Performance of SC-FDMA with diversity techniques over land mobile satellite channel

Universidad De Málaga Escuela Técnica Superior de Ingeniería de Telecomunicación Tesis Doctoral PERFORMANCE OF SC-FDMA WITH DIVERSITY TECHNIQUES OVER LAND MOBILE SATELLITE CHANNEL Autora Jyoti Ramesh Gangane Master in Electronics and Telecommunication Master Universitario en Technologias de Telecommunicacion Directores Mª Carmen Aguayo Torres y Juan Jesús Sánchez Sánchez Doctores Ingenieros de Telecomunicación Año 2016 AUTOR: Jyoti Ramesh Gangane http://orcid.org/0000-0003-3883-6906 EDITA: Publicaciones y Divulgación Científica. Universidad de Málaga Esta obra está bajo una licencia de Creative Commons Reconocimiento-NoComercialSinObraDerivada 4.0 Internacional: http://creativecommons.org/licenses/by-nc-nd/4.0/legalcode Cualquier parte de esta obra se puede reproducir sin autorización pero con el reconocimiento y atribución de los autores. No se puede hacer uso comercial de la obra y no se puede alterar, transformar o hacer obras derivadas. Esta Tesis Doctoral está depositada en el Repositorio Institucional de la Universidad de Málaga (RIUMA): riuma.uma.es Table of Contents Table of Contents iii List of Tables vi List of Figures viii Abstract xvii 1 Introduction 3 1.1 Motivation..................................... 3 1.2 Contributions of this Dissertation . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Outline of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 RelatedPublications ............................... 9 2 OFDM and SC-FDMA Fundamentals 11 2.1 Introduction.................................... 11 2.2 Orthogonal Frequency Division Multiplexing . . . . . . . . . . . . . . . . . . 12 2.3 Single Carrier Frequency Division Multiple Access . . . . . . . . . . . . . . . 13 2.4 Conclusions .................................... 15 3 Wireless Channel Models 17 3.1 Introduction to Wireless Channel . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2 Rayleigh and Rice Fading . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.2.1 RayleighFading.............................. 18 3.2.2 RiceFading ................................ 18 3.3 FrequencySelectivity............................... 19 3.3.1 Flat Fading Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.3.2 Frequency Selective Channels . . . . . . . . . . . . . . . . . . . . . . 19 3.4 Single-Input Multiple-Output Channel Model . . . . . . . . . . . . . . . . . 19 3.5 Satellite Channel Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.5.1 Shadowed Rice Land Mobile Satellite Channel Model . . . . . . . . . 22 3.5.2 Channel Frequency Response Models . . . . . . . . . . . . . . . . . . 23 iii 3.6 Conclusions .................................... 24 4 SC-FDMA BER Analysis over Shadowed Rice LMS Channel 25 4.1 Introduction.................................... 25 4.2 SystemModel................................... 25 4.3 SINRanalysis................................... 28 4.3.1 ZFFDE.................................. 28 4.3.2 MMSEFDE................................ 30 4.4 NumericalResults................................. 35 4.4.1 CDF of β................................. 36 4.4.2 BERResults ............................... 37 4.5 Conclusions .................................... 38 5 SC-FDMA Spectral Efficiency over Shadowed Rice LMS Channel 45 5.1 Introduction.................................... 45 5.2 Overview of Adaptive Modulation and Coding . . . . . . . . . . . . . . . . . 45 5.3 Spectral Efficiency Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.3.1 ZF-FDEAnalysis............................. 47 5.3.2 MMSE-FDEAnalysis........................... 48 5.4 Simulation Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 48 5.5 Conclusions .................................... 50 6 SC-FDMA Performance over Receiver Diversity Techniques 57 6.1 Introduction.................................... 57 6.2 Receiver Diversity Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . 58 6.2.1 Maximal Ratio Combining (MRC) . . . . . . . . . . . . . . . . . . . 60 6.2.2 Equal Gain Combining (EGC) . . . . . . . . . . . . . . . . . . . . . . 61 6.3 SC-FDMA SIMO System Model . . . . . . . . . . . . . . . . . . . . . . . . . 61 6.4 Simulation Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 63 6.4.1 Results and Discussion of SC-FDMA performance with MRC Diversity 64 6.4.2 Results and Discussion of SC-FDMA Performance with EGC Diversity 71 6.5 Conclusions .................................... 71 7 Conclusions and Future Scope 77 7.1 Synthesis of Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 7.2 Contributions ................................... 78 7.3 FutureScope ................................... 79 A Rendimiento de SC-FDMA con t´ecnicas de diversidad para canales m´oviles satelitales 81 A.1 Introducci´on.................................... 81 A.2 Descripci´on de OFDM y SC-FDMA . . . . . . . . . . . . . . . . . . . . . . . 84 iv A.3 Modelosdecanal ................................. 86 A.3.1 Modelo de canal LMS Rice sombreado . . . . . . . . . . . . . . . . . 86 A.3.2 Modelos de respuesta en frecuencia . . . . . . . . . . . . . . . . . . . 87 A.3.3 Modelo de canal para diversidad de antenas en recepci´on . . . . . . . 88 A.4 Probabilidad binaria de error para canales LMS . . . . . . . . . . . . . . . . 89 A.4.1 Modelodesistema ............................ 89 A.4.2 An´alisisdelaSINR............................ 90 A.4.3 Resultados................................. 95 A.5 Eficiencia espectral de SC-FDMA para el canal LMS . . . . . . . . . . . . . 98 A.5.1 Revisi´on de la modulaci´on adaptativa . . . . . . . . . . . . . . . . . . 99 A.5.2 An´alisis de la eficiencia espectral . . . . . . . . . . . . . . . . . . . . 100 A.5.3 Resultados................................. 103 A.6 Rendimiento de SC-FDMA para t´ecnicas de diversidad en recepci´on . . . . . 103 A.6.1 T´ecnicas de diversidad en recepci´on . . . . . . . . . . . . . . . . . . . 103 A.6.2 Modelo de sistema SC-FDMA SIMO . . . . . . . . . . . . . . . . . . 105 A.6.3 Resultados................................. 109 A.7 Conclusiones y l´ıneas futuras . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 v vi List of Tables 3.1 Parameters for considered fading channels [19]. . . . . . . . . . . . . . . . . . 23 4.1 Parameters for SC-FDMA over LMS channel . . . . . . . . . . . . . . . . . . 36 5.1 Switching thresholds of the SNR {γi}for adaptive modulation and coding . . 49 6.1 Simulation Parameters for SIMO SC-FDMA . . . . . . . . . . . . . . . . . . 63 A.1 Par´ametros para los canales con desvanecimientos considerados [19] . . . . . 87 A.2 Par´ametros de simulaci´on para SC-FDMA en canales LMS. . . . . . . . . . . 95 A.3 Umbrales entre regiones de modulaci´on de SNR y eficiencia espectral por constelaci´on.................................... 103 A.4 Par´ametros de simulaci´on para SIMO SC-FDMA . . . . . . . . . . . . . . . 110 vii viii List of Figures 2.1 OFDMATransmitter............................... 13 2.2 OFDMAReceiver................................. 13 2.3 Transmission scheme in SC-FDMA . . . . . . . . . . . . . . . . . . . . . . . 14 2.4 Reception scheme in SC-FDMA . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.1 Receiver Diversity combining scheme. . . . . . . . . . . . . . . . . . . . . . . 20 4.1 Transmission scheme in SC-FDMA . . . . . . . . . . . . . . . . . . . . . . . 26 4.2 Reception scheme in SC-FDMA . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.3 CDF of βfor heavy shadowing with MMSE and ZF equalizer for uncorrelated responses at different subcarriers . . . . . . . . . . . . . . . . . . . . . . . . 39 4.4 CDF of βfor light shadowing with MMSE and ZF equalizer for uncorrelated responses at different subcarriers . . . . . . . . . . . . . . . . . . . . . . . . 39 4.5 CDF of βfor heavy shadowing with MMSE and ZF equalizer for correlated responses at different subcarriers and interleaved allocation (IFDMA) . . . . 40 4.6 CDF of βfor light shadowing with MMSE and ZF equalizer for correlated responses at different subcarriers and interleaved allocation (IFDMA) . . . . 40 4.7 CDF of βfor heavy shadowing with MMSE and ZF equalizer for correlated responses at different subcarriers and localized allocation (LFDMA) . . . . . 41 4.8 CDF of βfor light shadowing with MMSE and ZF equalizer for correlated responses at different subcarriers and localized allocation (LFDMA) . . . . . 41 4.9 BER of OFDM, IFDMA, LFDMA, and independent SC-FDMA with 16QAM for light shadowing, Nc= 64 and ZF equalizer . . . . . . . . . . . . . . . . . 42 4.10 BER of OFDM, IFDMA, LFDMA, and independent SC-FDMA with 16QAM for light shadowing, Nc= 64 and MMSE equalizer . . . . . . . . . . . . . . . 42 ix xvi Laboratory, at the University of Malaga, Spain. I am also thankful to all the teaching and nonteaching staff of ETC Dept., SIT, Lonavala. Lastly, I would like to thank my Parents, Inlaws, and my husband Ramesh Gangane, daughter, Tejali for their strong support. Their loving and caring words were sources of encouragement to me, and I am deeply grateful to them. Abstract The major expansion seen in wireless technologies over last two decades is a direct result of the increasing demand for high data rate transmissions. Third Generation Partnership (3GPP) developed Long Term Evolution (LTE) mobile communication to meet high data rate requirement. Besides cellular communications, Land Mobile Satellite (LMS) systems is a significant part of the wireless systems. The LMS system provides services that are not feasible via Land Mobile Terrestrial (LMT) systems over a wide area of the network with low cost. Next generation of wireless systems is expected to be formed by the convergence of what now are considered independent systems. The demand for high data rate results in significant Inter-Symbol Interference (ISI) for single carrier systems over bandwidth and power limited channels. Overcoming the time and frequency selective nature of the propagation channel requires the use of powerful signal processing techniques. Recent examples include different transmitter/receiver diversity techniques for high data rate transmissions as well as including multiple antennas at transmitter/receiver known as Multiple-Input Multiple-Output (MIMO). Multiple antenna communication technologies provide significant advantages over single antenna systems. These advantages include extended range, improved reliability in fading environments and higher data throughputs. In certain environments (such as the uplink of a mobile link) usually only one antenna is available at the transmission. Thus, only Single-Input-Single-Output (SISO) or Single-Input Multiple-Output (SIMO) transmissions are feasible. Orthogonal Frequency-Division Multiplexing (OFDM) has been a more widely used modulation technique due to its robustness against frequency selective fading channels, scalability, and MIMO compatibility. However, it suffers from a high Peak-to-Average Power ratio (PAPR) which may be particularly troublesome in uplink cellular and satellite downlink transmissions as costly high-power linear amplifiers are needed for transmitting terminals. xvii xviii Single Carrier Frequency-Division Multiple access (SC-FDMA) has become an alternative to OFDM techniques, specifically used as the uplink multiple access schemes in 3GPP LTE. It is able to reduce the PAPR in the transmission, resulting in a relaxation of the constraints regarding power efficiency needed in user terminals and satellite units. The SC-FDMA can be described as a version of OFDMA in which pre-coding and inverse pre-coding stages are included at the transmitter and receiver ends respectively, thus symbols are transmitted in time but after processing in the frequency. Even with the use of OFDMA or SC-FDMA, inter-symbol interference has to be compensated by equalization, which is usually performed in frequency domain. The aim of this dissertation is to provide a mathematical analysis of the performance of SC-FDMA over land mobile satellite channel. For this purpose, the channel will be modelled as a shadowed Rice channel such that its Line Of Sight (LOS) follows Nakagami distribution. We first describe OFDMA and SC-FDMA multicarrier modulation techniques. Then we undertake an analysis of OFDMA and SC-FDMA based on enhanced complex noise at the detector stage. We evaluate the Bit Error Rate (BER) and spectral efficiency performance of SC-FDMA for different depths of shadowing. Finally, SC-FDMA performance with receiver diversity techniques, such as Maximal Ratio Combining (MRC) and Equal Gain Combining (EGC), is also evaluated. Resumen La expansi´on vista en las tecnolog´ıas inal´ambricas durante las ´ultimas dos d´ecadas es el resultado directo de la creciente demanda de las transmisiones de datos a alta velocidad. La asociaci´on Third Generation Partnership (3GPP) desarroll´o el est´andar Long Term Evolution (LTE) para cumplir los requisitos de alta velocidad de datos. Adem´as de las comunicaciones celulares, los sistemas m´oviles terrestres por sat´elite (Land Mobile Satellite, LMS) son una parte importante de los sistemas inal´ambricos. El sistema LMS proporciona servicios que no son factibles a trav´es de sistemas terrestres a trav´es de una amplia red con un bajo costo. Se espera que la pr´oxima generaci´on de sistemas inal´ambricos sea formada por la convergencia de lo que ahora se consideran sistemas independientes. La demanda de la alta velocidad de datos resulta en una importante interferencia entre s´ımbolos (Inter-Symbol Interference, ISI) para los sistemas monoportadora en canales de ancho de banda y potencia limitada. Superar la selectividad en el tiempo y la frecuencia del canal de propagaci´on requiere el uso de potentes t´ecnicas de procesamiento de se˜nales. Ejemplos recientes incluyen diferentes t´ecnicas de diversidad de transmisi´on/recepci´on para alta velocidad de datos, as´ı como el uso de m´ultiples antenas en el transmisor/receptor, en la t´ecnica conocida como Multiple-Input Multiple-Output (MIMO). Estas tecnolog´ıas proporcionan ventajas significativas sobre los sistemas de antenas individuales que incluyen mejor fiabilidad en entornos de desvanecimiento y caudales de datos m´as altas. En ciertos entornos (tales como el enlace ascendente de un enlace m´ovil) ´unicamente una antena est´a disponible en la transmisi´on. Por lo tanto, s´olo son factibles esquemas con entrada individual y salida ´unica (Single Input Single Output, SISO) o transmisiones con entrada ´unica y m´ultiples salidas (Single Input Multiple Output, SIMO). La multiplexaci´on por divisi´on ortogonal en frecuencia (Orthogonal Frequency-Division Multiplexing, OFDM) es una t´ecnica de modulaci´on ampliamente utilizada por su robustez xix xx frente a la selectividad en frecuencia de los canales, su escalabilidad y su compatibilidad MIMO. Sin embargo, sufre de una alta relaci´on de potencia de pico a promedio (Peak-to-Average Power Ratio, PAPR) que puede causar muchos problemas en el enlace descendente de las transmisiones por sat´elite y en el enlace ascendente de las comunicaciones celulares. La raz´on es que se necesitan amplificadores de alta potencia muy lineales, lo que resulta costoso energ´eticamente para la transmisi´on. La t´ecnica monoportadora con acceso m´ultiple por divisi´on de frecuencia (Single Carrier Frequency-Division Multiple Access, SC-FDMA) se ha convertido en una alternativa a OFDM que se utiliza espec´ıficamente en el enlace ascendente de LTE. SC-FDMA es capaz de reducir la PAPR en la transmisi´on, dando lugar a una relajaci´on de las limitaciones en cuanto a la eficiencia de potencia necesaria en los terminales de usuario y las unidades sat´elite. SC-FDMA puede ser descrito como una versi´on de OFDMA en el que se incluyen una etapa de pre-codificaci´on y de pre-codificaci´on inversa en el transmisor y el receptor respectivamente. As´ı, los s´ımbolos se transmiten en tiempo, pero despu´es de ser procesados en la frecuencia. Incluso con el uso de OFDMA o SC-FDMA, la ISI tiene que ser compensada por la igualaci´on, que normalmente se realiza en el dominio de frecuencia. El objetivo de esta tesis es proporcionar un an´alisis matem´atico del comportamiento de SC-FDMA en un canal m´ovil terrestre por sat´elite (Land Mobile Satellite, LMS). Para este prop´osito, el canal se modela como un canal Rice sombreado tal que la l´ınea de visi´on (Line of Sight, LOS) sigue la distribuci´on de Nakagami. En primer lugar, se describen las t´ecnicas de modulaci´on multiportadora OFDMA y SC-FDMA. A continuaci´on, se lleva a cabo un an´alisis de OFDMA y SC-FDMA basado en el ruido complejo recibido a la entrada del detector. Se eval´ua la probabilidad de error de bit (Bit Error Rate, BER) de SC-FDMA para diferentes profundidades del desvanecimiento y de la diversidad de antena en el receptor. Tambi´en se eval´ua su eficiencia espectral para el canal LMS. Por ´ultimo, se abordan las t´ecnicas de diversidad y se eval´uan las t´ecnicas conocidas como Maximal Ratio Combining (MRC) y Equal Gain Combining (EGC). Chapter 1 Introduction 1.1 Motivation Orthogonal Frequency Division Multiplexing (OFDM) has become a very popular transmission technique for wireless communication for two main reasons: a) it provides a lower complexity solution in the presence of severe frequency selectivity at the transmission channel; b) it provides good spectral efficiency. OFDM has been adopted as physical layer technology for important broadband wireless interface standards, such as: IEEE 802.11/Wi-Fi [1], IEEE 802.16/WiMAX [2], 3GPP Long Term Evolution (LTE) downlink [3], as well as Digital Video Broadcasting-Second Generation Terrestrial (DVB-T2) [4]. Recently, OFDM has been also adopted in standards for communications that make use of satellite links, such as the standard for Digital Video Broadcasting-Satellite services to Handhelds (DVB-SH) [5]. The DVB-SH is a broadcast standard for delivering multimedia services over hybrid satellite/terrestrial networks to a variety of small mobile and fixed terminals with compact antennas. It has very limited directivity. In satellite channels, which are typically not highly frequency selective, the use OFDM is mainly motivated by the need to keep commonalities with the terrestrial systems. The Standard DVB-SH adopted OFDM transmission with the same signal format defined in Digital Video Broadcasting-Handheld (DVB-H) [6] for terrestrial systems in such a way that the satellite and terrestrial transmitters form a Single Frequency Network (SFN). It is worth noting that for many years the application of OFDM has been considered unsuitable for satellite communication. An OFDM signal has high amplitude fluctuations 3 1.1. MOTIVATION that produce a large Peak-to-Average Power-Ratio (PAPR). It makes OFDM sensitive to non-linear distortion caused by transmitter power amplifier. Hence transmit power amplifiers need to operate in with large Input Back Off (IBO) from their peak power, which leads to poor power efficiency. Several solutions have been suggested to reduce the impact of the signal distortion due to the non-linearities of the power amplifier such as the adoption of strong channel coding techniques in conjunction with the use of non-linear distortion compensation techniques [7] or the use of the so-called Constant Envelope OFDM (CE-OFDM) [8]. On the other hand, an interesting alternative is represented by using Single Carrier Frequency Division Multiple Access (SC-FDMA) signal format. SC-FDMA has been adopted in the up-link of the Long Term Evolution standard [3]. The SC-FDMA has been described as a single carrier transmission scheme with frequency domain equalization. Its performance is close to that of a multi-carrier system but with lower PAPR, which makes the user terminal more power efficient. Therefore, SC-FDMA transmission scheme is better candidate for satellite communication with respect to OFDM, also having the advantage of being compatible with OFDM-based transmitter-receiver. It also uses Cyclic Prefix (CP) and the equalization is performed in the frequency domain. The term cyclic prefix refers to the repetition of the end of the symbol at the beginning, which makes the linear convolution of the channel appearing as though it were circular convolution. Thus, it preserves this property in the part of the symbol after the cyclic prefix. SC-FDMA can be seen as Discrete Fourier Transform (DFT) spread OFDM. It performs DFT spreading in the transmitter, previous to the Inverse Discrete Fourier Transform (IDFT) block. Using this DFT block, it spreads the energy of individual modulation symbols over a number of subcarriers in the DFT block. The main difference between OFDMA and SC-FDMA is the DFT and inverse DFT (IDFT) blocks at the transmitter and receiver, respectively. There are two types of SC-FDMA on the basis of how DFT spread symbols are mapped on to the subcarriers in the IDFT block: Localized SC-FDMA (LFDMA) and Interleaved SC-FDMA (IFDMA). In Localized SC-FDMA, successive subcarriers are occupied by the DFT outputs of the input data resulting in continuos spectrum that occupies a small part 4 CHAPTER 1. INTRODUCTION of the of the total available bandwidth. In case of interleaved SC-FDMA, the DFT outputs of input data are spread over the entire bandwidth [9]. To alleviate fading, many communication system make use of space diversity techniques [10]. Receiver diversity combining techniques such as Maximal Ratio Combining (MRC), Equal Gain Combining (EGC) and Selection Combining (SC) are commonly used to tone down the effect of multipath fading in. wireless communication systems. The maximum diversity gain takes place when diversity branches experience independent and identical fading. On the other hand, in reality, fading is not independent due to inadequate spacing between the antennas. In MRC, signals from different antenna elements are weighted and combined to make the most of the output Signal to Noise Ratio (SNR). In an EGC combiner, the output of different diversity branches are first co-phased to avoid signal cancellation and simply combined to give the resultant output. The most commonly described performance indicator for digital transmission is Bit Error Rate (BER) under certainly employed modulation. In last few years, Quadrature Amplitude Modulation (QAM) is the most widely used constellation. However, quality of received signal in wireless communication system depends on many factors such as short-term fading, path loss exponent, lognormal shadowing, and noise. To improve system capacity, data rate and coverage reliability, the signal transmitted to and by a particular user can be continuously modified to track the signal quality variation through a process commonly referred to as Adaptive Modulation and Coding (AMC). Thus, it is possible to match the modulation scheme to the channel conditions for each user (at the expense, of course, of the need for feedback and certain throughput reduction due to signaling). Those users with good channel conditions are assigned with higher order modulation (e.g. 64 QAM) whereas the modulation order is lowered for users with worse channel conditions to keep the bit error rate under certainly given the target. In this case, spectral efficiency (as bits per second per Hertz) is the performance metric of interest. The SC-FDMA transmission scheme performance has already motivated some studies over Land Mobile Satellite (LMS) channels [11]-[12]. In [11], a first attempt to evaluate the performance of an air interface similar to the uplink of the LTE-standard has been performed 5 1.1. MOTIVATION to identify proper modifications and parameters optimization for the satellite scenario. A detailed comparison between SC-FDMA schemes versus OFDMA for a satellite uplink in S and K band have been presented in [12]. In particular, in [12] it is shown the performance sensitivity of the considered techniques to typical non-idealities of satellite such as inter block interference due to the not perfect synchronization among users and nonlinearities introduced by the High Power Amplifiers (HPAs). The study confirmed that the SC-FDMA transmission schemes is less sensitive to the amplifier non-linearity with respect to OFDMA. All the previously described studies on SC-FDMA for satellite systems rely on simulation results of a specific transmission chain. The analysis difficulties due to the non-linear behaviour of the detection stage avoid the attempts to find expressions for SC-FDMA performance that could be easily evaluated. In [13], the author studied PAPR characteristics of SC-FDMA by using analytical and numerical methods. A BER performance study of SC-FDMA based on semi-analytical methods is presented in [14]. In [15], closed-form expressions for the BER of M-ary Quadrature Amplitude Modulation (M-QAM) SC-FDMA transmission on Nakagami-m fading channels are provided. Even fewer works can be found exploring the spectral efficiency reachable by SC-FDMA [13], and only a few attempts for AMC techniques on SC-FDMA can be found [16]. In [17], the author proposed a coded SC-FDMA for a satellite scenario. They also state that SC-FDMA spectral efficiency increases with respect to present second generation DVB interactive satellite system (DVB-RCS2) standard [17], which is based on single carrier system with a Joint Multi-User (JMU) estimation technique, and a Low-Density Parity Check (LDPC) coding scheme. All these previous analytical or semi-analytical approaches have considered Rayleigh or Nakagami-m channels. However, for satellite communications, Loo’s model [18] is most widely accepted and used extensively in the literature as LMS channel. This model assumes that the received signal envelope is described by a Rician model with logormal shadowing on the Line-of-Sight (LOS) component. Although the Rician-lognormal model provides good match to experimental data, its assumption results in complicated expressions for analytical performance evaluation. A new simple LMS channel model was proposed in [19] as a Rician channel where the amplitude of LOS component is assumed to follow a 6 CHAPTER 1. INTRODUCTION Nakagami distribution. This Rician-Nakagami model is shown to provide a similar fit to the experimental data as the model in [18] but with less computational complexity and more mathematically tractability. Bibliography consider different levels of fading for satellite communications, commonly named as light, average and heavy shadowing. Heavy shadowing is close to a simple Rayleigh channel, while light shadowing assumes a strong LOS along with weak multipath components. Average shadowing indicates Rician fading with about 4 dB Rice Factor. In a wideband transmission, Inter Symbol Interfernce (ISI) appears due to the short symbol duration as compared to the multipath delay spread. The Power Delay Profile (PDP) of the multipath also defines correlation between channel responses at different subcarriers as ISI can be also seen as selectivity in the frequency domain. In case of frequency selectivity, equalization is needed, defined as any signal processing technique used at the receiver to alleviate the ISI problem caused by delay spread. Linear equalization techniques are generally simpler to implement compared to nonlinear mechanisms and avoids error propagation. Zero Forcing (ZF) equalizer is a type of linear equalization algorithm used in communication system, which applies the inverse of the frequency response of the channel. In Minimum Mean Square Error (MMSE) equalization, the main aim of the equalizer is to minimize the average mean square error (MSE) between the transmitted symbol and its estimate at the output of the equalizer. The ZF compensates completely the channel but can enhance noise, while MMSE reaches a compromise. Both OFDM and SC-FDMA techniques are able to cope with frequency selective channels by ZF or MMSE equalization performed in the frequency domain. Although many works can be found in literature, those bounds of SC-FDMA performance over an LMS channel are unexplored yet. This dissertation aims at filling this gap by providing a semi-analytical approach to the BER and spectral efficiency evaluation. In particular, this work provides the statistical characterization of the enhanced noise at the detection stage for Interleaved SC-FDMA. Linear frequency domain equalizations such as ZF and MMSE are applied. The Rician-Nakagami model for LMS channel proposed in [19] has been considered in the analysis. Performance metrics under simulation are also given for 7 2.3. SINGLE CARRIER FREQUENCY DIVISION MULTIPLE ACCESS c N S s M S Nc-DFT S/P M-IDFT P/S CP ()tS () CP st s Mapping Figure 2.3: Transmission scheme in SC-FDMA r  c N R c N R ()tR M R P/S Nc-IDFTM-DFT S/P CP () CP rt r Demap FDE Figure 2.4: Reception scheme in SC-FDMA deep fades at all allocated subcarriers. On the other hand, LFDMA allows using adaptive modulation techniques able to track channel variations. Moreover, SC-FDMA can be seen as a precoded version of OFDM whose symbols travel in time. Processing in the frequency domain allows reduced power distortion. Also, SC-FDMA complexity focuses on the receiver side. Therefore, it is an appropriate technology for cellular uplink transmission and satellite downlink transmission, since complexity at the base station or the terrestrial receiver is not an issue. Both OFDMA and SC-FDMA techniques transmit the same amount of data symbols in the same period using the same bandwidth, and users are orthogonally multiplexed and demultiplexed in the frequency domain. However, SC-FDMA modulated symbols are transmitted sequentially over the air, and the final transmitted signal is a single carrier one. Figures 2.3 and 2.4 show the transmission chain for SC-FDMA. As a previous step, the sequence of the bits to be transmitted are mapped into a constellation of complex symbols by using digital modulation techniques such as QAM. The resulting complex vector sof length Ncis pre-coded by means of a DFT operation before being mapped onto the subset of Nc allocated subcarriers. The pre-coded column complex vector SNcis obtained by multiplying with the unitary Fourier matrix Fto perform the Nc-DFT operation as SNc=Fs. The pre-coded sequence SNcis then mapped onto a different subset of allocated subcarriers per user, i.e., Ncout of the Msubcarriers. As previously described, the 14 CHAPTER 2. OFDM AND SC-FDMA FUNDAMENTALS subset may consist of a group of adjacent, Localized SC-FDMA or Interleaved SC-FDMA subcarriers [9] and it is determined by the M×Ncmapping matrix Pfor which Pi,j = 1 if the pre-coded symbol jis transmitted over the subcarrier iand zero otherwise. The bandwidth expansion factor is U=M/Nc, where Mis the total number of subcarriers used for transmission and Ncis the number of allocated subcarriers. The frequency-domain symbol is defined as SM=PSNcand, consequently, non-allocated subcarriers are forced to zero. From this point on, transmission is similar to that of OFDMA, and an M-IDFT operation converts each frequency-domain symbol SMinto a time-domain symbol. These time domain symbols are transmitted sequentially after adding a cyclic prefix. The CP is inserted as the guard time, which prevents ISI due to multipath propagation. The length of CP is made longer than the maximum delay spread of the channel or duration of the channel impulse response. Since CP is the copy of the last part of the block, it converts discrete time linear convolution into discrete time circular convolution. At the receiver end, the cyclic prefix is removed, and the received signal is converted into the frequency domain by taking DFT. After subcarrier demapping, equalization can be done by using ZF or MMSE. The MMSE is preferred over ZF due to robustness against noise. Later, symbols are converted back to the time domain by an Nc-IDFT, and the sequence is given to the detector. 2.4 Conclusions In this chapter, we discussed SC-FDMA, a multiple access scheme that makes use of single carrier modulation, and orthogonal frequency division multiplexing. The main drawback of OFDM and OFDMA is that they have high PAPR. As SC-FDMA has low PAPR, it simplifies the transmitter circuit in the handset and reduces power consumption. The receiver structure with SC-FDMA is more complex than that with OFDMA. 15 2.4. CONCLUSIONS 16 Chapter 3 Wireless Channel Models 3.1 Introduction to Wireless Channel The behaviour of a wireless communication system depends on the wireless channel environment. As compared to wired channel, the wireless channel is dynamic and unpredictable. So, to make an exact analysis of the wireless communication system is often complicated. In wireless communication, radio propagation refers to the actions of radio waves when they travel from transmitter to receiver. During propagation, radio waves are mainly affected by three different modes of physical phenomena: reflection, diffraction, and scattering [26]. Reflection is the physical phenomenon that occurs when a propagating electromagnetic wave impinges upon an object. It makes the transmit signal power to be reflected back to its origin. Diffraction is the phenomena that occur when the radio path between the transmitter and receiver is blocked by a surface with sharp irregularities. The secondary waves generated by diffraction makes a path between the transmitter and receiver, even when an LOS path is not present. Scattering is a process that forces the radiation of an electromagnetic wave to deviate from a straight path by local obstacles. In short, the propagation of a radio wave is a complicated and less predictable process, the intensity of which varies with different environments at different instances. The deviation of the signal amplitude over time and frequency is called as fading. It is a unique characteristic of the wireless channel. In this dissertation, the frequency selective Rayleigh and Rice fading channel models are based on simple satellite model defined in [27]. This chapter describes channel models to be 17 3.2. RAYLEIGH AND RICE FADING employed in the rest of the thesis. 3.2 Rayleigh and Rice Fading 3.2.1 Rayleigh Fading Small-scale fading occurs due to small changes in position. It is often named it as Rayleigh fading since the signal envelope is often statistically characterized with a Rayleigh Probability Density Function (PDF). Rayleigh fading in the propagation channel is a major impairment in wireless communications and it significantly degrades the link performance. The PDF for signal envelope in Rayleigh fading is given by fX(x) = x σ2exp −x2 2σ2, x ≥0 (3.2.1) Being σthe standard deviation of the underlying Gaussian distribution. 3.2.2 Rice Fading In some environments, such as satellite communication, LOS between transmitter and receiver is present in the communication link. In this case, the amplitude gain is often described as a Rician distribution. It represents the envelope of a Gaussian signal whose average is different from 0. The PDF of the Rice distribution is given by fX(x) = x σ2exp −x2+c2 2σ2Ioxc σ2x≥0 (3.2.2) Where I0(.) is the modified zeroth-order Bessel function of the first kind. The Rician factor Kis defined as a ratio of the power present through the LOS path to the power contribution through non-LOS paths: K=c2 2σ2(3.2.3) If there does not exist an LOS component (i.e., K= 0), Eq. (3.2.2) reduces to the Rayleigh PDF Eq. (3.2.1) as in the non-LOS environment. As Kincreases, Eq. (3.2.2) tends to be Gaussian PDF. It is assumed that K=−40 dB stands for a Rayleigh fading channel and K > 15 dB for a Gaussian channel. 18 CHAPTER 3. WIRELESS CHANNEL MODELS 3.3 Frequency Selectivity 3.3.1 Flat Fading Channels It is said that a signal transmitted over a channel suffers flat fading if the coherence bandwidth of the channel is higher than the bandwidth of the signal. Therefore, all frequency components of the signal will experience the same magnitude of fading. The response of the channel is thus given by: h(t, τ) = ai(t)δ(τ),(3.3.1) Where aiis called amplitude of the channel, which may be Rayleigh or Rice distribution, t stands for time variability and τfor a delay. 3.3.2 Frequency Selective Channels If the coherence bandwidth of the channel is smaller than the bandwidth of the signal then it is called frequency selective fading. The various frequency components of the signal, therefore, experience different fading. Channel is commonly modelled as a set of Nuncorrelated taps, ak(t), each of them equivalent to the channel response of a flat fading channel. Finally the channel response for frequency selective channel is given by the well-known tapped-delay-line model: h(t, τ) = L X k=1 ak(t)δ(τ−τk) (3.3.2) Where akis the variant amplitude of each echo, which may be Rayleigh or Rice distribution as explained in Sec. 3.2, tstands for time variability and τfor the delay. Power Delay Profile (PDP) is the average intensity of a signal received through a multipath channel as a function of time delay, that is, the set of power averages of ak(t). In general, most powerful echoes are received first while power is decreasing with τ. 3.4 Single-Input Multiple-Output Channel Model The multiple receiving antenna Single-Input Multiple-Output (SIMO) link is shown in Figure 3.1 for two antennas. The channel system with NRantennas, i.e. diversity branches, 19 3.4. SINGLE-INPUT MULTIPLE-OUTPUT CHANNEL MODEL 1 h x TX combiner 2 h 1 y 2 y Figure 3.1: Receiver Diversity combining scheme. at the receiver can be represented by the channel vector: h= [h1, h2, .., hNR]T(3.4.1) Where hjis the channel coefficient between the transmit antenna and the jth receive antenna. Moreover, hjcan be described by multiple paths, as described in the previous section, that is, if there are Ljdistinct paths from the transmitter to the receiver, the impulse response of this channel will be: hj(t, τ) = Lj X k=1 ak,j(t)δ(τ−τk) (3.4.2) With tstanding for time variability and τfor the delay. If there is less spacing between multiple antennas at receiver, then there exists certain correlation between antennas given by the correlation coefficient ρ. The values of ρstand between 0 and 1. The value of ρ= 0 means no correlation between the antennas and ρ= 1 indicates perfect correlation, i.e., similar signal is received by both antennae. The correlated channel model His given as: H=R1/2 rHwR1/2 t(3.4.3) Where Rtis the correlation matrix, reflecting the correlation between the transmit antennas (i.e., the correlations between the column vector of H), Rris the correlation matrix reflecting the correlation between receiver antenna (i.e., the correlation between the row vectors of H), and Hwdenotes the i.i.d fading channel gain matrix. The diagonal entries of Rrand Rtare considered to be unity. 20 CHAPTER 3. WIRELESS CHANNEL MODELS For SIMO the channel model H is simplified to: H=R1/2 rHw(3.4.4) 3.5 Satellite Channel Model There are two types of propagation between a satellite and a mobile receiver, unshadowed and shadowed [28][29]. When the mobile receiver has an unobstructed LOS path to the satellite, it is called unshadowed propagation. In the case of shadowed propagation, the LOS path of the satellite is obstructed by terrain, vegetation, or human made structures. The unshadowed signal received at the mobile receiver is composed of direct and diffuse components. The diffuse component is a phase incoherent multipath wave due to reflections and scattering from outside the first Fresnel zone of the vehicle of little variations with directions of arrival [30]. The direct LOS component is essentially constant. The shadowed direct component can be modelled as the sum of attenuated LOS signal and a random forward scattered field. Generally speaking, the available models for narrowband LMS channels can be placed into two categories: single model and mixture model. In the single model, the channel is characterized by a single statistical distribution, while a combination model refers to a combination of several statistical distributions. Single models are valid for stable conditions, where the channel statistics remain approximately constant over the period of interest in the small area. On the other hand, the mixture models are developed for nonstationary channels, where the signal statistics vary significantly over the observation interval. A list of proposed single and mixture models is given in [19]. Among the proposed models for LMS channels, the shadowed Rice model proposed originally by Loo [18] has found wide applications in different frequency bands such as the UHF-band, L-band, S-band, and Ka-band. In this dissertation, an LMS model proposed in [19] is employed. This model can gather in a single distribution both for shadowed and unshadowed LMS channels by a different parameter set. 21 3.5. SATELLITE CHANNEL MODEL 3.5.1 Shadowed Rice Land Mobile Satellite Channel Model In an ideal land mobile satellite channel, there is a clear line of sight between the satellite and the land user, thus the envelope is a non-random constant. However, after multipath fading is caused by the weak scatter components propagated via different non-LOS paths, together with the non blocked LOS component, the envelope becomes a Rice random variable. Moreover, LOS shadow fading comes from the complete or partial blockage of the line of sight which makes the amplitude of the LOS component also a random variable. Finally, the low pass-equivalent complex envelope of the channel can be written as; z(t) = A(t) exp (jα(t)) + B(t) exp(jα0) (3.5.1) Where α(t) is the stationary random phase process of multipath fading with uniform distribution over [0,2π), while α0is the phase of the LOS component. The independent stationary random processes A(t) and B(t), which are also independent of α(t), represents the amplitudes of the scatter and the LOS components. As previously stated, we follow the LMS channel model described in [19], through which A(t) and B(t) follow Rayleigh and Nakagami distributions, respectively, as given bellow; fA(a) = a b0 exp −a2 2b0(3.5.2) fB(b) = 2mm Γ(m)Ωmb2m−1exp −mb2 Ω(3.5.3) Where 2b0=E[A2] is the average power of scatter component, Γ ( ) is the gamma function, m=E[B2] V ar [B2]≥0 is the Nakagami parameter with V ar [ ] as the variance, and Ω = E[B2] is the average power of the LOS component. The shadowed Rice probability density function (PDF) for the signal envelope Z=|z(t)| in an LMS channel can be derived from: PZ(z) = EBz b0 exp (z2+B2) 2b0 I0(Bz b0 )(3.5.4) Where EB[ ] is the expectation with respect B, and In( ) is the n-th order modified Bessel’s function of the first kind. Averaging [19] yields to: fZ(z) = 2b0m 2b0m+ Ωmz b0 exp−z2 2b01F1m, 1,Ωz2 2b0(2b0m+ Ω)(3.5.5) 22 CHAPTER 3. WIRELESS CHANNEL MODELS The shadowed Rice PDF for the instantaneous power |z(t)|2can be then written as (Eq. (6) in [19]): fZ2(q) = 2b0m 2b0m+ Ωm1 2b0 exp −q 2b01F1m, 1,Ωq 2b0(2b0m+ Ω)(3.5.6) Different values of mare able to model distinct scenarios. While m= 0 corresponds to areas with complete obstruction of the LOS, any value 0 <m<∞represents areas with partial obstruction of the LOS. The most common values considered for those parameters, which fully characterize the fading model, are presented in Table 3.1. Finally, the expressions of gamma Γ( ) and the confluent hyper geometric function 1F1(, , ) can be found in [31]. Table 3.1: Parameters for considered fading channels [19]. Shadowing type b0mΩ Light fading 0.158 19.4 1.29 Heavy fading 0.063 0.739 8.97 ×10−4 Average fading 0.126 10.1 0.835 3.5.2 Channel Frequency Response Models The channel frequency response, represented by the M×Mdiagonal matrix HM, is the Fourier transform of the variable impulse response of the channel. Two different models have been employed for dependence among the channel frequency responses at those allocated subcarriers, h=diag(PHHMP), with Pis the mapping matrix as described in Sec. 2.3. As first model, elements of hare assumed to be independent identically distributed random variables. Each element hjfollows the shadowed Rice distribution given by Eq. (3.5.5) and the PDF for the instantaneous power at each subcarrier |hj|2can be then written as (Eq. (6) in [19]): f|hj|2(q) = 2b0m 2b0m+ Ωm1 2b0 exp−q 2b01F1m, 1,Ωq 2b0(2b0m+ Ω)(3.5.7) As the second model, a wideband LMS tapped delay line model is used. The first echo is considered to follow the narrow band shadowed Rice model while the rest are considered 23 4.3. SINR ANALYSIS Function (CDF) of βZF using Gil-Pel´aez’s inversion theorem expression [34] which can be written in this particular case as FβZF βZF =1 2−1 2πZ∞ 0ΦβZF (ω) ıω e−ıβZF ω−ΦβZF (−ω) ıω eıβZF ωdω (4.3.11) Once we have characterized βZF , we might use it to evaluate the instantaneous SNR conditioned to channel as they are related by the following expression γZF =ES/N0 βZF (4.3.12) 4.3.2 MMSE FDE In order to avoid ZF noise enhancement, the design of Wcan be accomplished under the MMSE criterion WMMSE =N0 ES I+HHH−1 HH(4.3.13) Where ESand N0are the signal and noise powers respectively. In this case, the recovered symbol is r=FHN0 ES I+HHH−1 HHHFx +FHN0 ES I+HHH−1 HHη(4.3.14) Each received symbol can be written as the transmitted symbol modified and contaminated by noise and interference. rk=Tk,ksk+ Nc X l6=k l=1 slTk,l + Nc X j=1 F∗ j,kh∗ j |hj|2+N0/ES ηj(4.3.15) The elements of the matrix T=FHWMMSEHF can be expressed as Tk,l = Nc X j=1 F∗ j,kFj,l |hj|2 |hj|2+N0/ES (4.3.16) Specifically, the term Tk,k, which multiplies the transmitted signal, is given by Tk,k = Nc X j=1 |Fk,j|2|hj|2 |hj|2+N0/ES =1 Nc Nc X j=1 |hj|2 |hj|2+N0/ES (4.3.17) 30 CHAPTER 4. SC-FDMA BER ANALYSIS OVER SHADOWED RICE LMS CHANNEL Please, note that really Tk,k is independent of k, that is, all symbols are received with the same power. After compensating it, the transmitted symbol is obtained as rk=sk+T−1 k,k Nc X l6=k l=1 slTk,l +T−1 k,k Nc X j=1 F∗ j,kh∗ j |hj|2+N0/ES ηj=sk+ Nc X l6=k l=1 δl,k + Nc X j=1 ˜ηj,k (4.3.18) That is, the transmitted symbol is contaminated by the equivalent noise ˜ηkand the interference δk rk=sk+δk+ ˜ηk=sk+ξk(4.3.19) The term ξkgathers the effects of noise and interference, which are inherently independent [14]. When the channel frequency response is such that N0 ES |hj|2, the effect of the interference term decreases and the noise term is equal to that of ZF in Eq. (4.3.3). The density of the noise ˜ηkconditioned to the channel frequency response h= [h1. . . hNc] is the addition of Ncterms, each one following a complex Gaussian distribution. Thus, the total equivalent noise follows also a complex Gaussian distribution p(˜ηk/h)∼ CN 0, σ2 ˜η(4.3.20) Whose variance σ2 ˜η, equal for all symbols, can be simply obtained by taking into account that |F∗ j,k|= 1 and that noise samples are independent as σ2 ˜η=N0 (Tk,k)2 1 Nc Nc X j=1 |hj| |hj|2+N0/ES!2 (4.3.21) Now, we are evaluating the interference component. Conditioned the channel frequency response, we can assume it also follows a complex circularly symmetric Gaussian distribution [14]: p(δk/h)∼ CN 0, σ2 δ(4.3.22) Please, note that this will only hold if transmitted symbols are independent and with equal energy ES, and it will be more realistic as more subcarriers are used. In this case, σ2 δcan be 31 4.3. SINR ANALYSIS expressed as σ2 δ=ES (Tk,k)2 Nc X l6=k l=1 |Tk,l|2(4.3.23) We can manipulate the expression as to write σ2 δ=ES (Tk,k)2 Nc X l=1 |Tk,l|2−(Tk,k)2!(4.3.24) or σ2 δ=ES 1 Nc Nc P j=1 |hj|2 |hj|2+N0/ES2− 1 Nc Nc P j=1 |hj|2 |hj|2+N0/ES!2 1 Nc Nc P j=1 |hj|2 |hj|2+N0/ES!2(4.3.25) As noise and interference components are independent random variables, their sum can be modelled with another zero-mean complex Gaussian random variable whose variance, given by σ2 ξ=σ2 δ+σ2 ˜η(4.3.26) Can be written as σ2 ξ= N0 Nc Nc P j=1 |hj| |hj|2+No/Es 2+Es Nc Nc P j=1 |hj|2 |hj|2+No/Es 2−Es 1 Nc Nc P j=1 |hj|2 |hj|2+No/Es !2 1 Nc Nc P j=1 |hj|2 |hj|2+No/Es !2.(4.3.27) We will name βMMSE to βMMSE =1 Nc Nc X j=1 1 |hj|2+N0/ES = Nc X j=1 βMMSE j(4.3.28) Please, note, that after this definition the range of variation for βMMSE is limited to 0 ≤ βMMSE ≤Es N0. 32 CHAPTER 4. SC-FDMA BER ANALYSIS OVER SHADOWED RICE LMS CHANNEL Manipulating the equations, as Es Nc Nc X j=1 |hj|2 |hj|2+N0/Es2 =Es Nc Nc X j=1 |hj|2 |hj|2+N0/Es |hj|2 |hj|2+N0/Es(4.3.29) =Es NcX|hj|2 |hj|2+N0/Es1−N0/Es |hj|2+N0/Es (4.3.30) =Es NcX|hj|2 |hj|2+N0/Es−No NcX |hj|2 (|hj|2+N0/Es)2!(4.3.31) Then σ2 ξ= Es Nc P|hj|2 |hj|2+No/Es −Es 1 Nc Nc P j=1 |hj|2 |hj|2+No/Es !2 1 Nc Nc P j=1 |hj|2 |hj|2+No/Es !2.(4.3.32) Taking into account that 1 Nc Nc X j=1 |hj|2 |hj|2+N0/Es=1 Nc Nc X j=1 1−N0/Es |hj|2+No/Es(4.3.33) =1 Nc Nc X j=1 1−1 Nc Nc X j=1 N0/Es |hj|2+N0/Es = 1 −N0 Es βMMSE (4.3.34) We can write σ2 ξ=Es 1−No Es βMMSE−Es 1−No Es βMMSE2 1−No Es βMMSE2=Es 1 1−No Es βMMSE −1!(4.3.35) And, finally, σ2 ξ=ES N0 ESβMMSE 1−N0 ESβMMSE (4.3.36) PDF of βMMSE jcan be easily obtained by a procedure similar to that in previous section for ZF except that we have to use first fX+K(t) = fX(t−K) (4.3.37) Thus f|hj|2+N0/ES(s) = 2b0m 2b0m+ Ωm1 2b0 exp−s−N0/ES 2b01F1m, 1,Ω(s−N0/ES) 2b0(2b0m+ Ω) (4.3.38) 33 4.3. SINR ANALYSIS Using now Eq. (4.3.5) f1/(|hj|2+N0/ES)(s) = 1 s22b0m 2b0m+ Ωm1 2b0 exp− 1 s−N0/ES 2b01F1m, 1,Ω(1 s−N0/ES) 2b0(2b0m+ Ω , (4.3.39) i.e. f1/(|hj|2+N0/ES)(s) = 2b0m 2b0m+ Ωm1 2b0 1 s2exp−1−sN0/ES 2b0s1F1m, 1,Ω(1 −sN0/ES) 2b0(2b0m+ Ω)s. (4.3.40) By using now Eq. (4.3.7) to divide by Nc fβMMSE j(s) = 2b0m 2b0m+ Ωm1 2b0 1 Ncs2exp−1−sNcN0/ES 2b0Ncs1F1m, 1,Ω(1 −sNcN0/ES 2b0(2b0m+ Ω)Ncs (4.3.41) We can compare it to that simpler expression for ZF (also repeated here for convenience) and note that both are equivalent for N0= 0 fβZF j(s) = 2b0m 2b0m+ Ωm1 2b0 1 Ncs2exp−1 2b0Ncs1F1m, 1,Ω 2b0Ncs(2b0m+ Ω) (4.3.42) As in the previous case, if independence among the channel frequency responses is assumed, the CHF of βMMSE yields ΦβMMSE (ω) = ΦβMMSE j(ω)Nc(4.3.43) With the CHF of βjgiven by ΦβMMSE j(ω) = Z∞ 0 fβMMSE j(s)eısωds (4.3.44) To our best knowledge, it is not possible to obtain an analytical expression for the CHF of βjbut only numerical calculation could be carried on, e.g. by means of successive computations of Eq. (4.3.44). Later on, it is possible to evaluate the CDF of βMMSE using Gil-Pel´aez’s inversion theorem expression [34] which can be written in this particular case as FβMMSE βMMSE=1 2−1 2πZ∞ 0ΦβMMSE (ω) ıω e−ıβM MSE ω−ΦβMMSE (−ω) ıω eıβMMSEωdω (4.3.45) 34 CHAPTER 4. SC-FDMA BER ANALYSIS OVER SHADOWED RICE LMS CHANNEL We might later evaluate the SNR by using Eq. (4.3.36) as γMMSE =Es σ2 ξ = Es N0 βMMSE −1 (4.3.46) Its CDF is also easy to evaluate from that of βMMSE as FMMSE γ(g) = Pr γMMSE < g=Pr "Es N0 βMMSE −1< g#=Pr β > Es/N0 g+ 1 (4.3.47) Thus FMMSE γ(g)=1−FMMSE βEs/N0 g+ 1 (4.3.48) Note that, being the range of βMMSE from 0 to Es/N0,FMMSE γ(0) = 1−FMMSE β(Es/N0)=0 as expected. For g→ ∞ it is clear that FMMSE γ(g) = 1. In [14] a similar procedure was followed but with a slight different expression as they name √αMMSE =1 Nc Nc X j=1 |hj|2 |hj|2+N0/ES (4.3.49) And from it, they evaluate the SNR conditioned to channel as 1 q1 αMMSE −1 (4.3.50) 4.4 Numerical Results As previously stated, the modulation scheme employed to convert bits into complex symbols given to the SC-FDMA system could be fixed to certain constellation or could track the quality of the received signal, that is, an Adaptive Modulation (AM) approach. In this chapter, fixed modulation schemes have been considered. Specifically, BPSK and square QAM performance in terms of BER has been evaluated. BER can be evaluated as the average of the BER conditioned to β, that is, BER . =ZBER(β)fβ(β)dβ (4.4.1) 35 4.4. NUMERICAL RESULTS Table 4.1: Simulation Parameters Parameter Value Total number of subcarriers M1024 Allocated subcarriers Nc1,64 Modulation scheme BPSK, 4QAM, 16QAM Equalizers ZF, MMSE Cyclic Prefix Length 144 Default Es/N0(dB) 10 ∆f15KHz τavg (Exponential PDP) 1.17µs With BER(β) evaluated from the bit error rate of QAM over AWGN channels [14]. For Gray coding and square M-QAM, it might be evaluated from [35] as BER(γ) = 2 √Mlog2√M log2√M X k=1 (1−2−k(√M−1) X i=0 ((−1bi2k−1 √Mci2k−1 √M+1 2Q s6(2i+ 1)2 2(M−1) γ!) (4.4.2) By simply substituting γby βas given Eq. (4.3.12) or Eq. (4.3.46) for ZF and MMSE FDE, respectively. 4.4.1 CDF of β Two different approaches have been followed to evaluate the PDF of βdepending on the channel model. For independent subcarrier responses, CDF of βcan be evaluated numerically following the previously described procedure with Eq. (4.3.9) and Eq. (4.3.10) for ZF and their equivalent ones for MMSE. Results for CDF in this case are shown in Figures 4.3 and 4.4 for heavy and light shadowing, respectively, and average SNR 10 dB. When the exponential PDP given in Eq. (3.5.8) is assumed, certain correlation between channel responses at different subcarriers avoids using that the CHF of the addition is the product of those of each term, making the analysis infeasible. In this case, a semi-analytical approach has been taken from the values of h, that of βhas been evaluated by using Eq. (4.3.4) or Eq. (5.3.7) and, by using ergodicity condition, its CDF is evaluated from time histogram. Results in this case are shown in Figures 4.5 and 4.8. As expected, in all cases, βtakes statistically 36 CHAPTER 4. SC-FDMA BER ANALYSIS OVER SHADOWED RICE LMS CHANNEL lower values that for light fading as fading is deeper for heavy shadowing. From results for independent subcarriers, it is clear that for ZF equalization βcould reach higher values due to the noise enhancement for low values of channel gain. By contrast, MMSE equalization forces the maximum value for β. It is also observed a completely different performance for ZF and MMSE equalization as the number of subcarriers Ncchanges. With ZF equalizer, reducing the number of subcarriers Nclower the value of the effective βZF , thus βvalues for OFDM (Nc= 1) are statistically lower. On the other hand, in case of MMSE equalizer, the effect is opposite. The effect of correlation in frequency can be seen from comparison among Figures 4.3, 4.5, and 4.7 for heavy shadowing, and from Figures 4.4, 4.6, and 4.8 for light shadowing. Only slight differences are shown for MMSE. However, for ZF equalization the use of subcarriers with highly correlated responses, as in LFDMA, reduces the range for β. This is the result of the fact that for ZF equalization the most faded subcarrier is forcing its value. The more uncorrelated the responses, the higher the probability of having at least one subcarrier which is highly faded. 4.4.2 BER Results Figures 4.9 and 4.10 show comparison of BER over light shadowed channel with ZF and MMSE equalizer. Heavy shadowing results are consistent with Rayleigh fading results. Result for AWGN channel is also shown for comparison. The BER has been numerically evaluated when possible and obtained as time averaging of the BER for the given SNR in another case. As previously stated, in the case of ZF equalization, for an SC-FDMA transmission scheme the value of the SNR (γ) is always lower than that of OFDMA; that causes a BER worse in the former. In the case of MMSE equalization, the SC-FDMA BER for uncorrelated frequency responses is below OFDMA. The consequence of the CDF of β, frequency correlation for ZF improves BER, that is, BER for LFDMA is better than that of IFDMA. On the contrary, MMSE enhance is clear for independent subcarrier responses. These results are consistent with SC-FDMA performance over Rician channel [36]. 37 4.5. CONCLUSIONS In Figures 4.11 and 4.12 the effect of the number of allocated subcarriers is analyzed. It observed that for the higher number of allocated subcarriers, BER decreases in case MMSE equalizer is used. In the case of ZF equalizer, this increase does not improve the performance. The BER performance of SC-FDMA over different shadowing is depicted in Figure 4.13. It is observed that for both with ZF and MMSE equalization light shadowing BER is below that of average and heavy shadowing. Obviously, this is due to its stronger LOS component. Figure 4.14 shows the performance of different constellations over one selected scenario, that is of light shadowing with ZF equalizer. It is obvious that the higher the constellation size, the worse is the BER. This is of course true for all considered scenarios. 4.5 Conclusions In this chapter we have analysed the performance of SC-FDMA for independent and correlated channel frequency responses. The OFDM BER response was compared to that of SC-FDMA over different shadowing. It is observed that OFDM BER results are lower than those of SC-FDMA with ZF equalization. In the case of MMSE equalizer, independence among channel frequency responses improves the BER compared to that of Interleaved SC-FDMA, Localized SC-FDMA and OFDM. Heavy shadowing results are consistent with Rayleigh fading results. Light shadowing results are equivalent to 5 dB Rice factor results of Rician channel. 38 CHAPTER 4. SC-FDMA BER ANALYSIS OVER SHADOWED RICE LMS CHANNEL 0 2 4 6 8 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 s F β MMSE (s) N c = 1 N c = 4 N c = 8 N c = 16 N c = 64 0 20 40 60 80 100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 s F β ZF (s) N c =1 N c = 4 N c = 8 N c = 16 N c = 64 Figure 4.3: CDF of βfor heavy shadowing with MMSE and ZF equalizer for uncorrelated responses at different subcarriers 0 2 4 6 8 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 s F β MMSE (s) N c = 1 N c = 4 N c = 8 N c = 16 N c = 64 0 3 6 9 12 15 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 s F β ZF (s) N c = 1 N c = 4 N c = 8 N c = 16 N c = 64 Figure 4.4: CDF of βfor light shadowing with MMSE and ZF equalizer for uncorrelated responses at different subcarriers 39 5.2. OVERVIEW OF ADAPTIVE MODULATION AND CODING 0 3 6 9 12 15 18 21 24 27 30 10-4 10-3 10-2 10-1 100 Instaneous SNR γ [n] dB BER 4QAM 16QAM 64QAM 1 3 γ γγ γ 1 γ γγ γ 2 γ γγ γ 3 2 BER T Figure 5.1: Examples of adaptive modulation regions for uncoded QAM AWGN approach. BERi(γ)≈BERAW GN i(γ) (5.2.1) Several criteria can be used to associate one constellation to each γin adaptive modulation. The approach considered in this dissertation is discretizing the range of channel fade levels in such a way that the BER is always kept under a target, that is, BERi≤BERT, being BERTa design parameter. As shown in Figure 5.1, the range of γis divided in to N fading regions <i={γi, γi+1},i= 0,1,2, ......N −1 where γ0= 0 and γN=∞. Within the fading region <i, a certain QAM constellation with Ribits/symbol is employed. Switching thresholds {γi, i = 1, .., N −1}are designed to maximize the spectral efficiency under that instantaneous BER constraints BERT. Being BERimonotonically decreasing, this finally means that at switching thresholds BER is exactly the target BER. BERi(γi) = BERT(5.2.2) 46 CHAPTER 5. SC-FDMA SPECTRAL EFFICIENCY OVER SHADOWED RICE LMS CHANNEL Whichever are the taken design criteria, the average spectral efficiency is written as R B= L−1 X i=1 log2(Mi)Pr{γi−1< γ ≤γi}(5.2.3) Note that stricter BERTvalues implies lower spectral efficiency values. For SNR values below a minimum threshold (i.e., in very poor channel conditions), there is an outage state in which there is no data transmission. To perform adaptation, the instantaneous SNR γhas to be estimated at the receiver to determine the current fading region <iand, consequently, the transmit rate Ri. Both the average and the instantaneous SNR are supposed to be perfectly known and the receiver suggestion assumed to be fed back to the transmitter without delay nor errors. 5.3 Spectral Efficiency Analysis In this section, the average spectral efficiency is evaluated for ZF and MMSE equalization. 5.3.1 ZF-FDE Analysis As we obtained in previous chapter, instantaneous SNR at the receiver for SC-FDMA with ZF equalization is given by γZF =ES βZF N0 (5.3.1) where βZF is defined as βZF =1 Nc Nc X k=1 1 |hk|2(5.3.2) For OFDM the same expression is valid if Ncis set to 1. In both cases, the constellation corresponding to the i-th region is applied when the SNR is between the corresponding boundaries, that is γi<Es βZF N0 < γi+1 (5.3.3) We can rewrite the above the equation and the threshold for βZF becomes βZF i+1 =1 γi+1 ES N0 < βZF <1 γi ES N0 =βZF i(5.3.4) 47 5.4. SIMULATION RESULTS AND DISCUSSION where βZF iand βZF i+1 are the modified thresholds. Please, note that they depend on Es/N0. The spectral efficiency can be averaged as R B= N−1 X i=1 log2(Mi)FβZF βZF i−FβZF βZF i+1 (5.3.5) where Nis the number of fading regions. Thus, in order to compute spectral efficiency values, we only require FβZF (βZF ) given by Eq. (4.3.11). 5.3.2 MMSE-FDE Analysis For MMSE-FDE the value of SNR at the receiver is determined by Eq. (4.3.46) repeated here for convenience γMMSE =Es σ2 ξ = Es N0 βMMSE −1 (5.3.6) where βMMSE =1 Nc Nc X j=1 1 |hj|2+N0/ES = Nc X j=1 βMMSE j(5.3.7) The parameter βMMSE decides the SNR and hence, the corresponding constellation to be used, that is βMMSE i+1 = ES N0 γi+1 + 1 < βMMSE < ES N0 γi+ 1 =βMMSE i(5.3.8) with βMMSE iand βMMSE i+1 , the modified thresholds that depend on the SNR. The average spectral efficiency can be computed as R B= N−1 X i=1 log2(Mi)FβMMSE βMMSE i−FβMMSE βMMSE i+1  (5.3.9) Again, in order to compute spectral efficiency values, we only require FβMMMSE(βMMSE) given by Eq. (4.3.45). 5.4 Simulation Results and Discussion In this section, we discuss the spectral efficiency results for OFDM, Independent SC-FDMA, Localized SC-FDMA, and Interleaved SC-FDMA and over different shadowing profiles. 48 CHAPTER 5. SC-FDMA SPECTRAL EFFICIENCY OVER SHADOWED RICE LMS CHANNEL Table 5.1: Switching thresholds of the SNR {γi}for adaptive modulation and coding γi(dB) 9 .7980 16.5290 22.5510 28.4170 log2(Mi) 2 4 6 8 The spectral efficiency is calculated using Eq. (5.3.5) and (5.3.9) for ZF and MMSE, respectively. We consider the channel frequency responses, which are generated for each allocated subcarriers, to calculate the effective noise at the detector stage for ZF and MMSE equalizer. All the results presented were obtained for a fix BERT= 10−3, which has chosen as a reference value. Stricter BER target BERTvalues implies lower spectral efficiency values. The set of switching thresholds {γi, i = 1, .., N −1}are given in Table 5.1. Figure 5.2 shows the spectral efficiency with ZF equalization over different shadowing profiles for independent subcarrier responses. The result for Nc= 1 refers to OFDM spectral efficiency. Equivalent results for MMSE are shown in Figure 5.3. It is observed that the noise amplification factor βis having low value for light shadowing as compared to average and heavy shadowing, so spectral efficiency is approaching close to OFDM. By observation of those figures, it is concluded that an increase in the number of allocated subcarriers reduces the spectral efficiency in SC-FDMA, whereas the spectral efficiency of OFDM is independent of this value (and equivalent to that of SC-FDMA with Nc= 1). The provided results for heavy shadowing are consistent with Rayleigh fading results [38]. It can be observed that SC-FDMA curves for spectral efficiency look like piecewise. This is more noticeable for light shadowing. The reason of this can be found in the behaviour of βPDF as shown for ZF equalization in Figure 5.4 and Figure 5.5 for 10 dB SNR over light and average shadowing channels, respectively. It can be observed that the PDF of β for Nc = 64 is very narrow, thus most values of βZF are concentrated in a unique fading region; so the same modulation scheme is applied under most instantaneous βZF values. In the case of OFDM (Nc= 1), βZF pdf is wider, occupying several fading regions. As a result of averaging, its spectral efficiency curve grows smoother [38]. Results for pdf of βMMSE for Nc= 64 are shown in Figure 5.6. As described in previous chapter, its range is much shorter, but conversion into γMMSE finally results in similar ZF and MMSE performance, despite the different approaches to equalization, especially for high 49 5.5. CONCLUSIONS SNR values. The results of spectral efficiency for correlated subcarrier responses are depicted as shown in Figures 5.7 and 5.8 for IFDMA and LFDMA over light shadowing with ZF and MMSE equalizer respectively. Results for independent subcarrier responses and OFDM are kept for comparison. It is observed that spectral efficiency for LFDMA results with ZF equalizer is approaching to OFDM with less allocated subcarriers as the spectral efficiency increases when there is strong correlation among channel frequency responses. In fact, for flat fading channels in which the channel frequency response is the same for all subcarriers, βbecomes the noise enhancement factor for OFDM. The same thing is observed for average shadowing with ZF and MMSE equalizer as shown in Figures 5.9 and 5.10. The results for heavy shadowing are shown in Figures 5.11 and 5.12. It is observed that spectral efficiency of independent subcarrier responses approaches to interleaved SC-FDMA, IFDMA. As in heavy shadowing the LOS component is weak, it is observed that the curves are lower compared to light shadowing. The existence of the strong line of sight component, as that of light fading, improves the performance through the weak multi-path component. 5.5 Conclusions In this chapter, spectral efficiency for SC-FDMA has been calculated under ZF and MMSE equalization. The CDFs of βevaluated in the previous chapter were enough to evaluate it. It is observed that spectral efficiency of OFDM is better than that of Interleaved and Localized SC-FDMA while independent SC-FDMA spectral efficiency is approaching to OFDM spectral efficiency. Spectral efficiencies reached by ZF and MMSE equalizer look like similar. Moreover, the higher is the number of allocated subcarrier, the lower is the spectral efficiency. 50 CHAPTER 5. SC-FDMA SPECTRAL EFFICIENCY OVER SHADOWED RICE LMS CHANNEL 0 5 10 15 20 25 30 35 40 45 0 1 2 3 4 5 6 7 8 SNR [dB] Spectral efficiency Heavy shadowing Light shadowing Average shadowing Indep.SC-FDMA N c = 64 N c = 1 OFDM Target BER = 10 -3 Figure 5.2: Spectral efficiency of OFDM and SC-FDMA for independent subcarrier responses over different shadowing profiles with ZF equalizer 0 10 20 30 40 0 1 2 3 4 5 6 7 8 SNR [dB] Spectral efficiency Heavy shadowing Light shadowing Average shadowing N c = 1 N c = 64 Target BER = 10 -3 OFDM Ind. SC-FDMA Figure 5.3: Spectral efficiency of OFDM and SC-FDMA for independent subcarrier responses over different shadowing profiles with MMSE equalizer 51 5.5. CONCLUSIONS 0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 s PDFβ ZF N c = 1 N c = 64 Figure 5.4: PDF of βZF under light shadowing for 10 dB SNR and different allocated subcarriers 0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1 1.2 s PDF β ZF N c = 1 N c = 64 Figure 5.5: PDF of βZF under average shadowing for 10 dB SNR and different allocated subcarriers 52 CHAPTER 5. SC-FDMA SPECTRAL EFFICIENCY OVER SHADOWED RICE LMS CHANNEL 0 1 2 3 4 0 0.5 1 1.5 2 2.5 3 s PDF of β MMSE Heavy shadow Average shadow Light shadow OFDM, Light shadow OFDM Average shadow Figure 5.6: PDF of βMMSE under different shadowing profiles for 10 dB SNR and 64 allocated subcarriers 53 5.5. CONCLUSIONS 0 10 20 30 40 50 0 1 2 3 4 5 6 7 8 SNR [dB] Spectral efficiency N c = 1, OFDM N c = 16, IFDMA N c = 16, LFDMA N c = 64, IFDMA N c = 64, LFDMA N c = 64, Ind.SC-FDMA Target BER = 10 -3 Figure 5.7: Spectral efficiency: OFDM, IFDMA, LFDMA and independent SC-FDMA for light shadowing with ZF equalizer 0 5 10 15 20 25 30 35 40 45 0 1 2 3 4 5 6 7 8 SNR [dB] Spectral efficiency N c = 1, OFDM N c = 16, IFDMA N c = 64, IFDMA N c = 16, LFDMA N c = 64, LFDMA N c = 64, Ind.SC-FDMA Target BER = 10 -3 Figure 5.8: Spectral efficiency: OFDM, IFDMA, LFDMA and independent SC-FDMA for light shadowing with MMSE equalizer 54 CHAPTER 5. SC-FDMA SPECTRAL EFFICIENCY OVER SHADOWED RICE LMS CHANNEL 0 5 10 15 20 25 30 35 40 45 0 1 2 3 4 5 6 7 8 SNR [dB] Spectral efficiency N c = 1, OFDM N c = 16, IFDMA N c = 64, IFDMA N c = 16, LFDMA N c = 64, LFDMA N c = 64, Ind.SC-FDMA Target BER = 10 -3 Figure 5.9: Spectral efficiency: OFDM, IFDMA, LFDMA and independent SC-FDMA for average shadowing with ZF equalizer 0 5 10 15 20 25 30 35 40 45 0 1 2 3 4 5 6 7 8 SNR [dB] Spectral efficiency Nc = 1, OFDM Nc = 16, IFDMA Nc = 64, IFDMA Nc = 16, LFDMA Nc = 64, LFDMA Nc = 64, Ind.SC-FDMA Target BER = 10-3 Figure 5.10: Spectral efficiency: OFDM, IFDMA, LFDMA and independent SC-FDMA for average shadowing with MMSE equalizer 55 6.3. SC-FDMA SIMO SYSTEM MODEL S(t) sP/S CP sCP(t) SNc Mapping sM M-IDFT Nc-DFT Figure 6.2: Transmission scheme in SISO SC-FDMA R1 RNR M-DFT CombinerNc-IDFT rNRCP(t) Demapping Remove CP S/P M-DFT r1CP(t) R(t) RNc Remove CP S/P Demapping Figure 6.3: Reception scheme in SIMO SC-FDMA The channel responses at each subcarrier can be written as a vector of the responses at each path, hi= [hi,j]T,j= 1..NR, to be combined to exploit diversity at the receiver as shown in Figure 6.3. Perfect channel estimation and synchronization avoiding interference from other users are assumed. The cyclic prefix is suppressed and Mpoint discrete Fourier transform (M-DFT) operation transfers the time domain symbol into a frequency-domain symbol at the receiver side. Later on, the demapping process is applied on received symbol at each antenna. The vector of received signals on the ith subcarrier at all antennas Ri= [Ri,j]T, j = 1..NRcan be expressed as Ri=hisi+n(6.3.1) Where nis a noise vector whose entries are i.i.d. complex Gaussian CN(0, N0). The NR signals at each subcarrier are frequency combined using the weight vector Wi= [Wi,j] and normalized to allow detection Ri=PNR j=1 Ri,jWi,j PNR j=1|Wi,j|2.(6.3.2) Specifically, the result for MRC is obtained with the weight vector as the channel 62 CHAPTER 6. SC-FDMA PERFORMANCE OVER RECEIVER DIVERSITY TECHNIQUES conjugated Wi,j =h∗ i,j, thus the vector of signals at all subcarriers can be written as RMRC i=PNR j=1 Ri,jh∗ i,j PNR j=1|hi,j|2(6.3.3) For EGC [46], the NRsignals are frequency combined by taking out the angle of the channel but with no modification in amplitude REGC i=PNR j=1 Ri,j exp (−·angle(hi,j)) PNR j=1|hi,j|(6.3.4) 6.4 Simulation Results and Discussion In this section, we provide the results of SIMO maximal ratio combining and equal gain combining SC-FDMA performance in different scenarios. We also compare BER performance of SIMO MRC with single-input-single-output SC-FDMA with zero forcing and minimum mean square error equalization. The SIMO MRC SC-FDMA performance also has been studied for different correlation factor (ρ), for various antennae at receiver and number of allocated subcarriers. The parameters used for channel modelling are depicted in Table 3.1. Simulation parameters for SIMO SC-FDMA are shown in Table 6.1. Table 6.1: Simulation Parameters for SIMO SC-FDMA Parameter Value Total number of subcarriers 1024 Allocated subcarriers ( Nc) 1, 8, 16, 64 Modulation scheme QPSK Carrier Frequency 2.00 GHz System Bandwidth 20.00 MHz Channel Model LMS with exponential PDP Number of Antennas used 2 by default, 4 Diversity Techniques Used MRC, EGC τavg 1.17µs 63 6.4. SIMULATION RESULTS AND DISCUSSION 6.4.1 Results and Discussion of SC-FDMA performance with MRC Diversity Figures 6.4 and 6.5 compare SIMO and SISO BER performance over different shadowing scenarios for both IFDMA and LFDMA, respectively. All results are plotted for two receiving antenna, NR= 2, except otherways said. In both cases, SIMO MRC performance is better than that of SISO IFDMA for both equalization techniques ZF and MMSE. The reason is that for SIMO MRC we have multiple antennas, and MRC combining technique can exploit their diversity as each signal arriving at receiver undergoes different fading. As in SISO, results with light shadowing are showing lower BER than with heavy shadowing because light shadowing is a Rice fading with 5 dB Rice factor. In the case of a Rician fading, the direct line of sight component is stronger than multipath component while the heavy shadowing is equivalent to Rayleigh fading. The effect of different allocated subcarriers is depicted in Figures 6.6 and 6.7 for interleaved and localized SC-FDMA, respectively. An increase in allocated subcarriers results in certain gain in SNR for LFDMA, about 1dB from Nc= 32 to 64. The effect is similar to that for the single antenna at the receiver. The effect of the different number of antennas at the receiver can be observed in Figures 6.8 and 6.9. As expected, BER improves with the increase in the number of antennae. However, it is shown that most gains were obtained for the use of two antennae as only BER improvement is slight for both shown shadowing. All the previous results were plotted assuming independence among signals received at different antennae. We now compare the effect of correlation among antennas on BER performance for different shadowing. Figures 6.10, 6.12 and 6.14 show the BER performance of IFDMA for various correlation factor and different shadowing. Equivalent results for LFDMA can be found in Figures 6.11, 6.13 and 6.15. It is observed that the BER grows with higher correlation factor. The reason is that less diversity exists between antennas as correlation increases. However, certain correlation (up to 0.5 in light shadowing and 0.3 for average and heavy shadowing) does not affect the BER significantly. 64 CHAPTER 6. SC-FDMA PERFORMANCE OVER RECEIVER DIVERSITY TECHNIQUES 0 5 10 15 20 25 30 10-4 10-3 10-2 10-1 100 SNR [dB] BER MRC, N R = 2, Light MRC, N R = 2, Average MRC, N R = 2, Heavy MMSE , Light ZF " Figure 6.4: BER of SIMO and SISO IFDMA for QPSK, Nc= 64, over different shadowed channel 0 5 10 15 20 25 30 10 -4 10 -3 10 -2 10 -1 10 0 SNR [dB] BER MRC, N R =2, Light MRC, N R = 2, Average MRC, N R = 2, Heavy MMSE, Light ZF, Light Figure 6.5: BER of SIMO and SISO LFDMA for QPSK, Nc= 64, over different shadowed channel 65 6.4. SIMULATION RESULTS AND DISCUSSION 0 5 10 15 20 25 30 10 -4 10 -3 10 -2 10 -1 10 0 SNR [dB] BER N c = 32 N c = 64 N c = 256 Figure 6.6: BER of SIMO MRC IFDMA with different Ncfor QPSK over average shadowing 0 5 10 15 20 25 30 10-4 10-3 10-2 10-1 100 SNR [dB] BER Nc = 8 Nc = 32 Nc = 64 Figure 6.7: BER of SIMO MRC LFDMA with different number of Ncfor QPSK over heavy shadowing 66 CHAPTER 6. SC-FDMA PERFORMANCE OVER RECEIVER DIVERSITY TECHNIQUES 0 4 8 12 16 20 10 -4 10 -3 10 -2 10 -1 10 0 SNR [dB] BER N R = 2, Average shadowing N R = 4 N R = 2, Light shadowing N R = 4 Figure 6.8: BER of SIMO MRC IFDMA with different NRat the receiver for QPSK, Nc= 64, over average and light shadowing 0 4 8 12 16 20 10 -4 10 -3 10 -2 10 -1 10 0 SNR [dB] BER NR = 2, Average shadowing NR = 4 NR = 2, Light shadowing NR = 4 Figure 6.9: BER of SIMO MRC LFDMA with different NRat the receiver for QPSK, Nc= 64, over average and light shadowing 67 6.4. SIMULATION RESULTS AND DISCUSSION 0 5 10 15 20 25 30 10 -4 10 -3 10 -2 10 -1 10 0 SNR [dB] BER ρ = 0 ρ = 0.3 ρ = 0.5 ρ = 0.7 ρ = 0.9 ρ = 1 Figure 6.10: BER of SIMO MRC IFDMA with different ρfor QPSK, Nc= 64 over heavy shadowing 0 5 10 15 20 25 30 10 -4 10 -3 10 -2 10 -1 10 0 SNR [dB] BER ρ = 0 ρ = 0.3 ρ = 0.5 ρ = 0.7 ρ = 0.9 ρ = 1 Figure 6.11: BER of SIMO MRC LFDMA with different ρfor QPSK, Nc= 64 over heavy shadowing 68 CHAPTER 6. SC-FDMA PERFORMANCE OVER RECEIVER DIVERSITY TECHNIQUES 0 5 10 15 20 25 30 10 -4 10 -3 10 -2 10 -1 10 0 SNR [dB] BER ρ = 0 ρ = 0.3 ρ = 0.5 ρ = 0.7 ρ = 0.9 ρ = 1 Figure 6.12: BER of SIMO MRC IFDMA with different ρfor QPSK, Nc= 64 over average shadowing 0 5 10 15 20 25 30 10 -4 10 -3 10 -2 10 -1 10 0 SNR [dB] BER ρ = 0 ρ = 0.3 ρ = 0.5 ρ = 0.7 ρ = 0.9 ρ = 1 Figure 6.13: BER of SIMO MRC LFDMA with different ρfor QPSK, Nc= 64 over average shadowing 69 6.4. SIMULATION RESULTS AND DISCUSSION 0 5 10 15 20 25 30 10 -4 10 -3 10 -2 10 -1 10 0 SNR [dB] BER ρ = 0 ρ = 0.3 ρ = 0.5 ρ = 0.7 ρ = 0.9 ρ = 1 Figure 6.14: BER of SIMO MRC IFDMA with different ρfor QPSK, Nc= 64 over light shadowing 0 5 10 15 20 25 30 10 -4 10 -3 10 -2 10 -1 10 0 SNR [dB] BER ρ = 0 ρ = 0.3 ρ = 0.5 ρ = 0.7 ρ = 0.9 ρ = 1 Figure 6.15: BER for SIMO MRC LFDMA with different ρfor QPSK, Nc= 64 over light shadowing 70 CHAPTER 6. SC-FDMA PERFORMANCE OVER RECEIVER DIVERSITY TECHNIQUES 6.4.2 Results and Discussion of SC-FDMA Performance with EGC Diversity In this section, we have investigated SC-FDMA performance for equal gain combining diversity over correlated shadowed Rice LMS channel. Figures 6.16 and 6.17 show the comparison of BER performance of SISO and SIMO 1 ×2 for IFDMA and LFDMA, respectively. As for MRC, it is observed that BER is lower with SIMO system as compared to SISO system with MMSE and ZF equalizer. Although in EGC all signals are co-phased to avoid signal cancellation, each diversity signal branch is weighted with the same factor, whatever may be the signal amplitude. As a result, the average SNR improvement of EGC is typically about 1 dB worse than that of MRC. On the other hand, implementation complexity is lower compared to that of MRC as only channel phase has to be estimated and compensated. A comparison between BER performance of both diversity techniques is shown in Figures 6.18 and 6.19 for interleaved and localized SC-FDMA, respectively. For different number of antennae, the BER performance of IFDMA and LFDMA is depicted in Figures 6.20 and 6.21. As expected, with increased number of antennae at the receiver, BER lowers although improvement is only slight compared to that of using two antennae instead of a SISO scheme. Different allocated subcarriers is shown in Figure 6.22. The comparison of IFDMA BER for different shadowing and with different correlation factor is shown in Figure 6.23. As explained with MRC IFDMA results, here also we observe with growing BER with antenna correlation, but influence below 0.5 is not great. 6.5 Conclusions In this chapter, we studied the MRC and EGC receiver diversity techniques. Results for MRC are better (about 1 dB) compared to those in EGC at the expense of higher complexity. It is observed that the results of IFDMA are better as compared to LFDMA in all kind of shadowing because subcarriers are distributed over entire bandwidth while with LFDMA they are adjacent to each other, so the correlation between them is high. Note that for a correlation between antenna signals below ρ= 0.5 the performance degradation is negligible. 71 7.2. CONTRIBUTIONS transmission spectral efficiency for shadowed Rice LMS channel. We provide a description of physical layer radio interface technologies, OFDM, and SC-FDMA. We discussed emphasizing the similarities and differences between them in Chapter 2. In Chapter 3, we have talked about the wireless channel and different fading models. We give details about shadowed Rice LMS channel whose LOS follows Nakagami distribution. In Chapter 4, we study SISO SC-FDMA performance for shadowed Rice LMS channel. SC-FDMA performance has been considered for different constellation size and different allocated subcarriers with ZF and MMSE equalization at the receiver side. SC-FDMA results has been compared to those of OFDM over channels with light, average and heavy shadowing. In Chapter 5, we study the analysis of the spectral efficiency of SC-FDMA transmission of shadowed Rice LMS channel for ZF and MMSE equalization. We compare SC-FDMA spectral efficiency for independent frequency channel responses at the allocated subcarriers and correlated channel frequency responses. We also compare IFDMA and LFDMA spectral efficiency to that of OFDMA. In Chapter 6, we study SC-FDMA performance with receiver diversity techniques, maximal ratio combining and equal gain combining over LMS channel. 7.2 Contributions The main purpose of this thesis is the analysis of an SC-FDMA systems for shadowed Rice land mobile satellite channels regarding BER, spectral efficiency, and receiver diversity techniques. The obtained results has been compared to those of OFDM. This comparison allows us to determine the difference in performance between both technologies. Other contributions made during the development of this thesis are listed below: •BER performance of SC-FDMA and OFDMA for shadowed Rice LMS channel with independence among channel frequency response and correlated channel frequency responses. •Spectral efficiency of SC-FDMA and OFDMA transmission for shadowed Rice LMS 78 CHAPTER 7. CONCLUSIONS AND FUTURE SCOPE channel with adaptive modulation and coding. •SC-FDMA performance with receiver diversity techniques for shadowed Rice LMS channel. The receiver diversity techniques used are maximal ratio combining and equal gain combining. 7.3 Future Scope The results of this dissertation give interesting directions for future work. Some of them are as follows: •SC-FDMA performance for shadowed Rice LMS channel where LOS follows a log-normal distribution. •SC-FDMA performance with multiple antennae at transmitter and receiver for shadowed Rice LMS channel •SC-FDMA transmit diversity with shadowed Rice LMS channel •SC-FDMA performance for multi-state LMS channel model •SC-FDMA performance with Massive MIMO •SC-FDMA channel capacity with LMS channel 79 7.3. FUTURE SCOPE 80 Appendix A Rendimiento de SC-FDMA con t´ecnicas de diversidad para canales m´oviles satelitales A.1 Introducci´on La modulaci´on por divisi´on en frecuencias ortogonales (OFDM, Orthogonal Frequency Divison Multiplexing) se ha convertido en una t´ecnica muy popular para la transmisi´on en canales m´oviles por dos razones: a) proporciona una soluci´on menos compleja en canales con selectividad en frecuencia; b) proporciona buena eficiencia espectral. OFDM ha sido adoptada como tecnolog´ıa de capa f´ısica por varios est´andares inal´ambricos como IEEE 802.11/Wi-Fi [1], IEEE 802.16/WiMAX [2], 3GPP Long Term Evolution (LTE) en el enlace descendente [3], y la televisi´on digital terrestre (Digital Video Broadcasting-Second Generation Terrestrial, DVB-T2) [4]. Recientemente, tambi´en ha sido elegida en est´andares que hacen uso de enlaces por sat´elite (Digital Video Broadcasting-Satellite services to Handhelds, DVB-SH) [5]. En enlaces por sat´elite, que habitualmente no son muy selectivos en frecuencia, el uso de OFDM se debe principalmente a la necesidad de mantener tecnolog´ıas similares con los sistemas terrestres. Durante muchos a˜nos, se ha considerado que OFDM no es apropiada para las comunicaciones por sat´elite. La raz´on son sus fluctuaciones de amplitud, que producen una raz´on entre potencia de pico y potencia media (Peak-to-Average-Power-Ratio, PAPR) elevada, lo que hace OFDM sensible a la distorsi´on no lineal causada por los amplificadores 81 A.1. INTRODUCCI ´ ON de potencia. Una alternativa interesante a OFDM es la t´ecnica de acceso m´ultiple por divisi´on en frecuencia de portadora ´unica conocida en ingls como Single Carrier Frequency Division Multiple Access (SC-FDMA) que ha sido adoptada en el enlace ascendente de la cuarta generaci´on de comunicaciones m´oviles Long Term Evolution (LTE) [3]. SC-FDMA se describe como un esquema de frecuencia ´unica con igualaci´on en el dominio de la frecuencia. Su rendimiento es similar al de OFDM pero con una PAPR menor, lo que hace m´as eficiente el terminal de usuario. As´ı, la t´ecnica SC-FDMA es una candidata mejor que OFDM para las comunicaciones por sat´elite, adem´as de que presenta la ventaja de que es compatible con los transmisores basados en OFDM. Tambi´en utiliza un prefijo c´ıclico y tambi´en la igualaci´on se realiza en el dominio de la frecuencia. Esta igualaci´on puede eliminar completamente las interferencias (Zero Forcing, ZF) o bien reducirlas pero limitando el ruido (Minimum Mean Square Error, MMSE). SC-FDMA puede verse como OFDM con una precodificaci´on en el transmisor a trav´es de una transformada de Fourier discreta (Discrete Fourier Transform, DFT) previa al bloque que implementa la transformada inversa de Fourier para la transmisi´on OFDM. Con esta DFT, la energ´ıa de los s´ımbolos modulados se reparte entre un conjunto de portadoras. Este mapeo puede realizarse a portadoras contiguas (Localized SC-FDMA, LFDMA) o equiespaciadas en todo el ancho de banda (Interleaved SC-FDMA, IFDMA) [9]. Para aliviar los desvanecimientos inherentes a las comunicaciones m´oviles, muchos sistemas hacen uso de t´ecnicas de diversidad espacial [10] como la combinaci´on m´axima (Maximal Ratio Combining, MRC) o igual (Equal Gain Combining, EGC) entre antenas. En MRC, las se˜nales que llegan a las antenas se pesan y combinan para obtener la relaci´on se˜nal a ruido (Signal to Noise Ratio, SNR) m´axima mientras que un combinador EGC ´unicamente corrige la fase de las se˜nales para evitar que se cancelen y directamente las combina. El indicador de rendimiento usado m´as habitualmente es la tasa binaria de error (Bit Error Rate, BER) para una cierta modulaci´on. En los ´ultimos a˜nos, la modulaci´on en cuadratura (Quadrature Amplitude Modulation, QAM) ha sido la constelaci´on m´as empleada. Para mejorar la capacidad del sistema, la velocidad binaria y la cobertura, la se˜nal 82 APPENDIX A. RENDIMIENTO DE SC-FDMA CON T´ ECNICAS DE DIVERSIDAD PARA CANALES M ´ OVILES SATELITALES transmitida a o desde un usuario puede ser modificada continuamente para seguir las variaciones de la calidad de la se˜nal a trav´es de un proceso habitualmente conocido como modulaci´on adaptativa (Adaptive Modulation and Coding, AMC). As´ı es posible adaptar el esquema de modulaci´on a las condiciones del canal para cada usuario (requiriendo, por supuesto, un canal de realimentaci´on y cierta reducci´on de la velocidad de transmisi´on debida a la se˜nalizaci´on necesaria). Aquellos usuarios con condiciones buenas del canal consiguen mayores ´ordenes de modulaci´on (por ejemplo, 64 QAM) mientras se reservan las modulaciones m´as robustas para los usuarios con peores condiciones para mantener la BER por debajo de un cierto objetivo. En este caso, la eficiencia espectral (en bits por segundos por hertzio) es la m´etrica de inter´es. En canales por sat´elite, el modelo de canal de Loo [18] es el m´as ampliamente aceptado y empleado en la literatura para canales m´oviles (Land Mobile Satellite, LMS). Este modelo asume que la envolvente del canal sigue la distribuci´on de Rice con una componente de visi´on directa que, a su vez, es aleatoria y sigue una distribuci´on lognormal. En [19] se propone un modelo m´as simple y anal´ıticamente tratable que considera que la l´ınea de visi´on directa sigue una distribuci´on de Nakagami. Este modelo de Rice-Nakagami se ajusta de manera similar al de Loo a los datos experimentales. La bibliograf´ıa considera diferentes niveles de desvanecimientos para las comunicaciones por sat´elite, com´unmente llamadas como sombreado ligero, medio y fuerte. El ´ultimo se refiere a un canal muy cercano a uno de Rayleigh mientras un sombreado ligero asume una l´ınea de visi´on directa potente con componentes multicamino d´ebiles. Al tratarse de una transmisi´on de banda ancha, aparece la interferencia entre s´ımbolos debido a que la duraci´on del s´ımbolo es corta comparada con el ensanchamiento del retardo. El perfil de potencia del canal define la correlaci´on entre las respuestas del canal a diferentes subportadoras ya que el mecanismo puede tambi´en verse como selectividad en frecuencia. La igualaci´on con fuerce a cero (ZF) aplica entonces la inversa de la respuesta del canal. MMSE es tambi´en una igualaci´on lineal pero al canal se le a˜nade una constante antes de la inversi´on para evitar un aumento del ruido incontrolado. 83 A.2. DESCRIPCI ´ ON DE OFDM Y SC-FDMA S/P M-IDFT CPP/S ()St () CP st s Figure A.1: Transmisor OFDMA S/P CP P/S M-DFT FDE X ()tR () CP rt r R  R Figure A.2: Receptor OFDMA A.2 Descripci´on de OFDM y SC-FDMA En OFDM [22], el ancho de banda disponible se divide entre subportadoras que son ortogonales en el sentido de que el pico de una portadora coincide con los nulos de las otras, evitando as´ı el uso de guardas en frecuencia y aumentando la eficiencia espectral. Normalmente, como el ancho de banda de la se˜nal transmitida sobre cada portadora es menor que el ancho de banda de coherencia del canal, la respuesta puede considerarse plana, y el canal puede modelarse entonces como un conjunto de canales de banda estrecha, cada uno de ellos monoportadora. El diagrama de bloques del transmisor y del receptor de un sistema OFDM se presentan en las figuras A.1 y A.2, respectivamente. Aunque no se muestra en la figura, en primer lugar la secuencia de bits se mapea a una secuencia de s´ımbolos complejos de acuerdo al esquema de modulaci´on usado, t´ıpicamente QAM. Los s´ımbolos complejos sse transmiten en paralelo empleando una transformada inversa de Fourier de tama˜no M. Para eliminar la dispersi´on temporal y la interferencia entre s´ımbolos, se a˜nade como guarda una extensi´on c´ıclica de longitud mayor que la respuesta del canal. Esta extensi´on transforma la convoluci´on lineal entre los s´ımbolos OFDM en el dominio del tiempo y la respuesta del canal en una convoluci´on circular que, en el dominio de la frecuencia, se presenta como una multiplicaci´on entre los s´ımbolos en cada suportadoras y la respuesta en frecuencia del canal. De esta forma, es posible realizar la igualaci´on en el dominio de la frecuencia. En el receptor, es elimina el prefijo c´ıclico y la trasformada discreta de Fourier devuelve el s´ımbolo OFDM al dominio 84 APPENDIX A. RENDIMIENTO DE SC-FDMA CON T´ ECNICAS DE DIVERSIDAD PARA CANALES M ´ OVILES SATELITALES c N S s M S Nc-DFT S/P M-IDFT P/S CP ()tS () CP st s Mapping Figure A.3: Esquema de transmisi´on en SC-FDMA r  c N R c N R ()tR M R P/S Nc-IDFTM-DFT S/P CP () CP rt r Demap FDE Figure A.4: Esquema de recepci´on en SC-FDMA de la frecuencia. La versi´on ruidosa de la se˜nal transmitida se entrega al igualador en el dominio de la frecuencia y, finalmente, al detector para recuperar los datos transmitidos. SC-FDMA y OFDMA usan los mismos bloques de transmisi´on, con la diferencia de que se incluye una trasformada de Fourier m´as en el transmisor (y su equivalente inversa en el receptor). En el modo entrelazado de SC-FDMA (Interleaved SC-FDMA, IFDMA) los s´ımbolos se sit´uan equiespaciados en todo el ancho de banda del canal. En el modo localizado (Localized SC-FDMA, LFDMA), los s´ımbolos se asignan a un grupo de portadoras adyacentes. OFDM y SC-FDMA pueden transmitir la misma cantidad de s´ımbolos en el mismo periodo pero en SC-FDMA la transmisi´on es monoportadora aunque distribuida en el ancho de banda asignado. Las figuras A.3 y A.4 muestran la cadena de transmisi´on-recepci´on para SC-FDMA. Como paso previo, la secuencia de bits a transmitir se mapean en s´ımbolos complejos usando QAM. El vector complejo resultante de longitud Nces precodificado por una transformada discreta de Fourier (Discrete Fourier Transform, DFT) antes de ser mapeado en un subconjunto de subportadoras asignadas. Los s´ımbolos precodificados son entonces mapeados al subconjunto de portadoras por usuario, Ncde las Mportadoras, contiguas o entrelazadas. Las portadoras no empleadas se fuerzan a cero. Desde este punto, la transmisi´on es similar a la de OFDMA. 85 A.3. MODELOS DE CANAL En el receptor, despu´es del demapeo de las subportadoras, la igualaci´on puede hacerse ZF o MMSE. Finalmente, los s´ımbolos vuelven al dominio del tiempo con una transformada discreta de Fourier inversa y son entregados al detector. A.3 Modelos de canal A.3.1 Modelo de canal LMS Rice sombreado En un canal LMS ideal, hay una l´ınea de visi´on directa clara entre el sat´elite y el usuario terrestre, de manera que la envolvente no es aleatoria. Sin embargo, existe un desvanecimiento producido en los caminos que no tienen l´ınea de visi´on directa, de manera que finalmente el canal puede considerarse que el canal sigue una distribuci´on de Rice. Adem´as, que la l´ınea se bloquee parcialmente hace aleatoria tambi´en la componente de visi´on directa. El equivalente paso bajo de la envolvente del canal puede finalmente escribirse como: z(t) = A(t) exp (jα(t)) + B(t) exp(jα0) (A.3.1) donde α(t) es la fase aleatoria que se distribuye uniformemente en [0,2π), y α0es la fase de la componente de visi´on directa. Los procesos estacionarios independientes A(t) y B(t), que tambi´en son independientes de α(t), representan las amplitudes de la componente multicamino y de visi´on directa respectivamente. El modelo propuesto en [19] usa una distribuci´on de Rayleigh para A(t) y de Nakagami para B(t) cuyas funciones de densidad de probabilidad (Probability Density Function, PDF) pueden escribirse como: fA(a) = a b0 exp −a2 2b0(A.3.2) fB(b) = 2mm Γ(m)Ωmb2m−1exp −mb2 Ω(A.3.3) donde 2b0=E[A2] es la potencia media de la componente multicamino, Γ ( ) es la funci´on gamma, m=E[B2] V ar [B2]≥0 es el par´ametro de Nakagami con V ar [ ] como la varianza y Ω = E[B2] es la potencia media de la componente de visi´on directa. La PDF de una distribuci´on de Rice sombreada para la envolvente de la se˜nal Z=|z(t)| 86 APPENDIX A. RENDIMIENTO DE SC-FDMA CON T´ ECNICAS DE DIVERSIDAD PARA CANALES M ´ OVILES SATELITALES en un canal LMS puede escribirse como [19]: fZ(z) = 2b0m 2b0m+ Ωmz b0 exp−z2 2b01F1m, 1,Ωz2 2b0(2b0m+ Ω).(A.3.4) Valores diferentes de mmodelan distintos entornos. Mientras m= 0 corresponden a ´areas con obstrucci´on completa de la l´ınea de visi´on directa, cualquier valor 0 < m < ∞ representa ´areas con obstrucci´on parcial. Los valores com´unmente considerados se presentan en la Tabla A.1. Table A.1: Par´ametros para los canales con desvanecimientos considerados [19] Tipo de sombreado b0mΩ Ligero 0.158 19.4 1.29 Fuerte 0.063 0.739 8.97 ×10−4 Medio 0.126 10.1 0.835 A.3.2 Modelos de respuesta en frecuencia La respuesta en frecuencia del canal, representada por la matriz diagonal HMde tama˜no M×M, es la transformada de Fourier de la respuesta al impulso variable del canal. Se han empleado dos modelos para la dependencia entre las respuestas en frecuencia del canal en las subportadoras asignadas, h=diag(PHHMP), con Pla matriz de mapeo. Como primer modelo, se han supuesto independientes e id´enticamente distribuidos los elementos de h. Cada uno de ellos sigue la distribuci´on de Rice ensombrecida dada por la ec. A.3.4. La PDF de la potencia instant´anea en cada subportadora, |hj|2, puede escribirse como (ec. (6) in [19]): f|hj|2(q) = 2b0m 2b0m+ Ωm1 2b0 exp−q 2b01F1m, 1,Ωq 2b0(2b0m+ Ω).(A.3.5) Como segundo modelo, se ha empleado un modelo en banda ancha formado por varios ecos retrasados una cantidad diferente. El primer eco se asume que sigue el modelo de Rice sombreado descrito anteriormente mientras que el resto de ecos siguen una distribuci´on de Rayleigh, es decir, no presentan l´ınea de visi´on directa. Con este modelo, es f´acil mostrar que 87 A.4. PROBABILIDAD BINARIA DE ERROR PARA CANALES LMS puede escribirse como σ2 ξ= N0 Nc Nc P j=1 |hj| |hj|2+No/Es 2+Es Nc Nc P j=1 |hj|2 |hj|2+No/Es 2−Es 1 Nc Nc P j=1 |hj|2 |hj|2+No/Es !2 1 Nc Nc P j=1 |hj|2 |hj|2+No/Es !2(A.4.25) Se nota como βMMSE a βMMSE =1 Nc Nc X j=1 1 |hj|2+N0/ES = Nc X j=1 βMMSE j.(A.4.26) Con esta definici´on, el rango de βMMSE est´a limitada a 0 ≤βMMSE ≤Es N0. Manipulando las ecuaciones, se puede escribir: σ2 ξ=ES N0 ESβMMSE 1−N0 ESβMMSE ,(A.4.27) La PDF de βMMSE jpuede obtenerse por un procedimiento similar al empleado en ZF para llegar a: fβMMSE j(s) = 2b0m 2b0m+ Ωm1 2b0 1 Ncs2exp−1−sNcN0/ES 2b0Ncs1F1m, 1,Ω(1 −sNcN0/ES 2b0(2b0m+ Ω)Ncs. (A.4.28) Si se compara esta expresi´on con la de ZF de la ec. (A.4.8), se pueden observar que ambas son equivalentes para N0= 0. Como en el caso anterior, si se asume independencia entre las respuestas a distintas frecuencias, la funci´on caracter´ıstica de βMMSE se puede calcular a partir de la de βj, que puede obtenerse con m´etodos num´ericos desde su PDF. En el caso de MMSE, la SNR puede expresarse desde ec. (A.4.27) como γMMSE =Es σ2 ξ = Es N0 βMMSE −1 (A.4.29) y su CDF se puede evaluar como: FMMSE γ(g)=1−FMMSE βEs/N0 g+ 1 .(A.4.30) 94 APPENDIX A. RENDIMIENTO DE SC-FDMA CON T´ ECNICAS DE DIVERSIDAD PARA CANALES M ´ OVILES SATELITALES A.4.3 Resultados Para ambos igualadores, la tasa binaria de error puede calcularse como la media de la BER condicionada a β, es decir, BER . =ZBER(β)fβ(β)dβ (A.4.31) con BER(β) evaluada a partir de la BER de QAM sobre canales AWGN [14]. Para codificaci´on Gray y QAM cuadrada, puede evaluarse esta BER [35] como: BER(γ) = 2 √Mlog2√M log2√M X k=1 (1−2−k(√M−1) X i=0 ((−1bi2k−1 √Mci2k−1 √M+1 2Q s6(2i+ 1)2 2(M−1) γ!) (A.4.32) simplemente sustituyendo γpor βcomo se describen en las ecuaciones (A.4.12) o (A.4.29) para ZF y MMSE FDE, respectivamente. Table A.2: Par´ametros de simulaci´on para SC-FDMA en canales LMS. Par´ametro Valor N´umero total de subportadoras M1024 Longitud del prefijo c´ıclico 144 Valor por defecto de Es/N0(dB) 10 ∆f15KHz τavg (perfil de potencia de retardo exponencial) 1.17µs CDF de β Se han seguido dos aproximaciones diferentes para evaluar la PDF de βdependiendo del modelo de canal. Para respuestas en frecuencia independientes, la CDF de βpuede evaluarse num´ericamente siguiendo el procedimiento descrito previamente. Los resultados se muestran en la figura A.5 para un desvanecimiento fuerte y SNR media 10 dB. Por su parte, cuando se considera el perfil exponencial de la ec. (A.3.6), la existencia de correlaci´on impide usar que la CHF de la suma es el producto de la de cada t´ermino por separado. En ese caso, se ha seguido una aproximaci´on semianal´ıtica al problema: desde los valores de hse ha 95 A.4. PROBABILIDAD BINARIA DE ERROR PARA CANALES LMS 0 20 40 60 80 100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 s F β ZF (s) N c =1 N c = 4 N c = 8 N c = 16 N c = 64 0 2 4 6 8 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 s F β MMSE (s) N c = 1 N c = 4 N c = 8 N c = 16 N c = 64 Figure A.5: CDF of βpara respuestas en frecuencia independientes entre subportadoras con sombreado fuerte para ZF (izquierda) y MMSE (derecha) 0 12 24 36 48 60 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 F β ZF (s) s N c = 1 N c = 4 N c = 8 N c = 16 N c = 64 0 2 4 6 8 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 F β MMSE (s) s N c = 1 N c = 4 N c = 8 N c = 16 N c = 64 Figure A.6: CDF of βpara respuestas en frecuencia correladas entre subportadoras con sombreado fuerte para ZF (izquierda) y MMSE (derecha) con IFDMA 96 APPENDIX A. RENDIMIENTO DE SC-FDMA CON T´ ECNICAS DE DIVERSIDAD PARA CANALES M ´ OVILES SATELITALES 0 12 24 36 48 60 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 s F β ZF (s) N c = 1 N c = 4 N c = 8 N c = 16 N c = 64 0 2 4 6 8 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 s F β MMSE (s) N c = 1 N c = 4 N c = 8 N c = 16 N c = 64 Figure A.7: CDF of βpara respuestas en frecuencia correladas entre subportadoras con sombreado fuerte para ZF (izquierda) y MMSE (derecha) con LFDMA evaluado los de βcon (A.4.7) o eq. (A.5.8). Usando la condici´on de ergodicidad, su CDF se eval´ua a partir del histograma. Los resultados en este caso se muestran en las figuras A.6 y A.7. Adem´as, βtoma estad´ısticamente valores menores que para el desvanecimiento medio o suave. Desde estos resultados para subportadoras independientes, est´a claro que para la igualaci´on ZF βpuede alcanzar valores m´as altos debido al aumento del ruido por valores bajos de ganancia del canal. Por el contrario, la igualaci´on MMSE fuerza un valor m´aximo de β. Esto tambi´en se observa en el comportamiento tan diferente para la igualaci´on ZF y MMSE seg´un Nc. Con la igualaci´on ZF, reducir el n´umero de subportadoras reduce el valor de βZF por lo que los valores para OFDM (Nc= 1) son menores. Por otra parte, en el caso de la igualaci´on MMSE, el efecto es el contrario. El efecto de la correlaci´on en frecuencia puede verse comparando A.5, A.6 y A.7. S´olo se observan diferencias peque˜nas para MMSE. Por el contrario, el uso de igualaci´on ZF cuando hay mucha correlaci´on en frecuencia, como en LFDMA, reduce el rango de β, forzado por la subportadora m´as desvanecida. 97 A.5. EFICIENCIA ESPECTRAL DE SC-FDMA PARA EL CANAL LMS 0 6 12 18 24 30 10 -4 10 -3 10 -2 10 -1 10 0 SNR [dB] BER AWGN Indep.SC-FDMA OFDM IFDMA LFDMA Numerical Results 0 6 12 18 24 30 10 -4 10 -3 10 -2 10 -1 10 0 SNR [dB] BER AWGN Indep. SC-FDMA OFDM IFDMA LFDMA Numerical Results Figure A.8: BER para 16QAM para respuestas en frecuencia independientes entre subportadoras con sombreado suave, Nc= 64 e igualaci´on ZF (izquierda) y MMSE (derecha) Resultados de BER La figura A.8 permite comparar la BER en canales con sombreado suave para igualaci´on ZF y MMSE. Como se ha descrito anteriormente, para ZF, el valor de SNIR γes siempre menor que el de OFDM y, como consecuencia, la BER es peor. En el caso de MMSE, la BER es, por el contrario, m´as peque˜na para SC-FDMA. La correlaci´on en frecuencia mejora el comportamiento de ZF mientras empeora el de MMSE. La figura A.9 permite analizar el efecto del n´umero de subportadoras. Se observa que aumentar el n´umero de portadoras asignadas mejora el comportamiento para MMSE pero no as´ı para ZF, limitada por la peor subportadora. A.5 Eficiencia espectral de SC-FDMA para el canal LMS La t´ecnica de modulaci´on y codificaci´on adaptativa (Adaptive Modulation and Coding, AMC) se emplea en la mayor parte de los sistemas de comunicaciones actuales para mitigar la selectividad temporal. B´asicamente, la modulaci´on adaptativa sigue el canal de manera que, cuando su calidad es peor, se emplean esquemas con constelaciones/codificaciones m´as robustas. Por otra parte, si la calidad del canal mejora, crece la velocidad de transmisi´on a 98 APPENDIX A. RENDIMIENTO DE SC-FDMA CON T´ ECNICAS DE DIVERSIDAD PARA CANALES M ´ OVILES SATELITALES 0 6 12 18 24 30 10 -4 10 -3 10 -2 10 -1 10 0 SNR [dB] BER N c = 1 N c = 8 N c = 64 AWGN Numerical Results 0 6 12 18 24 30 10 -4 10 -3 10 -2 10 -1 10 0 SNR [dB] BER N c = 1 N c = 8 N c = 64 AWGN Numerical Results Figure A.9: BER para 16QAM para respuestas en frecuencia independientes entre subportadoras con sombreado suave, diferentes Nce igualaci´on ZF (izquierda) y MMSE (derecha) coste de c´omo de protegidos est´an los bits transmitidos. A.5.1 Revisi´on de la modulaci´on adaptativa Se considera una modulaci´on adaptativa con potencia constante y con un conjunto de constelaciones limitadas cuyos tama˜nos son R0= 0, R1= 2, and Ri= 22(i−1),i=2 .. N. Pueden emplearse varios criterios para asociar una constelaci´on a cada SNR γen la modulaci´on adaptativa. En esta tesis se ha discretizado el nivel de desvanecimientos de tal manera que la BER se mantiene siempre bajo una objetivo, BERT, que es un par´ametro de dise˜no. Como se muestra en la figura A.10, el rango de γse divide en Nregiones <i={γi, γi+1},i= 0,1,2, ......N −1 con γ0= 0 y γN=∞. En la regi´on <ise emplea una constelaci´on QAM con Ribits/s´ımbolo. Para valores de SNR por debajo de un umbral m´ınimo (es decir, condiciones de canal muy pobres), hay un estado en el que no hay transmisi´on de datos. Los umbrales se han dise˜nado para maximizar la eficiencia espectral bajo la limitaci´on de que la BER sea menor que BERT. Dado que la BER es mon´otona decreciente con la SNR, los umbrales son aquellos valores que hacen la BER exactamente igual a la objetivo. 99 A.5. EFICIENCIA ESPECTRAL DE SC-FDMA PARA EL CANAL LMS 0 3 6 9 12 15 18 21 24 27 30 10-4 10-3 10-2 10-1 100 Instaneous SNR γ [n] dB BER 4QAM 16QAM 64QAM 1 3 γ γγ γ 1 γ γγ γ 2 γ γγ γ 3 2 BER T Figure A.10: Ejemplos de regiones de modulaci´on adaptativa para QAM sin codificaci´on Sea cual sea el criterio tomado, la eficiencia espectral media puede escribirse como: R B= L−1 X i=1 log2(Mi)Pr{γi−1< γ ≤γi}.(A.5.1) Valores m´as estrictos de BERTimplican eficiencias espectrales menores. Para llevar a cabo la adaptaci´on, la SNR instant´anea tiene que ser estimada en el receptor, que determina la regi´on de modulaci´on <ie informa al transmisor. Ambas SNRs, media e instant´aneas, se suponen perfectamente conocidas en el transmisor y en el receptor. A.5.2 An´alisis de la eficiencia espectral ZF Como se ha obtenido anteriormente, la SNR instant´anea en el receptor para SC-FDMA con igualaci´on ZF viene dada por γZF =ES βZF N0 ,(A.5.2) donde βZF se defini´o como βZF =1 Nc Nc X k=1 1 |hk|2.(A.5.3) 100 APPENDIX A. RENDIMIENTO DE SC-FDMA CON T´ ECNICAS DE DIVERSIDAD PARA CANALES M ´ OVILES SATELITALES Para OFDM la misma expresi´on es v´alida haciendo Nc= 1. En ambos casos, la constelaci´on correspondiente a la i-´esima regi´on se aplica entre los umbrales, es decir, γi<Es βZF N0 < γi+1 (A.5.4) Puede reescribirse la ecuaci´on anterior en funci´on de unos umbrales para βZF : βZF i+1 =1 γi+1 ES N0 < βZF <1 γi ES N0 =βZF i(A.5.5) donde βZF iand βZF i+1 son los umbrales. Puede observarse que los umbrales as´ı descritos dependen de Es/N0. La eficiencia espectral puede promediarse entonces como: R B= N−1 X i=1 log2(Mi)FβZF βZF i−FβZF βZF i+1.(A.5.6) donde Nes el n´umero de regiones de modulaci´on. Obs´ervese que s´olo es necesaria la CDF FβZF (βZF ) dada por la ec. (A.4.11). An´alisis para MMSE Para MMSE, el valor de la SNR en el receptor viene determinada por la ec. (A.4.29), repetida aqu´ı por conveniencia: γMMSE =Es σ2 ξ = Es N0 βMMSE −1.(A.5.7) donde βMMSE =1 Nc Nc X j=1 1 |hj|2+N0/ES = Nc X j=1 βMMSE j.(A.5.8) El par´ametro βMMSE determina la SNR y la constelaci´on que usar, esto es, βMMSE i+1 =1 γi+1 ES N0 < βMMSE <1 γi ES N0 =βMMSE i(A.5.9) con βMMSE iand βMMSE i+1 , los umbrales modificados que, de nuevo, dependen de la SNR. La eficiencia espectral puede calcularse como R B= N−1 X i=1 log2(Mi)FβMMSE βMMSE i−FβMMSE βMMSE i+1 .(A.5.10) que, de nuevo, solo requiere la CDF FβMMMSE (βMMSE). 101 A.5. EFICIENCIA ESPECTRAL DE SC-FDMA PARA EL CANAL LMS 0 5 10 15 20 25 30 35 40 45 0 1 2 3 4 5 6 7 8 SNR [dB] Spectral efficiency Heavy shadowing Light shadowing Average shadowing Indep.SC-FDMA N c = 64 N c = 1 OFDM Target BER = 10 -3 Figure A.11: Eficiencia espectral de OFDM y SC-FDMA para respuestas en frecuencia independiente entre subportadoras y diferentes sombreados con igualador ZF 0 10 20 30 40 0 1 2 3 4 5 6 7 8 SNR [dB] Spectral efficiency Heavy shadowing Light shadowing Average shadowing N c = 1 N c = 64 Target BER = 10 -3 OFDM Ind. SC-FDMA Figure A.12: Eficiencia espectral de OFDM y SC-FDMA para respuestas en frecuencia independientes entre subportadoras y diferentes sombreados con igualador MMSE 102 APPENDIX A. RENDIMIENTO DE SC-FDMA CON T´ ECNICAS DE DIVERSIDAD PARA CANALES M ´ OVILES SATELITALES Table A.3: Umbrales entre regiones de modulaci´on de SNR y eficiencia espectral por constelaci´on γi(dB) 9 .7980 16.5290 22.5510 28.4170 log2(Mi) 2 4 6 8 A.5.3 Resultados Los resultados presentados aqu´ı se refieren a una BER de referencia BERT= 10−3. Valores m´as estrictos implican menor eficiencia espectral. El conjunto de umbrales {γi, i = 1, .., N −1}se presenta en la tabla A.3. Las figuras A.11 y A.12 muestran la eficiencia espectral de OFDM y SC-FDMA para respuestas en frecuencia independientes entre subportadoras y diferentes sombreados para los igualadores ZF y MMSE, respectivamente. El resultado para Nc= 1 se refiere a OFDM. Se observa que un aumento del n´umero de subportadoras reduce la eficiencia espectral en SC-FDMA. En cuanto a la forma que presentan las figuras, se observa que parecen escalones. Este comportamiento se nota m´as para un sombreado suave. La raz´on para esto puede encontrarse en el comportamiento de la pdf de β, que es muy estrecha. Esto implica que la mayor parte del tiempo se emplea un ´unico esquema de modulaci´on. En el caso de OFDM (Nc= 1), la pdf de βocupa un rango m´as amplio, lo que, a trav´es del promediado, resulta en un crecimiento m´as suave de la eficiencia espectral. A.6 Rendimiento de SC-FDMA para t´ecnicas de diversidad en recepci´on A.6.1 T´ecnicas de diversidad en recepci´on Esta tesis doctoral se centra en dos t´ecnicas habituales de diversidad en recepci´on: combinaci´on por selecci´on (Selection Combining, SC) y combinaci´on de m´aximo cociente (Maximal Ratio Combining, MRC). Para las dos t´ecnicas, la meta es encontrar un conjunto de pesos para cada antena tal que el impacto de los desvanecimientos sea m´ınima para el usuario mientras se limita la complejidad tanto por la estimaci´on como por el procesado. Se 103 A.7. CONCLUSIONES Y L´ INEAS FUTURAS Table A.4: Par´ametros de simulaci´on para SIMO SC-FDMA Tama˜no de la FFT 1024 Esquema de modulaci´on QPSK Frecuencia de portadora 2.00 GHz Ancho de banda del sistema 20.00 MHz Modelo de canal LMS con perfil exponencial N´umero de antenas 2pordefecto, 4 T´ecnicas de diversidad MRC, EGC τavg 1.17µs factor de correlaci´on para varios canales. Se observa que, a medida que crece el factor de correlaci´on, la BER crece. La raz´on es que existe menos diversidad. Se muestra en la figura A.19 una comparaci´on entre EGC y MRC con dos antenas receptoras. EGC es ligeramente peor que MRC pero, aun as´ı, mejor que la igualaci´on con una ´unica antena. A.7 Conclusiones y l´ıneas futuras En esta tesis, se ha presentado el trabajo desarrollado para el an´alisis de rendimiento en t´erminos de BER, eficiencia espectral y diversidad de un receptor SC-FDMA para el canal sombreado Rice m´ovil terrestre por sat´elite (LMS). En primer lugar, se han introducido las tecnolog´ıas de capa f´ısica utilizadas y diferentes tipos de modelos de canal para llevar a cabo an´alisis anterior mencionado. A continuaci´on, se ha evaluado el rendimiento de SC-FDMA para canales con sombreado fuerte, medio y ligero con igualaci´on ZF y MMSE. Se ha analizado tambi´en la eficiencia espectral de SC-FDMA, y se ha estudiado la ganancia por usar varias antenas receptoras. Los resultados obtenidos se compararon con los de OFDMA. Algunas l´ıneas futuras posibles son estas: •An´alisis de SC-FDMA en canales LMS con la l´ınea de visi´on directa siguiendo una distribuci´on lognormal. 110 A.7. CONCLUSIONES Y L´ INEAS FUTURAS Table A.4: Par´ametros de simulaci´on para SIMO SC-FDMA Tama˜no de la FFT 1024 Esquema de modulaci´on QPSK Frecuencia de portadora 2.00 GHz Ancho de banda del sistema 20.00 MHz Modelo de canal LMS con perfil exponencial N´umero de antenas 2 por defecto, 4 T´ecnicas de diversidad MRC, EGC τavg 1.17µs factor de correlaci´on para varios canales. Se observa que, a medida que crece el factor de correlaci´on, la BER crece. La raz´on es que existe menos diversidad. Se muestra en la figura A.19 una comparaci´on entre EGC y MRC con dos antenas receptoras. EGC es ligeramente peor que MRC pero, aun as´ı, mejor que la igualaci´on con una ´unica antena. A.7 Conclusiones y l´ıneas futuras En esta tesis, se ha presentado el trabajo desarrollado para el an´alisis de rendimiento en t´erminos de BER, eficiencia espectral y diversidad de un receptor SC-FDMA para el canal sombreado Rice m´ovil terrestre por sat´elite (LMS). En primer lugar, se han introducido las tecnolog´ıas de capa f´ısica utilizadas y diferentes tipos de modelos de canal para llevar a cabo an´alisis anterior mencionado. A continuaci´on, se ha evaluado el rendimiento de SC-FDMA para canales con sombreado fuerte, medio y ligero con igualaci´on ZF y MMSE. Se ha analizado tambi´en la eficiencia espectral de SC-FDMA, y se ha estudiado la ganancia por usar varias antenas receptoras. Los resultados obtenidos se compararon con los de OFDMA. Algunas l´ıneas futuras posibles son estas: •An´alisis de SC-FDMA en canales LMS con la l´ınea de visi´on directa siguiendo una distribuci´on lognormal. 110 APPENDIX A. RENDIMIENTO DE SC-FDMA CON T´ ECNICAS DE DIVERSIDAD PARA CANALES M ´ OVILES SATELITALES •An´alisis de SC-FDMA con varias antenas transmisoras y receptoras. •SC-FDMA con diversidad en transmisi´on. •SC-FDMA para un canal multiestado LMS. •SC-FDMA con muchas antenas transmisoras y receptoras (Massive MIMO). •Capacidad de SC-FDMA. 111 A.7. 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