation . . . . . . . . . . . . . . . . . . . . . 30 3.1 System block diagram for generation of the kth subcarrier signal of a circularly pulse-shaped waveform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.2 CP appended circular pulses and their respective amplitude responses in a C-FBMC data block and in a single subcarrier band. Here, a rectangular window is used to time limit the generated block. . . . . . . . . . . . . . . . . . . . . . . . . 37 3.3 CP appended circular pulses and their respective amplitude responses in a C-FBMC data block and in a single subcarrier band. Here, a raised-cosine window is used to time limit the generated block. . . . . . . . . . . . . . . . . . . . . . . . . 37 3.4 Transmitter window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.5 Interference gain of an asynchronous data symbol (shown by *) to other synchronous data symbols for C-FBMC. Rectangular v[n]. . . . . . . . . . . . . . . . . . . . . 44 3.6 Interference gain of an asynchronous data symbol (shown by *) to other synchronous data symbols for C-FBMC. Raised cosine v[n]. . . . . . . . . . . . . . . . . . . . 44 3.7 Interference gain of a fully synchronous data symbol (shown by *) to other synchronous data symbols for GFDM. Rectangular v[n]. . . . . . . . . . . . . . . . . . . . 46 3.8 Interference gain of an asynchronous data symbol (shown by *) to other synchronous data symbols for GFDM. Raised cosine v[n]. . . . . . . . . . . . . . . . . . . . . 46 4.1 Expected values of I (y; d) and I (d̂; d) for ZF and MMSE equalizers as a function of SNR and for three choices of (a) M = 1, (b) M = 3, and (c) M = 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.2 Expected value of information loss versus M at SNR = 10 dB. . . . . . . . . . . . . . . 56 6.1 OTFS transmitter and receiver structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 6.2 OTFS transmitter and receiver structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 x LIST OF TABLES 1.1 Comparisons between 4G and 5G performances . . . . . . . . . . . . . . . . . . . . . . . . . 3 NOTATION AND SYMBOLS Throughout this dissertation, matrices, vectors, and scalar quantities are denoted by boldface uppercase, boldface lowercase, and normal letters, respectively. A Matrix A a Vector a [A]i,j The element at the ith row and jth column of A A−1 Inverse of A I Identity matrix 0 Null vector F Discrete Fourier transform (DFT) matrix (·)T Transpose operation (·)H Conjugate transpose operation ∗ (·) Conjugate operation (·)R Real parts (·)I Imaginary parts ( a) b a modulo b operator √ −1 j ⋆ Convolution x [n ] Discrete time signal xn Discrete time signal xc Complex-valued signal ⊗ Kronecker product ℜ{·} Real part operator ℑ{·} Imaginary part operator |·| Determinant operator E [·] Expectation δ(·) Kronecker delta function Loewner matrix order M C Set of all complex-valued vectors with M rows M× N C Set of all complex-valued matrices with M rows and N columns N (0, I) Zero-mean Gaussian distribution CN (0, I) Zero-mean circularly symmetric complex Gaussian distribution I ( a; b) Mutual information between a and b ACKNOWLEDGEMENTS I would like to express my profound gratitude to my supervisor Professor Behrouz Farhang-Boroujeny for his patience, encouragement, and immense knowledge. His guidance helped me in all the time of research and writing of this dissertation. I could not have imagined having a better advisor and mentor for my PhD study. I am deeply appreciative to my PhD committee members Prof. Tolga Tasdizen, Prof. Rong Rong Chen, Prof. John Belz, and Prof. Mingyue Ji for their insightful comments and encouragement, which helped me to widen my research from various perspectives. I am also grateful to other members in our research team with whom I have interacted during my PhD who helped me at different points with helpful comments and scientific discussions. I would like to thank my parents and my wife for their continuous support throughout my study. They are the motivation and inspiration of my life. CHAPTER 1 INTRODUCTION Motorola made the first real mobile phone call on April 3, 1973 in New York City [1]. It was the beginning of the first generation (1G) of cellular networks that would provide analog voice calls. Ever since, wireless communications have steadily evolved, resulting in four generations of wireless networks with exponential performance improvement. Each generation is characterized by a new network service like digital voice call, short message texting, and mobile internet. Empowered by improved wireless connectivity combined with substantial processing power, smart phones and tablets revolutionized our societies to the extent that the use of smart devices and their applications are an indispensable part of our personal and professional lives. Nowadays, multitudes of data hungry applications are driving up the demands for a faster, safer, and smarter wireless networks. In future, an enormous number of devices, with different capabilities ranging from very simple sensors to smart devices, will be served by wireless networks. Long Term Evolution (LTE), the fourth generation (4G) of cellular networks, is the most recent wave of wireless network development that has been commercially deployed for about a decade. It was primarily designed for superfast mobile data connection providing voice call and broadband internet as its main network services. Several classes of data traffic with different quality of services (QoS) are provided to support diverse applications like video streaming, web browsing, Voice over IP (VoIP) calls, etc. Mobile phones are the main stream devices that the 4G networks have been designed to serve. Although the first releases of 4G LTE have already offered remarkable data rates on both uplink and downlink connections, the Third Generation Partnership Project (3GPP), the association that develops and maintains mobile network standards, continuously upgrades the LTE standard through new releases like LTE-Advanced and LTE-Advanced Pro, to satisfy ever increasing demands. 2 Mobile data consumption has been increasing for years and it is predicted that it will rise up to ten-fold of what it is today by the year 2022 [2]. While it has been possible so far to cope with this rising demand and industry is dealing with this through traffic optimization and offloading to the other networks like WiFi, these solutions cannot fulfill the future demands. Moreover, the future wireless networks have to serve user devices with a vast range of capabilities. However, the required services are not implemented nor predicted in design of the current generation. It is clear that different approaches and innovations are needed to create a fundamental improvement to some of the core wireless technologies. The fifth generation (5G), targeted to be deployed by 2020, aims to be the revolutionary generation. It not only serves the needs of a typical mobile consumer, but opens up a new prospect and enables an extremely vast diversity of applications within a unified single network. International Telecommunication Union (ITU) defines different classes of applications of 5G and their performance requirements under the International Mobile Telecommunications (IMT)-2020 standard [3] . The first category, Enhanced Mobile Broadband (eMBB) service, encompasses the applications that require an ultra high speed connection, for instance, to watch ultra high-definition videos (4K and 8K), wireless streaming of virtual or augmented reality applications, and data synchronization of cloud computing. The second group is called Massive Machine Type Communications (mMTC) that represents all Internet of Things (IoT)-related applications. The mMTC applications are characterized by a huge volume of connected devices, broad coverage, low energy consumption, and relatively slow connection speed. Finally, the Ultra-reliable and Low Latency Communications (uRLLC) category represents the mission-critical Machine to Machine (M2M) applications where high reliability and low latency are essential. The uRLLC applications aim to enable applications like Vehicle-to-Vehicle communications, intelligent transportation, industry automation, and telesurgery. ITU established eight key performance indicators (KPI) to specify, quantify, and measure the characteristics of IMT-2020 5G systems. Table 1.1 summarizes the target performances for 5G and those currently available with 4G. An overview of 5G requirements reveals a significant enhancement in different KPIs from 4G to 5G. To meet these performance goals, enormous research challenges have been placed in front of wireless 3 Table 1.1. Comparisons between 4G and 5G performances Key Performance Indicators Peak data rate (Gbit/s) User experience data rate (Mbit/s) Spectrum efficiency Speed (km/h) Latency (ms) Connection density (objects/km2 ) Network energy efficiency Area traffic capacity (Mbit/s/m2 ) 4G 1 10 1x 350 10 105 1x 0.1 5G 20 100 3x 500 1 106 100x 10 engineers, the very first of which is the largely increased diversity in the performance requirements of 5G. Traditionally, the transition from one generation to the next takes place gradually. It is expected that 4G LTE will continue to evolve alongside with the 5G development and these two standards will likely be very complementary initially. Thus, there is an active R&D to gradually improve performance of 4G LTE and some of the objectives set for 5G could be achieved thanks to technologies introduced in LTE-Advanced and LTE-Advanced Pro. Examples of those features that will help achieve the required performance improvements include Wider Radio Channels, advanced Multiple Input Multiple Output (MIMO) techniques, Carrier Aggregation, Device-to-Device Communication, self-backhauling, Licensed Assisted Access, Heterogeneous Network (Microcells/Picocells/Femtocells), and higher-order modulation schemes. Even though the techniques developed for 4G LTE/LTE-Advanced pave the path toward 5G, in order to provide the user experience expected of 5G networks, revolutionary technologies are needed to be introduced over and above evolved 4G features. Both academia and industry have directed a significant research effort towards the development of key technology enablers for 5G, which include new modulation and waveforms, millimeter wave utilization (mm-wave) [4, 5], massive MIMO [6], Multi-Radio Access Technology (Multi-RAT) [7], small cells, Software Defined Network (SDN), and Network Function Virtualization (NFV) [8]. From the physical layer (PHY) perspective, modulation and waveform design, new 4 coding schemes, antenna beam-forming, nonorthogonal multiple access schemes, massive MIMO, and higher frequency use and mm-wave are the most critical aspects that play a major role in determining the system throughput, complexity, and reliability. In this dissertation, we focus on the question of waveform design for 5G air interface. 1.1 Candidate Waveforms for 5G The high level performance goals of 5G impose a set of requirements on the underlying radio access technology (RAT). The first 5G use case, eMBB, requires 5G-RAT to support a very high rate of data connection. The inevitable solution is to increase the connection bandwidth through the use of wider channel bandwidth and/or carrier aggregation. Since wireless spectrum is a scarce and costly resource, improved spectral efficiency is a key requirement for 5G-RAT. To enable carrier aggregation, the underlying waveform should have low out-of-band (OOB) emissions so that the guard bands between different carriers can be minimal. Having a lower guard band ultimately increases the effective spectral efficiency too. Moreover, RAT should support MIMO techniques, which are powerful methods to pack further data into a given spectral resources. On the other hand, uRLLC demands a very low latency, which puts a constraint on the latency of the radio access technique, so that the lower latency required at higher layers can be attained. Moreover, mMTC applications have scenarios where the devices are not connected to the base station (BS) at all times. The power constraints of these devices prevent the devices from having full synchronization with the BS. Hence, the underlying waveform should be able to operate with loose or, more preferably, no synchronization, potentially leading to some interuser interference. Relaxed synchronization requirements also improve the resource utilization efficiency of the wireless networks because less resources are devoted to device synchronization. Orthogonal frequency division multiplexing (OFDM) is a powerful modulation technique that is adopted in 4G LTE [9, 10] as well as other broadband wired [11, 12] and wireless systems [9, 13]. OFDM succeeded code division multiple access (CDMA), employed in third generation (3G) networks. It has been adopted in the broad class of digital subscriber line (DSL) standards, as well as in the majority of wireless standards, for example variations of IEEE 802.11 and IEEE 802.16, 3GPP-LTE, and LTE-Advanced. With the use of cyclic 5 prefix (CP), multipath fading can be effectively tackled with a one tap frequency domain equalizer, as long as the duration of CP is longer than the time domain span of the channel impulse response. CP-OFDM offers several attractive properties such as simple channel estimation, low-complexity equalization, efficient hardware implementation thanks to the use of fast Fourier transform (FFT) and inverse FFT (IFFT) blocks, good bandwidth efficiency, orthogonality of subcarriers that eliminates intercarrier interference, possibility of adapting the transmit power and the modulation cardinality, and easy combination with MIMO transmission. OFDM is known to be a perfect choice for point-to-point and downlink communications. Despite all these advantages, OFDM suffers from a lot of disadvantages and its adoption in the forthcoming generation of wireless networks is not taken for granted. The spectral efficiency of OFDM is limited by the need of a CP and by its large OOB emissions, caused by its rectangular pulse shape, which require some null guard tones at the spectrum edges. For instance, in an LTE system operating at 10 MHz bandwidth, only 9 MHz of the band is used. In addition, the loss of the CP is around 7%, so the accumulated loss totals at 16%. Its large OOB emissions create some interference issues and, as a result, limit utilization of noncontiguous spectrum chunks. This can be a major drawback for future applications where the use of noncontiguous spectrum may be considered in applications in which very wide bandwidths are required. In addition, OFDM is highly sensitive to synchronization errors, specifically carrier frequency offset (CFO). In a multiuser uplink, where a combination of frequency division multiple access (FDMA) and OFDM is used, presence of multiple CFOs as well as timing offsets of the users can cause detrimental ramifications on the received signal at the base station (BS) [14]. In such scenarios, multiuser interference can increase up to a level that hampers communication. Hence, a very stringent synchronization is required. Such synchronization has been found very difficult to establish, especially in mobile environments where Doppler shifts of different users are hard to predict/track. This results in a large amount of network overhead, which is obviously undesirable in the context of 5G networks where low latency is a pivotal requirement. Finally, OFDM signals may exhibit large peak-to-average-power ratio values. These drawbacks, which invalidate many of the above-mentioned OFDM advantages, form the basis of an open and intense debate on what the modulation format and multiple 6 access strategy should be in next-generation cellular networks and several alternative candidates have been intensively studied in the literature in the past few years [15–20]. To overcome the shortcomings of OFDM, a new waveform should be considered for 5G. On one hand, the waveform candidate should inherit all the advantages of CP-OFDM. On the other hand, the new waveform should be able to provide the flexibility required for efficient support of diverse services and deployment scenarios. Furthermore, the new waveform needs to have sufficiently good spectrum confinement, yielding higher spectrum efficiency. In the journey towards finding an appropriate waveform for the PHY of 5G networks, a number of signaling methods are suggested [21–26]. These waveforms can be categorized in multiple ways. From an orthogonality perspective, they can be classified as complex orthogonal, real orthogonal, and nonorthogonal waveforms. On the other hand, based on type of filtering, we have linearly filtered and circularly filtered groups. In the following, we present a review of a few major candidate waveforms that have recently appeared in the literature. 1.1.1 OFDM-based waveforms 1.1.1.1 Weighted Overlap and Add OFDM (WOLA-OFDM) CP-OFDM uses a time domain rectangular window to truncate the duration of each OFDM symbol. The rectangular window translates to a sinc pulse, with the undesirable high OOB emissions, in the frequency domain. WOLA-OFDM resolves this problem by replacing the rectangular window (with sharp edges) with a window with smooth roll-offs at the beginning and the end [27]. Figure 1.1 visualizes the properties of the rectangular window and a window with smooth (raised cosine) edges in both time and frequency domain. Here, also, we have shown the duration of each OFDM symbol. A point to be noted here is that, while in the case of CP-OFDM (rectangular window), there is no overlap among successive OFDM symbols, in WOLA-OFDM, the window size is wider than each OFDM symbol and as a result, successive OFDM symbols overlap over the roll-off periods; hence, the name “WOLA”. It should be also noted that a window with smooth edges may be applied at the receiver to avoid signal leakage among different portions of the spectrum that may have been used with different unsynchronized users. 7 Raised Cosine Rectangular Amplitude 1 OFDM Symbol 0.5 CP Frequency Response (dB) 0 0 Cp T Time 0 Raised Cosine Rectangular -50 -100 -150 0 10 20 30 40 50 60 Subcarrier Index Figure 1.1. Time and frequency response of Rectangular window (red) and Raised Cosine window (blue). 1.1.1.2 Filtered-OFDM (F-OFDM) F-OFDM is an approach similar to WOLA-OFDM to reduce OOB emissions of OFDM [28]. At the transmission, the poor OOB emissions of CP-OFDM is improved by using a filter at the output of a CP-OFDM transmitter. At the reception, interference from time and frequency asynchronous adjacent users is removed by a similar filter at the input of receiver. Filtering operation adds a ramp-up and ramp-down to the CP-OFDM symbol and increases the effective symbol length. This incurs some spectral efficiency loss of the system. A practical approach to solve this issue is hard truncating at both burst edges with an appropriate length in order to reduce the burst size. This results in a slightly higher OOB emissions. Another solution is to allow some overlap between adjacent symbols. This incurs interblock interference. 1.1.1.3 Universal Filtered MultiCarrier (UFMC) UFMC [29–31], also referred to as UF-OFDM [32], takes a similar approach to that of F-OFDM. However, in order to prevent any intersymbol interference due to the filtering operation, CP is replaced by a zero prefix (ZP) [33]. The typical filter length of UFMC is less 8 than, but close to, the length of ZP. As a result, in presence of a channel, successive symbols may overlap, but such overlap is usually small and, thus, negligible. In the absence of a multipath channel, while subcarriers of each user are perfectly orthogonal, different users may experience a small amount of interference due to time/frequency offset. Due to the use of ZP, UFMC can deliver higher spectral efficiency compared to F-OFDM. This comes at the cost of some level of intersymbol interference. 1.1.1.4 N-Continuous OFDM N-continuous OFDM [34] takes a different approach to solve the OOB emissions problem of CP-OFDM. Time domain discontinuity between adjacent CP-OFDM symbols has a considerable contribution in high OOB emissions of the waveform. Thus, the idea is to build successive OFDM symbols that are continuous in time domain to improve poor OOB emissions. The construction of a transmit signal results in a signal whose first N derivatives are continuous. This is accomplished by adding a precoding matrix before the OFDM modulator. The main disadvantage of N-continuous OFDM is that in order to successfully recover the transmitted data, receivers have to know the precoding matrix so a side channel is needed in the system. 1.1.2 FBMC-based waveforms 1.1.2.1 FBMC-OQAM The key idea of FBMC is to extend the filter duration in time domain to multiple times of a symbol duration and design a filter that is well-localized in the frequency domain [35– 39]. In contrast to F-OFDM and UFMC that filter the entire band, a bank of well-designed filters, a synthesis filterbank, at the transmitter is used to confine energy of each subcarrier individually. In order to maintain Nyquist rate, adjacent symbols in time domain and adjacent subcarriers in the frequency domain overlap. To ensure orthogonality of adjacent symbols, real and imaginary components of complex data symbols are separated by half a symbol duration. At the same time, a phase shift of π/2 is applied to the adjacent subcarriers. This technique is known as offset-QAM (OQAM). At the receiver, the data recovery step includes the use of another bank of filters, which is called an analysis filterbank, and removal of π/2 phase shifts. The intrinsic/imaginary interference between symbols in the time-frequency grid is suppressed by discarding the imaginary part and simply taking the 9 real part at the receiver. Thus, unlike CP-OFDM, FBMC-OQAM is orthogonal in the real domain only, not in the complex field. FBMC-OQAM utilizes a prototype filter with an excellent frequency localization, which results in a very low OOB emissions. Thus, no more than one subcarrier is needed as a guard band between adjacent users. Another benefit of super low OOB emissions of FBMC-OQAM is very low sensitivity to interference of time and frequency asynchronous users. This makes FBMC-OQAM a suitable candidate for asynchronous scenarios like uplink or loosely synchronized mMTC transmission. In addition to these benefits, FBMCOQAM eliminates the need for CP, which results in improved spectral efficiency. On the other hand, the FBMC-OQAM transmission block has a ramp-up and rampdown that, by design, can extend to multiple symbol duration. Even though high spectral efficiency can be realized when the number of transmitted symbols is very large, this can be problematic for the scenarios with a short burst transmission like mMTC applications. Moreover, communication channels are inherently complex and the real orthogonal nature of FBMC-OQAM results in challenges in some aspects of communication systems like pilot design. A more important bottleneck of FBMC-OQAM is that applying MIMO techniques to FBMC-OQAM is not straightforward. 1.1.2.2 FBMC-QAM Despite excellent OOB emissions and sensitivity to asynchronous interference, disadvantages of FBMC-OQAM come from its real orthogonality. Thus, FBMC-QAM has been proposed to transmit QAM symbols using FBMC where spacing between QAM symbols is restored to symbol duration [40–42] . Thus, it becomes possible to integrate with MIMO techniques while enjoying a very low OOB emissions. However, as Balian-Low theorem asserts , if FBMC-QAM utilizes a well-designed prototype filter, it will become a nonorthogonal waveform. Nonorthogonality degrades bit error rate (BER) performance of the system and necessitates the use of more complex receivers. To solve this issue, FBMC-QAM applies different filterbanks to different subcarriers, for example even and odd subcarriers, where these filterbanks are optimized to suppress self-generated interference and improve system performance. However, FBMC-QAM suffers from nonnegligible residual interference. 10 1.1.2.3 Filtered Multi Tone (FMT) FMT is a variant of the FBMC technique that has been specifically developed for DSL applications [43, 44]. In contrast with FBMC-OQAM and FBMC-QAM, in FMT, adjacent subcarriers do not overlap. Thus, each subcarrier carries a single carrier transmission based on Nyquist ISI-free criteria. The use of nonoverlapping subcarriers degrades spectral efficiency of the waveform; however, FMT presents a very high resilience to time and frequency asynchronicity. As a result, it has become an interesting candidate for spontaneous mMTC communications. Recently, a waveform called cyclic block filtered multitone (CB-FMT) is proposed that replaces linear filtering of FMT with a circular one that makes the use of CP possible [22, 45, 46]. As a result, the techniques developed for CP-OFDM can be applied to CB-FMT too. 1.1.2.4 Lapped-OFDM Lapped-OFDM [47], despite its name, is a variant FBMC-OQAM waveform that is built based on Lapped Orthogonal Transforms. It can be thought of as an FBMC-OQAM waveform that utilizes a half-Sine prototype filter with overlapping factor equal to two. 1.1.3 Circularly pulse shaped waveforms 1.1.3.1 Generalized Frequency Division Multiplexing (GFDM) GFDM can be considered as a combination of ideas from CP-OFDM and FBMC-QAM. On one hand, GFDM generalizes CP-OFDM by replacing the rectangular pulse shape of CP-OFDM with a more proper filter and performing a per-subcarrier filtering. This results in an improved OOB emissions. On the other hand, GFDM can be thought of as a variant of FBMC-QAM that replaces linear convolution with a circular one and appends a CP to the transmit block. Thus, it eliminates ramp-up and ramp-down of the transmission block, which results in an improved spectral efficiency. GFDM is a block transmission technique in which data symbols are spread on a set of subcarriers and time slots. While CP-OFDM appends a CP to each symbol, GFDM only needs to append a single CP to a block of GFDM that consists of multiple consecutive symbols. This results in an improved spectral efficiency for GFDM with respect to OFDM. Since GFDM is built based on the similar principles to those with which CP-OFDM is built, which is the use of circular convolution and CP, techniques that are already developed for CP-OFDM can be readily applied to 11 GFDM. GFDM is a nonorthogonal waveform due to the fundamental limitation of Balian-Low theorem. As a result, self-generated interference of GFDM degrades its BER performance. Even though it is possible to use a prototype filter that lowers the interference, this choice comes at the cost of increased OOB emissions, which goes against the initial intent of using a well-localized prototype filter. Moreover, a more complex receiver is needed to mitigate the impact of self-generated interference. Application of GFDM in 5G setup is vastly studied in the literature [48–54]. GFDM is presented with more details in Chapter 2. 1.1.3.2 Windowed CP-Circular OQAM (WCP-COQAM) WCP-COQAM [55, 56], also know as C-FBMC-OQAM or , simply, C-FBMC, is proposed to solve limitations of GFDM. It restructures FBMC-OQAM to use the circular convolution and CP. This results in a block-wise real orthogonal waveform. Due to removal of ramp-up and ramp-down and the use of a single CP for each block, C-FBMC has an improved spectral efficiency compared to its linearly filtered counterpart. At the same time, while GFDM has to deal with BER performance versus the OOB emissions dilemma due to its nonorthogonal nature, C-FBMC simultaneously minimizes OOB emissions and improves BER performance. Moreover, C-FBMC achieves a superior BER performance with a much simpler receiver. Interestingly, in contrast to FBMC-OQAM, extension of C-FBMC to MIMO is found straightforward. Since C-FBMC is a block waveform, there is a discontinuity between adjacent blocks that increases the OOB emissions of C-FBMC despite the use of a sophisticated prototype filter. Windowing is proposed to smooth the block boundaries and lower the OOB emissions. More details on C-FBMC waveform are presented in Chapter 2. 1.1.4 Orthogonal Time-Frequency Space (OTFS) One of the main requirements of 5G is to provide a reliable data connection to the users with up to 500 km/h speed. Besides time dispersion of wireless multipath channels, relative motion of the user and BS results in frequency domain dispersion, which is known as the Doppler effect. Multicarrier waveforms lead to a poor performance in doubly dispersive channels because channel Doppler spread generates considerable intercarrier interference that degrades the system performance. OTFS is a novel signaling technique 12 that is capable of handling the doubly dispersive channels [57–59]. OTFS modulation is a generalized signaling framework where precoding and postprocessing units are added to the modulator and demodulator of a multicarrier waveform, allowing for taking advantage of full time and frequency diversity gain of doubly dispersive channels. In contrast to other multicarrier waveforms that transmit from time-frequency domain to time-frequency domain, OTFS operates on a set of data symbols in the Doppler-delay domain. This modulation has two stages. First, complex data symbols in the Dopplerdelay domain are converted to the time-frequency domain through an inverse symplectic finite Fourier transform. In the second stage, the resulting time-frequency samples are fed into a multicarrier modulator to form the time domain transmit signal. After passing through the time varying channel, estimates of data symbols in the time-frequency domain are generated by multicarrier demodulator. Finally, a symplectic finite Fourier transform maps those samples to the Doppler-delay domain. OTFS delivers an improved BER performance because it converts a time varying channel to a time-invariant one. The cost to pay is the need to utilize a more complex receiver to recover the transmitted data. Chapter 6 presents OTFS with more details. 1.2 Contributions and Organization of The Dissertation In this dissertation, we present a detailed analysis of a number of major candidate waveforms proposed for 5G networks. We focus our analysis on GFDM, C-FBMC, and OTFS. On one hand, these waveforms present a potential to deliver a considerable performance improvement. Therefore, detailed analyses are essential for their comprehension and improvement. On the other hand, they are different from their predecessor to the extent that the prior analysis does not reveal every aspect of these waveforms. This dissertation tries to shed some light on undiscovered aspects of these waveforms. First, in Chapter 2, we describe transceiver structures of GFDM and C-FBMC and present novel efficient implementations of transmitter and receiver for both schemes. We show that while the transmitter complexity for both schemes is almost the same, due to nonorthogonality of GFDM, the GFDM receiver has a higher computational complexity when compared to the C-FBMC receiver. We also study BER performance comparison of both schemes by numerical simulations and we show that C-FBMC has a superior BER 13 performance with respect to GFDM. These studies lead to the conclusion that C-FBMC is a superior modulation to GFDM. In Chapter 3, we analytically examine the OOB emissions of the C-FBMC waveform and show that OOB emissions of the C-FBMC signal is mainly generated by sharp transition of pulse shapes at the packet boundaries. Our analysis allows us to quantify the proposed methods for improving OOB emissions. In addition, we derive equations for interference between any pair of data symbols in a time/frequency asynchronous multiuser scenario. The results reveal that some data symbols are more prone to generate or receive asynchronous multiuser interference. Then, impact of windowing at transmitter and receiver as a method to decreases MUI are discussed. Finally, we numerically extend our results to GFDM and show that both waveforms follow a similar trend for OOB emissions and MUI. In Chapter 4, we study FBMC-OQAM transceiver from an information theoretic perspective. We derive equations that measure information content of signals at different stages of a typical FBMC-OQAM receiver. We prove that MMSE and ZF equalizers maintain information content of the received signal. Furthermore, we show that the data recovery process in FBMC-OQAM where the intrinsic interference part of the processed signal is ignored incurs some information loss. However, through numerical results, the amount of information loss is found to be very small and approaches a constant as packet length increases. Finally, we prove that the data recovery process in C-FBMC does not incur any information loss. Comparing the analytical results of FBMC-OQAM and C-FBMC reveals that the performance loss in the data recovery process of the former corresponds to the ramp-up and ramp-down parts of the corresponding waveform. Chapter 5 presents a thorough study of the matrices that characterize the signal processing steps in the C-FBMC. Our study reveals a number of interesting properties of these matrices whose application may prove useful in further study and development of the C-FBMC waveforms. One interesting result that we show is that the real and imaginary parts of the demodulated signal at the receiver output are related through an orthonormal transformation, then, from a pure signal processing perspective, we prove our finding on the capacity of C-FBMC waveform, which confirms our prior conclusion in Chapter 4. In Chapter 6, we turn our focus on the OTFS waveform where we present a novel 14 concise, vectorized input-output relationship for OTFS that is applicable to general timevarying channels with arbitrary Dopplers and windowing functions. We provide an accurate characterization of the ergodic capacity of OTFS. This study shows that both OFDM and OTFS achieve the same ergodic capacity despite great benefits of the latter in practical receiver design. Finally, we conclude the dissertation in Chapter 7 and discuss possible future works to extend the results of this dissertation. Results of our analysis on the candidate waveforms have been published in [60–64]. CHAPTER 2 CIRCULARLY PULSE SHAPED WAVEFORMS: OPTIONS AND COMPARSONS Presence of multipath in wireless communication channels results in intersymbol interference (ISI) and necessitates the uses of channel equalizers, which increases receiver complexity. Moreover, equalization may cause noise enhancement over the portion of the frequency band where the channel gain is small. Such noise enhancement may result in a significant performance degradation of the receiver. On the other hand, time duration of channel impulse response does not depend on transmission rate and as the transmission rate increases, a larger portion of the received signal is polluted by channel ISI. Multicarrier is a technique that combats channel ISI by multiplexing a high-rate stream of data symbols into K parallel substreams, each of a rate K times slower than the original stream. These substreams modulate a set of carriers and are combined to build the transmit signal. Due to lower symbol rate, each substream suffers less from channel ISI. At the receiver, the same set of carriers are used to demodulate each stream and demultiplexing demodulated streams results in original data stream. The wireless frequency spectrum is a scarce and expensive resource. In order to maximize spectral efficiency, a multicarrier technique has to pack as many substreams as possible in a given frequency bandwidth. In other words, spacing between carriers has to be shrunk. However, as carrier spacing becomes smaller, data symbols of one substream may leak to other substreams due to channel or transceiver impairments, like transmitter and receiver timing or frequency mismatch, or relative movement of transmitter and receiver, which cause Doppler effect, etc. Such leakage is called intercarrier interference (ICI). The main reason for this ICI leakage is that the energy of each substream spreads beyond its assigned subband, and thus interferes with other substreams. Filterbank multicarrier (FBMC) utilizes a bank of filters to confine energy of each sub- 16 stream to its subband to mitigate ICI. As a result, energy leakage of each substream is at most limited to only neighboring substreams and overall performance of the communication system improves. The set of filters that is used at the FBMC transmitter is called a synthesis filterbank. At the receiver, another set of filters, called an analysis filterbank, is used to extract the transmitted substreams. ICI level improvement of FBMC comes at a cost of extended transmission duration. When the transmit substream has a finite duration, filtering operation on each substream expands its time duration with filter transient response at the beginning and the end of the substream. This duration expansion results in spectral efficiency loss and its impact for a short-duration transmission becomes nonnegligible. Circularly pulse shaped Filterbank Multicarrier replaces linear filtering operation of FBMC with a circular filtering operation and as a result, eliminates transient response of synthesis and analysis filterbanks. Orthogonal Frequency Division Multiplexing (OFDM) is the simplest form of circular waveforms, which uses rectangular pulse as its synthesis and analysis filter. It is well known that rectangular pulse has a poor frequency domain energy confinement property due to its Sinc shaped frequency response. Generalized Frequency Division Multiplexing (GFDM) [21, 65, 66] and Circular Filterbank MulticarrierOffset QAM (C-FBMC) are two novel waveforms that utilize properly designed pulses to circularly filter each substream. GFDM has a number of appealing properties that has stirred a great amount of interest among researchers. As it is shown in [67], out-of-band emissions of GFDM can be greatly reduced compared to OFDM. This clearly removes the limitations of OFDM in carrier aggregation. It is also more bandwidth efficient than OFDM as only one CP is used for each data packet. Through filtering each subcarrier band with a well-designed prototype filter, GFDM gains resiliency against carrier frequency offset as only its adjacent subcarriers overlap in frequency domain and hence, the leakage among subcarriers is reduced. The price to pay for gaining all the aforementioned advantages is some bit error rate (BER) performance loss compared with OFDM when linear detectors like zero forcing (ZF) or minimum mean square error (MMSE) equalizers are used [68]. This performance loss is due to the fact that GFDM is a nonorthogonal waveform and thus suffers from self-generated intercarrier interference. In order to avoid this performance degradation, interference cancellation techniques have to be utilized that demand a large amount of 17 computational burden compared with the linear detectors [66]. To tackle the performance degradation of GFDM, some researchers have recently proposed a multicarrier technique that is inspired by GFDM, but is able to preserve the orthogonality among the subcarriers [23]. These systems that are called circular filter bank multicarrier make use of a filter bank multicarrier offset QAM technique to satisfy the orthogonality in real domain [36]. Hence, simple detectors with a very low complexity, such as matched filter (MF) detectors, lead to the same BER performance as that of OFDM. C-FBMC may be thought of as a modified version of GFDM. In the following sections of this chapter, we elaborate on GFDM and C-FBMC with more details and present transceiver structures of these waveforms. We also investigate GFDM and C-FBMC systems in terms of receiver complexity and BER performance. This study is important as it reveals whether GFDM or C-FBMC is superior from an implementation point of view. As mentioned earlier, GFDM can get close to OFDM in terms of BER performance only through utilization of nonlinear interference cancellation techniques. Hence, we only consider nonlinear GFDM receiver techniques in our study. Our investigation reveals that for small constellation sizes, both GFDM and C-FBMC systems can achieve the same BER performance as that of OFDM. However, as the constellation size increases, a gap between BER performance of GFDM and that of OFDM and C-FBMC appears. Based on our computational complexity analysis, interference cancellation GFDM receivers have a substantially higher complexity than C-FBMC receivers. Accordingly, we realize that both performance-wise and complexity-wise, C-FBMC is a superior choice to GFDM. 2.1 Generalized Frequency Division Multiplexing In GFDM, a block of data symbols dk,m , for 0 ≤ k ≤ K − 1 and 0 ≤ m ≤ M − 1, are jointly synthesized to form a signal vector x of length MK. The synthesis is done such that x is a single period of a periodic signal, xp , with the following properties. 1. For each choice of k, the data symbols dk,0 , dk,1 , · · · , dk,M−1 are mapped to a subband of the transmission bandwidth as indicated in Figure 2.1. 2. As demonstrated, the adjacent subbands overlap. Such overlap is necessary to ensure high efficiency of spectral use, as in OFDM. 18 K K K Figure 2.1. Frequency Overlap 3. The periodic signal xp , obviously, has a Fourier series expansion. Hence, it can be written as a summation of the respective spectral tones (the fundamental and harmonics). 4. When xp is passed through a channel, after a transient period, these spectral lines will be affected by the channel gains at the respective frequencies. Channel effect can thus be removed at the receiver by equalizing the received tones through a single tap (equal to the inverse of the channel gain) per tone. 5. To deal with the channel transient, a cyclic prefix (CP) is attached to x before transmission and is removed at the receiver input before any processing. This, clearly, follows the same principle as OFDM. Figure 2.2 presents a block diagram that shows the steps that one should follow to synthesize the signal vector x in a GFDM transmitter. The inputs to the synthesis filter bank in Figure 2.2 are the rows of the K × M data matrix    D=  d0,0 d1,0 .. . d0,1 d1,1 .. . ··· ··· .. . dK−1,0 dK−1,1 · · · d0,M−1 d1,M−1 .. . dK−1,M−1    .  (2.1) These are denoted by d0 through dK−1 . The K-fold up-samplers insert K − 1 zeros after each symbol. The up-sampled symbol sequences are circularly convolved with a prototype 19 Circ. Conv. d0 K gn 2π ej K k Circ. Conv. d1 K gn . . . . . . . . . x ej 2π ( K −1) k K Circ. Conv. d K −1 K gn Figure 2.2. GFDM Transmitter block diagram filter gn of length MK. The results are then modulated to the respective subbands and added together. The result is the desired synthesized vector x, which should be cyclic prefixed before transmission. Here, for simplicity of discussion, we assume an ideal channel. In that case, after removing the CP at the receiver, the received signal vector, y, will be equal to the transmit signal vector x. Figure 2.3 presents a block diagram of a GFDM receiver analyzes the received signal vector y = x to detect the transmitted data vectors d0 through dK−1 . For each subband, the received signal is demodulated to baseband, and then circularly convolved with the matched version of the prototype filter gn . In practice, where gn is real-valued and is symmetric with respect to its center, the matched version of gn is gn 20 Circ. Conv. _ d̂0i K gn ICI Generator d̂iK−−11 d̂1i −1 2π e− j K k Circ. Conv. gn _ y ICI Generator d̂1i K d̂0i −1 d̂2i −1 . . . e− j . . . . . . . . . 2π ( K −1) k K Circ. Conv. _ gn ICI Generator K d̂iK −1 i −1 d̂K −2 d̂0i −1 Figure 2.3. GFDM Receiver block diagram itself. The outputs of the matched filters, after K-fold decimation, give estimates of the data symbols. However, because of overlapping of adjacent symbols, such estimates suffer from ICI. To resolve this problem, the interference over each subband is calculated and removed based on the current tentative decisions of the symbols from the adjacent subbands. In Figure 2.3, ‘i’ indicates the iteration index. Note that the reconstruction of ICI at the kth branch of Figure 2.3 is based on the decisions d̂ik−1 and d̂ik−+11 . This detection process is run over a few iterations to converge. 21 2.2 Circular Filterbank Multicarrier In C-FBMC, the transmission is established through offset QAM (OQAM) modulation; for example, see [69] and [36]. In OQAM, the real and imaginary parts of each QAM symbol are separated and transmitted with a time offset of T/2, as two pulse amplitude modulated (PAM) symbols. In addition, a π/2 phase shift is added between the adjacent PAM symbols. This setup assures the orthogonality of the modulated PAM symbols, with the orthogonality defined according to a definition that is commonly referred to as real orthogonality; for instance, see [69]. The real orthogonality implies that, in the absence of noise, perfect recovery of data symbols is possible after taking the real part of the match filters outputs. In C-FBMC, after separating the real and imaginary parts of the data symbols dk,m = ak,m + jbk,m to the respective PAM symbols, we obtain the real-valued symbol matrix    D=  a0,0 a1,0 .. . b0,0 b1,0 .. . ··· ··· .. . aK−1,0 bK−1,0 · · · a0,M−1 a1,M−1 .. . b0,M−1 b1,M−1 .. . aK−1,M−1 bK−1,M−1    .  (2.2) The transmitter and receiver structure of C-FBMC are presented in Figure 2.4 and Figure 2.5, respectively. The input vectors d0 through dK−1 in Figure 2.4 are the rows of the matrix D in (2.2). The phase toggle block at the transmitter adds the required phase changes of π/2 to the adjacent symbols. The added phase shifts are removed at the receiver output. 2.3 Implementations and Complexity Analysis Direct hardware implementation of transmitter and receiver structures that are presented in Section 2.1 requires ( MK )2 complex multiplications and about the same number of additions. This obviously is not efficient for practical use cases where MK may be very large. Gaspar et al. [70] have presented efficient structures for both transmitter and receiver of GFDM. Here, we present a lower complexity receiver structure for GFDM, show how this new structure can be extended to C-FBMC, and discuss how the receiver structure presented in [70] can be extended to C-FBMC. 22 Circ. Conv. d0 K gn 2π ej K k Circ. Conv. d1 Phase Toggle . . . K gn . . . . . . x ej d K −1 2π ( K −1) k K Circ. Conv. K gn Figure 2.4. C-FBMC Transmitter block diagram Circ. Conv. gn K ℜ{·} d̂0 K ℜ{·} d̂1 2π Circ. Conv. gn . . . y e− j . . . Undo Phase Toggle e− j K k . . . . . . 2π ( K −1) k K Circ. Conv. gn Figure 2.5. C-FBMC Receiver block diagram K ℜ{·} d̂K −1 23 2.3.1 A new GFDM transmitter Following Figure 2.3, one finds that the elements of the signal vector x, { xl , 0 ≤ l ≤ MK − 1}, are given as xl = M −1 K −1 ∑ ∑ dk,m g(l−mK) m =0 k=0 Next, we rearrange (2.3) as xl = M −1 K −1 m =0 k=0 ∑ ∑ dk,m e j 2πkl K ! MK ej 2πkl K . g(l −mK) MK (2.3) (2.4) and note that the summation within the parenthesis on the right-hand side, as l varies from 0 to MK − 1, is M repetitions of the DFT of mth column of the data matrix D of (2.1). This in turn means the elements of the GFDM signal vector x can be generated as follows. 1. Find the DFT of the columns of D and repeat each result M times to construct a set of vectors of length MK. 2. Point-wise multiply the length MK vectors obtained in 1) by the circularly shifted versions of the prototype filter coefficients signified by g(l −mK) MK and add the results. This implementation has a complexity of M FFTs of size K plus M point-wise vector multiplications and adding up the results, as noted in point 2, above. Noting that gn is real-valued and counting each real value by complex multiplication as one half of a complex multiplication, one finds the following complexity number for the proposed GFDM transmitter C1 = MK (log2 K + M ) operations. 2 (2.5) Here, the unit of operations refers to one complex multiplication and addition. Taking the formula (7) of [70] and assuming that the number of active subcarriers is equal to K to match our assumption here, the reported result in [70] will be1 C2 = 3MK log2 MK + 2MK operations. 2 (2.6) A quick comparison of (2.5) and (2.6) reveals that, for typical situations where M ≥ 4, the implementation proposed here is three times or more less complex than that of [70]. 1 In [70], the number of operations to perform an FFT of size K is assumed to be K log K. Here, we have 2 replaced it with the more accurate figure (K/2) log2 K [71]. 24 2.3.2 C-FBMC transmitter The GFDM transmitter that was developed above can also be extended to C-FBMC with a simple modification. Here, the data matrix D to be considered has double the number of columns (compare (2.1) and (2.2)). This may immediately imply that the C-FBMC transmitter has a complexity that is twice that of GFDM. However, if one takes into account the fact that the elements of D in (2.2) are real-valued, while the elements of D in (2.1) are complex-valued, and FFT of two real-valued vectors can be combined together to reduce the complexity by one half, he will find that a C-FBMC transmitter can be implemented with a complexity that remains about the same as that of GFDM. 2.3.3 GFDM receiver Low complexity GFDM receiver structures with and without interference cancellation are proposed in [70] and [72], respectively. As mentioned earlier, linear GFDM receivers such as the one proposed in [72] cannot get close to or achieve the same BER performance as that of OFDM for any constellation size. Thus, in this chapter, we only consider nonlinear GFDM receivers based on interference cancellation such as the one proposed in [70]. The low complexity GFDM receiver structure of [70] before successive interference cancellation (SIC) can be explained as follows. First, the received GFDM signal which has MK number of samples is transformed to frequency domain using a fast Fourier transform (FFT) block of the size MK. In the next step, the corresponding frequency domain samples for each subband2 are selected and multiplied to the frequency domain response of the prototype filter, gn . The result is then aliased and an inverse FFT (IFFT) of the size M is applied to the resulting signal in order to recover M symbols that are transmitted over each subband. This procedure is equivalent to matched filtering. Accordingly, matched filtering demands an FFT of the size MK, M complex multiplications of the frequency domain samples for each subband to those of the prototype filter. In addition to matched filtering, the GFDM receiver includes successive interference cancellation, which needs 2MK log2 M + MK complex multiplications for each SIC iteration [70]. Putting all together, the GFDM receiver complexity is 2 We consider K total number of subbands or subcarriers in our analysis. 25 C3 = MK log2 MK + MK + MK log2 M + J (2MK log2 M + MK ) operations (2.7) where J is the number of iterations. 2.3.4 C-FBMC receiver According to the similarities between GFDM and C-FBMC, almost the same receiver structure to that of the matched filter GFDM receiver with small modifications can be used for reconstruction of transmitted symbols in C-FBMC [73]. Hence, the C-FBMC receiver includes an FFT of size MK, 2M number of complex multiplications per subband, which includes multiplications of frequency domain samples of the prototype filter to the frequency domain signal samples in each subcarrier position. Ultimately, operation of an IFFT of size 2M per subband recovers the transmitted real data symbols. Consequently, the C-FBMC receiver has the total computational load of C4 = MK log2 MK + 2MK + MK log2 2M operations. 2 (2.8) Based on the above discussion, the GFDM receiver with only one SIC iteration is more complex than the C-FBMC receiver, for M > 4. However, according to our numerical simulation results that are presented in Section 2.4, for 16-QAM and larger constellation sizes, the GFDM receiver has to perform at least 3 iterations to completely remove ICI and reach the same BER performance as that of OFDM. Hence, the GFDM receiver is always more complex than its C-FBMC counterpart, and this complexity difference increases when data symbols with larger constellation sizes are transmitted. 2.4 Analysis of Bit Error Rate Performance of Circularly Pulse Shaped Waveforms In this section, we evaluate the BER performance of different receivers for GFDM and C-FBMC through numerical simulations. It is worth noting that the BER performance of OFDM is chosen as a reference. Simulated receivers include the GFDM matched filter (MF) receiver, the GFDM matched filter receiver with 1 iteration of successive interference cancellation (GFDM-SIC-1), the GFDM matched filter receiver with 3 iterations of successive interference cancellation (GFDM-SIC-3), and the C-FBMC matched filter receiver (C-FBMC-MF). Each point of the following BER performance plots is based on simulation of sufficient data, which results in 1000 error bits. 26 For C-FBMC modulation, adjacent subbands are real orthogonal. Thus, it is desirable to choose a prototype filter that fully extends up to the center of the adjacent subbands. This choice minimizes stop band response of the prototype filter, which is the key factor to decrease interference from far subbands. We have chosen the prototype filter based on the method presented in [74], which is widely known as PHYDYAS design. A point to make, here, is that choosing a prototype filter for GFDM with the same frequency span as C-FBMC causes a very high level of ICI among adjacent subbands. Figure 2.6 presents BER performance of all receivers for the prototype filter of [74] and 16-QAM modulation. According to Figure 2.6, one notices that while C-FBMC-MF has the same BER performance as that of OFDM, all GFDM receivers have very high error levels. Based on a GFDM-SIC-3 curve where 3 iterations were involved in the SIC procedure, the BER curve of GFDM cannot get even close to that of OFDM. Increasing the constellation size makes SIC inefficient and a residual ICI remains. As a result, the BER will be higher. Therefore, the prototype filter for GFDM modulation is chosen such that each subband has a small overlap with the adjacent subbands. The consequence of this selection is sacrificing good stop band response of the prototype filter, which is presented in Figure 2.7. Figure 2.8 through Figure 2.11 present BER performance of all receivers for different constellation sizes. In these simulations, K = 64 subcarriers and an overlapping factor of M = 5 are considered. A root raised cosine (RRC) prototype filter with the roll-off factor α = 0.3 is utilized for GFDM and the prototype filter of [74] is employed for C-FBMC. According to Figure 2.8 through Figure 2.11, we conclude that the C-FBMC-MF receiver has the same performance as that of the OFDM receiver, while the GFDM-SIC receivers can approach C-FBMC-MF performance by incorporating a higher number of SIC iterations that comes at a cost of increasing the complexity of the receiver. The above conclusion is the case only for odd numbers of overlapping factor M. For even numbers of M, BER is higher than OFDM and the SIC algorithm is not effective as for odd value of M. Figure 2.12 presents the receivers performance for M = 6, and according to the results, the GFDM-SIC receiver is not able to reach an acceptable BER performance, while C-FBMC-MF with lower computational complexity has the performance similar to that of OFDM. [70] and [68] reported that for even values of M, the modulation matrix A of ZF and MMSE receivers are ill-conditioned. However, no analysis on the BER performance 27 10 10 BER 10 10 10 10 10 0 −1 −2 −3 −4 OFDM GFDM−MF GFDM−SIC−1 GFDM−SIC−3 C−FBMC−MF −5 −6 0 5 10 15 E /N b 20 25 30 0 Figure 2.6. BER performance for PHYDYAS prototype filter, 16-QAM 0 Frequency Response (dB) RRC filter (α = 0.3) Full span prototype filter −50 −100 −150 0 0.2 0.4 ω/π 0.6 Figure 2.7. Frequency response comparison of prototype filters 0.8 1 28 10 10 BER 10 10 10 10 10 0 OFDM GFDM−MF GFDM−SIC−1 GFDM−SIC−3 C−FBMC−MF −1 −2 −3 −4 −5 −6 0 5 10 15 E /N b 20 25 30 25 30 0 Figure 2.8. BER performance for M=5, and 4-QAM Modulation 10 10 BER 10 10 10 10 10 0 −1 −2 −3 −4 OFDM GFDM−MF GFDM−SIC−1 GFDM−SIC−3 C−FBMC−MF −5 −6 0 5 10 15 Eb/N0 20 Figure 2.9. BER performance for M=5, and 16-QAM Modulation 29 10 10 BER 10 10 10 10 10 0 −1 −2 −3 −4 OFDM GFDM−MF GFDM−SIC−1 GFDM−SIC−3 C−FBMC−MF −5 −6 0 5 10 15 E /N b 20 25 30 25 30 0 Figure 2.10. BER performance for M=5, and 64-QAM Modulation 10 10 BER 10 10 10 10 10 0 −1 −2 −3 −4 OFDM GFDM−MF GFDM−SIC−1 GFDM−SIC−3 C−FBMC−MF −5 −6 0 5 10 15 Eb/N0 20 Figure 2.11. BER performance for M=5, and 256-QAM Modulation 30 10 10 BER 10 10 10 10 10 0 −1 −2 −3 −4 OFDM GFDM−MF GFDM−SIC−1 GFDM−SIC−3 C−FBMC−MF −5 −6 0 5 10 15 E /N b 20 25 30 0 Figure 2.12. BER performance for M=6, and 16-QAM Modulation of GFDM-SIC receiver is reported in the literature while even values are reported for the SIC receiver in the implementations that are presented in [66]. 2.5 Summary Generalized frequency division multiplexing (GFDM) and circular filter bank multicarrier (C-FBMC) that are two candidate waveforms for the fifth generation of wireless communication networks (5G) were compared and analyzed in this chapter. We discussed efficient implementations for the transmitter of both systems. We showed that these structures lead to about the same complexity for both GFDM and C-FBMC. Nonorthogonality of the GFDM signal results in a large amount of intercarrier interference (ICI) that needs to be compensated through successive interference cancellation techniques, demanding a great amount of computational burden. In contrast, C-FBMC preserves the orthogonality through a simple modification to GFDM structure. Hence, a simple matched filter (MF) receiver leads to a bit error rate (BER) performance that is the same as that of orthogonal frequency division multiplexing (OFDM). Our BER performance analysis revealed that GFDM has a serious limitation with respect to the number of symbols in each data block. 31 It does not allow having an even number of symbols in each data block. C-FBMC, on the other hand, does not suffer from this limitation and works for any arbitrary number of symbols in each data block. Moreover, we showed that for large constellation sizes, GFDM suffers from a BER performance loss compared with OFDM while C-FBMC keeps the same performance. In addition, according to our complexity analysis, C-FBMC was shown to be simpler than GFDM. We thus conclude C-FBMC is superior to GFDM both from practical implementation and performance viewpoints. C-FBMC may be thought as a modified GFDM, when QAM symbols are replaced by offset-QAM (OQAM) ones. This replacement leads to an orthogonal modulation and, as a result, resolves the limitations of GFDM all originating from its nonorthogonal design. CHAPTER 3 OUT-OF-BAND EMISSIONS AND MULTIUSER INTERFERENCE OF CIRCULARLY PULSE SHAPED WAVEFORMS Generalized frequency division multiplexing (GFDM), and circular filter bank multicarrier (C-FBMC), are two recent candidate waveforms that have been proposed as possible replacements to orthogonal frequency division multiplexing (OFDM) in multiuser environments. OFDM is highly sensitive to time/frequency mismatch and requires strict synchronization. In addition, OFDM generates high out-of-band (OOB) emissions, which makes carrier aggregation difficult. Both GFDM and C-FBMC are proposed to resolve limitations of OFDM and they operate based on the same principle and closely follow that of OFDM. In particular, a cyclic prefix (CP) is used to absorb channel transient. However, instead of adding a CP to each OFDM symbol, consisting of a cluster of data symbols that are distributed over K subcarriers, a single CP is added to a full packet consisting of MK data symbols that are spanned over K subcarriers and M instants in time. Also, each subcarrier sequence is filtered to confine its respective signal to a limited bandwidth. Moreover, to allow the use of CP, the filtering operations are done through a set of circular convolutions. Subcarrier filtering results in a lower OOB emissions in GFDM/C-FBMC when compared to OFDM. However, due to block structure of transmit signal, their OOB emissions may still remain nonnegligible. In [65], [49], and [66], it has been recognized that the symbols at packet boundaries are a major contributor to high OOB emissions. Therefore, they suggested the use of one or more zero-valued guard symbols/subcarriers at GFDM packet time/frequency boundaries to decrease OOB emissions. Another method is to use some virtual carriers at edge frequencies with values that effectively cancel OOB emissions of the GFDM/C-FBMC signals [67]. Recently, linear FBMC with periodically extended input 33 data symbols is used to build a C-FBMC waveform with smooth boundary transitions, which results in very low OOB emissions [75, 76]. These methods clearly lead to some loss of spectral resources and thus make GFDM/C-FBMC inefficient. This goes against the initial intent that the designers of GFDM and C-FBMC had [21, 23]. On the other hand, [49] and [77] improved OOB emissions by using sinc function or a short pulse as the transmitter prototype filter. This method has some side effects, for instance sensitivity to timing offset. Others have proposed the use of windowing methods to reduce OOB emissions, as in windowed OFDM [36]. Sample publications that elaborate on this approach are [66], [23], [73], [78], and [79]. The subject of multiuser interference (MUI) has recently been brought up in [80] and [81]. MUI decreases effective signal to interference and noise ratio (SINR) of the received data symbols, which results in system level bit error rate (BER) performance loss. The authors of [80] study a GFDM setup in the uplink of a wireless sensor network (WSN), and [81] compare the MUI for a few 5G candidate waveforms. The conclusions drawn in all the above papers are mostly based on intuitions and computer simulations. In this chapter, we introduce a novel analysis that allows us to get deeper into the details of the sources that contribute towards OOB emissions and MUI in circularly pulseshaped waveforms, specifically, GFDM and C-FBMC. We present a mathematical framework that facilitates our analysis. Our analysis quantifies some of the suggestions made in the previous publications for reducing OOB emissions and/or reducing MUI. In addition, we borrow some ideas from the digital subscriber lines literature [82] to reduce MUI in GFDM/C-FBMC. We note that both GFDM and C-FBMC operate based on the same principle. Filtering in each subcarrier band is performed through a circular convolution to allow the use of a CP for combating channel frequency selectivity. However, while GFDM is a nonorthogonal modulation, C-FBMC can be classified as a orthogonal one (though, in the real domain). The nonorthogonality of GFDM adds some complexities when it comes to its study. Such complexities do not exist in C-FBMC [60]. Noting this, in the rest of this chapter, we limit our study to C-FBMC, but we note that all of our findings are extendable to GFDM with some minor changes. To back up this statement, we present a numerical example of GFDM OOB emissions along with its C-FBMC counterpart. 34 3.1 Alternate C-FBMC Transmitter Formulation A C-FBMC waveform, which is a single packet carrying a block of data symbols, is constructed as follows. A set of 2MK real-valued data symbols, arranged in an K × 2M matrix D, are built into a waveform for transmission. The alternate elements in a row of D are in-phase and quadrature parts of QAM symbols, following the offset QAM principles [36]. The elements in each row of D are transmitted through a subcarrier band. The filtering operation that is used to limit the spectrum of each subcarrier band is performed through a circular convolution [21, 23]. This allows the use of a CP to absorb the channel transient, as done in OFDM. Here, we present a novel formulation of the waveform construction for C-FBMC that facilitates our analysis in the rest of the chapter. This presentation is not intended to introduce any efficient implementation and thus, its more complex structure than those reported in the literature [73, 83] should not be a concern. The goal here is to only dig into the details of OOB emissions and MUI and suggest methods of reducing them. To present our formulation, we first concentrate on the contribution of the kth row of D to the generated C-FBMC signal, x[n]. Figure 3.1 presents the steps that may be taken to generate the contribution of the kth row of D. The result is the output signal xk [n]. The C-FBMC signal, x[n], is obtained by adding the contributions from all the rows of D. In Figure 3.1, the input signal dk [n] is defined as 2M dk [n ] = ∑ j k+ m m =1 K dk,m δ n − m 2 (3.1) where dk,m , for m = 1, 2, · · · , 2M, are the elements of the kth row of D, and the additional factor jk+m is to introduce a phase shift of 90◦ among adjacent symbols, following the OQAM multicarrier modulation [36]. The impulse response h[n] is a periodic signal that is constructed by periodically repeating the impulse response of the prototype filter g[n] of C-FBMC waveform. It is assumed that g[n] has a length of MK samples and this is also equal to the length of a period of h[n]. Thus, g[n] and h[n] are related as ∞ h[n ] = ∑ i=− ∞ g[n − iMK ]. (3.2) The modulator in Figure 3.1 shifts the generated baseband signal to the respective subcarrier band. It is worth noting that the generated signal at the modulator output, by 35 ej w [n ] 2πk K n dk [n ] xk [n ] h[n ] modulation windowing Figure 3.1. System block diagram for generation of the kth subcarrier signal of a circularly pulse-shaped waveform. construction, is periodic and has a period of MK samples. The windowing block truncates one period of this periodic signal plus an additional segment prior to it for the CP part of the packet. Roll-offs may be also added to both sides of w[n] to improve on the OOB emissions of the generated C-FBMC signal. 3.2 Out-of-Band Emissions Analysis of C-FBMC This section is devoted to a study of energy spectral density (ESD) of a C-FBMC signal at the transmitter output. Before we dive into detailed equations, we note that the ESD of x[n] can be written as the summation of contributions from the individual elements dk,m of D, assuming the data symbols dk,m are independent of one another. We also note the contributions of the elements dk,m to the ESD of x[n] differ, depending on their position in the matrix D. Noting these, in the rest of this section, we concentrate on derivation of the ESD of the signal (3.3) Ek,m (ω ) = | Xk,m (ω )|2 (3.4) xk,m [n] = j k+ m K j 2πk n dk,m h n − m e K w [ n ]. 2 We note that the ESD of xk,m [n] is given by where Xk,m (ω ) is the discrete-time Fourier transform (DTFT) of xk,m [n]. To facilitate our derivations as well as interpretation of the developed results, we assume that the prototype filter g[n] is the one proposed by Martin and Mirabbasi [84] 36 and [74]. This design has been widely accepted in the FBMC community and is often referred to as PHYDYAS filter [85]. A PHYDYAS filter g[n] with the length of MK, which is designed to be a square-root Nyquist (N) with overlapping factor M, has the Fourier series expansion M −1 2π cr e j MK rn . ∑ g[n ] = (3.5) r =−( M −1) Also, the coefficients cr are real-valued and even-symmetric around r = 0, that is, cr = c−r . With the above choice of g[n], the DTFT of the periodic signal h[n] is obtained as H (ω ) = 2πr . c δ ω − ∑ r MK r =−( M −1) M −1 (3.6) Next, we note that (3.3) implies 2πk mK 2πk ) ⋆ W (ω ) Xk,m (ω ) = jk+m dk,m e− j( ω − K ) 2 H (ω − K (3.7) where W (ω ) is the DTFT of w[n]. Substituting (3.6) in (3.7) and the result in (3.4), we obtain Ek,m (ω ) = σd2 M −1 ∑ r =−( M −1) cr e − jπ mr M W 2π (r + kM ) ω− MK 2 (3.8) where σd2 = E [|dk,m |2 ], and E [·] denotes expectation. Figure 3.2 presents the CP appended circular pulse shapes for all values of the parameter m in a C-FBMC data block and their respective amplitude responses in a single subcarrier. Here, there are K = 16 subcarriers and there are 2M = 6 real symbols across time. The length of CP is equal to 10 and the generated data block is windowed using a rectangular window. The rectangular window is the dashed-line red plot. The selected subcarrier is the subcarrier number 2. The results show that while the pulse shapes near the center of the block have a well-contained spectrum within the respective subcarrier band, the pulse shapes that are closer to the edges of the block have a significant OOB emissions. Such OOB emissions are clearly due to the sharp transitions at the beginning and the end of the block. One may also note that such OOB emissions can be reduced significantly by extending the length of the block and introducing a roll-off to its edges as demonstrated in Figure 3.3. This solution, which follows the approach of windowed OFDM [36], has also been mentioned in [23, 66, 73, 78, 79]. Others have suggested nulling 37 1.2 1 0.8 0.6 0.4 0.2 0 -0.2 -20 -10 0 10 20 30 40 10 12 50 60 Time (sample) 0 -20 -40 -60 -80 -100 0 2 4 6 8 14 16 Subcarrier index Figure 3.2. CP appended circular pulses and their respective amplitude responses in a C-FBMC data block and in a single subcarrier band. Here, a rectangular window is used to time limit the generated block. 1.2 1 0.8 0.6 0.4 0.2 0 -0.2 -20 -10 0 10 20 30 40 10 12 50 60 Time (sample) 0 -20 -40 -60 -80 -100 0 2 4 6 8 14 16 Subcarrier index Figure 3.3. CP appended circular pulses and their respective amplitude responses in a C-FBMC data block and in a single subcarrier band. Here, a raised-cosine window is used to time limit the generated block. 38 the symbols that are near the edges of the block [49, 65–67]. This latter solution is not a desirable one, as it incurs a significant loss in spectral efficiency. It is also instructive to pay some attention to the mathematical details in (3.8) and relate our observation there to the energy spectral densities in Figure 3.2 and Figure 3.3. When w[n] is a rectangular window (Figure 3.2), W (ω ) is a sinc-like function of ω, clearly, with side-lobes that are relatively large. For m = M, the choice of the coefficients cr has been made such that the side-lobes of the terms under the summation in (3.8) cancel each other, and thus, lead to a minimal OOB emissions. mr As m deviates from M, the added phase rotations in the coefficients cr e− jπ M in (3.8) introduce some imbalance in the said side-lobes cancellation and as a result, the OBB emissions deteriorate. On the other hand, the introduction of roll-offs to the ends of the window function w[n] reduces the side-lobes of W (ω ) and as a result, OBB emissions will have less dependency on the parameter m, that is, the symbol position along the time. 3.3 Multiuser Interference Analysis of C-FBMC In the up-link of a multiuser network, signals from different users may reach the base station out of sync. In this section, we quantify MUI by looking at interference between an arbitrary pair of data symbols dk,m and d p,l that are transmitted asynchronously from two mobile terminals. We consider the case where the receiver is synchronized to detect d p,l and study the interference introduced by dk,m . We assume the signal that carries dk,m is received with a time offset of ∆n and a frequency offset of ∆ f . The associated received signal is thus given by y[n] = xk,m [n − ∆n]e j2π∆ f n . (3.9) The receiver selects a block of the received signal with a window v[n], and demodulates the pth subcarrier to the baseband. Accordingly, the portion of the demodulated signal that arises from dk,m is obtained as z[n ] = y[n ]v[n ]e− j 2π K pn . (3.10) k,m Next, we define I p,l as the leakage gain between the transmit data symbol dk,m and the receiver output that delivers an estimate of d p,l . This leakage gain can be calculated by 39 using the following formula. k,m I p,l = = 1 dk,m ( ℜ j−( p+l ) z[n] ⋆ h[n] n = l K2 ) Z 2π K 1 1 ℜ j−( p+l ) Z (ω ) H (ω )e jωl 2 dω dk,m 2π (3.11) 0 where Z (ω ) is the DTFT of z[n]. Substituting (3.6) in (3.11) and rearranging the results, we obtain k,m I p,l = 1 dk,m ( ℜ j M −1 −( p+ l ) ∑ cr Z r =−( M −1) 2πr MK e j πlr M ) . (3.12) k,m To develop some insight into the values of I p,l under different conditions, we substitute (3.3) in (3.9) and the result in (3.10) to obtain K − j 2πk ∆n j2π k −K p +∆ f n k+ m K u [n ] e e z[n ] = j dk,m h n − ∆n − m 2 (3.13) where u[n] is the combined window function u[n] = w[n − ∆n]v[n]. Applying DTFT to both sides of this result, we arrive at the result that is 2π R(ω ) = dk,m jk+m e− j MK k∆n M −1 2πr ′ ∑ r ′ =−( M −1) cr ′ e− j MK (∆n+ mK 2 ) U ω − 2π (k − p) + 2π∆ f − 2π r′ . (3.14) K MK where U (·) is the DTFT of the combined window function u[n]. Also, substituting (3.14) in (3.12), we obtain k,m I p,l ( = ℜ jk− p+ m − l e 2π − j MK k∆n M −1 ∑ cr e j πlr M Q (r ) r =−( M −1) ) (3.15) where Q(r) is defined as Q (r ) = M −1 ∑ r ′ =−( M −1) cr ′ e ′ mK − j 2πr MK ( ∆n + 2 ) U 2π (r − r′ − (k − p) M ) + 2π∆ f MK . (3.16) Figure 3.4 shows the transmitter window function w[n] and the receiver window function v[n] and their respective positions. We note that in a synchronized case, u[n] = v[n]. The roll-offs of the window functions, shown in Figure 3.4, as noted earlier, help in reducing OOB emissions and MUI. Also, an important property of the receiver window v[n] that will become useful in the rest of our discussion is the following: 2π 1, r = 0 r = V 0, r 6= 0. MK In the sequel, we dig into the specific details of (3.15) under various conditions. (3.17) 40 Ncp MK Figure 3.4. Transmitter window, w[n](red line), and receiver window, v[n](blue line) 3.3.1 Fully synchronized In a fully synchronized C-FBMC network, by design, there will be no intercarrier interk,m ference and, thus, MUI is avoided. In other words, I p,l = 0, for ( p, l ) 6= (k, m). Here, we present a proof of this fact by looking into details of (3.15) for the case where ∆n = ∆ f = 0. This proof/presentation, although it may seem unnecessary, plays the roll of an instructive introduction to the rest of our study in this chapter. We recall that when ∆n = ∆ f = 0, u[n] = v[n] and thus the property (3.17) will also be applicable to U (ω ) and may be used to simplify (3.15). Such simplification leads to ( k,m I p,l = ℜ jk− p+ m − l M −1 ∑ r =−( M −1) cr cr −(k− p) M e j πr M (l −m) ) . (3.18) In the case where the subcarriers k and p are nonadjacent to each other, |k − p| ≥ 2 and under this condition, the set of coefficients cr and cr −(k− p) M, for −( M − 1) ≤ r ≤ M − 1 are nonoverlapping. This, in turn, implies that all the terms under the summation in (3.18) are zero, hence, k,m I p,l = 0. (3.19) In the case where the subcarriers k and p are adjacent, k − p = ±1. For this case, we study (3.18) for two scenarios where l − m is odd and when it is even. When l − m is an odd number, jk− p+m−l = ±1 and (3.18) simplifies to k,m I p,l =± M −1 ∑ r =−( M −1) cr cr ± M cos πr M (l − m) . (3.20) 41 Next, we introduce a change of variable r to r ∓ k,m I p,l =± M 2 −1 ∑ r =− M 2 +1 M 2 to rearrange (3.20) as cr + M cr − M sin 2 2 πr M (l − m) = 0. (3.21) Here, the second identity follows since the expression under summation in the first line of (3.21) is an odd series in r. When l − m is an even number, jk− p+m−l = ± j and (3.18) simplifies to k,m I p,l =± M −1 ∑ r =−( M −1) cr cr ± M sin πr M (l − m) = 0. (3.22) Here, the second identity follows since the expression under summation can be made an odd series in r through a change of variable r to r ∓ M 2 . To summarize, the key factors that guarantee interference-free operation of C-FBMC in a synchronous scenario are • A phase toggle of π/2 between adjacent time/frequency data symbols. • Design of a prototype filter such that DTFT coefficients cr are nonzero only for − M + 1 ≤ r ≤ M − 1 and are even symmetric around r = 0. As we will see below, both timing offset and frequency offset ruins these properties, hence, MUI will be unavoidable in asynchronous networks. Windowing method greatly helps in reducing MUI. 3.3.2 Timing offset only When ∆ f = 0, but ∆n 6= 0, two scenarios can happen. First, the window v[n] falls over the flat part of the transmitter window, w[n]. This can be due to the presence of a sufficiently long CP and/or a cyclic suffix (CS). In this scenario, u[n] = v[n] and (3.15) reduces to ( k,m I p,l = ℜ jk− p+ m − l e − j 2πk MK ∆n M −1 ∑ r =−( M −1) cr cr −(k− p) M e j πr M (l −m) ) . (3.23) 2π This is similar to (3.18) with the addition of the phase factor e− j MK k∆n . The presence of this k,m phase factor has no impact on the leakage gain I p,l being equal to zero when |k − p| ≥ 2. 42 However, it introduces some leakage when k − p = ±1. In this case, one finds that when l − m is an odd number, (3.23) simplifies to k,m I p,l =± M −1 ∑ r =−( M −1) cr cr ± M cos (3.24) 2πk πr (l − m) − ∆n . M MK (3.25) πr 2πk (l − m) − ∆n M MK and when l − m is an even number, (3.23) simplifies to k,m I p,l =± M −1 ∑ r =−( M −1) cr cr ± M sin With presence of phase − 2πk MK ∆n, the expressions inside the summations in (3.24) and (3.25) k,m no longer hold the odd symmetry property that was mentioned above, and thus, I p,l may not be zero. Hence, MUI will be present. The second scenario is when the window v[n] does not fall over the flat part of w[n]. In this scenario, u[n] 6= v[n] and, thus, U (ω ) no longer holds the clean property that is stated by (3.17). The samples of U (ω ) that appear under the summation in (3.18) are all nonzero. k,m This results in nonzero values for I p,l for all pairs of ( p, l ) and (k, m). The addition of proper roll-offs to both w[n] and v[n] helps in moderating the power leakage, but cannot remove it completely. 3.3.3 Frequency offset only The presence of a frequency offset ∆ f shifts the spectrum of the received signal. Therefore, samples of U (ω ) in (3.15) are taken at frequencies other than multiple integers of 1 MK . k,m All these samples are nonzero. Thus, (3.15) results in a nonzero value for I p,l . Extending the window function v[n] with proper roll-off decreases OOB emissions of U (ω ). As a result, samples of U (ω ) in (3.15) have smaller values, and the resulting interference gains become smaller. This method reduces the interference gains but their values still remain nonzero. 3.4 Simulation Results k,m The interference gain I p,l varies with the position of the asynchronous data symbol (k, m), the position of the estimated data symbol ( p, l ), the time offset, ∆n, and the frequency offset, ∆ f . In this section, we present a few case studies to gain more insight to the analytical results that were presented in the previous section. 43 We simulate a C-FBMC system with K = 16 subcarriers and 2M = 16 real-valued symbols along the time. The length of CP is set equal to 8. The channel is assumed to be k,m an ideal one. We present color map plots of I p,l for three choices of (k, m) = (8, 1), (8, 3), and (8, 7) and three sets of the time and frequency offsets (∆n, ∆ f ) = (4, 0), (−4, 0), and (0, 0.05/K ). For the cases where roll-off windows are applied, the packet length is extended as indicated in Figure 3.4. The roll-off lengths at both the transmitter and receiver are set equal to 8. k,m Figure 3.5 presents the color map plots of I p,l for the case where rectangular windows are used at both the transmitter and receiver. The same set of color map plots are repeated in Figure 3.6 when windows with raise-cosine roll-offs have been used at both the transmitter and receiver. The following observations (that all match our theoretical analysis in the previous section) are made from the color maps: • The asynchronous data symbols that are closer to the center of packet generate less interference. • For (∆n, ∆ f ) = (4, 0), only data symbols in the adjacent subcarriers are subject to interference. This is the case where the selected window of the asynchronous user falls over the flat part of its transmit window. The results are very different when (∆n, ∆ f ) = (−4, 0). • The use of windows with smooth corners has a significant impact on the level of leakage/interference. In particular, for the case where (k, m) = (8, 1) and (∆n, ∆ f ) = (−4, 0) or (0, 0.05/16), in Figure 3.5 we see a lot of orange and yellow color boxes, indicating leakage gains of −10 to −25 dB. These mostly drop below −40 dB in Figure 3.6. 3.5 Extension to GFDM We can derive energy spectral density and MUI of GFDM following similar steps we have for C-FBMC. GFDM uses the same block diagram as Figure 3.1. GFDM transmit signal contains a set of MK complex data symbols where each subcarrier carries M data 44 (∆n, ∆ f ) = (4, 0) (∆n, ∆ f ) = (−4, 0) (∆n, ∆ f ) = (0, 0.05 K ) 0 1 8 Subcarrier Index −20 16 1 −40 8 −60 16 1 −80 8 16 1 8 16 1 8 Time Index 16 1 8 16 −100 Figure 3.5. Interference gain of an asynchronous data symbol (shown by *) to other synchronous data symbols for C-FBMC. Rectangular v[n]. (∆n, ∆ f ) = (4, 0) (∆n, ∆ f ) = (−4, 0) (∆n, ∆ f ) = (0, 0.05 K ) 0 1 8 Subcarrier Index −20 16 1 −40 8 −60 16 1 −80 8 16 1 8 16 1 8 Time Index 16 1 8 16 −100 Figure 3.6. Interference gain of an asynchronous data symbol (shown by *) to other synchronous data symbols for C-FBMC. Raised cosine v[n]. 45 symbols with K sample spacing. Unlike C-FBMC, the GFDM transceiver does not apply any phase factor between adjacent data symbols. Therefore, the input signal is equal to M ak [n ] = ∑ m =1 dk,m δ [ n − mK ] . (3.26) Moreover, the GFDM matched filter receiver performs the same procedure as of CFBMC except GFDM estimate of transmit data symbols are complex valued output of the matched filter. In addition, GFDM may use a different prototype filter and (3.5) has be to replaced by Fourier series expansion of MK samples repeated version of prototype filter of interest. Here, we do not provide derivation of ESD and MUI for GFDM, but one can trivially repeat ESD and MUI derivation for GFDM with aforementioned changes in mind. Note that due to the nonorthogonal structure of GFDM, even a fully synchronous user induces k,m interference to other users and leakage gain I p,l is always nonzero and GFDM does not have equations equivalent to (3.21) and (3.22). This fact is demonstrated in Figure 3.7 k,m where leakage gain I p,l of a fully synchronized symbol of GFDM is presented. It is clear that even a fully synchronized GFDM user leaks some level of interference to the neighboring users due to nonorthogonal nature of GFDM. Synchronization violation of a user increases interference leakage gain of GFDM. Here, we demonstrate this fact, by repeating the results of Figure 3.6 for GFDM in Figure 3.8. Note that since in GFDM, unlike C-FBMC, the QAM data symbols are transmitted without introducing any time offset between the phase and quadrature parts, the number of time slots has reduced from 16 to 8. However, as seen, there is no noticeable difference between the interference levels in Figure 3.6 and in Figure 3.8. 3.6 Summary Two candidate waveforms of the next generation of wireless networks, GFDM and CFBMC, are studied in this chapter. Our focus was on C-FBMC, but the result are extendable to GFDM with some minor changes. We analytically examined the out-of-band (OOB) emissions of C-FBMC waveform. We showed that OOB emissions of C-FBMC signal is mainly generated by sharp transition of pulse shapes, which carry the first and last data symbols in the packet. Our analysis allowed us to quantify the proposed methods for improving OOB emissions. In addition, we derived equations for interference between 46 0 1 Subcarrier Index -20 -40 8 -60 -80 16 -100 1 8 Time Index Figure 3.7. Interference gain of a fully synchronous data symbol (shown by *) to other synchronous data symbols for GFDM. Rectangular v[n]. (∆n, ∆ f ) = (4, 0) (∆n, ∆ f ) = (0, 0.05 K ) (∆n, ∆ f ) = (−4, 0) 0 1 8 Subcarrier Index −20 16 1 −40 8 −60 16 1 −80 8 16 1 8 1 8 1 8 −100 Time Index Figure 3.8. Interference gain of an asynchronous data symbol (shown by *) to other synchronous data symbols for GFDM. Raised cosine v[n]. 47 any pair of data symbols in asynchronous multiuser scenario. We discussed how different scenarios for time/frequency offset of an asynchronous user generate interference to other users. The results revealed that some data symbols are more prone to generate or receive asynchronous multiuser interference. We discussed the impact of windowing at transmitter and receiver as a method to decreases MUI. Derived equations can be used to quantify the MUI when any arbitrary window functions are used at transmitter and receiver. CHAPTER 4 CAPACITY ANALYSIS OF FILTERBANK MULTICARRIER In filter bank multicarrier offset quadrature amplitude modulation (FBMC-OQAM), data symbols are chosen from a real-valued alphabet and distributed at a density of two per unit area in a time-frequency phase space. Moreover, the adjacent data symbols are π/2 phase-shifted with respect to one another. This arrangement along with a correctly designed prototype filter leads to a transceiver system in which perfect data recovery at the receiver involves the use of an analysis filter bank, removal of the π/2 phase shifts from the analyzed signal samples, and taking the real parts. The imaginary parts are simply ignored. However, one may realize that these imaginary parts, which are referred to as intrinsic interference, are also dependent on the transmitted real-valued data symbols and thus may carry some information beyond what exists in the real parts. The following question thus arises. Can one design a better receiver that benefits from the intrinsic interference? In other words, is there any information in the intrinsic interference that does not exist in the preserved real parts? Razavi et al. [86] have made an attempt to answer this question. This work, which contradicts our findings in this chapter, claims that the intrinsic interference contains a significant amount of new information beyond what is present in the preserved real parts, and hence, can be utilized to improve the FBMC-OQAM receiver performance. The authors have claimed the addition of intrinsic interference to the receiver detection process can lead to a SNR gain of about 3 dB. Our study in this chapter shows only a very marginal gain exists. Such gain is in the order of a small fraction of 1 dB. This difference appears to be in the presentation of the capacity formula (10) in [86]. This formula accounts for the intrinsic interference power without realizing the noise terms in the real part and in the intrinsic interference part are highly correlated. This correlation when accounted for will 49 result in loss of most of the capacity gain that has been reported in [86]. Other published works that have made an attempt to take advantage of the intrinsic interference in improving the receiver performance are [87] and [88]. In [87], the authors make a clever use of intrinsic interference to improve on channel estimation. Reference [88] presents a capacity analysis of the circular FBMC (C-FBMC). C-FBMC is a modified form of FBMC in which linear convolutions are replaced by circular convolutions. This improvement first appeared in [79] under the name ‘FBMC-OQAM with weighted circularly convolved filtering’. The shorter name C-FBMC was later suggested [15]. The analysis presented in [88] preserves the intrinsic interference terms for capacity calculation, and therefore, does not provide any insight to whether the intrinsic interference carries any information that is not contained in the real parts. Our analysis that separates the real and intrinsic interference parts reveals that in the particular case of C-FBMC, the intrinsic interference does not carry any additional information beyond what could be extracted from the real parts. 4.1 Filterbank Multicarrier-Offset QAM 4.1.1 Synthesis filter bank and perfect reconstruction In FBMC-OQAM, a set of real-valued information symbols dk,m are spread across frequency (represented by the k index) and time (represented by the m index) and combined together through a synthesis filter bank to produce the time domain baseband signal x(n) = K −1 2M −1 ∑ ∑ k=0 m =0 K − j 2π kn dk,m jm+k g n − m e K 2 (4.1) where K is the number of subcarriers, 2M is the number of data symbols along time, and g(n) is a length NK prototype filter that is designed to satisfy the perfect reconstruction property that is discussed below. Taking into account the ramp-up and ramp-down periods that are introduced by the transient of the prototype filter, one will find that the signal sequence x(n) has a length of L x = (2M − 1) K2 + NK. The reader should also be reminded that in FBMC-OQAM, each pair of real-valued symbols are the real and imaginary parts of an OQAM symbol. To facilitate the derivation of our analytical results, we define the column vectors 50    xc =    x ( 0) x ( 1) .. . x ( L x − 1)     (4.2) and    d=  d0 d1 .. . dK −1      (4.3) where dk = [dk,0 , dk,1 · · · , dk,2M−1 ]T , and note that (4.1) has the compact form xc = Ac d (4.4) where Ac is an L x × 2MK matrix whose n-th row and (m + 2Mk)-th column element 2π is equal to jm+k g n − m K2 e− j K kn . The subscript ‘c’ is added to the vector xc and the matrix Ac to emphasize that they are complex-valued. The same will be followed for other complex-valued vectors and matrices that appear in the sequel. The perfect reconstruction property of the prototype filter p(n) leads to the following equality ℜ{AH c Ac } = I (4.5) where I is the identity matrix. 4.1.2 Channel impact When x(n) is passed through a channel with a baseband equivalent impulse response that is represented by the length Lch column vector hc , the channel output is given by yc =Hc xc + nc (4.6) where Hc is an ( L x + Lch − 1) × L x Toeplitz matrix whose first column is hc appended with L x − 1 zeros, and nc is the channel noise. We assume that nc is a vector of independent and identically distributed (i.i.d.) zero-mean circularly symmetric complex Gaussian random variables with variance 2σ2 . 51 4.1.3 Channel equalization and data recovery We consider a generic linear equalizer that takes the received vector yc and converts it to an estimate of the transmit vector xc according to the equation x̂c = Ec yc (4.7) where Ec is an L x × ( L x + Lch − 1) equalization matrix. This estimate is then passed through a bank of analysis filters whose operation is compactly written as zc = AH c x̂c . (4.8) The real part of zc provides an estimate of the data vector d. That is d̂ = ℜ{zc }. (4.9) The imaginary part of zc is called the intrinsic interference. d̂′ = ℑ{zc }. (4.10) 4.1.4 Real-valued signal models To set the ground for the analytical results of the next section, we define the real-valued vectors ℜ{xc } ℜ{yc } x= , y= , ℑ{xc } ℑ{yc } ℜ{x̂c } ℜ{nc } x̂ = , n= . ℑ{x̂c } ℑ{nc } We also define the real-valued matrices ℜ{Ac } A= , ℑ{Ac } ℜ{Hc } − ℑ{Hc } H= , ℑ{Hc } ℜ{Hc } ℜ{Ec } − ℑ{Ec } E= . ℑ{Ec } ℜ{Ec } (4.11) (4.12) Using these definitions, (4.4), (4.6), (4.7), and (4.9) can, respectively, be rearranged as x =Ad (4.13) y =Hx + n (4.14) x̂ =Ey (4.15) d̂ =AT x̂ (4.16) 52 We also note that the perfect reconstruction property (4.5) implies AT A = I. (4.17) 4.2 Capacity Analysis We look at different vectors in the signal path of an FBMC-OQAM system and study the information content of each. As a measure of information content of a signal, we use mutual information between the signal and the transmitted vector d. To simplify the derivations, we assume that the entries of the information vector d are i.i.d and follow a normal distribution with variance of unity. Hence, E [dT d] = I, where E [·] denotes statistical expectation. We recall that the differential entropy of an n-element Gaussian vector q with covariance matrix Cq is [89] h(q) = 1 log2 (2πe)n |Cq | 2 (4.18) where | · | is the determinant operator. Moreover, the mutual information between two Gaussian vectors q and r is the difference between the differential entropies h(q) and h(q|r). This is written as I (q; r) = h(q) − h(q|r) | Cq | 1 = log2 . 2 | Cq | r | (4.19) Using (4.19) and (4.14), the mutual information between the received signal vector y and the transmit data vector d is obtained as |HAAT HT + Σ| 1 I (y; d) = log2 2 |Σ | (4.20) where Σ = σ2 I is the covariance matrix of the noise vector n. We use I (y; d) as a reference and compare information content of subsequent vectors at the receiver chain to it. 4.2.1 Channel equalization In this part, we study the impact of channel equalization on the information content of the received signal. We focus on a class of equalizers that have the form of E = (HT H + αI)−1 HT . This covers both the commonly used minimum mean squared error (MMSE) and zero forcing (ZF) equalizers. 53 Considering (4.15), the mutual information between the equalizer output and the transmit data vector d is obtained as 1 I (x̂; d) = log2 2 EHAATHT ET + EΣET EΣET ! . (4.21) Using the singular value decomposition H = U H D H VTH , the equalizer matrix E can be rearranged as E = V H PDTH UTH (4.22) where P = (DTH D H + αI)−1 . Substituting (4.22) in (4.21) and recalling that Σ = σ2 I, the identities |XY| = |X| × |Y| (4.23) |I + XY| = |I + YX| (4.24) for size compatible matrices X and Y, and also |U| = |V| = 1, UT U = I, and VT V = I, (4.21) reduces to (4.20). Hence, I (x̂; d) = I (y; d). (4.25) That is, MMSE and ZF equalizers preserve all the information content of the received signal. 4.2.2 Data recovery Substituting (4.15) in (4.16) and using (4.19), we obtain 1 I (d̂; d) = log2 2 AT EHAAT HT ET A + AT EΣETA ! AT EΣET A −1 1 T T T T T T = log2 I + A EΣE A A EHAA H E A 2 −1 1 T T T T T T A EHA = log2 I + A H E A A EΣE A 2 (4.26) where to obtain the second line, we have used the identity |X | / |Y | = |Y − 1 X | (4.27) and the third line follows by using the identity |I + XYZ| = |I + ZXY|. (4.28) 54 Moreover, to facilitate comparison of I (x̂; d) and I (d̂; d), we note that following a similar line of derivation, (4.21) can be rearranged as I (x̂; d) = 1 log2 I + AT HT ET (EΣET )−1 EHA . 2 (4.29) Next, we compare (4.26) and (4.29) to see if the matched filtering process (4.16) or, to be more exact, matched filtering followed by taking the real part, results in any loss of information. This comparison makes use of the Schur complement inequality [90, pp. 92-93] , which says for any positive semi-definite matrix X and any matrix A, we have X−1 A(AT XA)−1 AT . (4.30) Here, X Y means X − Y is a positive semi-definite matrix. Using this inequality, we get EΣET −1 −1 A AT EΣET A AT . (4.31) Also, noting that X Y implies ZT XZ ZT YZ, the latter result implies −1 −1 EHA I + AT HT ET A AT EΣET A AT EHA. I + AT HT ET EΣET (4.32) Moreover, making use of the positive definite ordering theorem [91, p. 495], which says “for any pair of positive semi-definite matrices that satisfy X Y, |X| ≥ |Y|,” it follows from (4.26), (4.29), and (4.32) that I (d̂; d) ≤ I (x̂; d). (4.33) This result shows the data recovery process, in which the intrinsic interference terms are ignored, incurs some loss of information. However, the above analysis does not provide the necessary insight into how much this loss will be. The numerical results that are presented below and further theoretical results that are developed in Section 4.3 lead to additional insights that give a better picture of the level of information loss under different conditions. 4.2.3 Numerical results Here, we present a set of numerical results that compare I (d̂; d) and I (x̂; d) for a particular transceiver setup. We study an FBMC transceiver with K = 8 subcarriers. The PHYDYAS prototype filter [74] with an overlapping factor of N = 4 is used. The channel 55 impulse response has a uniform power delay profile that is distributed over a length of Lch = 7. We calculate the expected value of information content of the signal vectors by evaluating the respective ensemble averages over a large number of realizations. Figure 4.1 shows the results for three choices of M = 1, 3, and 10. The results of both ZF and MMSE equalizers are presented. As seen, the relative difference between I (d̂; d) and I (x̂; d) is small and gets diminishingly smaller as M (equivalently, the packet length) increases. To get further insight to this deference, ∆I = I (x̂; d) − I (d̂; d) as a function of M is presented in Figure 4.2. The interesting observation here is that ∆I converges to a fixed value for M ≥ 4. This matches the length of ramp-up/ramp-down of FBMC in our implementation here which is equal to 4K. Accordingly, we argue, the loss of capacity that is observed should be attributed to the first few data symbols at the beginning and the end of each packet. For data symbols in the rest of the packet, the information loss due to removal of intrinsic interference should be zero or, at least, close to zero. The additional analysis presented in the next section further confirms this observation. The reader should also be reminded that the channel that we have used to obtain the results of Figure 4.1 is an unrealistically long channel, when compared to the FBMC symbol spacing. While OQAM symbols are K = 8 samples apart, we have set Lch = 7. Moreover, the uniform power-delay profile of the channel is an exaggeration. Typical channels have a exponentially decaying power-delay profile. The choice of this simulation setup was to show some difference between I (d̂; d) and I (x̂; d). Our study of more typical practical scenarios with K ≥ 64 and Lch ≪ K (not presented here, for brevity) shows much smaller values of ∆I. 4.3 C-FBMC C-FBMC is a variant of FBMC that uses circular convolution for filtering operation at each subcarrier band [15, 79]. In this case, also, a similar equation to (4.4) relates the data vector d and the synthesized signal vector xc ; however, the length of xc reduces to MK. Moreover, Ac has a size of MK × 2MK and its n-th row and (m + 2Mk)-th column element 2π is jm+k g (n − m K2 )KM e− j K kn , where ‘mod’ indicates the modulo sign. Capacity analysis of C-FBMC follows the same line of derivations as those of FBMC. The real-valued signal models that were presented in Section 4.1.4 can be similarly pre- 56 50 500 150 I(y; d) I(d̂; d):ZF I(d̂; d):MMSE 40 I(y; d) I(d̂; d):ZF I(d̂; d):MMSE I(y; d) I(d̂; d):ZF I(d̂; d):MMSE 400 100 bits bits 300 bits 30 20 200 50 10 0 100 0 5 10 15 20 0 0 SNR (dB) (a) 5 10 15 0 20 0 5 SNR (dB) (b) 10 15 20 SNR (dB) (c) Figure 4.1. Expected values of I (y; d) and I (d̂; d) for ZF and MMSE equalizers as a function of SNR and for three choices of (a) M = 1, (b) M = 3, and (c) M = 10. 4 ∆I (bits) 3 2 1 ZF MMSE 0 1 2 3 4 5 6 7 8 9 10 M Figure 4.2. Expected value of information loss versus M at SNR = 10 dB. sented. Furthermore, we note that the derivations that led to (4.25) work the same way for C-FBMC and hence (4.25) is a valid result for C-FBMC as well. However, the capacity analysis of the data recovery step leads to a different result. This difference is a by-product of the fact that in C-FBMC, the real-valued matrix A is a square matrix of size 2MK × 2MK. At the output of data recovery of C-FBMC, noting that A is a square matrix, information content of the recovered data d̂ can be written and rearranged as 57 |AT EHAAT HT ET A + AT EΣET A| |AT EΣET A| T |A (EHAAT HT ET + EΣET )A| 1 = log2 2 |AT (EΣET )A| |EHAAT HT ET + EΣET | 1 . = log2 2 |EΣET | 1 I (d̂; d) = log2 2 (4.34) Noting that this result is the same as (4.21), we conclude: the C-FBMC data recovery process does not incur any loss of information. This interesting result may be used to further explain the observations made in the previous section. The numerical results that were presented in the previous section suggest that the information loss ∆I comes from signal transients at the beginning and the end of each packet. In other words, the information content of the middle symbols whose corresponding signal has reached a steady state should not incur any loss of information. This matches the theoretical result of this section, if one realizes that a C-FBMC packet may be thought of as a single period of a periodic FBMC signal of length infinity and such a period is free of any transient. 4.4 Summary In this chapter, we studied the FBMC-OQAM transceiver from an information theoretic perspective. We derived equations that measure information content of signals at different stages of a typical FBMC-OQAM receiver. We proved that MMSE and ZF equalizers maintain information content of the received signal. However, the data recovery process in FBMC-OQAM where the intrinsic interference part of the processed signal is ignored incurs some information loss. Through numerical results, we found that the amount of information loss is very low and approaches a constant as packet length increases. We thus conjectured that this information loss should be attributed to the transients at the beginning and the end of each FBMC packet. We confirmed this conjecture further by proving that the data recovery process in C-FBMC does not incur any information loss and a C-FBMC packet may be thought of as a single period of a periodic FBMC signal of length infinity. CHAPTER 5 CAPACITY ANALYSIS OF CIRCULAR FILTERBANK MULTICARRIER Filter bank multicarrier with offset quadrature amplitude modulation (FBMC-OQAM) is a powerful signal processing method that allows data transmission with a maximum bandwidth efficiency. Data symbols are chosen from a real-valued alphabet and distributed at a density of two per unit area in a time-frequency phase space. This is equivalent to one QAM symbol per unit area. In addition, to allow straightforward separation of the real-valued data symbols, the adjacent symbols are π/2 phase-shifted with respect to one another. This arrangement along with a correctly designed prototype filter leads to a transceiver system in which, in the absence of channel, perfect data recovery at the receiver involves the use of an analysis filter bank, removal of the π/2 phase shifts from the analyzed signal samples, and taking the real parts. The imaginary parts are simply ignored. However, one may realize that these imaginary parts, which are referred to as intrinsic interference, are also dependent on the transmitted symbols and thus may carry some information beyond what exists in the real parts. The following question thus arises. Can one design a better receiver that benefits from the intrinsic interference to improve on the detection performance? In other words, is there any information in the intrinsic interference that does not exist in the preserved real parts? In Chapter 4, the above questions have been answered through an information theoretic analysis of FBMC-OQAM. To allow such analysis, the real-valued data symbols were assumed to be i.i.d. and from a normal distribution. The finding in Chapter 4 was that the excess information carried by the intrinsic interference terms, in general, is very minimal and, thus, is not worth the effort of recovering it. It was further revealed that this minimal excess information belongs to the transient parts (ramp-up and ramp-down portions) of the FBMC-OQAM waveform. As a consequence, the capacity analysis of 59 circularly pulse-shaped FBMC-OQAM (C-FBMC-OQAM or C-FBMC, for short) in Chapter 4 revealed that the excess information carried in the intrinsic interference part of C-FBMC is zero. This observation can be contributed to the fact that in C-FBMC, the portion of the signal that is analyzed (after the removal of the cyclic prefix (CP)) is free of any transient period. In this chapter, we revisit the C-FBMC waveform and provide an answer to the above questions from a pure signal processing point of view. In particular, we show that in the case of the C-FBMC waveform, the vectors of the real and imaginary parts of the equalized and demodulated signal samples are related through an orthonormal linear transformation. This development is not limited to any form of symbol alphabets and/or characteristics of the channel noise plus any interference. Hence, what is presented in this chapter provides the most comprehensive proof that under all conditions, ignoring the intrinsic interference terms from the output of a C-FBMC receiver does not incur any loss of information. It is worth noting that prior to our analysis published in [61], Razavi et al. [86] had made an attempt to take advantage of the intrinsic interference and arrived at a conclusion that the use of intrinsic interference could lead to a gain of as much as 3 dB. The reasons behind this contradictory conclusion to that of Chapter 4 and what is discussed here is explained in Chapter 4. Nevertheless, it should be noted that the use of the intrinsic interference for other means of improving the FBMC transceivers capabilities should not be ignored. For instance, a few authors have made clever use of intrinsic interference to develop a number of effective channel estimation methods when pilot symbols are scattered along a data packet, for instance [92], [93], [94], and [87]. 5.1 C-FBMC Tranceiver In C-FBMC, a set of real-valued data symbols {dk,m ; k = 0, 1, · · · , K − 1; m = 0, 1, · · · 2M − 1} are combined together to construct the length KM signal vector x = [ x(0) x(1) · · · x(KM − 1)]T , where x(n) = K −1 2M −1 ∑ ∑ k=0 m =0 2π dk,m jm+k g(n−m K )KM ej K kn 2 (5.1) and gn is the filter bank prototype filter. This construction, that is, the use of a circular convolution instead of a linear one, is to allow the addition of CP for absorbing the channel 60 transient. Straightforward inspection of (5.1) reveals that x = Ad where    d=   d0 d1 .. .   ,  dK −1 (5.2)    dk =   dk,0 dk,1 .. . dk,2M−1 and A is a KM × 2KM matrix with the (m + 2kM )-th column am+2kM     ,  2π g(0−m K )KM ej K k×0    = jm + k    2 2π g(1−m K )KM ej K k×1 2 .. . 2π g(KM−1−m K )KM ej K k×(KM−1) 2     .   (5.3) To construction A, we let k = 0, 1, · · · , K − 1 and m = 0, 1, · · · , 2M − 1. Accordingly, the element at n-th row and m + 2Mk-th column of A is 2π [A]n,m+2Mk = jm+k g(n−m K )KM ej K kn . 2 (5.4) A CP is added to the vector x before its transmission. At the receiver, after removing the CP, the received signal vector can be expressed as y = HAd + n (5.5) where n is the channel noise and H is the KM × KM circulant matrix of the channel impulse response. To obtain an estimate of d from y, the following steps are taken at the receiver: 1. The vector y is converted to the frequency domain through a DFT. Mathematically, this is written and rearranged as yf = Hf Gd + nf (5.6) where, nf = F n, Hf = F HF H is a diagonal matrix whose diagonal elements are the DFT of the first column of H, and G = F A. 61 2. A per tone linear equalizer is applied to yf to mitigate the channel effect. The generic −1 H form of the equalizer, here, is Ef = (HH f Hf + αIKM ) Hf . This form reduces to a zero- forcing (ZF) equalizer, when α = 0, and to a minimum mean square error (MMSE) equalizer, when α is equal to the variance of the channel noise. The equalized signal is x̂ = Ef yf . (5.7) It is an estimate of the transmitted vector x. 3. The equalized signal x̂ is passed through a bank of matched filters, which mathematically is expressed as z = GH x̂. (5.8) 4. Finally, the estimate of d is obtained by taking the real part of z. That is, d̂ = zR . (5.9) It is worth noting that an alternative estimate of d can be obtained through a widely linear (WL) design [95]. The WL design takes note of the fact that d is real-valued and derives an estimate of d by linearly combining the outputs of two linear filters that separately process the real and imaginary parts of y. The WL design, in general, may lead to a better estimator. However, we note that the special form of the matrices A and G that are presented in the next section leads to a WL estimator that is not different from d̂ of (5.9). 5.2 Properties of A and G Here, we list a number of interesting properties of the modulation matrix A and its frequency domain equivalent G. Some of these properties are well-known, but others, to our knowledge, have not yet been reported. The properties of interest here are: 1. The well-known perfect reconstruction property of FBMC-OQAM implies that [15, 96] ℜ{AH A} = I. (5.10) 62 Making use of G = F A, one will find that AH A = GH G. (5.11) ℜ{GH G} = I. (5.12) Hence, from (5.10), we get These relationships are often referred to as the real orthogonality of the respective columns of A and G. 2. Considering (5.10) and (5.12), one may note that the off-diagonal elements of the matrices AH A and GH G are either zero or pure imaginary. 3. The rows of A and G satisfy a stronger orthogonality condition that is expressed by equations AAH = 2I (5.13) GGH = 2I. (5.14) A proof of (5.13) is presented in the proof section of this chapter. Moreover, making use of G = F A, (5.14) can be obtained from (5.13). 4. Another pair of useful relationships are AAT = 0 (5.15) GGT = 0. (5.16) The identity (5.15) is also proven in the proof section of this chapter. The identity (5.16) is obtained by multiplying both sides of (5.15) from left and right by F and F T , respectively. 63 5. Defining G = GR + jGI , and substituting in (5.14) and (5.16), we get the pair of equations GR GTR + GI GTI + j GI GTR − GR GTI = 2I (5.17) GR GTR − GI GTI + j GI GTR + GR GTI = 0 (5.18) whose simultaneous solution leads to GR GTR = GI GTI = I (5.19) GI GTR = GR GTI = 0. (5.20) 6. Defining B = ℑ{AH A} = ℑ{GH G}, straightforward application of (5.19) and (5.20) leads to BT B = I (5.21) which means B is an orthonormal matrix. We also note that B is a (2KM ) × (2KM ) real-valued matrix and may be expressed as B = GTR GI − GTI GR (5.22) or a similar equations in terms of AR and AI . 5.3 Relating zR and zI In this section, we make use of the properties of the modulation matrix G that were developed in the previous section to show that removing the imaginary part of the vector z for the information recovery in C-FBMC (specifically, Step 4 of the data recovery procedure that was discussed in Section 5.1) does not incur any loss of information. This is done by showing that the real and imaginary parts of the vector z (which are zR and zI ) are related through a linear orthonormal transformation. Hence, any information that could be recovered from zI is already present in zR . 64 Recalling (5), (6), and (7) and the general form of Ef that was mentioned in Section 5.1, we note that z = GH DGd + GH nf (5.23) where D = Ef Hf is a real-valued diagonal matrix. Recalling that the data vector d is also real-valued, from (5.23), we obtain zR = (GTR DGR + GTI DGI )d + (GTR nf,R + GTI nf,I ) (5.24) zI = (GTR DGI − GTI DGR )d + (GTR nf,I − GTI nf,R ). (5.25) and Next, recalling (5.22) and using (5.19) and (5.20), straightforward manipulations lead to zI = BzR (5.26) which, in light of (5.21), shows zR and zI are related through a linear orthonormal transformation. Accordingly, any information that could be extracted from zI is already present in zR . 5.4 Proofs of Properties of A and G Matrices In this section, we go through the mathematical steps that lead to a proof of the identities (5.13) and (5.15). In this development, we make use of some interesting properties of underlying prototype filter gn . 5.4.1 Properties of the prototype filter We first note that the prototype filter is commonly designed to be zero-phase. This design when applied to a C-FBMC modulator implies the identity gn = g(−n)KM . Also, gn in all designs is real-valued. These assumptions are made use of in the following derivations. We first note that the ith element of the first row of the matrix AH A may be expressed as [AH A]0,i = aH 0 ai . (5.27) 65 Recalling (5.4), for i = m + 2Mk, this can be rearranged as [AH A]0,m+2Mk = jm+k KM −1 ∑ n =0 2π g(n)KM g(n−m K )KM ej K kn . 2 (5.28) With the change of variable n = v + K2 u, (5.28) can be rearranged as [AH A]0,m+2Mk = jm+k β m,k (5.29) where β m,k = K 2 −1 ∑ 2π f m,k (v)ej K kv (5.30) v =0 and f m,k (v) = 2M −1 ∑ u =0 g(v+u K )KM g(v+(u−m) K )KM ejπku . 2 2 (5.31) We note that f m,k (v) is a real-valued function of v and f m,k ( Making the change of variable v → K − v) = e−jπk f m,k (v). 2 K 2 (5.32) − v in the right-hand side of (5.30) and using (5.32), one will find that β∗m,k = β m,k , hence, β m,k is real-valued. Next, considering the above results and recalling (5.10), we conclude that 1, m = k = 0 β m,k = 0 m + k even. (5.33) For odd values of m + k, β m,k may be nonzero, but its value is irrelevant to the results that we wish to develop. In the sequel, we consider a few combinations of m and k that will allow us to prove the identities (5.13) and (5.15). 1. For m = 0, from (5.31), one will find that f0,k (v) = 2M −1 ∑ u =0 g2(v+u K ) 2 KM , for k even. (5.34) Moreover, since the right-hand side of (5.34) is independent of k, this result implies that f 0,k (v) = f 0,0 (v) for all even values of k. In addition, using (5.33), one will get the set of K/2 simultaneous equations K 2 −1 ∑ v =0 f 0,0 (v)e j 2π K kv = ( 1 0 k=0 k = 2, 4, · · · , K − 2 (5.35) 66 for the unknowns f 0,0 (0), f0,0 (1), · · · , f0,0 ( K2 − 1). The solution to this set of equations leads to the following set of identities 2M −1 ∑ g2(v+u K ) 2 KM u =0 = 2 , 0≤v≤ K K 2 − 1. (5.36) 2. For m = 2r, r = 1, 2, · · · , M − 1 and k even, also, the same line of derivations lead to f 2r,k (v) = f2r,0 (v) for all even values of k. Moreover, using (5.33), one will get the set of K/2 simultaneous equations K 2 −1 ∑ v =0 2π f2r,0 (v)ej K kv = 0, k = 0, 2, · · · , K − 2. (5.37) For each value of r in the range 1 to M − 1, we have one set of equations and the solution to this set leads to the following identities 2M −1 ∑ u =0 g(v+u K )KM g(v+(u−2r ) K )KM = 0, 0 ≤ v ≤ 2 2 K 2 − 1. (5.38) 3. For m = 2r + 1, r = 0, 1, · · · , M − 1 and k odd in the range 1 to K − 1, a similar line of derivations to those that led to (5.38), lead to the following identities 2M −1 ∑ u =0 g(v+u K )KM g(v+(u−2r −1) K )KM (−1)u = 0 2 (5.39) 2 for r = 0, 1, · · · , M − 1 and v = 0, 1, · · · , K2 − 1. 5.4.2 Proof of (5.13) Noting that element of AAH at i-th row and j-th column is equal to [AAH ]i,j = 2MK −1 ∑ [A]i,l [A]∗j,l (5.40) l =0 and making use of (5.4), simple manipulations lead to [AAH ]i,j = 2M −1 ∑ m =0 g(i−m K )KM g( j−m K )KM 2 2 K −1 ∑ ej 2π K k( j− i) . (5.41) k=0 In (5.41), the second summation reduces to K when j = i + ℓK, for −2M + 1 ≤ ℓ ≤ 2M − 1. For ℓ = 0, which means i = j, the first summation reduces to the summation on the left-hand side of (5.36). As a result, we get [AAH ]i,i = 2 ×K K = 2. (5.42) 67 For ℓ 6= 0, the first summation in (5.41) reduces to the summation on the left-hand side of (5.38) and thus leads to [AAH ]i,i+ℓK = 0. Other off-diagonal elements of AAH will also be zero, because when j 6= i + ℓK, the second summation in (5.41) will be always zero. These complete the proof of (5.13). 5.4.3 Proof of (5.15) Here, we get [AAT ]i,j = 2M −1 ∑ m =0 g(i−m K )KM g( j−m K )KM (−1)m × 2 2 In (5.43), the second summation will be zero when i + j + K. For the cases where i + j + K 2 K −1 ∑ ej 2π K K k( i+ j+ 2 ) . (5.43) k=0 K 2 is not an integer multiple of is an integer multiple of K, it is not difficult to show that the terms “i − m K2 ” and “j − m K2 ” in the subscripts of gn differ by an odd factor of K 2. Accordingly, the first summation in (5.43) reduces to zero, because of (5.39). With these observations, we conclude that [AAT ]i,j = 0 (5.44) for all pairs of i and j, hence, (5.15) is proven. 5.5 Summary In this chapter, we presented a thorough study of the matrices that characterize the signal processing steps in the circularly pulse-shaped FBMC-OQAM waveforms. Our study revealed a number of interesting properties of these matrices whose application may prove useful in further study and development of the C-FBMC-OQAM waveforms. One interesting result that we showed in this chapter was that the real and imaginary parts of the demodulated signal at the receiver output are related through an orthonormal transformation, hence, these two parts carry exactly the same information. We thus draw the conclusion that ignoring the imaginary part of the demodulated subcarrier signals does not incur any loss of information. CHAPTER 6 CAPACITY ANALYSIS OF ORTHOGONAL TIME-FREQUENCY SPACE WAVEFORM While orthogonal frequency division multiplexing modulation scheme achieves a performance near the capacity limits in linear time-invariant channels, it leads to a poor performance in doubly dispersive channels. This is due to the large amount of interference that is imposed by the channel Doppler spread. The common approach to cope with this issue in the existing wireless standards such as IEEE 802.11a and digital video broadcasting systems is to shorten OFDM symbol duration in time so that the channel variations over each OFDM symbol are negligible. However, this reduces the spectral efficiency of transmission since the cyclic prefix (CP) length should remain constant. A thorough analysis of OFDM in such channels is conducted in [97]. Another classical approach for handling the time-varying channels is to utilize filtered multicarrier systems that are optimized for a balanced performance in doubly dispersive channels [98]. Orthogonal time-frequency space (OTFS) modulation has recently been proposed as an effective waveform that takes advantage of the time diversity (that is, variation of channel with time) to improve on the reliability of wireless links [57]. OTFS was first introduced in the pioneering work of Handani et al. [57] where the two-dimensional (2D) Doppler-delay domain was proposed for multiplexing the transmit data. OTFS modulation is a generalized signaling framework where precoding and postprocessing units are added to the modulator and demodulator of a multicarrier waveform allowing for taking advantage of full time and frequency diversity gain of doubly dispersive channels. This process also converts the time-varying channel to a time-invariant one. In OTFS, the transmit data symbols are treated as values of the grid points in a Dopplerdelay space. A transformation step takes each data symbol and spreads it over the entire space of time-frequency points. The result of this transformation is then passed to a 69 multicarrier system for modulation and transmission. In this way, all data symbols are equally affected by the channel frequency selectivity and time diversity and, as a result, the time-varying channel, within a good approximation, converts to a unified time-invariant impulse response for all the data symbols. To recover the transmitted (Doppler-delay space) data symbols, at the receiver, the respective multicarrier demodulation followed by a transformation takes the received signal back to the Doppler-delay space. In [57], it is proposed that an inverse symplectic finite Fourier transform (SFFT−1 ) be used for transformation from the Doppler-delay space to the time-frequency space, at the transmitter, and the corresponding symplectic finite Fourier transform (SFFT) be used for the reverse operation, at the receiver. The OTFS modulation and demodulation process is shown in Figure 6.1. In the above setup, the equivalent channel that connects the transmitted data symbols and the received signal samples, both in the Doppler-delay space, is modeled by a time-invariant two-dimensional impulse response. Soft detectors/equalizers, like some extensions to those presented in [99], may thus be used for near-optimal recovery of the transmitted information. In order to simplify such detectors, [57] has proposed that proper windows should be applied to the time-frequency signals at both the transmitter and receiver sides to improve on the sparsity of the OTFS channel. The transformation of data symbols from the Doppler-delay space to time-frequency at the transmitter and the corresponding inverse transformation at the receiver, clearly, leads to full diversity gain across both time and frequency. In multiple-input multipleoutput (MIMO) channels, the space diversity gain will naturally be present because in a MIMO setup, any signal going out of each antenna reaches all the receiver antennas with statistically similar gains. The goal of this chapter is to examine the details of OTFS in terms of the reliability of transmission brought up as a result of the addition of time diversity in the modulator. To allow simple derivations, OFDM is used for multicarrier transmission of time-frequency signals in OTFS. To this end, we present a novel discrete-time end-to-end formulation of an OFDM-based OTFS setup. Such a formulation provides a concise representation of the effect of each signal processing unit on the input-output relationship of an MIMO OFDM-based OTFS system. Based on this formulation, we show that the proposed MIMO 70 dm,n dˆm,n SFFT−1 SFFT Windowing Windowing x̃k,l OFDM Modulator ỹk,l s [i ] r [i ] LTV Channel OFDM Demodulator Figure 6.1. OTFS transmitter and receiver structure. OFDM-based OTFS system achieves the same ergodic capacity as that of an OFDM system, under the assumption that channel state information is known at the receiver and most importantly, with the use of cyclic prefix, a block of OTFS transmission can be implemented as N consecutive, noninterfering OFDM transmissions. Since this capacity analysis is derived assuming perfect channel knowledge, optimal receiver, and an infinite code length, it does not violate the known fact that in practical systems with higher Dopplers, OTFS outperforms OFDM and has lower receiver complexity due to simpler channel estimation and equalization in the Doppler-delay domain. This work is inspired by the landmark paper [57], which was the first to propose a continuous-time formulation of a single-antenna OTFS modulation. Recently, [100] presents a discrete-time formulation of a single-antenna OTFS system by sampling the continuoustime channel for the case when the transmit and receive window functions are rectangular. In [101], a matrix-form discrete-time formulation of a single-antenna transceiver with rectangular windows is presented. However, this formulation is mostly based on twodimensional signal matrices instead of vectors. Moreover, [102] studies a single-antenna 71 transceiver when the transmit and receive window functions are separable. In comparison, our formulation here uses a general form of window functions in an MIMO OFDM-based OTFS setup and presents a vectorized formulation of signals, which is amicable to analytical analysis and practical implementation of MIMO OTFS systems. 6.1 Orthogonal Time-Frequency Space An OTFS transmitter combines a set of complex-valued data symbols {dm,n }, m = 0, 1, · · · , M − 1 and n = 0, 1, · · · N − 1, in the Doppler-delay domain to construct an OTFS signal. First, the transmitter maps data symbols on the Doppler-delay lattice, {dm,n }, to a lattice in the time-frequency domain through an inverse symplectic finite Fourier transform operation (SFFT−1) xk,l = √ 1 M −1 N −1 MN m =0 n =0 ∑ ∑ mk nl dm,n e− j2π ( M − N ) , (6.1) where k = 0, 1, · · · , M − 1 and l = 0, 1, · · · N − 1. Let D ∈ C M× N denote the data matrix that contains elements {dm,n }. A closer look into (6.1) reveals that the SFFT−1 transform of D can be obtained by applying an M-point discrete Fourier transform (DFT) and an N-point inverse DFT (IDFT) to the columns and rows of the matrix D, respectively. Accordingly, (6.1) can be written in a compact form as X = F M DF H N, (6.2) where X ∈ C M× N . Subsequently, the transmitter applies a transmit window, uek,l , to the time-frequency signal as xek,l = xk,l uek,l . (6.3) Finally, the transmitter packs the time-frequency signal, xek,l , to the transmitted signal, s[i ], using a set of time-frequency basis functions, gk,l [i ], s [i ] = M −1 N −1 ∑ ∑ k=0 l =0 xek,l gk,l [i ]. (6.4) The received signal samples after transmission over a linear time varying (LTV) channel can be obtained as r [i ] = L −1 ∑ h[i, l ]s[i − l ] + w[i], l =0 (6.5) 72 where h[i, l ] is the instantaneous channel impulse response with length L at time instant l, and w[i ] is the additive channel noise. To obtain an estimate of dm,n from the received signal r[i ], first, we map r[i ] to the time-frequency lattice by projecting it onto another set of time-frequency basis functions f k,l [i ] as yek,l =< f k,l [i ], r[i ] > . (6.6) < f k,l [i], gk′ ,l ′ [i] >= δ(k − k′ )δ(l − l ′ ). (6.7) The sets of transmit and receive basis functions are designed to satisfy the perfect reconstruction criterion Then, the receiver performs the windowing operation through multiplication of the receive window function coefficients vek,l to the time-frequency samples as yk,l = yek,l vek,l . (6.8) Finally, signals on the time-frequency lattice are transformed back to the Doppler-delay domain by applying SFFT as 1 dˆm,n = √ MN M −1 N −1 ∑ ∑ mk nl yk,l e j2π ( M − N ) . (6.9) k=0 l =0 Similar to SFFT−1, the SFFT transform can be split into an M-point IDFT and an N-point DFT on rows and columns of its operand. Thus, (6.9) can be rearranged as D̂ = F H M YF N , (6.10) where Y ∈ C M× N and D̂ ∈ C M× N contain elements {yk,l } and {dˆm,n }, respectively. Typically, an OFDM-based OTFS modulation with M subcarriers and a cyclic prefix (CP) of length Mcp utilizes the following transmit and receive time-frequency basis functions 2π 1 gk,0 [i ] = √ e j M ki , − Mcp ≤ i < M M gk,l [i ] = gk,0 [i − M + Mcp l ] 2π 1 f k,0 [i ] = √ e− j M ki , 0 ≤ i < M M f k,l [i ] = f k,0 [i − M + Mcp l ]. (6.11) (6.12) 73 6.2 OFDM-based OTFS: Single Antenna In this section, we present a vectorized formulation for the OFDM-based OTFS single antenna transceiver. The first step of the OTFS modulation in (6.2), that is the SFFT−1 operation, using (A.7) can be rearranged as x = FH ⊗ F M d, N (6.13) where x = vec(X) and d = vec(D). Then, the vector x is partitioned into blocks of length M, denoted by xn , n = 0, 1, · · · , N − 1 and each block is multiplied to the corresponding transmit window as e xn = U n xn , (6.14) where the transmit window Un ∈ C M× M is a diagonal matrix consisting of diagonal elements uek,n , for k = 0, 1, · · · , M − 1. Stacking the results as e x, we have e x = Ux (6.15) where U ∈ C MN × MN is a diagonal matrix whose (lM + k)-th diagonal element is uek,l . Each partition of e x, which is e xn , is fed into an OFDM modulator. The OFDM modulator multiplies an IDFT matrix F H M to each block and then appends a CP to each block as xn , sn = F H Me e sn = Acp sn , (6.16) (6.17) where Mcp is the CP length, Acp = [Gcp , I M ]T is the CP addition matrix, and Gcp ∈ C M× Mcp includes the last Mcp columns of the identity matrix I M . Note that Acp appends last Mcp samples of each OFDM block to its beginning. Stacking the results as vectors, (6.16) and (6.17) can be written as x, s = IN ⊗ F H M e e s = IN ⊗ Acp s. as (6.18) (6.19) After the signal e s is passed though the LTV channel, the received signal can be written er = He s + w, (6.20) 74 where H ∈ C N ( M+ Mcp )× N ( M+ Mcp) is the channel impulse response matrix, and w is the channel additive noise vector. The receiver partitions the received vector into N blocks, ern and removes CP from each received block as rn = Rcpern , (6.21) where Rcp ∈ C M×( M+ Mcp ) is the CP removal matrix, which can be obtained by removing the first Mcp rows of I M+ Mcp . Stacking the output vectors rn into a length NM vector r, (6.19) to (6.21) can be written as r = IN ⊗ Rcp H IN ⊗ Acp s + w e + w, = Hs (6.22) e = IN ⊗ Rcp H IN ⊗ Acp is an MN × MN block diagonal matrix where H  e 0 0M · · · H ¯e 0  ¯ M H1 · · · e H= . .. ..  .. . . 0M 0M · · · ¯ ¯  0M ¯ 0M  ¯.   ..  e N −1 H (6.23) e n is the channel impulse response matrix of the n-th OFDM symbol. and H The received signal is fed into an OFDM demodulator and the output signal is y = (IN ⊗ F M ) r, e (6.24) where e y is a length MN time-frequency signal vector. Finally, after performing the receiver windowing operation, the time-frequency signal is mapped to the Doppler-delay domain and an estimate of the transmitted vector can be obtained as d̂ = F N ⊗ F H M y, (6.25) where y = Ve y, and the receive window V ∈ C MN × MN is a diagonal matrix defined similarly to that of the transmit window. 75 We note that equations (6.13) to (6.25) describe the end-to-end relationship for a general OFDM-based OTFS setup, which is summarized in a compact form as Remove CP d̂ = F N ⊗ | {z FH M SFFT z }| { window z}|{ V (IN ⊗ F M ) IN ⊗ Rcp |{z} H {z } | } Channel OFDM Demod. z Add CP }| { Window z}|{ IN ⊗ Acp IN ⊗ F H U FH M N ⊗ F M d + ŵ. | | {z } {z } OFDM Mod. (6.26) Inv. SFFT Equation 6.26 clearly demonstrates the contribution of each signal processing step of an OFDM-based OTFS setup from Doppler-delay domain to Doppler-delay domain where each step corresponds to one section of the transceiver block diagram shown in Figure 6.1. It is clear that this compact form of (6.26) will allow simplified analysis and implementation of OFDM-based OTFS systems. Next, we will present how these equations simplify for some special setup considered below. 6.2.1 Separable windows In general, transmit and receive window functions can be arbitrary functions that are designed for various purposes. These include, for instance, zero-forcing or minimum mean squared error equalization, or shortening the channel response in the Doppler-delay domain. For a wide range of applications, however, one can treat windowing across time axis and across frequency axis separately because of the independency between channel delay spread and Doppler spread. This separate windowing can lead to simplified transmitter and receiver structures as shown in [102]. Assume that the transmit and receive windows are separable functions, that is uek,l = al bk and vek,l = pl qk . We can write transmit and receive window matrices as U = A ⊗ B and V = P ⊗ Q where A ∈ C N × N , P ∈ C N × N , B ∈ C M× M , and Q ∈ C M× M are diagonal matrices with elements {al }, { pl }, {bk }, and {qk }, respectively. Substituting these window forms in (6.15) to (6.25) and after simplifications we have e d̂ = IN ⊗ F H Q F M (F N ⊗ I M ) (P ⊗ I M ) H M (6.27) IN ⊗ F H (A ⊗ I M ) F H M BF M d + ŵ, N ⊗ IM where ŵ = F N ⊗ F H M V (I N ⊗ F M ) w. We note that (6.27) can be used for the simplifi- cation of OFDM-based OTFS transceiver. 76 6.2.2 Rectangular window When rectangular windows are used as the transmitter and receiver window functions, which is U = V = I MN , substituting (6.15) to (6.24) in (6.25), it simplifies to e FH d̂ = (F N ⊗ I M ) H ⊗ I M d + ŵ, N (6.28) where we used the mixed-product property of Kronecker product, that is, (A ⊗ B) (C ⊗ D) = (AC) ⊗ (BD) . (6.29) 6.2.3 Frequency domain representation e f denote the block diagonal frequency domain channel impulse response matrix. Let H We see that e IN ⊗ F H e f = (I N ⊗ F M ) H H M . (6.30) e we obtain Solving this for H, e e = IN ⊗ F H H M H f (I N ⊗ F M ) . (6.31) e in (6.26), after simplification, it boils down to Substituting H H e V H U F ⊗ F d̂ = F N ⊗ F H M d + ŵ, f N M (6.32) which shows that transmit and receive windows directly change the frequency domain channel impulse response matrix. In other words, the effective frequency domain channel e f U. is VH When the LTV channel varies slowly such that it is approximately invariant over each e n } become circulant. Accordingly, H e f is an MN × MN OFDM symbol, then the matrices {H e f ,n = diagonal matrix that contains N diagonal channel frequency response matrices, H e FH M Hn F M . In this case, (6.32) is equivalent to the 2D circular convolution of the transmit data matrix with the effective 2D channel impulse response matrix. 77 6.3 OFDM-based OTFS: MIMO Consider a nt × nr MIMO setup that utilizes OFDM-based OTFS modulation. Let us stack transmit data matrices of all antennas to form an Mnt × N matrix D as   D0  D1    D= , ..   . (6.33) Dnt −1 where Dt is the M × N data matrix of t-th antenna. Applying SFFT−1 to each submatrix Dt , we obtain X = ( I n t ⊗ F M ) DF H N, (6.34) where F M is repeated as Int ⊗ F M to cover all data matrices of all antennas. Using (A.7), (6.34) can be written as x = FH N ⊗ Int ⊗ F M d, where d is the vectorized version of D defined as follows.     d0n d0  d1   d1n      d =  .  , dn =  ..  , dtn =  ..   .  dnnt −1 d N −1 (6.35)      t d0,n t d1,n .. . dtM−1,n    ,  (6.36) and dtm,n ; t = 0, 1, · · · , nt − 1 is the data symbol of t-th transmit antenna. After partitioning x to N blocks, xn , the window function is multiplied to each block. Assuming that all transmit antennas use the same window function, we get x̆n = Un xn , (6.37) where Un = Int ⊗ Un . We stack these results to obtain x̆ = Ux, (6.38) where  U0 0 Mnt ¯ 0  ¯ Mnt U1 U= . ..  .. . 0 Mnt 0 Mnt ¯ ¯ ··· ··· .. . 0 Mnt ¯ 0 Mnt ¯ . .. ··· U N −1    .  (6.39) 78 The OFDM modulator transforms the time-frequency domain signal of each antenna to the time domain signal as sn = In t ⊗ F H M x̆n , s = INnt ⊗ F H M x̆. (6.40) (6.41) The transmitter appends the CP to the transmit signal and sends the result through an LTV channel. At the receiver, after removing the CP, the received signal can be written in a compact form as rn = Inr ⊗ Rcp Hn Int ⊗ Acp sn + wn = H n sn + w n , where Hn = Inr ⊗ Rcp Hn Int ⊗ Acp can be expanded as  0,0 0,1 0,n −1 Hn Hn ··· Hn t  1,0 1,1 1,n −1  Hn Hn ··· Hn t  Hn =  . .. .. .. .  .. . . n −1,0 Hnr n −1,1 Hnr ··· n −1,nt −1 Hn r r,t    .   (6.42) (6.43) Here, Hn is the n-th channel impulse response matrix between t-th transmit and r-th receive antenna. We stack the output vectors to obtain r = Hs + w, where   0 Mnr × Mnt · · · 0 Mnr × Mnt ¯ ¯ 0 H1 · · · 0 Mnr × Mnt   ¯ Mnr × Mnt  ¯ . H= . .. .. .. .   . . . . H N −1 0 Mnr × Mnt 0 Mnr × Mnt · · · ¯ ¯ The received signal is fed to an OFDM demodulator and the output signal is (6.44) H0 y̆ = (INnr ⊗ F M ) r, (6.45) (6.46) where y̆ is the length MNnr time-frequency signal vector. Finally, after multiplying by the receiver window, the time-frequency signal is mapped to the Doppler-delay domain and the estimate of the transmitted vector can be obtained as d̂ = F N ⊗ Inr ⊗ F H M Vy̆, (6.47) 79 where  V0 0 Mnr ¯ 0  ¯ Mnr V1 V= . ..  .. . 0 Mnr 0 Mnr ¯ ¯ ··· ··· .. . 0 Mnr ¯ 0 Mnr ¯ . .. ··· V N −1      (6.48) e n . Using (6.35) to (6.47), end-to-end vectorized relationship for the and Vn = Inr ⊗ V general MIMO OFDM-based OTFS setup can be obtained as Window Remove CP }| z z}|{ { I ⊗ R V I ⊗ F H d̂ = F N ⊗ Inr ⊗ F H ) ( cp Nn Nn M M r |{z} | r {z } | {z } Channel OFDM Demod. SFFT z Window Add CP }| z}|{ { H ⊗ I ⊗ F U F INnt ⊗ Acp INnt ⊗ F H nt M d+w N M | {z } {z } | OFDM Mod. Inv. SFFT (6.49) 6.4 Ergodic Capacity Analysis of MIMO OFDM-based OTFS In this section, we examine the ergodic capacity of MIMO OFDM and MIMO OFDMbased OTFS systems. While the analysis here applies to general time-varying channels, it is important to note that due to the use of cyclic prefix, the transmission of an OTFS block of symbols consists of N consecutive transmissions of OFDM blocks, each with a block length M + Mcp in the time domain. Assuming that the receiver knows the channel perfectly, one (larger) block of OTFS transmission is equivalent to N parallel transmissions of OFDM blocks. Furthermore, we assume that the channel is independent from one OTFS block to the next and the channel is ergodic. These form the underlining key assumptions of the capacity analysis below. Consider an arbitrary OTFS transmission block. Since there is a one-to-one mapping between Doppler-delay domain data symbols d and time-frequency data samples x, the mutual information between the OTFS received signal vector r and transmit data vector d, which is I (d; r), can be written as I (d; r) = I (x0 , x1 , · · · , x N −1 ; r0 , r1 , · · · , rN −1 ). (6.50) 80 Since the received signal rn only depends on xn , that is rn = H n xn + wn , where H n represents all signal processing steps that relate xn to rn including modulation and channel impact. Then, (6.50) can be written as I (d; r) = N −1 ∑ I ( xn ; rn ) , (6.51) n =0 due to the fact that each OTFS transmission is realized by transmissions over a set of N parallel channels. Furthermore, each parallel channel is an MIMO channel rn = H n xn + wn with known channel state information at the receiver. Since the capacity-achieving input distribution for such a channel is the zero-mean circularly symmetric complex Gaussian distribution CN (0, I M ) [103], we conclude that such distribution is indeed capacityachieving for the proposed MIMO OFDM-based OTFS system and the resulting ergodic capacity equals to the average sum capacities of individual parallel channels, which equals with the ergodic capacity of the OFDM system considered here [103]. In the following, for an OFDM-based OTFS setup, we calculate I (d; r) and show that (6.51) holds for OFDM-based OTFS modulation and we proceed to calculate the capacity of OFDM-based OTFS setup. We recall that the differential entropy of an n-element complex Gaussian vector q with covariance matrix Cq is [89] h(q) = log2 (2πe)n |Cq | (6.52) where | · | is the determinant operator. Moreover, the mutual information between two complex Gaussian vectors q and r is the difference between the differential entropies h(q) and h(q|r). This is written as I (q; r) = h(q) − h(q|r) | Cq | . = log2 | Cq | r | (6.53) Substituting (6.37) and (6.40) in (6.42), the received signal of the n-th OFDM transmission can be obtained as rn = H n In t ⊗ F H M U n xn + w n . (6.54) Then, the mutual information between the OFDM received signal vector rn and the transmit data vector xn is I (rn ; xn ) = log2 2 |Kn I Mnt KH n + σ I Mnr | |σ2 I Mnr | (6.55) 81 where Kn = Hn Int ⊗ F H M Un . Similarly, the received signal of OFDM-based OTFS can be obtained by substituting (6.35) and (6.38) in (6.44) as H r = H INnt ⊗ F H M U F N ⊗ Int ⊗ F M d + w. (6.56) The mutual information between the OFDM-based OTFS received signal vector r and the transmit data vector d is given as |KI MNnt KH + σ2 I MNnr | (6.57) |σ2 I MNnr | H where K = H INnt ⊗ F H M U F N ⊗ In t ⊗ F M . H H I ⊗ F ⊗ I ⊗ F is a unitary matrix and Note that F H n Nn M M U is a block diagonal N t t I (r; d) = log2 matrix, thus we have   K0 KH ··· 0 Mnr 0 H ¯ 0  0 Mnr  ¯ Mnr K1 K1 · · ·  H ¯ KK =  . . .. .. ..  ..  . . . H 0 Mnr 0 Mnr · · · K N −1 K N −1 ¯ ¯ Substituting (6.58) in (6.57), we have I (r; d) = = N −1 ∑ n =0 N −1 ∑ log2 2 | K n KH n + σ I Mnr | |σ2 I Mnr | (6.58) I ( rn ; xn ) . (6.59) n =0 Hence, the ergodic capacity can be obtained as COTFS = COFDM 2 | K n KH 1 n + σ I Mnr | E log2 . = M + Mcp |σ2 I Mnr | (6.60) While in the above analysis we reached the conclusion that OTFS has the same ergodic capacity as OFDM, it is important to note that such analysis assumes perfect knowledge of the (time-varying) channel, an optimal detector, and an infinite code length. In a practical receiver design, however, the sparsity and lower variability of the OTFS channel in the Doppler-delay domain yield great benefits over OFDM, especially for higher Doppler channels. In such scenarios, an OFDM receiver needs to track rapid channel variations 82 in the time domain. Here, because of the channel variation over each OFDM symbol, keeping track of such variations turns out to be a difficult task. In OTFS, on the other hand, channel variation in time averages out and translates to a much slower variation in the Doppler-delay domain. In addition, it results in a sparse channel that will be easier to estimate. These benefits of OTFS, clearly, enable a simpler channel estimation and equalization design and hence reduce overhead and complexity of the receiver. 6.5 Numerical Simulation Section 6.4 claims that performance of OFDM and OFDM-based OTFS are the same when an optimum receiver is utilized. On the other hand, when a low complexity receiver is used, OTFS outperforms OFDM. In order to verify this claim, we evaluate the codeword error rate (CER) performance of two receivers for OFDM and OFDM-based OTFS through numerical simulations. MMSE-based turbo equalizer [99, 104] with a convolutional code is used as a low complexity receiver. Moreover, since complexity of the optimum detector is prohibitive, we use a Genie-aided Successive Interference Cancellation (G-SIC) detector [105] with Turbo code as our near-optimum receiver. For both receivers, detector and decoder exchange soft information to improve receiver performance based on Turbo principles. We examine a system with M = 32 subcarriers, 15 kHz carrier spacing, and N = 8 OFDM blocks. We simulate the wireless fading channel according to the 3GPP standardized channel models, TDL-C channel model, with 300 ns delay spread and maximum Doppler spread of 500 Hz [106]. Channel state information at the receiver is ideal. Both receivers use 1/3 code rates and detector and decoder perform five outer turbo iterations. Each codeword contains 30720 bits (30 OTFS blocks). Figure 6.2 presents BER performance of both receivers for 16-QAM modulation. For the low complexity receiver (MMSE-based turbo equalizer with convolutional code), OTFS outperforms OFDM by 2 dB. On the other hand, the near-optimum detector (Genie-aided Successive Interference Cancellation) improves CER performance of both OFDM and OTFS and decreases the performance gap to 1 dB. These observations match our results in the previous section that for the optimum receiver, OTFS and OFDM have the same performance. 83 10 OFDM MMSE Convolutional OFDM G-SIC Turbo OTFS MMSE Convolutional OTFS G-SIC Trubo 0 10 -1 10 -2 10 -3 4 5 6 7 8 9 10 Figure 6.2. OTFS transmitter and receiver structure. 6.6 Summary In this chapter, we conducted a discrete-time analysis for MIMO OFDM-based OTFS modulation. Such analysis led to a concise, vectorized input-output relationship that is applicable to general time-varying channels with arbitrary Dopplers and windowing functions. We provided an accurate characterization of the ergodic capacity that shows that both OFDM and OTFS achieve the same ergodic capacity despite great benefits of the latter in practical receiver design. The analysis developed here provides a strong theoretical foundation for the design of practical detectors/equalizers for OTFS systems. CHAPTER 7 CONCLUSIONS AND FUTURE RESEARCH The fifth generation of cellular networks is going to redefine the concept of wireless networks by providing a vast range of new services. These novel services necessitate innovations in different parts of the network to fulfill the expected requirements. One of the challenging aspects of 5G networks design is finding a suitable waveform for its physical layer. Several candidate waveforms have been proposed to replace the currently dominant waveform of 4G LTE, the CP-OFDM. In this dissertation, we focused on three major signaling techniques and studied different aspects of these waveforms. The waveforms were Generalized Frequency Division Multiplexing, Windowed Circular FBMC-OQAM (C-FBMC, for short), and Orthogonal Time-Frequency Space (OTFS). We presented the signal processing steps needed for building GFDM and C-FBMC transmit signal and recovery of the transmitted data symbols at the receiver. We presented a novel low complexity implementation of transmitter for both waveforms and derived computational complexity of the proposed structures. Our results showed that the complexity of implementation of both waveforms is almost the same. Then, we focused on receiver structure of GFDM and C-FBMC. We discussed that C-FBMC is able to extract the data symbols by utilizing a low complexity matched filter receiver, while, GFDM, being a nonorthogonal waveform, cannot rely on a simple matched filter for data symbol recovery. Thus, a computationally complex successive interference cancellation procedure is needed to remove the interference and bring BER performance of GFDM to an acceptable range. We thus concluded that C-FBMC is a superior waveform over GFDM. 5G waveforms must be able to successfully deliver data symbols in asynchronous scenarios. We studied OOB emissions of C-FBMC and identified the main sources of its OOB emissions. We derived equations that quantify its OOB emissions for any arbitrary window function and we looked at a few OOB suppression techniques to improve its spec- 85 tral confinement. Then, we quantified interference leakage of an arbitrary asynchronous data symbol and determined the important contributors to that interference leakage. We recognized that data symbols near the packet boundaries generate the majority of the asynchronous interference. At the same time, they are the most vulnerable symbols to be influenced by interference from other asynchronous symbols. We showed that the use of windowing can significantly reduce OOB emissions and MUI of C-FBMC. We extended our finding to GFDM through numerical simulations where we showed that GFDM follows the same trend. We inspected FBMC-OQAM from an information theoretic perspective to study the possibility of improving reception performance by using its imaginary interference. The imaginary interference is normally discarded. We derived equations for information content of the signals in the transceiver chain of an FBMC-OQAM setup. By carefully inspecting these signals, we concluded that channel equalization does not incur any information loss; however, discarding imaginary interference results in some information loss. Our numerical simulations showed that this information loss is very marginal and we attributed this to transient response of the FBMC-OQAM waveform. Then, we proved that information loss of the circular version of FBMC-OQAM, that is, C-FBMC, is zero and as a result, the real part of the C-FBMC matched filter receiver preserves all of the received signal information. Next, we developed a more general proof for the capacity analysis of the C-FBMC waveform where we analyzed C-FBMC from a pure signal processing point of view. Our analysis revealed a number of interesting properties of the matrices involved in C-FBMC transmission and reception. We proved that regardless of the nature of wireless channel and statistical properties of additive noise, the real and imaginary parts of C-FBMC matched filter output are related through an orthonormal transformation. As a result, they carry the same information content and discarding imaginary interference does not incur any performance loss. Finally, we focused on the OTFS waveform and presented a novel vectorized formulation that describes all of the signal processing steps of a MIMO OFDM-based OTFS transceiver. This formulation streamlines analysis and detector design of the OTFS waveform. We also derived the ergodic capacity of the MIMO OFDM-based OTFS setup for a 86 time varying channel and we proved that the ergodic capacity of the MIMO OFDM-based OTFS and MIMO OFDM are the same. This clarified that even though OTFS outperforms OFDM when suboptimum receivers are utilized, with optimum receivers, they deliver the same performance. 7.1 Future Research The analysis presented in this dissertation clarifies some important aspects of GFDM, C-FBMC, and OTFS waveforms. However, more analyses are needed to fully comprehend the performance of these waveforms in 5G networks. Moreover, the methodology used in the dissertation can be applied to other waveforms too. Here, we briefly highlight some possible future research topics. 7.1.1 Detector design for OTFS The end-to-end formulation that we presented in this dissertation lays the foundation for detector design of the OFDM-based OTFS setup. At the OTFS transmitter, all data symbols are mixed by inverse symplectic finite Fourier transform. To achieve a near optimum performance, the receiver must jointly detect all the symbols, which results in a very high computational complexity. Recently, a few low complexity suboptimum detectors have been proposed in the literature [100, 101, 107]. However, no near optimum receiver with an acceptable computational complexity has been reported yet. Moreover, since most of the available designs assume ideal channel estimation, a detailed study on different channel estimation methods and their impact on receiver performance seems necessary. 7.1.2 OOB and MUI analysis of other waveforms In Chapter 3, OOB and MUI analysis of C-FBMC and GFDM were presented and impact of OOB and MUI improvement methods was quantified. The approach taken there can be generalized to cover all OFDM-based candidate waveforms like F-OFDM, WOLA-OFDM, and UFMC. Thus, extension of the presented analysis to those waveforms is straightforward, which can provide more analytical insight on their performance. Since, 5G standard committees recently decided to use OFDM-based waveforms, such analysis is essential for successful implementation of these waveforms. 87 7.1.3 OTFS waveform with different multicarrier In this dissertation, we focused on the OFDM-based OTFS waveform, however, OTFS signaling can be combined with many other multicarrier waveforms to provide a broad range of capabilities. Such a combined waveform can benefit from both the diversity extraction nature of OTFS and improved spectral confinement of the underlying multicarrier waveform. To the best of our knowledge, such combinations are not studied in the literature and we believe such merger has a potential to solve many problems in time varying and time invariant scenarios. APPENDIX A KRONECKER PRODUCT PROPERTIES In this dissertation, we make frequent use of the Kronecker product to arrive at a concise and elegant representation of the input-output relationship in OTFS systems. Hence, in this section, we first review some of the relevant Kronecker properties. For an m × n matrix A and a p × q matrix B, the Kronecker product A ⊗ B is a mp × nq block matrix   a11 B a12 B · · · a1n B  a21 B a22 B · · · a2n B    A⊗B =  . .. ..  . . . .  . . . .  am1 B am2 B · · · amn B (A.1) The Kronecker product is an associative but noncommutative operator A ⊗ B 6= B ⊗ A, (A.2) (A ⊗ B) ⊗ C = A ⊗ (B ⊗ C) . (A.3) The mixed-product property of the Kronecker product is (A ⊗ B) (C ⊗ D) = (AC) ⊗ (BD) . (A.4) Moreover, under transpose and Hermitian operations, order of the Kronecker product operands does not change ( A ⊗ B )H = AH ⊗ B H . (A.5) Finally, consider a matrix equation AXB = C, where A, X, B, and C are proper size matrices. We can rewrite this equation as BT ⊗ A vec(X) = vec(C), (A.6) (A.7) where vec(X) denotes the vectorized version of the matrix X formed by stacking the columns of X into a single column vector. APPENDIX B ACRONYMS AND ABBREVIATIONS 1G 2G 3G 3GPP 4G 5G ADSL AWGN BER BS CDMA CER C-FBMC CFO CP CS CSI DFT DSL DTFT eMBB ESD F-OFDM FBMC FDMA FDM FFT FMT GFDM G-SCI ICI IDFT IFFT i.i.d IMT IoT ISI ITU First generation of wireless communication systems Second generation of wireless communication systems Third generation of wireless communication systems Third Generation Partnership Project Fourth generation of wireless communication systems Fifth generation of wireless communication systems Asymmetric Digital Subscriber Line Additive White Gaussian Noise Bit Error Rate Base Station Code Division Multiple Access Codeword Error Rate Circular Filter Bank MultiCarrier Carrier Frequency Offset Cyclic Prefix Cyclic Suffix Channel State Information Discrete Fourier Transform Digital Subscriber Line Discrete-Time Fourier Transform Enhanced Mobile Broadband Energy Spectral Density Filtered-OFDM Filter Bank MultiCarrier Frequency Division Multiple Access Frequency Division Multiplexing Fast Fourier Transform Filtered MultiTone Generalized Frequency Division Multiplexing Genie-aided Successive Interference Cancellation Intercarrier Interference Inverse Discrete Fourier Transform Inverse Fast Fourier Transform independent and identically distributed International Mobile Telecommunications Internet of Things Intersymbol Interference International Telecommunication Union 90 KPI LTE LTI LTV M2M MF MIMO ML mMTC MMSE MUI NVF OFDMA OFDM OOB OQAM OTFS PAM PAPR PHY QAM QoS RAT RRC SC-FDMA SDN SIC SINR SIR SFFT−1 SFFT SNR SVD TV UFMC UF-OFDM uRLLC VoIP WCP-COQAM WOLA-OFDM WSN ZF ZP Key Performance Indicators Long Term Evolution Linear Time Invariant Linear Time Varying Machine to Machine Matched Filter Multiple Input Multiple Output Maximum Likelihood Massive Machine Type Communications Minimum Mean Square Error MultiUser Interference Network Function Virtualization Orthogonal Frequency Division Multiple Access Orthogonal Frequency Division Multiplexing Out of Band Offset Quadrature Amplitude Modulation Orthogonal Time-Frequency Space Pulse Amplitude Modulation Peak to Average Power Ratio Physical Layer Quadrature Amplitude Modulation Quality of Service Radio Access Technology Root Raised Cosine Single Carrier - Frequency Division Multiple Access Software Defined Network Successive Interference Cancellation Signal to Interference plus Noise Ratio Signal to Interference Ratio Inverse Symplectic Finite Fourier Transform Symplectic Finite Fourier Transform Signal to Noise Ratio Singular Value Decomposition Time Varying Universal Filtered MultiCarrier Universally Filtered OFDM Ultra-reliable and Low Latency Communications Voice over IP Windowed CP-Circular OQAM Weighted Overlap and Add OFDM wireless sensor networks Zero Forcing Zero Prefix REFERENCES [1] B. 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