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Bluetooth receiver design based on Laurent’s decomposition Ibrahim, Noha 2005-12-31

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BLUETOOTH RECEIVER DESIGN BASED ON LAURENT'S DECOMPOSITION by NOHA IBRAHIM B.Eng, American University of Beirut, 2003 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES (Electrical and Computer Engineering) December, 2005 © Noha Ibrahim, 2005 Abstract Bluetooth is a widely used communication standard for wireless personal area networks (WPAN). The Bluetooth transmit signal is Gaussian frequency shift keying (GFSK) modulated. GFSK belongs to the family of continuous-phase modulation (CPM) sig nals, which achieve a good trade-off between power and bandwidth efficiency and, due to constant envelope modulation, allow for low-complexity transmitter implementa tion. Bluetooth devices often employ a simple discriminator receiver, which is highly suboptimum in terms of power efficiency compared to the optimum receiver. Other ap proaches proposed in the literature consider trellis-based detection using the Viterbi or forward-backward algorithm. These schemes achieve significant performance improve ments over discriminator detectors while entailing a considerably higher computational complexity. The main challenges faced when designing a Bluetooth sequence detector is the varying modulation index, which results in a varying trellis structure, and the time-variant channel phase, making coherent detection which assumes perfect channel phase estimation an almost impossible task. In this research work, we present a receiver design for Bluetooth transmission based on Laurent's decomposition of the Bluetooth transmit signal. The main features of this receiver are its low-complexity compared to alternative solutions, its excellent performance close to the theoretical limit, and its high robustness against frequency offsets, phase noise, and modulation index variations, which are characteristic for low-cost Bluetooth devices. In particular, we show that the devised noncoherent decision-ii m feedback equalization receiver achieves a similar performance as a recently proposed 2-state noncoherent sequence detector, while it is advantageous in terms of complexity. The new receiver design is therefore highly attractive for a practical implementation. Contents Abstract ii Contents v List of Tables vii List of Figures viiGlossary xii Acknowledgements xvi1 Introduction 1 1.1 The Evolution of the Bluetooth Technology 2 1.2 Challenges and Motivation 4 1.3 Contributions 6 1.4 Thesis Outline 7 2 Background 9 2.1 The Bluetooth System 9 2.1.1 Radio Front End 10 2.1.2 Physical Channel and Packet Definition 11 2.1.3 Physical Links 2 iv Contents v 2.1.4 Modulation Scheme 13 2.1.5 Transmission Model 6 2.1.6 Error Correction 20 2.2 Bluetooth Receivers Literature Review 21 2.2.1 Optimum Receiver 22 2.2.2 Suboptimum Receivers 7 2.3 MLSD Lower Bound 31 3 Noncoherent Decision Feedback Equalizer Receiver Structure 33 3.1 Laurent's Representation of CPM Signals 34 3.1.1 Laurent's Decomposition and Its Application to Bluetooth ... 35 3.1.2 Optimum Coherent Receiver Based on Laurent's Decomposition 39 3.1.3 Main Pulse 41 3.2 Filter Design 3 3.3 Discrete-time Transmission Model 47 3.4 Reduced-State Sequence Estimation 8 3.5 Decision-Feedback Equalizer Structure 50 3.5.1 Minimum Phase Channel 1 3.5.2 DFE Decision Rule 53 3.6 Noncoherent Detection 4 3.6.1 NDFE Decision Rule 57 3.6.2 Frequency Offset Estimation 58 3.6.3 Adaptive NDFE .3.7 Summary of Proposed Receiver Structure 60 4 Performance Results and Discussion 2 4.1 Coherent Detection 63 4.1.1 Filter Selection4.1.2 State Reduction 5 Contents vi 4.2 Noncoherent Detection 66 4.2.1 Performance with Constant Channel Phase 67 4.2.2 Performance Using the Modified Phase Reference Under Con stant and Varying Channel Conditions 75 4.2.3 Adaptive NDFE ' 80 4.2.4 Performance in the Presence of Interference . : 83 5 Conclusions 86 Bibliography 9 List of Tables 2.1 Interference performance [51] 18 2.2 The values of relatively-prime integers m and p corresponding to differ ent modulation indices h within the range specified by the Bluetooth standard 25 3.1 Duration of Laurent components.Cp(t) 38 vii List of Figures 1.1 The required 101ogi0(iViVo) ^or B^R = ^ 3 as a function of the mod-. ulation index h 5 2.1 General basic rate packet format [51] 11 2.2 General enhanced data rate packet format [51] 12 2.3 Block diagram of a CPM modulator 14 2.4 Frequency pulse shape g(t) and phase pulse q(t) for Gaussian frequency shift keying with time-bandwidth product BT = 0.5 15 2.5 Block diagram representing the ECB system model 19 2.6 Illustration of Repetition code 21 2.7 Phase state trellis structure for Bluetooth with h = 1/3 26 2.8 LDI detector block diagram 29 2.9 Block diagram of the GFSK modulator using the Rimoldi/Huber&Liu decomposition approach 31 3.1 Laurent pulses C0(t) and Ci(t), and the main pulse P(t) for Bluetooth GFSK signals with L = 2 and h = 1/3 39 3.2 Block diagram of optimum coherent receiver based on Laurent's decom position 40 3.3 The imaginary and real components of an exact Bluetooth signal s(t) and an approximate signal s(t) generated using only the main pulse P(i). 42 viii List of Figures ix 3.4 Impulse response of SRC, hSRc{t), before Hamming window (a), and after Hamming window (b) 45 3.5 Magnitude frequency response for WMF, HWMF(f), assuming h = 0.28 and h = 0.35, and for SRC filter, HSRC{f), with p = 0.3 46 3.6 The CIR of the (a) ISI channel hc[k] compared to (b) its minimum phase equivalent ho[k] 48 3.7 Block diagram of decision-feedback equalizer 51 3.8 The roots of Hc(z) compared to the roots of its minimum phase equiv alent H0(z) 53 3.9 Structure of the proposed Bluetooth receiver 60 4.1 Performance comparison of the SRC and WMF as receiver filters. Co herent detection with h = 1/3 is assumed 64 4.2 Evaluation of the state reduction in the proposed receiver. Coherent detection with h = 1/3 is assumed 65 4.3 Performance of NDFE with a = 0.8 with FBF and FFF fixed at the receiver, or varying according to the value of h at the transmitter. ... 66 4.4 Evaluation of the performance of the proposed receiver with state re duction in case of noncoherent detection 67 4.5 Performance of the NDFE using the /V-metric compared to MLSD, LDI, and coherent DFE, and h = 1/3 for all cases 69 4.6 Performance of the NDFE using the a-metric compared to MLSD, LDI, and coherent DFE, and h = 1/3 for all cases 69 4.7. Performance of the NDFE using the iV-metric h = 1/3 in the presence of phase jitter 70 4.8 Performance of NDFE using the a-metric h = 1/3 in the presence of phase jitterList of Figures x 4.9 Performance of the NDFE with a = 0.8 for different values of the mod ulation index compared to the MLSD bound, LDI, and MLM-LDI. ... 72 4.10 Performance of the NDFE for varying h with a = 0.4, 0.6, 0.8, and 0.9. 73 4.11 Performance of NDFE for varying h with a = 0.6 and 0.8 compared to that of NSD for a = 0.6 74.12 The required 10\ogw{Eb/N0) for BER = 10~3 for NDFE with a = 0.8 with varying modulation index 75 4.13 lOlogio (Eb/N0) required for BER = 10-3 for different combinations of a and (3 as a function of the modulation index with constant channel phase and h known 76 4.14 The required l0\ogw(Eb/N0) for BER = 10"3 for NDFE Wlth a = 0.6,(3 = 0.9 for varying h 8 4.15 The required mog10(Eb/NQ) for BER = 10"3 for NDFE with a = 0.8, f3 = 0.9 for varying h 74.16 The required 10\og10(Eb/N0) for BER = 10-3 in the presence of fre quency offset and phase jitter 79 4.17 The required 101ogi0(^b/^Vo) for BER = 10"3 in the presence of fre quency offset and phase jitter 79 4.18 101ogio(£yiVo) required for BER = 10~3 for ANDFE with phase refer ence (3.46). The channel phase is constant and the notation ANDFE(JVe, H) applies. 81 4.19 101bgio(£&//V0) required for BER = 10~3 for ANDFE with various com binations of a and /3 with phase reference (3.53). The channel phase is constant and the notation ANDFE(Are, H) applies 82 4.20 Mog10(Eb/N0) required for BER = lO"3 for ANDFE and ANSD with the favorable pair (a = 0.6, (3 = 0.9) and with constant channel phase. 82 List of Figures xi 4.21 Performance of NDFE in the presence of interference for the WMF and the SRC filter. The channel phase is constant and the notation IF(SIR,A/Cii) applies 84 4.22 10\og10(Eb/N0) required for BER = 1(T3 for NDFE and ANDFE in the presence of interference. The channel phase is constant and the notation IF(SIR,A/C|i) applies 84 Glossary List of Abbreviations (In alphabetical order) ACL Asynchronous connectionless link ANDFE Adaptive noncoherent decision-feedback equaliz* AMP Amplitude modulated pulses ARQ Automatic retransmission query AWGN Additive white Gaussian noise BER Bit error rate CPM Continuous phase modulation CRC Cyclic redundancy check DFE Decision-feedback equalizer DM Medium data rate DV Data voice ECB Equivalent complex baseband EDR Enhanced data rate EV Extended voice eSCO extended synchronous connection-oriented FBF Feedback filter FCC Federal Communications Commission FEC Forward error correction xii Glossary xiii FHS Frequency hop synchronization FIR Finite impulse response FFF Feedforward filter GFSK Gaussian frequency shift keying HEC Header error check HV High-quality voice ISI Intersymbol interference ISM Industrial, scientific, medical LDI Limiter-discriminator integrator MLS(D/E) Maximum-likelihood sequence (detector/estimator) NDFE Noncoherent decision-feedback equalizer NRSSE Noncoherent reduced-state sequence estimation NSD Noncoherent sequence detector PSK Phase shift keying PSP Per-survivor processing RSSE Reduced-state sequence estimation SCO Synchronous connection-oriented SIG Special interest group SIR Signal-to-interference power ratio SD Sequence detector SRC Square-root raised cosine UWB Ultra-wide band VA Viterbi algorithm WLAN Wireless local area network WMF Whitened matched filter Glossary xiv Operators and Notation t continuous time [k] Discrete time a Input binary data vector of a[fc] € {±1} b[k] Zeroth complex symbols bp[k] Complex symbols B 3-dB bandwidth Cp(t) Laurent pulses exp Exponential function Ef, Received energy per bit Es Signal energy per modulation interval f[k] (N)DFE feedforward filter fc Carrier frequency fd Frequency deviation with respect to the carrier frequency g(t) Frequency shaping pulse h Modulation index h(t) Gaussian low-pass filter h0[k] Minimum phase equivalent of hc[k], the (N)DFE feedback filter hc[k] Discrete-time channel impulse response (ISI channel) hsRc{t) Impulse response of the SRC filter hwMF(t) Impulse response of the WMF HsRc{f) Frequency response of the SRC filter HWMF(I) Frequency response of the WMF H Number of modulation index hypotheses used in ANDFE i(t) Equivalent complex baseband representation of interfering signal SRF(S>) Passband representation of transmitted signal io(-) Modified Bessel function of first kind and zero order Glossary xv K Rician factor log Base-10 logarithm In Natural logarithm L CPM memory m the relatively prime numerator integer such that h = m/p n{t) Additive white Gaussian noise No One-sided noise power spectral density of the passband noise process Ne Length of estimation period used in ANDFE p the relatively prime denominator integer such that h = m/p P(t) Laurent main pulse q(t) Normalized phase pulse qTef Phase reference used in noncoherent detection Q(t) Gaussian Q-function r(t) Equivalent complex baseband representation of received signal 7"DFE [k] output of feedforward filter rect(i/T) rectangular pulse s(t) Equivalent complex baseband representation of transmitted signal Sn(t) Special sin functions used in Laurent's derivation process T Symbol period a Phase estimation forgetting factor (3 Frequency offset estimation forgetting factor A/ Frequency offset between the transmitter and receiver oscillator A/C)j frequency offset of the desired signal compared to the interfering signal #0 Constant phase p Roll-off factor of the SRC filter 0o Constant phase rotation as a result of fading 4>(t) Time-varying phase tp(t) Phase function <p(t) = 2irhq(t) Glossary xvi ifj(t,a) Information-carrying phase ty(t) Generalized phase pulse function Acknowledgements I would like to express my gratitude to my research supervisors, Dr. Lutz Lampe and Dr. Robert Schober, for their guidance and encouragement, and their valuable contribution to this research work. Their productive supervision throughout the past two years not only helped me broaden my scope of knowledge, but also increased my appreciation and dedication to this challenging research project, which has proven to be highly rewarding. I also gratefully acknowledge the financial support in the forms of RA from Dr. Lutz Lampe and Dr. Robert Schober through NSERC grants STPGP 257684 and RGPIN 283152-04, and TA from the Electrical and Computer Engineering Department. I would like to thank my colleagues in the communication theory group for sustaining a friendly work environment. Finally, I would like to acknowledge the love and consistent support my family has provided me throughout my graduate studies in UBC. I would also like to thank my friends, especially Axel Davidian, for always being there, while being physically present at the other side of the globe. xvii Chapter 1 Introduction The pervasive use of mobile computing devices such as laptops and personal digi tal assistants (PDAs), and the evident success of cellular phones called for a wire less technology to connect these devices together. This technological vision became a reality with the introduction of the Bluetooth standard for wireless personal area networks (WPANs) which enables wireless communication among various electronic devices. Bluetooth has revolutionized the wireless world, for it provides low-power, low-cost, and short-range radio links with secure and reliable transmissions and global compatibility. Bluetooth is an open standard, which enables manufacturers to take full advantage of the capabilities of the technology and build products according to its specifications, thus expanding the Bluetooth applications to diverse market areas. With seamless voice and data connections to virtually all mobile devices, the human imagination remains the only limit to application options. The Bluetooth technology specification, currently in its fourth version of the core specification, is developed by an industry-based association, the Bluetooth Special Interest Group (SIG), and will be outlined in detail in the next chapter. The following section gives a brief history of the Bluetooth technology, its current 1 1.1 The Evolution of the Bluetooth Technology 2 status, and its future path. Section 1.2 states the motivation behind the present work and the challenges faced, and a summary of the various contributions is provided in Section 1.3. Finally, a brief description of the contents of the thesis concludes this chapter. 1.1 The Evolution of the Bluetooth Technology In 1994, as the sales of cellular phones were increasing, Ericsson, one of leading telecom munications manufacturers, was investigating ways to add value to its phones in the crowded market. Consequently, Ericsson mobile research lab in Lund, Sweden, initi ated a study to investigate the feasibility of a low-power, low-cost wireless technology to connect mobile phones and their accessories without the need of cumbersome cables. The study rapidly progressed, and as researchers realized the developed technology can be used to exchange data among numerous wireless and wired devices, the study quickly turned into a large project, which was given an internal code name,"Bluetooth". Blue tooth is the English derivative of the Viking word Blatand, and was named after the 10th century Danish Viking King Harald Blatand who united Denmark and Norway during a time of fighting. Bluetooth developers considered the name appropriate for the technology since they anticipated it will unite the telecommunications world, just as Blatand united his world [5]. To allow Bluetooth to be an accepted industry standard, five major companies from three diverse business areas formed the Bluetooth Special Interest Group (SIG). The group was formally announced on May 20, 1998, and included two leading companies in mobile telephony, Ericsson and Nokia, two leading companies in laptop computing, IBM and Toshiba, and one leading company in digital signal processing, Intel. Today, the Bluetooth SIG has over 3,400 member companies all over the world [21]. In its mission statement, the Bluetooth SIG affirms that it "will support a collaborative 1.1 The Evolution of the Bluetooth Technology 3 environment and drive programs to develop and advance Bluetooth wireless technology in order to exceed personal connectivity expectations and meet the needs of a changing world". Versions 1.0, 1.0b, and 1.1 of the Bluetooth specifications were released in 1999, 2000, 2001, respectively. Bluetooth received additional support when the IEEE Standards Association approved the IEEE Std. 802.15.1, derived from the Bluetooth Specification, in March, 2002. The standard was published three months later, and introduced minor changes to the physical and medium access control (MAC) layers. Version 1.2 of the Bluetooth specification was formally ratified on November 2003, while the latest Bluetooth Core Specification Version 2.0+ Enhanced Data Rate (EDR) was released a year later, and provided increased data rates and lower power consumption. The market for Bluetooth devices has been rapidly growing, and, as a result, the tech nology has met the significant milestone of five million Bluetooth units shipped per week. The number of Bluetooth-enabled devices doubled from 2003 to 2004, and is expected to reach 500 million units by the end of 2005. Moreover, a recent study con ducted in Japan, United states, and United Kingdom has shown an increased customer awareness of Bluetooth. Another emerging technology, ultra-wide band (UWB) appears to have great potential for the wireless applications which currently use Bluetooth. UWB transmits with very low power at extremely large bahdwidths, but there are still some challenges before this technology can be realized. To meet the future market demands, and take advantage of the high data rates that UWB offers, the Bluetooth SIG announced in May, 2005 the collaboration of the Bluetooth and UWB developers. The goal is to employ UWB in the next-generation Bluetooth products, while maintaining backward compatibility with the existing products [1]. 1.2 Challenges and Motivation 4 1.2 Challenges and Motivation The objective of the present work is to design a cost-effective, power-efficient, and structurally-simple Bluetooth receiver. The optimum Bluetooth receiver has very high structural and computational complexity, for it consists of a bank of matched filters followed by a coherent1 maximum likelihood sequence detector (MLSD) with a number of states varying according to the modulation index. Since the modulation index in Bluetooth systems is allowed to vary in a relatively wide range (0.28 < h < 0.35), the corresponding number of states in the optimum receiver broadly varies with a minimum of 12 (as will be detailed in the next chapter), making it unfeasible. The practical and simple alternative, namely, the limiter-discriminator integrator (LDI) receiver, currently used for Bluetooth devices is a simple, low-cost receiver. However, it is highly suboptimal, for it suffers a performance loss of more than 6 dB over the optimum receiver, as shown in Fig. 1.1. The 6 dB gap suggests that the LDI receiver consumes four times as much signal power as the optimum receiver to achieve the same bit error rate (BER). In the present work, we investigate the feasibility of an alternative Bluetooth receiver design, motivated by the large possible power efficiency gains that may be achieved over the conventional LDI receiver. This requires overcoming the following challenges faced when designing a Bluetooth receiver. The varying modulation index results in a varying trellis structure with a large number of states, which poses a serious challenge when considering trellis-based detection. In addition, the metrics required for the detection of the maximum likelihood symbol se quence are also dependent on h. However, the trellis-based receivers achieve significant performance improvements over the conventional LDI receiver. Hence, we investigate 1 Coherent detection assumes perfect channel phase estimate at the receiver. 1.2 Challenges and Motivation 5 18 17 T CO o 16 1— II CC UJ 15 CD O T3 14 a '5 CT <U 13 12 LU 11 O O 10 9 • LDI receiver -*-• Optimum receiver] t • k : >6dB ! 0.29 0.3 0.31 0.32 0.33 0.34 0.35 Figure 1.1: The required 101ogi0(£,6/A/o) for BER = 10-3 as a function of the modula tion index h. the design of a reduced-state sequence detector independent of the modulation index. A carrier frequency deviation of up to ±100 KHz is allowed in Bluetooth systems [51], resulting in fast channel phase variations. Several coherent detectors which achieve significantly higher power efficiency compared to the LDI are available in the litera ture. However, perfect channel phase estimation is assumed at the receiver, which is impractical in Bluetooth systems. Therefore, we consider the suboptimum noncoherent detection schemes which perform implicit channel phase estimation. Since Bluetooth operates in the license-free ISM band, it is vulnerable to interference from other Bluetooth and wireless local area network (WLAN) devices operating in the same frequency band. The design of the optimal receiver filter is dependent on the modulation indices, and, moreover, it is not guaranteed to accomplish strong out-of-band interference suppression. The present work studies the design of a practical input receiver filter independent of h and capable of accomplishing strong adjacent and co-channel interference suppression. 1.3 Contributions 6 1.3 Contributions In the present work, we consider noncoherent sequence detection for Bluetooth signals based on Laurent's decomposition which models the GFSK signal as a superposition of amplitude modulation pulses, as will be described in Chapter 3 [32]. We then make the following contributions. • As with Laurent's decomposition the actual nonlinear modulation scheme is transformed into a linear modulation over an intersymbol-interference (ISI) chan nel, we develop reduced-state trellis-based equalizers using the concepts of reduced-state sequence estimation (RSSE) [16] and per-survivor processing (PSP) [39], which will be presented in Chapter 3. We also devise noncoherent versions of the RSSE and DFE receivers, which we refer to as noncoherent RSSE (NRSSE) and noncoherent DFE (NDFE). • We propose the use of an off-the-shelf square-root raised cosine (SRC) filter as receiver input filter preceding symbol-rate sampling and RSSE, and we argue that this filter provides an almost sufficient statistic for the transmitted data. • Using a seven-tap feedforward filter to obtain a minimum-phase channel im pulse response, we show that decision-feedback equalization (DFE) achieves a performance close to the lower bound of maximum-likelihood sequence estima tion (MLSE)2 [17]. This is a remarkable result, since complete state reduction incurred considerable performance degradation in case of an alternative nonco herent sequence detector (NSD) in the literature [31], [27]. We also note that a similar approach based on Laurent's decomposition required a 4-state trellis decoder to achieve a similar performance [43]. 2In the context of the present work, "sequence estimation" and "sequence detection" are used synonymously. 1.4 Thesis Outline 7 • We develop a simple frequency estimation method which provides high robustness to (extreme) frequency offsets and phase jitter. • NRSSE and NDFE are extended to adapt the decision metric to the modulation index h used at the transmitter side. This adaptive algorithm offers a simple solution to the problem of varying modulation index, and provides acceptable performance for all values of h. • By means of simulation results for various transmission scenarios, we show that NDFE achieves almost the same performance as 2-state NSD. Due to its com plexity advantage over NSD, the NDFE receiver is an attractive solution for a practical implementation. 1.4 Thesis Outline This thesis is organized as follows. Chapter 2 introduces the background information required for the implementation and evaluation of a Bluetooth receiver. Moreover, a literature review on the optimum and suboptimum Bluetooth receivers, as well as the MLSE lower bound are presented in order to provide solid grounds for comparison. Chapter 3 describes in detail the structure of the proposed receiver. It starts by introducing Laurent's decomposition, which forms the basis of the present work. It points out the decomposition's major strengths and presents the optimum Bluetooth receiver based on the decomposition. After discussing the input receiver filter design, the corresponding discrete-time transmission model is introduced. The decision metric for coherent RSSE is then given, leading to the description of the structure of the extreme case of RSSE, the DFE. The minimum phase channel needed to achieve the maximum performance in reduced-state detectors is also discussed. The following 1.4 Thesis Outline 8 section applies noncoherent detection to RSSE and DFE, through which phase and frequency estimation schemes are devised. The adaptive detector which estimates the modulation index at the receiver is then presented. We conclude this chapter by giving a brief summary of the proposed receiver structure. Chapter 4 analyzes the performance of the proposed receiver in various scenarios and for various design parameters. After the evaluation of the proposed input receiver filter, the receiver performance with coherent and noncoherent detection is presented. Specifically, we study the effect of state reduction, and show the performance of the phase and frequency offset estimation methods under constant and varying channel conditions. We also show the effect of the varying modulation index, before the adaptive modulation index estimator is evaluated. The simulation of the receiver performance achieved in the presence of adjacent and co-channel interference concludes the chapter. Chapter 5 concludes this dissertation by giving a brief summary of the key character istics of the proposed receiver. Chapter 2 Background This chapter provides the background information necessary for the design, analysis, and evaluation of the Bluetooth receiver presented thereafter. Bluetooth was designed to be a universal wireless technology, which operates at low power and can be imple mented at low cost. Accordingly, the Bluetooth specifications [51] described in the first section of this chapter were carefully chosen so as to satisfy these requirements. Since the present work involves a receiver design, the second section provides a literature sur vey on the optimum and suboptimum Bluetooth receivers. These, in addition to the MLSE lower bound presented in Section 2.3 will serve as benchmarks when evaluating the proposed solution. 2.1 The Bluetooth System This section elaborates on the Bluetooth specifications relevant for the present work. The radio specifications will be described first, followed by the physical channel and packet definition. The types of physical links available for data transmission as well as the modulation scheme used will then be presented. We conclude this section with an illustration of the channel model and an overview of the error correction schemes offered by Bluetooth. 9 2.1 The Bluetooth System 10 2.1.1 Radio Front End Bluetooth operates in the 2.4 GHz Industrial Scientific Medical (ISM) band, a fre quency band that is globally available, license-free, and open to any radio system. The Bluetooth standard specifies a 1 MHz bandwidth for each RF channel. However, there are variations in the range of frequencies available in the ISM band, and the Bluetooth specifications were defined to accommodate these differences. In most countries, where at least 80 MHz of bandwidth is allocated to the ISM band, Bluetooth defines 79 RF channels located at (2402 + k) MHz, k = 0,1,... 78. In other countries, such as France, Spain, and Japan, which allow the usage of only a reduced spectrum for the ISM band, 23 Bluetooth channels are defined, located at (2454 + k) MHz, k = 0,1,..., 22. Since Bluetooth devices do not have exclusive use of the frequency band, possible in terference from other devices (garage door openers, baby monitors, microwave ovens, cordless phones, etc.) can be experienced. To minimize the effect of interference, a frequency-hopping spread spectrum approach is used. The radio hops through all the RF channels following a pseudo-random sequence [9], [13], [51]. Federal communica tions commission (FCC) regulations part 15.247 require that a device operates on a given channel for a maximum of 0.4 seconds within a 30-second interval [34]. The Bluetooth core specification defines three transmit-power classes: 100 mW (20 dBm), 2.5 mW (4 dBm), and 1 mW (0 dBm), corresponding to Class 1, 2, and 3 devices, re spectively. With Class 1 devices, the communication range may increase up to 100 m. Power-save modes including Sniff, Hold, and Park (in decreasing order of power require ments) modes are used to reduce power consumption. Power control is used for devices with transmitter power higher than 4 dBm, and thus, necessary for Class 1 devices. For devices with lower transmitter power, power control is optional for optimizing the power consumption and the interference level [51]. 2.1 The Bluetooth System 11 ACCESS CODE HEADER PAYLOAD Figure 2.1: General basic rate packet format [51]. 2.1.2 Physical Channel and Packet Definition Bluetooth devices communicate with each other by forming piconets, which can ac commodate up to eight devices. Several piconets can establish interconnections using bridge devices to form a larger network known as a scatternet. A piconet channel is divided into slots, each having a duration of 625 /usee. Users sharing the same channel are each assigned a time slot, and are time- and hop-synchronized to the channel to ensure reliable communication. Information between Bluetooth devices is exchanged using packets. A packet may oc cupy one, three, or five time slots, and a frequency hop occurs only once the packet is complete. The hopping rate varies based on the number of slots occupied per packet, and reaches a maximum rate of 1600 hops/sec. The Bluetooth Standard defines sixteen different types of packets depending upon the application. In the Bluetooth specifi cation Version 2.0 + EDR, higher transmission rates (relative to the older Bluetooth specification versions) and corresponding packet formats were introduced. In addition to the basic rate of 1 Mbps, which uses the Gaussian frequency shift keying (GFSK) modulation scheme, an enhanced data rate with two new modulation modes was also defined. The primary modulation mode, 7r/4-DQPSK (differential encoded quaternary phase shift keying), provides a data rate of 2 Mbps, while the secondary mode, 8-DPSK (differential encoded 8-ary phase shift keying), increases the data rate to up to 3 Mbps. The general basic rate packet format is shown in Fig. 2.1. Each packet begins with an access code, unique for the channel. The access code consists of 4 Preamble bits, a 64-bit sync word and an optional 4-bit trailer appended to the sync word only if a header packet follows. The access code is used for synchronization, direct current (DC) 2.1 The Bluetooth System 12 ACCESS CODE HEADER GUARD SYNC ENHANCED DATA RATE PAYLOAD TRAILER GFSK • ~* DPSK • Figure 2.2: General enhanced data rate packet format [51]. offset compensation, and identification. A 72-bit header may follow the access code. The header contains important control information such as packet type, flow control, and header error control (HEC), and is encoded with a 1/3 forward error correction (FEC) code. The payload field has a length varying between 0 and 2745 bits [51]. The general enhanced date rate packet format is shown in Fig. 2.2. Each packet consists of an access code, a header, a guard period, a synchronization sequence, an enhanced data rate payload, and a trailer. As noted in the figure, the access code and the header use GFSK, the same modulation scheme used in basic rate packets. The guard time then allows for transition into the higher data rate modulation scheme. Since this dissertation only involves the basic data rate modulation scheme, only GFSK will be elaborated in the following section. 2.1.3 Physical Links Bluetooth uses two types of links for voice and data transmission, the synchronous, connection-oriented (SCO) links and the asynchronous connectionless links (ACL). Bluetooth devices can use either link, depending on the type of packets being trans mitted. Further, a device can alternate between the two types of links during trans mission, as long as an ACL link is established before using an SCO link. ACL links support point-to-multipoint connections with a maximum data rate of 732.2 Kbps in an asymmetrical mode or 433.9 Kbps in a symmetrical mode and are primarily used for file and data transfers. SCO links support point-to-point connections with a data rate of 64 Kbps and are typically reserved for voice traffic. The SCO link was further improved in Version 1.2 of the Bluetooth specification, by defining a new extended 2.1 The Bluetooth System 13 SCO (eSCO) link. New packet types were defined for this link as extended voice (EV) packets, which are the original high quality voice (HV) packets with cyclic redundancy check (CRC) enabled to achieve higher reliability. The eSCO may be used for trans parent synchronous user data and audio transmissions, and data rates of 384 Kbps and 564 Kbps can be achieved. These data rates are the same as those enabled by the third generation (3G) Universal Mobile Telecommunications Service (UMTS) systems, making the eSCO the most suitable link for audio and video streaming applications over 3G networks [6], [22]. 2.1.4 Modulation Scheme Gaussian frequency shift keying (GFSK), a special case of continuous phase modula tion (CPM), is the modulation scheme used in the basic data rate mode of Bluetooth (cf. Section 2.1.2). CPM has been widely used in communications due to its power and bandwidth efficiency as well as its constant envelope. Constant envelope mod ulation schemes are known to allow for efficient power amplification. Further, CPM provides better spectral utilization than frequency shift keying (FSK) by introducing a continuous phase to smooth variations between symbols. The block diagram of a CPM modulator is shown in Fig. 2.3, where the frequency modulator consists of a voltage controlled oscillator (VCO). The resulting CPM passband signal is represented by [2] SRF(t) = Cos (27r/ct + ib(t, a) + d0) , (2.1) where Es denotes the signal energy per modulation interval T, fc is the carrier fre quency, #o is a constant phase which may be ignored without loss of generality, and ip(t,a) is the information-carrying phase oo ip(t, a) = 2irh ^ a[k]q{t - kT) . (2.2) k=—oo Here, a[k] € {±1} is the random binary data, h is the modulation index, and q(t), t commonly referred to as the normalized phase pulse, is defined as q(t) = J g(r)6.T. \ 2.1 The Bluetooth System 14 a[k] e {±1} g(t) vco ) * SRF{t) FM Modulator 2nh Figure 2.3: Block diagram of a CPM modulator. The frequency shaping pulse g(t) determines the smoothness of ip(t, a) and is assumed to be of finite duration LT, where L is known as the CPM memory. A time-limited g(t) leads to an appropriate representation of q(t) as 9(t) 0 t < 0 t J g{r)dT 0 < t < LT o 1/2 t > LT (2.3) When L > 1, the frequency pulse exceeds the symbol interval T, leading to inter-symbol interference (ISI). However, this approach, known as partial response signaling, is desirable in CPM since it introduces additional memory and allows for a narrower power spectrum with low spectral sidelobes. Varying the parameter h, and choosing different frequency shaping pulses, g(t), results in various CPM schemes. Continuous phase frequency shift keying (CPFSK) signals, for example, use a rectangular pulse shape of duration T, but this causes sudden frequency transitions, resulting in a large bandwidth [42]. To achieve a more compact spectrum and smoother frequency transi tions, the frequency shaping pulse in GFSK consists of a rectangular pulse pre-filtered by a Gaussian low-pass filter h(t) as [40] g(t) = h(t) * rect(*/T) , where * denotes the convolution and h(t) is given by 1 ' / -t2 h(t) = 2ixaT exp 2a2T2 (2.4) 2.1 The Bluetooth System 15 Figure 2.4: Frequency pulse shape g(t) and phase pulse q(t) for Gaussian frequency shift keying with time-bandwidth product BT = 0.5. with a = yj\n{2)/ (2vr) BT and BT is the 3-dB bandwidth-time product (B: 3-dB bandwidth of the Gaussian filter). The result of the convolution of the Gaussian filter with the rectangular pulse that is defined as rect I — I = is the frequency pulse shape given by 1/T for |t| < T/2 0 otherwise (2.5) 9(t) = ± Q(f(t + T/2J)-Q(£(t-T/2J) (2.6) where Q(t) = (l/\/27r) J e~T /2dr is often referred to as the Gaussian Q-function. The t bandwidth of the GFSK spectrum and the resulting bit error rate at the receiver are both affected by two parameters: BT and the modulation index h = 2faT, where fd is the frequency deviation (maximum frequency shift with respect to the carrier frequency). It is desirable to have a low BT product and a small modulation index, for it results in a narrower spectrum. However, a low BT product produces higher ISI. In Bluetooth systems, T — 1 psec, BT = 0.5, and fa may vary between 140 KHz and 175 KHz, resulting in 0.28 < h < 0.35. The low modulation index values are due to 2.1 The Bluetooth System 16 the restrictions imposed by the FCC section 15.247 rule governing frequency hopping spread spectrum in ISM devices [51]. The GFSK frequency pulse shape g(t) used in Bluetooth and the resulting phase pulse q(t) are shown in Fig. 2.4. 2.1.5 Transmission Model For short-range wireless systems, such as Bluetooth, communication often occurs in residential homes, office buildings, or commercial areas (factories, shopping centers, etc.). The signal propagation depends on the indoor environment and its topography, and varies with time due to the motion of people and equipment. In such settings, Doppler and delay spreads are minimal, resulting in a stationary or a slow-varying flat-fading channel. The envelope of such channels is comparable to a Rician distribution. The probability density function of the Rician distribution is defined as [40] [o for(0<O) where g is the envelope of the channel gain, A, (A > 0), is the amplitude of the domi-nant signal, 2cr2 is the variance of the diffuse path, and IQ(X) — l/2ir J exp(xcosc/!>)d(/> o is the modified Bessel function of first kind and zeroth order. The Rician distribution is described by a Rician factor K — A2/2a2, which is the ratio of the power of the dominant path to the power of the diffuse path. In the special case of a weak dominant signal (A —> 0), a Rayleigh distribution is obtained, while as cr2 —> 0, the result is an additive white Gaussian noise (AWGN) channel. The indoor propagation model in the 2.4 GHz unlicensed ISM band in which Bluetooth operates (cf. Section 2.1.1) was investigated in [29]. The root mean square (rms) of the delay spread was found to be below 70 nsec, with an average value of approximately 50 nsec. These values are very low compared to the symbol duration of 1 /isec in Bluetooth. The channel propagation model was classified into two major categories. The first category includes channels with a line-of-sight path, which may be approximated by a Rician distribution with 2.1 The Bluetooth System 17 K — 5 (~ 7 dB). The second category describes channels with an obstructed path, and was found to have the same distribution with K = 2 (~ 3 dB), which is very close to the Rayleigh distribution. In a slow-fading channel, and after being affected by noise, the equivalent complex baseband (ECB) representation of the received signal is given by r(t) = g^*°s(t) + n(t) , (2.8) where <f>0 is a phase rotation resulting from fading and is constant over time, s(t) denotes the complex envelope of the transmitted signal that is written with a normalized unit amplitude as1 s(i)=exp(jV(t,a)) , (2.9) and n(£) is additive white Gaussian noise (AWGN) with two-sided power spectral den sity A^o/2. We note that the passband signal sRF(t) (2.1) can be written in terms of s(i) (prior to amplitude normalization) as SRp(t) = 3f?{s(£)ej27r/c*}, where 3?(.) represents the real part of the signal. Oscillators are used to frequency modulate the Bluetooth information-carrying signal (cf. Section 2.1.4). The frequency stability of an oscillator is defined as the measure of the degree to which an oscillator maintains the same value of frequency over a given time. Oscillator instability results in a phenomenon known as phase noise or timing jit ter, which introduces an additional time-varying phase rotation to the signal, A0(£,r), where A(p(t,r) is a zero-mean Gaussian variable with a variance CF\(T) depending on the employed oscillator and linearly increasing with time. Further, frequency fluctua tions of a signal occur due to the frequency offset (A/) between the transmitter and receiver oscillator, contributing to a greater phase variation [14], [33]. The resulting time-varying phase is represented as a function of the aforementioned parameters as 0(t + r) = <f>{t) + 2TTA/T + A(f>{t, T) . (2.10) The constant phase term 0Q has been ignored. 2.1 The Bluetooth System 18 Table 2.1: Interference performance [51]. Frequency of Interference (A/C)i) SIR 0 MHz (Co-channel interference) 1 MHz (adjacent interference) 2 MHz (adjacent interference) > 3 MHz (adjacent interference) 11 dB 0 dB -30 dB -40 dB Consequently, the received signal in the presence of frequency offset and phase jitter is expressed as r(t) = Se''(*(t)+*o)s(t) + n(t) . (2.11) Due to operation in the license-free ISM band and frequency reuse, Bluetooth can suffer from interference from other Bluetooth and WLAN devices. Co-channel interfer ence occurs when two or more piconets occupy the same frequency at the same time, while adjacent channel interference occurs when two piconets operate at neighboring frequencies in the same band. The performance in the presence of interference is de termined by measuring the signal-to-interference (SIR) power ratio. The Bluetooth standard requires that the bit error rate be less than or equal to 0.1% in the presence of an interfering signal at fc + A/C>i for all the SIR ratios shown in Table 2.1 [51]. A/Cii represents the frequency offset of the interfering signal compared to the desired signal. After considering the interfering signal, the ECB representation of the received signal may be given by r(t) = geKM+^slt) + n(t) + i(t) , (2.12) where i(t) denotes the ECB representation of the interfering signal. In the present work, we investigate and implement a novel receiver design for Bluetooth systems, as mentioned in Section 1.3. Since the channel gain can be assumed constant for the duration of one packet, and since detection for different packets is performed indepen-2.1 The Bluetooth System 19 a[k] G {±1} ai[k] e {±1}-CPM s(t) Modulator eMt) n(t) -0 CPM Modulator i(t) r(t) Figure 2.5: Block diagram representing the ECB system model. dently, channel fading is irrelavant for receiver design. Consequently, in our design and after ignoring channel fading, the ECB representation of the received signal is r(t) = e>*WS(t) + n{t) + i{t) . (2.13) A block diagram of the channel model is shown in Fig. 2.5, were ai[k] represents randomly-generated binary data symbols which are independent of a[k]. The CPM modulator was described in Section 2.1.4 and illustrated in Fig. 2.3. A path loss model for the channel was defined in the IEEE standard 802.15.1 based on Bluetooth and approved by the IEEE in 2001 [26]. The path loss is represented as a function of the separation distance d between the transmitter and receiver as follows {40.2 + 201og(d) d < 8 m (2-14) 58.5 + 331og(d/8) d > 8 m where Lp is the path loss in decibels (dB). The corresponding received power PR is then obtained by subtracting the path loss from the transmitted power. The receiver sensitivity threshold in Bluetooth is -70 dBm with a nominal 0 dBm transmit power. 2.1 The Bluetooth System 20 2.1.6 Error Correction Bluetooth packets are checked for errors or wrong delivery using the channel access code, the HEC in the header, and the CRC in the payload. In addition, three error-correction schemes are offered: • 1/3 rate FEC • 2/3 rate FEC • Automatic retransmission query (ARQ) The above schemes are used to transmit different parts of the packet. Bluetooth de vices may switch between these schemes as the signal level improves or degrades, thus maintaining performance at different noise levels. The three schemes are described in the following. The 1/3 FEC code is used for the header in all packets, as well as the synchronous data fields in the high quality voice 1 (HV1) packets. A (3,1) repetition code is employed, as shown in Fig. 2.6. A conventional FEC decoder uses hard-decision decoding where the most frequent bit value is taken as correct. This results in a single-error correcting code. It is clear that this code may cause unnecessary overhead and, thus, a decrease in efficiency if not required, for we transmit two extra bits with no additional infor mation. For this reason, the Bluetooth developers kept the 1/3 FEC code optional. The presence or absence of the code depends on the packet type, defined in the 4-bit 'TYPE' field in the packet header. However, this code is always used to protect the header, for it contains important link information, and bit errors would have a greater impact on performance [51]. The 2/3 FEC code is used in medium rate data (DM) packet, the data field of the data voice (DV) packets, the frequency hop synchronization (FHS) packet, the high data rate voice 2 (HV2) packet, and in the extended voice 4 (EV4) packet. It accepts 10 bits as input, and outputs 15 bits by using a (15,10) shortened Hamming code with 2.2 Bluetooth Receivers Literature Review 21 as, a\ Ol 02 o2 02 03 03 03 Figure 2.6: Illustration of Repetition code. the generator polynomial g(D) = (D5 © D4 © D2 © 1). If a packet contains less than 10 bits, zeros are appended to the codeword before applying the code. All single and double adjacent errors can be corrected through the 2/3 FEC code [51]. The ARQ scheme is used only on the payload of packets that have CRC. These include DV, DM, DH, and EV packets. At the start of a new connection, the ARQN bit is initialized to zero (NACK). After transmission starts, the ARQN bit is set to one (ACK) upon the successful reception of a CRC packet. If the HEC or CRC of any packet fails, or if no access code is detected, the ARQN bit is again set to NACK. The data payload is retransmitted until a positive acknowledgement is received or a time-out is exceeded [51]. 2.2 Bluetooth Receivers Literature Review CPM is considered an attractive modulation choice for wireless systems. This is partly due to the continuous phase and memory inherent in the signal, providing protection from channel errors. However, these properties that make CPM attractive also present difficulties for receiver designs. The next sections discuss several approaches for Blue tooth receiver designs in literature. The optimum receiver is presented first to serve as a lower bound for the suboptimum receivers discussed in the following section. 2.2 Bluetooth Receivers Literature Review 22 2.2.1 Optimum Receiver The optimum receiver discussed here assumes coherent detection and an AWGN chan nel. It uses the maximum-likelihood sequence detector (MLSD), which determines the most likely sequence of received symbols given all possible combinations of transmitted symbols. The optimum algorithm for performing (MLSD), the Viterbi algorithm (VA) [18], is a trellis search algorithm originally proposed in 1967 for decoding convolutional codes [55]. The optimum CPM receiver exploits the general state trellis structure of CPM to perform MLSD. Due to the memory inherent in a CPM signal, the phase at one time instant is dependent on the data symbol at that time instant in addition to all the previous data symbols. This can be shown in the general CPM state trellis structure obtained by using the assumption that the frequency shaping pulse g(t) is of finite duration LT (cf. Section 2.1.4). The following trellis structure exist only when the modulation index h is a rational number, and constitutes the basis for performing MLSD for CPM using the VA, as will be described later in the section. Introducing the CPM memory, L, into Eq. (2.2), the information-carrying phase can be appropriately represented as The first term of the right-hand side of Eq. (2.15) constitutes the accumulated phase up to time (n — L)T. It is referred to as the phase state and denoted hereafter by 9[n — L]. Using the fact that 9[n — L] and 9[n — L}mod(2ir) are physically indistinguishable, and conveniently representing the modulation index h in terms of two relatively prime integers m and p as h = m/p, the phase state becomes n—L n k=n-L+l a[k]q(t - kT) nT <t<{n+\)T . (2.15) (2.16) (2.17) 2.2 Bluetooth Receivers Literature Review 23 The second term in Eq. (2.15) is a sum of the phase contribution due to the most recent symbol a[n] and the state vector (a[n — 1], a[n — 2],a[n — L + 1]). Hence, the state of the CPM signal at time t = nT can be uniquely defined by the combination of the phase state and the state vector as Sn = {9[n — L],a[n — L + 1],a[n — L], ...,a[n — 1]}. Consequently, the number of states Ns in the trellis is {pML~Y (even m) (2.18) 2pML~1 (odd m) We have shown that a CPM signal can be represented by a trellis consisting of a finite number of states, which form what is referred to as a finite state machine (FSM)2. To perform detection through MLSD, the VA decides on the path metric which maxi mizes the log-likelihood function log[p(r(£)|s(£))], where p(r(t)\s(t)) is the conditional probability density function (pdf) defined for AWGN channels by - p.(r(t)\s(t))cxexp^J \r(t)-s(t)\2d^ . (2.19) Consequently, the MLSD decision rule is given by +oo a = argmax^- [ \r(t) - s(t)\2dt \ (2.20) a • —oo argmax < +oo +oo +O0 - f \r J \r(t)\2dt- J \s{t)\2dt+ J 2M{r{t) • §*{t)}2dt \ (2.21) -oo where a denotes the estimated symbol sequence, d represents the trial symbol sequence, and s(t) represents the hypothetical transmitted signal sequence associated to the hypothetical symbol sequence d. We notice that in Eq. (2.21) the decision rule is 2A FSM is a model of computation consisting of a set of states, having an input alphabet and a transition function which maps the input symbols and current states to a next state. 2.2 Bluetooth Receivers Literature Review 24 independent of |r(£)|2, while \s(t)\2 is a constant term since it is expressed in terms of an exponential (2.9). Hence, we can eliminate these terms and disregard the factor of 2 in the remaining term since it has no effect on the decision, resulting in the following MLSD decision rule a = argmax j J fH{r(t) • s*(t)}dt| . (2.22) The above equation shows that the complexity of the MLSD increases exponentially with the length of the symbol sequence a. This can be avoided by using the VA, which introduces a recursively calculated metric An(d) as follows (n+l)T An(a)= J M{r(t)-?(t)}dt (2.23) —oo (n+l)T = An_1(d)+ J $l{r(t)-s*(t)}dt (2.24) nT where An_i(d) is the accumulated metric of the surviving sequence up to time t = nT and the second term on the right-hand side the equation, known as the branch metric An(d), is given by (n+l)T An(d) = J 3f? J r(t) • exp I -j nT d[n -L} + 2nh ]T a[k]q(t - kT) k=n-L+l dt (2.25) We note that the branch metric is interpreted as filtering the received signal r(t) through a bank of 2ML matched filters and sampling to obtain outputs which form sufficient statistics for detection. We notice that the complexity of the MLSD using the VA increases only linearly with the length of the symbol sequence. The VA can be best described with the illustration of its key steps: add, compare, and select (ACS). The decision rule as given by Eq. (2.22) shows that we have a maximum problem. Therefore, the ACS consists of the following steps [8]. 2.2 Bluetooth Receivers Literature Review 25 Table 2.2: The values of relatively-prime integers m and p corresponding to different modulation indices h within the range specified by the Bluetooth standard. h 0.28 0.29 0.30 0.31 0.32 0.33 0.34 0.35 m 7 29 3 31 8 1 17 7 P 25 100 10 100 25 3 50 20 1. Add: At time t — (n + 1)T, we examine the two branches which lead to a common state. For each of these branches, we add the calculated branch metric to the corresponding accumulated metric. 2. Compare: The result of the sums generated in step 1 are compared. 3. Select: The branch (and corresponding data symbol) associated with the maximum path metric is selected. After performing the ACS step, the branch metric corresponding to the selected sym bol, also known as the surviving metric, is retained. The VA repeats the ACS step till the end of the symbol sequence, with one out of M branch metrics selected at each state. Hence, at each time instant, the VA computes all possible pML (or 2pML) metrics An(d) corresponding to all ML possible sequences and p (or 2p) possible phase states. This results in pML~x (or 2pML~l) surviving sequences at each time instant of the VA. The value of the final accumulated path metric corresponds to the maximum path metric value for all possible sequences of symbols. The symbol sequence esti mate corresponding to this maximum path metric d constitutes the output of the VA. Considering the special case of Bluetooth with binary GFSK and L = 2, the received vector is required to pass through 8 (= 2ML) filters prior to sampling. The state vector becomes Sn = {9[n — 2],a[n — 1]}. The variables needed to construct the trellis depend on the integers m and p, which are given in Table 2.2 for various values of h (in the range specified in the Bluetooth standard). Obviously, the minimum values for m and 2.2 Bluetooth Receivers Literature Review 26 Input symbol = -1 Input symbol = 1 [5TT/3,-1] •' i = 0 i = T p^states {*,3,*,5*/3} {0,2^/3,4^/3} i = 2T i = 3T {TT/3, 7r/57r/3} {0,27r/3,47r/3} Figure 2.7: Phase state trellis structure for Bluetooth with h = 1/3. p result in the least complex receiver. To illustrate the complexity of the optimum receiver even in the simplest scenarios, we choose the case of h = 1/3. For Bluetooth with h = 1/3, there would be 12 states in the VA; each state is a unique combination of a[n — 1] G {±1} and 8[n — 2] G {0,7r/3, 27r/3, 7T, 47r/3,57r/3}. At each state, two trial symbols d[n] = +1 and a[n] = -1 are considered and the VA decides on the corresponding metric that maximizes (2.22). Hence, at each time instant of the VA, 24 metrics are calculated and 12 surviving metrics remain. The trellis structure for the optimum Bluetooth receiver for h = 1/3 is shown in Fig. 2.7. For simplicity, the figure illustrates the VA paths at only three phase states at i = 0. These phase states were carefully chosen, for they lead to three 2.2 Bluetooth Receivers Literature Review 27 different phase states at the next symbol interval. From Table 2.2, we notice that even a slight variation of h can lead to a totally different trellis structure. Therefore, assuming a fixed nominal h is not an option in this design. Moreover, the complexity of the optimum receiver, defined by the number of states pML~l (or 2pML~1), increases exponentially as M and L increase, and tremendously varies with h. It also assumes perfect channel phase estimation, which is not valid in practice (cf. Section 2.1.5). We conclude that the optimum receiver performance may be considered as a benchmark, but it is definitely not a feasible solution for the Bluetooth receiver design. 2.2.2 Suboptimum Receivers Since the optimum receiver has an extremely high complexity, Bluetooth devices often employ a simple discriminator receiver to recover the GFSK modulated data. This receiver uses a limiter-discriminator integrator, and, therefore, is referred to hereafter as the LDI receiver. It is adopted in Bluetooth due to its low computational and structural complexity, making it simple to implement at a low cost. However, the LDI suffers from a considerable performance loss (> 6 dB) over the optimum receiver, as illustrated in Section 1.3. Since the introduction of Bluetooth devices in the market, several receiver designs have been proposed in literature. The LDI-based designs in [47], [48], have only a slightly increased complexity; however, they offer only a slight increase in performance (< 1 dB). These include a receiver based on zero-crossing demodulation with a decorrelating matched filter [48], termed BT-ZXMF. Another design proposed by the same authors uses least squares-based post-integration filtering [47]. This approach is further ex tended in [49] by employing a max-log-maximum likelihood (MLM) symbol detection which involves a forward-backward algorithm on a 4-state trellis, and the resulting de tector is termed MLM-LDI [49], [50]. Although a significant performance improvement 2.2 Bluetooth Receivers Literature Review 28 (~ 3.5 dB) is observed, the design complexity generated by the postprocessing and the 4-state trellis is not acceptable for the low-cost Bluetooth devices. As was observed in [49], sequence detection (SD) introduces tremendous performance gains since the memory of the CPM is properly taken into account. Consequently, many authors have considered trellis-based detection using the forward-backward al gorithm [49] or the Viterbi algorithm [53], [43], [31]. A sequence detector based on Laurent's decomposition, termed Maximum aposteriori probability (MAP) receiver, was proposed in [43]. It consists of a filter matched to the first Laurent pulse, fol lowed by a 4-state trellis decoder. It performs better than all the previously-described receivers, reaching performance gains of approximately 6 dB. However, it employs co herent detection, where a perfect channel phase estimate is assumed, and, therefore, it is impractical. In another SD design, [53], the trellis states in the Viterbi decoder were reduced to two by eliminating the effect of additional phase states. The main disadvantage of these trellis-based designs is that they assume a certain nominal value for the modulation index h. Since in Bluetooth h is allowed to vary in a relatively wide range (cf. Section 2.1.4), assuming an incorrect h at the receiver tremendously degrades the performance. Recently, a noncoherent sequence detector (NSD) which uses an adaptive algorithm that estimates the modulation index h has been proposed [31]. It achieves a performance gain similar to the MAP receiver with a 2-state trellis. However, none of the sequence detectors proposed in literature offer performance gains (relative to the LDI) with complete state reduction. Specifically, a one-state trellis in the NSD receiver was shown to lead to a performance loss of more than 4 dB over the LDI receiver [27]. The two receivers relevant for comparison purposes are the LDI and NSD receivers, and, therefore, will be described in detail in the following. The LDI is considered a benchmark for all designed Bluetooth receivers since it is employed in state-of-the-art Bluetooth devices. The NSD receiver is relevant to the present work since, similar to 2.2 Bluetooth Receivers Literature Review 29 Pre-detetion TF{t) Li miter- Integrate and A*d(f Hard filter Discriminator Dump Decision a[k] Figure 2.8: LDI detector block diagram. the proposed receiver, it uses noncoherent sequence detection and a modulation index estimator. LDI detector The conventional limiter-discriminator integrator (LDI) detector is illustrated in Fig. 2.8. The received signal is first filtered by a pre-detection bandpass Gaussian filter with im pulse frequency response [54] where Br is the 3-dB bandwidth of the filter, with an optimum value of 1.1 MHz in an AWGN channel [52]. The output of the Gaussian filter, rp(t), consists of a signal envelope, a distorted signal phase and a filtered noise term. The limiter-discriminator then outputs the derivative of the phase of rp(t), ip'jj,). The integrate and dump filter reintegrates the differentiated phase, producing a phase difference A$<f(£) , which represents the change over a symbol time of the signal phase plus the change in phase noise contributed by the AWGN. Hard decision is then performed on A$d(i), where a bit decision of '1' results if A$d(t) > 0, and '0' otherwise. The LDI detector was simulated in [27], and the obtained results will be used in this work for comparison purposes. NSD This receiver, described in [27], [31], is based on the Rimoldi/Huber&Liu decomposi tion approach to CPM [41], [25]. It is shown in [41] that a CPM modulator can be 2.2 Bluetooth Receivers Literature Review 30 decomposed into a trellis encoder and a signal mapper. The trellis encoder consists of shift registers with L delay elements and a recursive phase integrator. A time-invariant trellis is achieved by introducing a slope function. Further, using a new reference fre quency fr and modified unipolar information symbols am[fc], a modified phase state is obtained as [25] n—L Vm[n-L]=[m Om[fc]]mod(p)e{0,l,..,p-1}. (2.26) fc=—oo From Eq. (2.26), we can deduce that the number of phase states is equal to p. Hence, the phase trellis for binary GFSK with L = 2 would consists of pML_1 = 2p states and pML = 4p possible trajectories in each time interval, resulting in 4p time-limited, constant-envelope signal elements Pm(am[k])(t)> where am[k] = {am[k], am[k — 1], tym[n — L]} is a unique address vector associated with each signal element. Applying this decomposition, the ECB transmit signal can be expressed as oo s(t) = Z)Pm(amii])(t-ir). (2.27) i=0 The resulting block diagram of the GFSK modulator using the Rimoldi decomposition approach is illustrated in Fig. 2.9. The filter design for this receiver uses time-limited exponential functions as described in [25], [30]. In case of GFSK signals in Bluetooth, it was found in [27] that the use of only one filter, hi(t) yields excellent performance. The final filter consisted of a cascade of hi(t) and a Gaussian pre-filter. In contrast to the previous designs, the concept of noncoherent detection [11] is adopted, eliminating the need for channel phase estimation. A decision metric which accounts for the time-varying phase and frequency offset was also developed. In addition, an adaptive sequence detector was designed to cope with the varying modulation index h. It was found that a 2-state Viterbi decoder provides significant performance improve ment over the conventional LDI detector. NSD is shown to be robust against phase variations and frequency offset. The main disadvantage is the complexity required for 2.3 MLSD Lower Bound 31 Signal mapper T m mod(p) e L •ZT] *m[n-2] T f » with 4p signal elements s(t) am[k] am[k - 1] Figure 2.9: Block diagram of the GFSK modulator using the Rimoldi/Huber&Liu decomposition approach. a 2-state trellis search. With complete state reduction, the NSD performs poorly in terms of power efficiency [27]. The results presented in [27] will be used in the present work to illustrate the similarity in performance between NSD and the proposed receiver inspite of the proposed receiver's lower complexity. A lower bound for the performance of maximum likelihood sequence detectors for CPM signals was developed in [2]. This theoretical bound will be used to evaluate the proposed receiver in the results presented in Chapter 4. For high signal energy per bit-to-noise ratio (SNR), the bit error probability of coherent MLSD in an AWGN channel is given by [2] where dmm is the minimum normalized Euclidean distance between two sequences of information symbols a, and Ef, is the received energy per bit. For CPM signals, the squared and normalized Euclidean distance measure is given by 2.3 MLSD Lower Bound (2.28) (2.29) 2.3 MLSD Lower Bound 32 oo where <p(t, 7) = 2nh ^ l[i]q(t - iT) and — a[i] — a[i] is the difference between the actual transmitted symbol a[i] and the hypothetical symbol a[i]. Therefore, for binary symbols G {0, ±2}. We notice that this distance depends on the modulation index h. It is clear from Eq. (2.29) that the Euclidean distance is a nondecreasing function of iV, the number of symbol intervals. In [2, Chapter 3], an upper bound for the minimum d, dm\n, is obtained as a function of h. It represents the distance between the pair of sequences of infinite length that merges the earliest for any h. A 'merger' occurs once the phases of two sequences merge at a certain point in time, and coincide for all subsequent time intervals. If the merger between these two sequences occurs after Nm symbol intervals, then the upper bound on the minimum squared Euclidean distance reads k 0 ; Using the above equation, a lower performance bound of MLSD is obtained by substi tuting (2.30) into (2.28) as Since the proposed receiver employs sequence estimation, this bound is used for com parison purposes in Chapter 4. 2=0 d2mm(h) = log2(M) \ i / [l-cos<K£,7min)]dt (2.30) (2-31) Chapter 3 Noncoherent Decision Feedback Equalizer Receiver Structure This chapter provides a description and analysis of the proposed receiver structure. The proposed receiver is designed to be structurally simple, yet power efficient and compliant with the Bluetooth specifications outlined in Chapter 2. As the nonlinear structure of the CPM signal presents various challenges for the receiver design, we adopt the linear representation of CPM, known as Laurent's decomposition. Section 3.1 explains the derivation process leading to Laurent's decomposition, which serves as the basis of receiver design. It further presents the corresponding optimum receiver and applies the linear representation to the Bluetooth GFSK signal. Section 3.2 discusses the proposed receiver input filter which, in addition to providing sufficient statistic for data detection, achieves strong interference suppression. In Section 3.3, the discrete-time model is presented, resulting in an overall intersymbol interference channel. To perform reliable detection, the proposed receiver must then employ an equalization scheme. Several equalization techniques have been proposed in literature, of which the MLSD and Maximum A Posteriori (MAP) are proven to be optimum. The MLSD is implemented using the Viterbi algorithm which determines the most likely symbol 33 3.1 Laurent's Representation of CPM Signals 34 sequence as was detailed in Section 2.2.1. The MAP is implemented using the Bahl, Cocke, Jelinek, and Raviv (BCJR) algorithm [4] which determines the most proba ble symbol at a given time. However, due to the high computational complexity of these optimum methods, alternative low-complexity suboptimum schemes have been developed. In the present work, we employ a well-known suboptimum equalization scheme, reduced-state sequence estimation (RSSE), described in Section 3.4. Further more, we adopt a special case of RSSE, decision feedback equalization (DFE), of which an illustration is given in Section 3.5. To obtain a high performance with DFE, a min imum phase channel with maximum energy concentration in the first taps, is essential. Therefore, we employ an all-pass prefilter that transforms the ISI channel into a min imum phase channel, which serves as the feedback filter in the proposed DFE. For a practical implementation which accounts for the channel phase variations in Bluetooth signals, noncoherent detection, where the phase is implicitly estimated, is employed to the proposed RSSE and DFE. Section 3.6 describes the techniques used for the channel phase and frequency offset estimation. Moreover, since the modulation index varies in a relatively wide range in Bluetooth, an adaptive /i-estimator is presented to tackle this problem. Finally, a brief summary of the proposed receiver structure is given in Section 3.7. 3.1 Laurent's Representation of CPM Signals There are two alternative representations of CPM in literature. One representation given by Rimoldi in [41] decomposes the CPM into a trellis encoder and a signal mapper (cf. Section 2.2.2). This approach was adopted for the NSD receiver design described in [27]. As mentioned in Section 2.2.2, the NSD receiver achieves high performance gains with a 2-state trellis, but performs worse than the conventional LDI in case of 3.1 Laurent's Representation of CPM Signals 35 complete state reduction (1-state trellis). Another decomposition approach, referred to as Laurent's decomposition, was pro posed by Pierre Laurent in 1986 [32]. It represents the CPM signal either exactly or approximately as a linear superposition of a finite number of amplitude modulated pulses (AMP). Laurent's decomposition has been the basis of a considerable amount of research on CPM, including developing noncoherent sequence detection [11] and capacity bounds [56] for CPM. Moreover, using Laurent's representation, the chal lenges faced in the CPM receiver design due to the nonlinear structure of the signal are eliminated. As a result, it has been shown to be an effective tool for constructing reduced-complexity coherent [28], [10] and noncoherent [24], [11] CPM receivers. More specifically, a Bluetooth receiver based on Laurent's decomposition was proposed in [43], and achieved considerable performance gains over the conventional LDI. However, this receiver employed coherent detection, and the varying modulation index, channel phase variations, and frequency offsets were not addressed (cf. Section 2.2.2). In the present work, we adopt Laurent's decomposition approach. The following sec tions describe the derivation which gives an exact representation of the CPM signal. Furthermore, we illustrate its application to the Bluetooth GFSK signal, discuss the corresponding optimum receiver, and present an approximate representation of the CPM signal. 3.1.1 Laurent's Decomposition and Its Application to Blue tooth Laurent developed an alternative representation for binary CPM signals, which ex presses the CPM signal as a sum of (2L~1 — 1) pulse amplitude modulated (PAM) 3.1 Laurent's Representation of CPM Signals 36 components1. The complete derivation process presented in [32] will be described in the following. For convenience, we define a new phase function <p(t) = 2irhq(t), where q(t) was given in (2.3), resulting in ip(t) = irh for t > LT. The expression of ip(t, a) in the nth interval (2.15) can now be written as n—L n iP{t,a) = nh ^ a[k]+ ^ a[k]ip{t - kT) , nT < t < (n + 1)T . (3.1) fc=—oo k=n—L+l Recalling the ECB representation of the transmitted signal formulated in (2.9), and observing one signaling interval only, we may use Eq. (3.1) to express s(t) in the nth interval as n—L n s(t) = exp [jirh Yl aW)' II exP (M^Mt ~ fcT)) . (3.2) fc=—oo k=n—L+l The next step in the derivation is referred to by Laurent as the 'most important step'. It involves the usage of Euler's formula in addition to exploiting the fact that cos(a[k]ip(t— kT)) = cos{<p(t - kT)) and sm(a[k]<p(t - kT)) = a[k] sin(v?(£ - kT)). After several mathematical manipulations, the complex exponential associated to the mth symbol can be defined as a sum of two terms sin[-7r/i — •</?(£ — mT)\ exp (ja[m]ip(t — mT)^ = sin(7r/i) / . , r n sin[^(* - mr)] { } + exp(jnh am) . , , sin(7m) n - L + 1 < m < n , nT < t < (n + 1)T . We may notice that this formulation is not valid for integer values of h since it would result in sin(7r/i) = 0. An alternative solution, proposed in [23], derives the AMP repre sentation of a CPM signal with integer modulation index. We continue the derivation process by defining a generalized phase pulse function as (ip(t) t < LT (3.4) irh-<p(t-LT) t> LT 1In this work, "Laurent pulses", and "Laurent components", and, in some cases, "AMP (or PAM) pulses (or components)", are used synonymously. 3.1 Laurent's Representation of CPM Signals 37 In order to express Eq. (3.3) in terms of one function, we introduce the functions Sn(t) sin (7m) Using these functions, Eq. (3.3) can be rewritten as exp (^ja[m]<f(t - mT)) = SL-m(t) + ex.p(jirh a[m])S-m(t) . (3.6) Introducing the above result in Eq. (3.2) leads to n—L n s(t) = exp [jivh J2 E[ lSL-k(t) + exp (jirh a[k]) S.k(t)] fc=—oo k=n—L+l n-L L-1 = exp (jirh ^2 aik]) H [Si+L-n{t) + exp (jirh a[n - i\) Si-n(t)} . (3.7) k——oo i=0 The product term on the right-hand side of Eq. (3.7) can be expanded into a sum of 2L terms. However, a close examination of the result reveals that only 2L_1 of these terms are distinct functions of time. These form the AMP components that constitute Laurent's representation. To obtain a general form of these impulses, we define the radix-2 representation of the index P L-1 P = Y12i_1 ' IK ' 0 < P < 21-1 - 1 (3.8) i=l where 7p;i 6 {0,1}. The AMP pulses are given by L-1 (t) . (3.9) The Laurent pulses Cp(t) are nonzero in the interval [0, mini=li2)...L_i (L(2 — 7^) — i)}. Referring to Table 3.1, which shows the duration of the Laurent components, we notice that Co(t) has the longest duration. Laurent exploits this fact to represent the CPM signal in terms of only one impulse as will be shown in Section 3.1.3. Finally, Laurent's representation of a CPM signal in terms of the AMP pulses reads +00 2L~1-l s(t)= bp[k]CP{t - kT) , (3.10) fc=-oo P=0 3.1 Laurent's Representation of CPM Signals 38 Table 3.1: Duration of Laurent components Cp{t). Laurent Component Component duration C0(t) (L + 1)T Ci(t) (L-l)T C2(t),C3(t) (L - 2)T C4(t),C5(t),C6(t),C7(t) (L - 3)T C(2i-l)/2, C(2i-l_i) T where the complex symbols bp[fc], referred to in [35] as pseudo-symbols, are expressed in terms of the binary data symbols as bp[k] = exp ^jnh ^ ^ a[l] — ^ a[k — i] • Jp,i^j ^ — exp (jirh AP[k]) . (3.11) Eq. (3.10) is an exact representation of Eq. (2.9). However, as mentioned earlier Lau rent's decomposition as derived above is restricted to binary CPM signals. The decom position approach was further extended in [35] to M-ary signals. Considering the special case of Bluetooth GFSK signals with L = 2 and applying Laurent's decomposition, the resulting linear representation is given by +oo s(t)= (bo[k}C0{t - kT) + b^C^t - kT)) (3.12) fc=—oo where C0(t) = S0(t)Si{t) and Cx(t) = S0{t)S3{t) are illustrated in Fig. 3.1 for h = 1/3. Since L = 2 for Bluetooth, we notice that C0(t) has a duration of 3T, while C\(t) has a duration of T and significantly less energy. The corresponding optimum receiver is described in the next section. 3.1 Laurent's Representation of CPM Signals 39 0.9; t/T-> Figure 3.1: Laurent pulses Co(t) and C\(t), and the main pulse P(t) for Bluetooth GFSK signals with L = 2 and h = 1/3. 3.1.2 Optimum Coherent Receiver Based on Laurent's De composition Using the linear representation of CPM given in Eq. (3.10), a new optimum coherent detector for binary CPM was developed in [28]. This receiver requires 2L-1 matched filters and a Viterbi sequence detector with p2L~l states, as shown in Fig. 3.2. To obtain the VA state vector, we conveniently express the complex symbols (3.16) as follows. When P = 0, b0[k] = b0[k - 1] exp (jirh a[k}), while for P = 1,2, ...2L~l - 1, the complex symbols are represented in terms of &o [k — L] as bP[k -L] = bQ[k -L)Y[ exp (jirh a[k - i\) (3.13) ieip where Ip is a nonempty subset of the set {0,1,L — 1}. Hence, the state of the CPM signal can be uniquely represented at every time instant by the vector {b0[k — L], a[k — L+l], ...,a[k — 2],a[k — 1]}. Recalling that the modulation index h may be represented as h = m/p, we deduce that bo[k — L] can take p values. Taking into account the remaining (L — 1) binary symbols, it is evident that p2L~l states are required for the 3.1 Laurent's Representation of CPM Signals 40 r(i) CJ(-t) Viterbi decoder with states Figure 3.2: Block diagram of optimum coherent receiver based on Laurent's decompo sition. Viterbi algorithm. For Bluetooth GFSK signals (3.12), the optimum receiver is implemented as follows. The received signal r(t) is fed into a bank of 2 filters CQ( —t) and Cl(—t) matched to the Laurent pulses Co(t) and Ci(t), which provide sufficient statistics for the VA decision metric. The output of each of the matched filters is sampled at every symbol interval and is given at time t = kT by zP[k] = (r(t) * Cp(-t)) \t=kT (for P = 0,1). The VA consists of 2p states, and searches for the symbol sequence that maximizes the total path metric given that the branch metric is expressed as [28] A[fc] = K{z0[fe]bS[fc] + ^i[^iW} (3.14) where z0[k] and zi[k] are the sampled outputs of the matched filters to CQ(t) and C\(t), respectively, and b0[k] and b\[k] are the trial complex symbols corresponding to the trial sequence of binary symbols a. This metric is used to search the trellis by performing MLSD and outputing the maximum-likelihood symbol sequence as described in Section 2.2.1. Comparing this receiver to the optimum receiver described in Section 2.2.1, we notice that in case of Bluetooth GFSK signals, the number of filters is reduced from 8 to 3.1 Laurent's Representation of CPM Signals 41 2 and the number of states required in the VA is reduced in case of odd m from 4p to 2p. Moreover, the complexity of this receiver may be easily reduced by approximating the CPM signal using fewer Laurent pulses, as shown in [28] for binary CPM and in [10] for multilevel CPM. However, the resulting performance would of course be suboptimum. Regardless, the number of trellis states are still dependent on the modulation index h, and coherent detection is assumed. Hence, this approach is not a feasible solution for the Bluetooth receiver design, and, therefore, will not be pursued in the present work. 3.1.3 Main Pulse For reduced-complexity receivers, it is desirable to develop a linear representation of a CPM signal using only one pulse. Co(t) is the most important component in (3.10), for it has the longest duration and contains most of the energy of the signal (as was illustrated in Fig. 3.1). Laurent used this fact to derive a 'main pulse', P(t), which has the same duration as Cn(£), and presents by itself the best approximation of s(t) as s(t)= b[k]P(t-kT) (3.15) k——oo where b[k] = bo[k] = exp j jirh ^ a[l] J (3.16) \ l=-oo J may be represented as differentially-encoded complex symbols b[k] = b[k - 1] exp [jnh a[k}), (3.17) as previously mentioned. The desired main pulse P(t) is obtained through an opti mization criterion which minimizes the average energy of the difference between the exact signal s(t) and its approximation s(t). P(T + mT), 0 < r < T, is given by 3.1 Laurent's Representation of CPM Signals 42 Real components 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15 t/T Imaginary components I - 1 1 I I ^—•—i—""--^^^^ Exact CPM Signal Approximate CPM Signal Using Main Pulse i i i ,1 1 . I I I ' 1 1 1 1 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15 t/T Figure 3.3: The imaginary and real components of an exact Bluetooth signal s(t) and an approximate signal s(t) generated using only the main pulse P{t). P(T + mT) = Pm(r) with [32] j m—2 oo Pm(r)= —r^- J] cos^r-HT)) J] cos (rm - ip(r - iT)) ^ ' i=—oo i=m+l (3.18) .[ sin (v?(r + mT)). cos (r+ (m - 1)T) — cos(7r/i). sin (<p(r + (m — 1)T)). cos (irh — <p(r + mT)) ] . Observing Eq. (3.18) closely, we may notice that P(t) is zero for t < 0 and t > (L + 1)T, which proves our previous statement that P(t) is the same duration, as Co(t). Laurent gives an alternative representation for the main pulse in [32] where it is shown to be a weighted superposition of time-shifted versions of the Laurent pulses Cp(t). In both representations, we notice that the main pulse depends on the modulation index h. Hence, varying h affects both the complex symbols b[k] as well as the main pulse P(i). Also note that the representation of CPM as given in (3.15) may be modelled as a 3.2 Filter Design 43 phase shift keying (PSK) signal constellation of p points, and the main pulse may be considered as the pulse shaping filter. Hence, the memory of the CPM manifests itself as intersymbol interference. Figure 3.3 shows the negligible difference in the imaginary and real components of an exact Bluetooth signal as formulated in (2.9) and an approximate signal using only the main pulse as given in (3.15), both simulated for the same set of binary data, a, and for h = 1/3. In the following sections, we assume that the Laurent's approximation given by Eq. (3.15) holds with equality, and, thus, will be denoted hereafter by s(t). The optimum demodulator for a linear system with a pulse shaping filter P(t) is known to be the matched filter P*(—t) [38]. The noise sequence at the output of the matched filter is zero-mean Gaussian distributed noise. Although the noise variance is reduced, the noise sequence is correlated since P(t) is not a VNyquist filter in general, and, therefore, the cascade of the main pulse and its matched filter does not satisfy the Nyquist criterion. To obtain uncorrelated additive white noise which is more convenient to deal with, a \/Nyquist receiver filter with the frequency response H(f) is required such that The optimum filter that satisfies this property is the whitened matched filter (WMF) [17]. The WMF is a cascade of a matched filter and a noise whitening filter. It provides sufficient statistics for the detection of the symbol sequence, and the resulting sampled noise sequence at the output of the filter is white Gaussian noise. The frequency response #WMF(/) of the WMF corresponding to the main pulse P(t) (3.18) is given 3.2 Filter Design (3.19) 3.2 Filter Design 44 by P*(-f) #WMF(/) = , +po 1 J) — • (3-20) J E |P(/-n/r)|2 y r»=—oo The WMF depends on the main pulse P(t), which is formulated in terms of the modu lation index h (cf. Section 3.1.3). Hence, the WMF would ideally need to be adjusted according to the modulation index of the transmitted signal. To avoid this, and to facilitate a practical implementation, we propose replacing the WMF with the widely-used square root-raised cosine (SRC) filter. The root-raised cosine filter is a v'Nyquist filter obtained by splitting the raised cosine filter into two parts, with the frequency response HsRc{f) of each part being the square-root of the original as [37] for |/| < HSRCU) = < -sf (|/|-^) for^<|/|<<^>. (3-21) 0 for |/| > ^ The corresponding impulse response is given by 4p(t/T) cos (TT(1 + p)t/T) + sin (TT(1 - p)t/T) hsRc{t) = 7Tt(i-(4Pt/m/T (3-22) where p denotes the roll-off factor of the raised cosine filter. The roll-off factor (0 < p < 1) determines the excess bandwidth occupied by the signal beyond the Nyquist frequency 1/(2T). In addition, we consider an important issue in the implementation of a finite impulse response in the following. To obtain an ideal frequency response, the impulse response has to be infinitely long. Practically, an infinite impulse response cannot be imple mented, and, thus, needs to be truncated. However, truncating in the time domain leads to overshoots and ripples in the frequency domain, which degrade the perfor mance, particularly in the presence of interference. To reduce these undesirable effects, the ideal impulse response may be multiplied by a suitable window function which 3.2 Filter Design 45 1 0.8 T 0.6; 5 0.4 0.2 0 -°-26 ^Hamming window 0 (a) I Figure 3.4: Impulse response of SRC, hSRC(t), before Hamming window (a), and after Hamming window (b). allows the resulting impulse response to smoothly decay to zero. Therefore, to increase interference suppression, which is mandatory for Bluetooth devices (cf. Section 2.1.5), we apply a windowing method and choose the widely used Hamming window given by 0.54 + 0.46cos(27rf/ATjj) - XL(NH) < t < Xu{NH) w{t) = <( . (3.23) 0 elsewhere where XL(NH) and XL(NH) represent the upper limit and lower limit of the Ham ming window, respectively, and the length of the Hamming window is denoted by NH. XL{NH) = ^ and Xu(NH) = ^ if NH is odd, and XL(NH) = XV(NH) = if NH is even. Fig. 3.4 depicts the root-raised cosine filter in the time domain (3.22) before and after the Hamming window. The Hamming window is illustrated by the dashed curve in Fig. 3.4 (a) for a predefined value of NH equal to 64, which is eight times the oversampling time, and is zero otherwise. In the following, we assume that a Hamming window with NH — 64 have been applied to both the WMF and the SRC. This requires the multiplication of the WMF and SRC filter in the time domain by the Hamming window given by (3.23). Fig. 3.5 depicts 3.2 Filter Design 46 10 5 0 1 -5 5f =| -10 6T o 8 -15 -20 -25 -30 -WMF, h = 0.28| -WMF, h = 0.35 - SRC, p = 0.3 -0.5 0 rr-0.5 Figure 3.5: Magnitude frequency response for WMF, #WMF(/), assuming h = 0.28 and h = 0.35, and for SRC filter, #SRC(/), with p = 0.3. the magnitude frequency response of the WMF (3.20) for h = 0.28 and h = 0.35, respectively, and that of the SRC (3.21) with p = 0.3. We observe that the differences of |#WMF(/)| for different h are rather small, and that the magnitude frequency response of the SRC filter well approximates that of the WMF for different h. Moreover, while closely resembling the WMF in the passband, the SRC also accomplishes strong out-of-band interference suppression. Since the SRC filter is widely used in practice with hardware implementations available, we adopt /ISRC(^) as input filter for the proposed receiver. 3.3 Discrete-time Transmission Model 47 3.3 Discrete-time Transmission Model The discrete-time received signal after filtering with /ISRC(0 and symbol-rate sampling can be well-approximated by r{k}=r(t)*hSRC(t)\t=kT (3.24) Qh = e>m hc[l}b[k - I] + i[k] + n[k] (3.25) with the discrete-time channel impulse response (CIR) hc[k] = P(t) * hSRC(t)\t=kT (3.26) of order qh. n[k] is the sampled noise signal, which is still AWGN, and i[k] represents the sampled interference signal. Furthermore, the time variance of the phase <f>(t) is assumed to be slow compared to the modulation interval T, and, thus, <p[k] = <p(t)\t=kT-Through Laurent's representation (3.15), the memory of the CPM manifests itself as intersymbol interference (ISI), and hc[k] is referred to hereafter as the ISI channel, which may be described by a finite state machine with a trellis diagram. Recalling that the complex symbols b[k] belong to a p-ary PSK constellation (3.16), and since the ISI channel was found to be of length qh +1, the trellis would consist of p^+i states Qh (since /ic|7]6[fc — I] can assume pqh+l different values) determined by the state vector 1=0 b[k] = ^b[k],b[k — 1], ...,b[k — qh] of hypothetical (or trial) symbols b[k]. Assuming the channel phase <f>[k] is known at the receiver, the VA searches for the symbol sequence which minimizes the path metric given that the branch metric at time k is expressed by m = e'jmr[k] - J2 hc[l]b[k - I] 1=0 [k]-m\ (3.27) 3.4 Reduced-State Sequence Estimation 48 0.6 T g 0.4 0.2 1 (a) T £ °-4t o 0.2 (b) Figure 3.6: The CIR of the (a) ISI channel hc[k] compared to (b) its minimum phase equivalent h0[k]. which may be easily derived from (2.20) and represents the Euclidean distance between the received signal e~j(t>^r[k] and the hypothetical transmitted signal y[k\. Observing Fig. 3.6 (a) which depicts the significant taps of the overall channel hc[k] (3.26), we can see that hc[k] is of order qh = 2. Taking into account that the appropriate p for modulation indices 0.28 < h < 0.35 can be significantly larger than 8 (cf. Table 2.2), the complexity of MLSD becomes prohibitive. Hence, state reduction is mandatory, and will be employed in the present work using the concepts of reduced-state sequence estimation (RSSE) [16] and per-survivor processing [39]. 3.4 Reduced-State Sequence Estimation Reduced-state sequence estimation (RSSE) [15], [16] is an increasingly popular tech nique used to perform sequence estimation with a reduced-complexity VA. It provides a good tradeoff between performance and complexity by combining the high perfor-3.4 Reduced-State Sequence Estimation 49 mance of MLSD with the low complexity of DFE. It truncates the number of states searched by the VA by determining the additional information no longer present in the current state from previous decisions. Moreover, it provides flexibility through a design parameter ns that controls the number of states in the VA, and, as a result, the complexity. In Bluetooth, and due to the varying modulation index, it is essential to achieve state reduction independent of the modulation index h = m/p with potentially large p (cf. Table 2.2). As observed in the previous section, the state vector of the trellis is defined in terms of the complex symbols b[k], which are expressed as (3.16), and are therefore dependent on p. To tackle this problem, we define a trellis state vector reduced to a certain number ns of hypothetical data symbols a[k] as a[k] = [a[k] ...a[k-na + 1]] . (3.28) Therefore, the number of states required for the VA is reduced to 2ns, which is inde pendent of p. A vector b[k] = [b[k — ns] ... b[k — qn + 1]] of estimated symbols b[k] is associated with each state (b[k] = [b[k — ns]] if ns > qh)- The symbols b[k] are determined by applying per-survivor processing [39], and are calculated as (3.17). The RSSE simply involves partitioning the ISI channel at each time k into two using the parameter ns, such that one part is associated with the hypothetical data symbols a[k] and the other retrieved from the survivor sequence as y[k] = ]T hc[l]b[k - 1} + J2 hc[l)b[k - I] (3.29) 1=0 l=ns+l where b[k -l]= b[k - ns - 1] exp (jirh a[n] ) . (3.30) \ n=k—ns / In other words, a state at time k does not carry all the necessary symbols, and the remaining (q^ — ns) required symbols b[k] are determined from the surviving sequence. 3.5 Decision-Feedback Equalizer Structure 50 A constant channel phase will be assumed here, but this constraint will be relaxed in the following sections. The VA branch metric at time k may be developed by applying RSSE in (3.27), resulting in X[k} = where d[k] - hc[l]b[k - I) 1=0 (3.31) .Qh d[k] = e-j*[k]r[k] - hc[l}b[k - 1} . (3.32) (=ns+l The RSSE branch metric (3.31) is controlled by the design parameter ns and the state trellis (3.28) is independent of p . It allows the equalizer to vary from a full VA (ns > qh) to the simplest case of a DFE for ns = 0. The DFE structure will be illustrated in the following section. 3.5 Decision-Feedback Equalizer Structure The decision-feedback equalizer (DFE), which may be interpreted as a one-state VA, is the simplest nonlinear equalizer that constitutes an attractive compromise between complexity and performance. The main idea behind the DFE is using the previous decisions to cancel ISI in the present symbol, a concept originally proposed by Austin [3] in 1967. The decision-feedback equalizer structure was fully developed by Monsen [36], of which a block diagram is shown in Fig. 3.7. Referring to the aforementioned block diagram, the received signal vector r is fed into a feedforward filter (FFF) fF = [/F[0], /F[1], /F[<?F]] of order qF which suppresses the contribution of the precursor ISI. To cancel the ISI in the present symbol, the feedback filter (FBF) fB = [/B[0], /S[1], /B[<7B]] of order qB uses previously-detected symbols, and the output of the FBF is then subtracted from the output of the FFF. Assuming correct previous decisions, the interference from previously detected symbols is removed 3.5 Decision-Feedback Equalizer Structure 51 FFF Decision device FBF a[k — ko] Figure 3.7: Block diagram of decision-feedback equalizer. through the FBF, and the constant k0, known as the decision delay, specifies the number of future measurements processed before any decision is made on the present symbol. The input z[k] to the decision device may be expressed as2 QF QB = J2 fF[l]r[k fB[l]I[k - Z - fed] - fB[0]I[k - fco] (3.33) 1=0 1=1 where / is the vector of previously estimated symbols. Since the DFE operates under the assumption that the past decisions are correct, which is almost always violated, the DFE is evidently a suboptimum equalization scheme. The coefficients of the FFF and FBF filters obviously have a great impact on the DFE performance. Several techniques for computing the FFF and FBF coefficients were proposed in literature, of which a comprehensive review was given in [7]. In the present work, we adopt the simple DFE structure considering both coherent and noncoherent detection and compute the coefficients of the corresponding filters as outlined in the next section. 3.5.1 Minimum Phase Channel A minimum-phase overall impulse response is essential to obtain a high performance with RSSE [16], especially in the extreme case of RSSE, DFE. The finite impulse 2Usually a threshold decision is made, where the input to the decision device does not include the last term on the right hand side of (3.33). 3.5 Decision-Feedback Equalizer Structure 52 response (FIR) transfer function Hmin(z) of a minimum phase channel is such that Hmm{z) nas roots only inside the unit circle. A minimum phase channel /imin[fc] has the fastest decay, ensuring that the energy concentration in the first samples is maximized where the /imjn[fc] is the minimum phase equivalent of h[k] and lh is the length of the CIR. Since the decision.in a DFE largely depends on the first tap (3.33), it is obvious that a minimum-phase channel is desirable. Ideally, a minimum phase channel is produced by applying a discrete-time prefilter f[k] with an allpass characteristic to the sampled received signal r[k]. The minimum phase equivalent of the Bluetooth ISI channel hc[k] (3.26) is obtained by prefiltering it by f[k] as where the minimum phase equivalent, h0[k], represents the last qh + 1 taps of the resulting impulse response. The allpass filter f[k] alters the phase without affecting the overall magnitude response, producing an AWGN noise sequence at the output. In order not to change the magnitude frequency response, the all-pass filter moves the roots of the FIR transfer function Hc(z) of hc[k] which are outside the unit circle simply by reflecting them about the unit circle, resulting in the roots of its corresponding minimum phase HQ(Z), as illustrated in Fig. 3.8. For a practical implementation, a finite-impulse response (FIR) approximation of the allpass filter is of interest. A comprehensive review of the state-of-the-art prefiltering strategies is given in [20]. In this research work, the computation of the prefilter is based on linear prediction, as was presented in [20]. This prefilter consists of the cascade of a matched filter to the channel and a prediction error filter, which is calculated via the Levinson-Durbin algorithm. It was found to be an efficient prefiltering method for the ISI channel hc[k] in addition to having low computational complexity. The prefilter (3.34) hc[k] * f[k] (3.35) 3.5 Decision-Feedback Equalizer Structure 53 0 Roots of Hc(z) x Roots of HQ(z) \ •O--: i : ! ®* -5 -4 -3 -2-1 0 1 Real Part Figure 3.8: The roots of Hc(z) compared to the roots of its minimum phase equivalent H0(z). is designed to be of order qp = 6, and was found to slightly vary with h, but only insignificant performance differences incur if it is designed for a fixed h. The overall channel h0[k] is of order qB = 2 since it is of the same length as hc[k], as was mentioned earlier, and is shown for a fixed h = 1/3 in Fig. 3.6 (b). 1.O 1 0.5 0 -0.5 -1 H C 3.5.2 DFE Decision Rule From the previous discussion, it is clear that in a DFE structure, f[k] would serve as an effective FFF, while h0[k] would be its corresponding FBF, and, hence, in the present work, we employ f[k] as the FFF and h0[k] as the FBF in the proposed DFE structure. To further illustrate the role of the FBF in reducing the effect of the ISI caused by previous symbols, we may rewrite r[k] * f[k] as r[k] * f[k] = e*l%[0] (b[k] + -p h0[l]b[k - l]j + i[k] * f[k] + n[k] * f[k] (3.36) where the term QB J2h0[l]b[k - I] represents the intersymbol interference. Assuming coherent detection, the DFE deci-3.6 Noncoherent Detection 54 sion rule reads a[k — k0] = argmax < §R{dDFE[A;']/io[0]6[/c a\k—ho] ko}}} (3.37) where QB dDFE[k) = e-^r[k] * f[k] -J2h0[l]b[k ko-l]. (3.38) We may notice that the FBF subtracts out the ISI term from the decision. This decision rule corresponds to the branch metric (3.31) with ns — 0. We have to note here that for the proposed receiver the ISI channel is prefiltered to obtain its minimum phase equivalent only in case of DFE although, theoretically, a minimum phase channel is essential for RSSE as well. However, since the ISI channel hc[k] only has 3 significant taps, a 4-state RSSE is interpreted as full state VA, and a 2-state RSSE is not severely affected if the channel is not minimum phase. Consequently, the all-pass filter f[k] is only employed in case of DFE. 3.6 Noncoherent Detection So far, we have successfully employed coherent detection using RSSE and DFE, and the results presented in Chapter 4 show remarkable performance results. However, Bluetooth signals encounter channel phase variations due to oscillator phase noise and frequency instabilities of Bluetooth transceivers (cf. Section 2.1.5), and, as a result, coherent detection which assumes reliable phase estimates <j)[k] of <p\k\ is not possible in practice. Noncoherent equalization algorithms based on sequence detection were proposed in [19] and [11]. These schemes are power efficient and highly robust against channel phase variations, for the channel phase is implicitly estimated. The optimum noncoherent MLSD branch metric is given at time k by [11] Evidently, this branch metric has unlimited memory, for at each time k it depends on the entire previous sequence of symbols. To limit the computational complexity, k k \[k] = ]T (|rM|2 + \y[v}\2) ~ 2£ IWMI • (3.39) 3.6 Noncoherent Detection 55 Colavolpe and Raheli [11] proposed reducing the branch metric memory by truncating the window of observations to a finite value N, N > 2. Hence, (3.39) becomes k k X[k}= IM2-2 E l«>]l- (3-40) v=k-N+l v=k-N+l The term |r[t>]|2 was eliminated in (3.40) since it is independent of the decision. A noncoherent DFE (NDFE) was further derived from (3.40) in [45], with the branch metric, now suboptimum, given by k k ANDFE[&] = E I2/DFEH|2-2 I^DFEHJ/DFEMI (3-41) v=k-N+l v=k-N+l where roFE[fe] is the output of the FFF and T/DFE^] is the output of the FBF. Using the FBF and FFF recommended in Section 3.5, rcFE[fe] and j/DFE[fe] at time k can be formulated as QF rDFE[k] = ^f[l]r[k-l] (3.42) 1=0 and QB yDFE[fc] = 5>°[»-fco-Z] (3.43) (=0 where the complex symbols b[k — ko] are calculated as (3.17). Hence, (3.41) becomes fe-i QB h0[l]b[k -ko-l] + ho[0]b[k - k0] i=i (3.44) ANDFEW = E I^DFEMI2 + v=k-N+l -2\C[k-l] + rDFE[k}y*DPE[k}\ where A:-l &-!]= E ^FEM^PEM. (3.45) v=k-N+l QB _ Here, yr)FE[k] = Y^/ ho[l}b[k — ko — l] + ho[0]b[k — ko], b[k — k0 — 1} represent the previously-i=i detected symbols, while b[k — k0] is the hypothetical complex symbol. The phase of q^,{[k — 1] constitutes the estimate of the phase distortion caused by the channel, where the number of phase reference contributions is controlled by the finite value N, and, 3.6 Noncoherent Detection 56 therefore, [k — 1] is referred to as the N-ph&se reference. For a time-invariant channel phase, the greater the N, the less noise variance in the phase estimate, and, as a result, the better the performance. However, the computational complexity increases as well due to the increasing number of terms in (3.45). Schober and Gerstacker [44], [45], suggest avoiding this problem by recursively calculating the phase reference as where the design parameter a, 0 < a < 1, acts as a forgetting factor, and q^ef[k — 1] is referred to as the a-phase reference. Similar to JV, in case of constant channel phase, as a increases, the performance improves. However, phase jitters and frequency offsets contribute to a time-varying phase which does not represent the actual channel phase (cf. Section 2.1.5). In this case, the noise variance of the phase estimate will be high, and will accumulate as N or a become larger, degrading the performance. Consequently, the optimum values of N and a vary and depend on the channel parameters, and the phase jitter and frequency offset variations. Assuming constant channel phase, the performance approaches that of coherent detection as a —> 1 and N —• oo. This is an expected result since these values lead to the optimal branch metric (3.39) with no memory truncation. In the following, the metric associated with q?ei[k — 1] is referred to as the A^-metric, while that associated with q"eS[k — 1] is referred to as the a-metric. The performance of the N- and the a-metric was found to be approximately equivalent, precisely according to the following equation [46] However, since the a-metric requires less number of arithmetic operations [46], we adopt the a-metric in the present work. To perform noncoherent RSSE (NRSSE) using the VA, the same concept which in volves estimating the channel phase is applied. However, in contrast to the NDFE, the q^\k - 1] = aq^[k - 2] + rDFE[/c - l]&FE[fc - 1] (3.46) (3.47) 3.6 Noncoherent Detection 57 previously estimated symbols are dependent upon the surviving sequence of the state being considered. The NRSSE decision rule involves minimizing the following branch metric X[k] = \y[k)\2 - 2\q?ef[k - 1] + r[k]y*[k]\ (3.48) where q^[k - 1] - aq^[k - 2] + r[k - l]y*[k - 1] (3.49) and y[k] is calculated as for coherent RSSE (3.29). Therefore, similarly, the NRSSE decision rule is controlled by the design parameter ns. Also similar to coherent RSSE, prefiltering is not employed, as was explained in Section 3.5.2. 3.6.1 NDFE Decision Rule The branch metric given in Eq. (3.44) was further developed in [45]. The limiting performance for N —> oo, or, equivalently, a —> 1, was investigated, and the obtained results were also applied to finite value of N and to a < 1. The result was a modified suboptimum NDFE decision rule expressed as b[k - k0] = argmin{|dNDFE[/c] - h0[0)b[k - k0}\2} (3.50) b[k-k0] = argmax m{d*iDFE[k]hQ[0}b[k-k0}}\ (3.51) 6[fc-fco) where WW = rDFE[k] - J2h0[l}b[k -k0-l]. (3.52) QB l9ref[fc-l]| i=1 Qref[k — 1] may be calculated using N (3.45) or a (3.46). As mentioned earlier, in the present work, we use the a-metric, and in the following, the NDFE receiver is the final adopted receiver design which refers to the receiver employing the decision rule given by (3.50). We may notice that this decision is similar to the coherent decision metric (see (3.32) and (3.31)) with ns = 0 (since this is a DFE), and with e~j,pWr[k] replaced 1» WEf 3.6 Noncoherent Detection 58 3.6.2 Frequency Offset Estimation The NDFE decision rule presented in the previous section has proven to be extremely power efficient and robust against channel phase variations (cf. Chapter 4). However, for Bluetooth-standard compliant devices, deviations from the center frequency as large as A/T = 0.1 are admissible [51], and these contribute to additional phase variations which are not accounted for in (3.50). Variations of the phase reference which explicitly account for a frequency offset have been proposed in [12],[31]. Applied to qre([k — 1] in (3.46), the modified update equation reads qTe([k] = (aqre{[k - 1] + rDFE[fc]yDFE[fc]) e^^"1' , (3.53) where the frequency offset estimate e>2wA^k\ = pref[fc]/|Pref[k]| follows from Pref[fc] = PPretik ~ 1] + '"DFE[k]2/DFEM('"DFE\k - l]yDFE[fc ~ 1])* • (3-54) Similar toa,/3,0</?<l, is a forgetting factor for the frequency offset estimation. The second term in the right hand side of Eq. (3.54) represents the estimate of the frequency offset over one symbol duration. The modified phase reference as given by (3.53) is still robust against phase variations, but now provides reliable performance in the presence of oscillator frequency offsets as well. This will be clearly illustrated in Chapter 4. The choice of a and (3 trades performance for constant phase and frequency offset and robustness against phase variations and oscillator drifts, respectively (see the results in Chapter 4). 3.6.3 Adaptive NDFE In the previous sections, we have tackled the problems of phase variations including the frequency offsets, and presented the corresponding solutions. There remains the problem of the varying modulation index h which affects several parameters in the proposed receiver. Although the receiver input filter design (cf. Section 3.2) as well 3.6 Noncoherent Detection 59 as the FBF and FFF filters (cf. Section 3.5) turned out to be almost unaffected by varying h, considerable deviations of the decision metrics will occur even for small mismatches between the modulation index h assumed at the receiver and the actual h at the transmitter. This is evident from the formulation of the complex symbols given by (3.30)'. Hence, even in case of a correct estimated binary symbol a[k], due to the mismatch in h, the estimated complex symbol will be incorrect. We may assume a nominal value h at the receiver regardless of the modulation index's actual value h. However, results presented in Chapter 4, show significant performance degradation since h at the transmitter is allowed to vary in a relatively large interval (0.28 < h < 0.35). This is especially the case for large values of a and (3 since the errors accumulated from previous estimates are of greater weight. To account for this problem, we propose to test a set 7i of modulation index hypothe ses h, and, after an estimation period, Ne, to select the "best" one. We calculate \H\ NDFE accumulated path metrics, each for a total of Ne symbols. A pragmatic ap proach to determine the best alternative is to decide for the h that yields the smallest accumulated path metric. Therefore, the performance as well as the complexity of this h adaptation depends on the number of hypothesis and the length of the estimation period. We employ this adaptation technique to the noncoherent decision feedback equalizer presented in the previous sections, and the resulting receiver is referred to as adaptive NDFE (ANDFE). This adaptation process is applied to the first packet received, and the estimated modulation index used to detect the rest of the symbol sequences, h is obtained by the following decision rule Clearly, the greater the set H and the estimation period Ne, the better the performance of ANDFE. From the results given in Chapter 4, we may deduce that four modulation index hypotheses and estimation over 50 symbol intervals is quite sufficient to obtain (3.55) 3.7 Summary of Proposed Receiver Structure 60 r(t)-Hamming-windowed SRC ~T=^kT k] [r[0],...,r[Ne-l\ h- Estimator UJ LL Q Figure 3.9: Structure of the proposed Bluetooth receiver. a performance close to the case of h being perfectly known to the receiver. With these parameters, the complexity increase of ANDFE compared to NDFE is negligible, but results show significant performance improvements under varying h conditions. 3.7 Summary of Proposed Receiver Structure The structure of the Bluetooth receiver designed in the present work is shown in Fig. 3.9. The proposed receiver is based on Laurent's decomposition which presents the CPM signal as a superposition of PAM pulses (3.10). We adopt the approxima tion which expresses the Bluetooth GFSK signal in terms of one 'main pulse'. The recommended receiver input filter is the SRC filter (3.21), which is a practical choice that highly resembles the optimal WMF (3.20). To increase interference suppression, a Hamming window (3.23) is applied to the SRC filter. The output of the filter is symbol period sampled and fed into a simple noncoherent decision-feedback equalizer, through which phase estimation is employed as given in (3.51). Furthermore, for robust detec tion in the presence of high frequency offsets, frequency offset estimation is employed using (3.53). Moreover, due to the varying modulation index (0.28 < h < 0.35) in Blue tooth systems, we propose using ANDFE (3.55), where an estimate of the modulation index of the transmitted signal is given. The chosen modulation index is the one which maximizes the path metric over a limited number of samples Ne, (25 < Ne < 50), and is selected from a predefined set of hypotheses, H, (\H\ = 2,4). This estimated h is then used to detect the rest of the received signal, and the final output is the sequence 3.7 Summary of Proposed Receiver Structure 61 of estimated binary symbols d. Chapter 4 Performance Results and Discussion This chapter presents and analyzes the performance results of the designed Bluetooth receiver described in Chapter 3, which was simulated using C++ and Matlab. We assume the channel model is as given by (2.13), and perform detection for a block of 1500 data points at one time. These blocks are generated until a minimum of 1000 errors is obtained, giving a high level of accuracy. To put the results into context, the proposed receiver is compared to that of the NSD and the LDI, which are considered relevant benchmarks, as discussed in Section 2.2.2. Furthermore, we consider the MLSD bound (cf. Section 2.3) as the theoretical performance limit [2]. The results for the LDI and NSD are obtained from [27], and the MLSD bound is calculated in the present work for different values of h. When comparing the performance of different receivers, we measure the signal-to-noise ratio required to achieve a BER of 10-3 since this is the required BER for Bluetooth receivers [51]. Different performance results were observed when the sampling start point was changed. Therefore, when obtaining the discrete-time channel model (3.25), we optimize our sam pling start point to get the best performance possible using the adopted SRC filter. We 62 4.1 Coherent Detection 63 use an oversampling time of 8, and the total length of the employed Hamming window, and as a result, the receiver input filter, is given by 8T. The first section of this chapter shows the coherent detection performance results for RSSE and DFE. In case of DFE, the FFF and FBF are of orders qF = 6 and qB = 2 (cf. Section 3.5.1), and are computed for a fixed modulation index h = 1/3 throughout the chapter. Moreover, the modulation index of the transmitted signal is also assumed to be 1/3 and known at the receiver and the receiver input filter is the SRC filter, unless stated otherwise. The second section is devoted to noncoherent detection were the channel phase is implicitly estimated in various scenarios. Most of the presented figures investigate the bit error rate (BER) performance as a function of the signal-to-noise ratio (SNR) Eb/N0 in decibels (dB), where Eb is the received energy per bit and No is the one-sided noise power spectral density of the underlying passband noise process. In other cases, the required SNR for BER = 10~3, which is representative in case of Bluetooth devices [51], is recorded for different values of the modulation index h. 4.1 Coherent Detection In this section, we present the results obtained for coherent detection; i.e, assuming a perfectly estimated channel phase at the receiver. These serve as the basis for our filter selection and our evaluation of the performance loss which the proposed receiver suffers as a result of state reduction. 4.1.1 Filter Selection We have selected the SRC filter as the input receiver filter for the proposed receiver. The similarity in the magnitude frequency responses for the WMF and SRC filters was shown in Chapter 3. Furthermore, in this section, to support the decision of selecting 4.1 Coherent Detection 64 6 8 10 12 14 10log10(Eb/N0) [dB]-> 18 20 Figure 4.1: Performance comparison of the SRC and WMF as receiver filters. Coherent detection with h = 1/3 is assumed. the SRC filter, we demonstrate the performance of the SRC filter [Eq. (3.21)] and compare it with that of the optimal WMF [Eq. (3.20)]. In Fig. 4.1, we show the BER as a function of the SNR for RSSE with 4 states and for DFE, using, in both cases, the WMF and the SRC, respectively. For comparison, the BER curves for the MLSD bound and the LDI receiver are also included. We notice that for the 4-state RSSE, as well as the DFE, the curves for the SRC filter almost coincide with those of the WMF. As a matter of fact, a closer observation would reveal that in case of a 4-state RSSE, there is ~ 0.05 dB performance improvement when the SRC is used, while in case of a DFE, there is ~ 0.01 dB performance improvement when the WMF is used. Ideally, of course, the optimal WMF filter would generate better performance in case of the full-state (4-state) RSSE. However, due to the fact that the sampling time is optimized for the SRC filter, as was previously mentioned, this is not the case here. Regardless, the performance differences are almost negligible. 4.1 Coherent Detection 65 10" 10 10 ' DC LU m 10" 10" 10 ! y i !• MLSD bound • LDI receiver x one-state NSD -a—DFE -e— 2-state RSSE -e>— 4-state RSSE 6 8 10 12 14 16 18 20 22 24 10log10(Eb/N0) [dB]-> Figure 4.2: Evaluation of the state reduction in the proposed receiver. Coherent de tection with h — 1/3 is assumed. . • Therefore, the proposed SRC filter is a practical and ideal choice for the input receiver filter, and will be used for the following simulations unless states otherwise. 4.1.2 State Reduction Observing the BER curves in Fig. 4.2 for 2-state RSSE [Eqs. (3.31)-(3.32)], 4-state RSSE and DFE [Eqs. (3.37)-(3.38)], all using the SRC filter, and comparing them to the LDI, MLSD, and one-state NSD, we make the following notes. First, the RSSE with only two states approaches very closely the MLSD bound. Second, the DFE incurs a loss of ~ 0.7 dB compared to the 4-state RSSE, and less than 0.6 dB compared to the 2-state RSSE, which is highly acceptable considering the reduction in complexity. Third, comparing the performance of the proposed receiver and NSD using a one-state trellis (or DFE), we notice tremendous differences with the NSD suffering a performance loss of more than 10 dB. The proposed receiver, on the other hand, outperforms the LDI by more than 5 dB, which is evidently a remarkable result using a one-state trellis. 4.2 Noncoherent Detection 66 12.2 a = 0.8, FBF and FFF set according to h at transmitter 10.4 10.2 0.31 0.32 h at transmitter -> 0.33 0.34 0.35 0.28 0.29 0.3 Figure 4.3: Performance of NDFE with a = 0.8 with FBF and FFF fixed at the receiver, or varying according to the value of h at the transmitter. Similar results are expected in case of noncoherent DFE, which will be considered in the following sections. 4.2 Noncoherent Detection In this section, we evaluate the performance of the proposed receiver in case of non coherent detection, where the channel phase is estimated using the phase reference. We briefly discuss and demonstrate the performance of NRSSE [Eqs. (3.48)-(3.49)], and focus on the performance of NDFE [Eqs. (3.51)-(3.52)], which constitutes the final receiver design (cf. Section 3.7). The results for MLSD, LDI, and coherent DFE are included for comparison purposes. The modulation index h = 1/3 and is known at the receiver unless stated otherwise. 4.2 Noncoherent Detection 67 10 LU m 10 •-*-• MLSD bound • LDI —e— 4-state NRSSE, a = 0.8 —±— 2-state NRSSE, a = 0.8 —e— NDFE, a = 0.8 -A - 2-state NSD, a = 0.8 6 8 10 12 14 16 18 20 10log10(Eb/N0) [dB]-> Figure 4.4: Evaluation of the performance of the proposed receiver with state reduction in case of noncoherent detection. 4.2.1 Performance with Constant Channel Phase In this section, the channel phase is assumed to be time-invariant. The performance under this condition is evaluated with state reduction, and for different values of N and a. Moreover, we discuss the effect of the varying modulation index at the transmitter compared to the results obtained when h is known at the receiver. We remind the reader that the results presented in this work assume the FBF and FFF filters are fixed according to h = 1/3. Only negligible, if any, performance loss incurs compared to the case where h is varied according to the actual value at the transmitter, as illustrated in Fig. 4.3. State Reduction We previously found that the DFE suffers fairly acceptable performance losses with respect to the 2-state and 4-state RSSE. In this section, we illustrate that a similar result is obtained in case of noncoherent detection and compare the performance of the 4.2 Noncoherent Detection 68 proposed receiver to the LDI and the NSD. Fig. 4.4 shows that NRSSE and NSD yield similar performance. In fact, the BER curve of the 2-state NRSSE almost coincides with that of the 2-state NSD. Moreover, we notice that NDFE suffers a performance loss of about 0.3 dB and 0.4 dB compared to 2-state NRSSE and 2-state NSD, re spectively, and achieves a performance gain of about 5.5 dB over the LDI. As further state reduction turned out impossible for the NSD approach of [27] based on Rimoldi decomposition, we can deduce that Laurent's decomposition is preferable for the design of low-complexity Bluetooth receivers. Finally, we conclude that NDFE provides an excellent tradeoff between performance and complexity, making it the ideal equaliza tion scheme for the proposed receiver. Hence, the following discussion will focus on the performance evaluation of NDFE. Performance with Different Values of N and a Starting with the N-metric (3.45), the BER curves for NDFE with different values of N are shown in Fig. 4.5. As expected, without channel phase variations, as the value of N increases, the performance closely approaches that of coherent DFE. We also observe that even for the minimum value of N = 2, we have a performance improvement of more than 2.5 dB over the LDI receiver. Fig. 4.6 shows the performance using the a-metric. With a = 0.95, the performance difference between coherent DFE and NDFE is a negligible 0.06 dB. Similar to N, as a increases, we get better power efficiency Moreover, recalling Eq. (3.47), which compares the performance of the NDFE using the N- and a-metric, it may be verified by a careful observation of Figs. 4.5-4.6. Specifically, we see that the performance for N = 10 is equivalent to that of a = 0.8, and a similar statement is true for N = 2,5 and a = 0,0.6, respectively. However, since the a-metric is computationally less complex (cf. Section 3.6), the following discussion will be solely based on the a-metric. 4.2 Noncoherent Detection 69 6 8 10 12 14 10log10(Eb/N0)[dB]^ Figure 4.5: Performance of the NDFE using the A^-metric compared to MLSD, LDI, and coherent DFE, and h = 1/3 for all cases. 6 8 10 12 14 10log10(Eb/N0)[dB]^ Figure 4.6: Performance of the NDFE using the a-metric compared to MLSD, LDI, and coherent DFE, and h = 1/3 for all cases. 4.2 Noncoherent Detection 70 10 f 10" m •o ~ 10"^ LU CD O o rx LU m 10" 10 "~' ~. frr 7 ~'.".T'.fri'. ~ \/~'.'fr:':~r~.'. ~. : fr r. . "fr : MLSD bound 4 5 oA -10 Figure 4.7: Performance of the NDFE using the TV-metric h — 1/3 in the presence of phase jitter. Figure 4.8: Performance of NDFE using the cv-metric h = 1/3 in the presence of phase jitter. 4.2 Noncoherent Detection 71 As mentioned above, increasing the value of a and N achieves better performance when the channel phase is constant. However, this is not the case in the presence of phase jitter, as can be seen in Fig. 4.7 which plots the BER with respect to the standard deviation of the phase jitter a&. In this case, the performance of the proposed receiver deteriorates as N increases since the assumption of constant channel phase is no longer valid. A similar observation can be made when using the a-metric, as shown in Fig. 4.8. Therefore, the optimal value of N or a depends on the channel conditions, and we may conclude that NDFE with N between 5 and 10 or a between 0.6 and 0.8 is power efficient and gives satisfactory performance in the presence of phase jitter. Performance with Different Modulation Indices In the previous sections, we have assumed h = 1/3 at the transmitter and receiver. However, since the GFSK signal depends on h [Eqs. (2.1)-(2.2)], even with the as sumption that h is known at the receiver, different performance results are obtained for different modulation indices. Specifically, the performance deteriorates as h de creases since fd (cf. Section 2.1.4) decreases. Fig. 4.9 shows the performance of NDFE with a = 0.8 for h = 0.28, 1/3, and 0.35, along with the corresponding MLSD bounds, the LDI, and the MLM-LDI1. The performance of NDFE, which is equivalent to a one-state trellis gives remarkable results when compared to the MLSD bound with a performance loss of only about 1.2 dB. Recall that the optimum receiver (cf. Section 2.2.1) requires 100 trellis states for h = 0.28 and 80 trellis states for h = 0.35, while the optimal receiver based on Laurent's decomposition (cf. Section 3.1.2) requires 50 and 40 states for h = 0.28 and h = 0.35, respectively. The performance gains compared to the LDI and MLM-LDI are more than 5 dB and 1 dB, respectively. JThe results for the MLM-LDI are taken from [49]. 4.2 Noncoherent Detection 72 10 CE UJ CD 10 MLSD bound NDFE with a = 0.8| LDI MLM-LDI * h = 0.28 • h = 1/3 • h = 0.35 16 18 20 2 4 6 8 10 12 14 10log10(Eb/N0) [dB]-» Figure 4.9: Performance of the NDFE with a = 0.8 for different values of the modula tion index compared to the MLSD bound, LDI, and MLM-LDI. Performance with Unknown Modulation Index So far, we have assumed the modulation index h to be known at the receiver. In Blue tooth devices, the varying modulation index (0.28 < h < 0.35) poses a serious challenge for the receiver design. To support this statement, we show the BER obtained for an SNR of 11 dB for different values of a = 0.4, 0.6, 0.8, and 0.9 when a fixed modulation index h = 0.28, 0.30, 1/3, and 0.35 is assumed at the receiver. The simulations are performed for different values of the actual modulation index of the transmitted signal, and the obtained results are shown in Fig. 4.10. It is clear that as h at the receiver deviates from the actual h of the transmitted signal, the performance degrades. We also notice that the performance loss with the h deviation is more severe for greater values of a. This is due to the fact that a mismatch of h at the receiver takes longer to forget in case of a greater forgetting factor a. Hence, a lower value of a provides more robustness against variations in the modulation index. Consider, for example, a = 0.4, a nominal value of h at the receiver would result in very slight changes from the case 4.2 Noncoherent Detection 73 • h = 0.28 • h = 0.30 -h = 1/3 • h = 0.35 - h known at receiver a = 0.4 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 h at transmitter -> a = 0.8 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 h at transmitter -» a =0.9 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 h at transmitter -> 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 h at transmitter -» Figure 4.10: Performance of the NDFE for varying h with a = 0.4, 0.6, 0.8, and 0.9. h = 0.28 at receiver h = 0.30 at receiver T m ^ 10 m 10 C T 10° m •o £ 10" 10" .O JJ "~o ra 10" o o 6 10" rr LU m 10" V - NSD, a = 0.6 •V NDFE, a = 0.6 -V— NDFE, a = 0.8 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 h at transmitter -» : 1/3 at receiver 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 h at transmitter -> h = 0.35 at receiver a - NSD, a = 0.6 •a NDFE,a = 0.6 s— NDFE, a = 0.8 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 h at transmitter -> 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 h at transmitter -» Figure 4.11: Performance of NDFE for varying h with a = 0.6 and 0.8 compared to that of NSD for a = 0.6. 4.2 Noncoherent Detection 74 when h is known at the receiver but still achieve more power efficiency than the LDI. To evaluate the proposed receiver's robustness against h variations with respect to the alternative noncoherent sequence detector in literature, namely, NSD, we show a relative comparison in Fig. 4.11. The BER curves obtained for SNR = 11 dB are shown for various modulation indices h = 0.28, 0.3, 1/3, and 0.35 at the receiver, and are plotted for a varying h at the transmitter. We choose the curves for NSD with a = 0.6 since it was found to be the optimal value of a for NSD [27]. The NSD curves are compared to of NDFE with a = 0.6 and 0.8. The presented results show that as h varies, a much greater deviation is observed in the BER curves for NSD compared to those of NDFE, both with a = 0.6. Considering the case when h = 0.28 at the receiver, as h varies at the transmitter, the corresponding BER is almost constant in case of NDFE, which is not the case for NSD. Specifically, the resulting percentage of error varies between 0.25% and 0.5% in NDFE, and between 0.4% and 12% in NSD. Moreover, we even observe less change in the BER of NDFE for a = 0.8 compared to NDFE with a = 0.6. We conclude that NDFE is more robust against h variations compared to NSD. A more representative evaluation of the effect of varying modulation index is obtained by considering the required SNR for a BER of 10~3. We adopt the same scenario as before. Fig. 4.12 shows the required SNR for a BER of 10~3 as a function of the modulation index h for NDFE with a = 0.8 and for the MLSD bound and the LDI for comparison purposes. The figure also illustrates the ideal case of h known at the receiver. As can be observed, severe power efficiency losses result when h is unknown at the receiver. The variations in the modulation index may be minimized by estimating h at the transmitter using ANDFE described in Section 3.6.3, and the corresponding results will be presented later in this chapter. 4.2 Noncoherent Detection 75 i i i 1 i i J 0.29 0.3 0.31 0.32 0.33 0.34 0.35 h at transmitter —> Figure 4.12: The required 101og10(Eb/N0) for BER = 10"3 for NDFE with a = 0.8 with varying modulation index. 4.2.2 Performance Using the Modified Phase Reference Un der Constant and Varying Channel Conditions The more realistic and thus more relevant transmission scenario considers a time-varying channel phase <p(t). In the present work, we account for channel phase varia tions through the modified phase reference [see Eqs. (3.53)-(3.54)] formulated in terms of two forgetting factors a and (3. Throughout this chapter, if qief[k] from (3.46) is used, only the value of a is specified, and if c7ref[fc] from (3.53) is applied, the values of both a and (3 are given. Performance With Constant Channel Phase Before considering a time-variant channel phase, we first show the performance of the modified phase reference when the channel phase is constant. Fig. 4.13 illustrates the performance for different values of a and /3 as a function of the modulation index, 4.2 Noncoherent Detection 76 0.28 0.29 0.3 0.33 0.34 0.35 0.31 0.32 h at transmitter -> Figure 4.13: 10iogi0 (Eb/N0) required for BER = 10-3 for different combinations of a and (3 as a function of the modulation index with constant channel phase and h known. which is assumed to be known at the receiver. We notice that there is a loss in power efficiency when (3.53) is used compared to when (3.46) is applied due to the phase noise introduced by the frequency estimation. More specifically, the gap between a = 0.6 and the pair (a = 0.6, j3 = 0.9) is approximately 0.4 dB. This loss for the ideal case of a constant channel phase has to be accepted to achieve a high performance in the more realistic scenario of a time-varying phase. Moreover, we observe that as (3 increases the performance of qief[k] from (3.53) closely approaches that of qTef[k] from (3.46). It was found that for various scenarios, the pair (a = 0.6, (3 = 0.9) appeared to be a favorable choice. We further observe that the robust NDFE with (a = 0.6, (3 = 0.9) outperforms the LDI detector by more than 4 dB, and it is still somewhat more power efficient than the MLM-LDI detector, which requires a four-state forward-backward algorithm. Finally, we note that NDFE with frequency-offset estimation performs very similar to the more complex NSD with the same values of a and (3. 4.2 Noncoherent Detection 77 Performance with Varying Modulation Index As for the effect of the (3 factor on the varying modulation index, the resulting per formances of NDFE with (a = 0.6,(3 = 0.9) and (a = 0.8,(3 = 0.9) are shown in Figs. 4.14 and 4.15, respectively. The curves represent the required SNR for BER = 10~3 for fixed values of the modulation index at the receiver as function of a varying modulation index at the transmitter. We notice that the performance degradation re sulting from a mismatch in h encountered using (3.46) persists when using (3.53). As expected, when a decreases the SNR differences between unknown h at the receiver and known h decreases. Regardless, using the modulation index estimator (cf. Section 3.6.3) is advisable, and will be presented in the Section 4.2.3. Performance With Varying Channel Phase The modified phase reference (3.53) was developed to account for the extreme (yet admissible) phase variations and frequency offsets of A/T = 0.1 in Bluetooth devices. We now evaluate the performance of NDFE using this modified phase reference in the presence of frequency offset and phase jitter. Fig. 4.16 shows the required SNR for BER = 10~3 as a function of the normalized frequency offset. The curves are parameterized with a standard deviation of a A = 0° (no jitter) and a A = 2°, 5° for the phase jitter. The LDI detector fails for offsets A/T > 0.03 ... 0.05, but we note that DC offset cancellation methods could be applied to mitigate the effect of frequency offset. We further observe that NDFE with phase reference (3.46) cannot cope with large frequency offsets of A/T > 0.02. However, considering the pair (a = 0.6,3 = 0.9), the NDFE with phase reference (3.53) allows power-efficient transmission even for extreme offsets. In addition, the maximum performance degradation when assuming an additional phase jitter with a A = 5° is not more than 1.5 dB. For the same scenario, Fig. 4.17 compares the performances of NDFE for the two pairs (a = 0.6,(3 = 0.9) and (a = 0.8,(3 = 0.9) and NSD with the pair (a = 0.6,(3 = 0.9), 4.2 Noncoherent Detection 78 0.31 0.32 0.33 h at transmitter -> 0.35 Figure 4.14: The required 101ogio(£6/iVo) for BER = 10"3 for NDFE with a = 0.6, B 0.9 for varying h. 18i 1 1 t 17 CO I O Ii 16 or LU ™ 15 o "O 2 14 — NDFE, a = 0.8, p = 0.9, fixed h -e— h known at receiver \ h = 0.32 V • - _ _ _ \h = 0.33 \h = 0.34 )l 0.31 0.32 0.33 h at transmitter -» 1-3 0.35 Figure 4.15: The required 101ogi0(£?6./JVo)'for BER = 10~3 for NDFE with a = 0.8,8 0.9 for varying h. 4.2 Noncoherent Detection 79 8 n 1 1 i r a = 0.6, p = 0.9, o = 0° _ a = 0.6, P = 0.9, OA = 2° 0 A a = 0.6, p = 0.9, o& = 5° a = 0.6,o = 2° •-• - *- MLSD bound -- * *. "0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 AfT —> Figure 4.16: The required 101ogi0(j5fe/Aro) for BER = 10~3 in the presence of frequency offset and phase jitter. - NDFE, a = 0.6, (3 = 0.9 NDFE, a = 0.8, P = 0.9 NSD, a = 0.6, P = 0.9 a =0" 16 15 14h 13 12, 11 0.04 0.05 AfT-» 0.06 0.07 0.08 0.09 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 AfT -> Figure 4.17: The required 101og10(£'b/^vo) for BER = ICT3 in the presence of frequency offset and phase jitter. 4.2 Noncoherent Detection 80 which was found most favorable in [27]. We observe that the performance deteriorates with increasing phase jitter as a increases (both with 8 = 0.9). The NDFE with the pair (a = 0.6,3 = 0.9) was found to have comparable performance with the 2-state NSD. The preferable performance for (a = 0.6,8 = 0.9) in the presence of extreme phase variations is one of the reasons why this pair is considered to be the favorable choice for NDFE. 4.2.3 Adaptive NDFE The performance of NDFE severely deteriorates when there is a mismatch between the value of h assumed at the receiver and the actual h at the transmitter, as was illustrated in the previous sections (cf. Figs. 4.12, 4.14, 4.15). We now evaluate the performance of ANDFE (cf. Section 3.6.3), where several NDFEs are executed for a number of hypothetical modulation indices, and the decision metrics are accumulated. We consider different combinations of an estimation period Ne = 25 or 50, and a set of 2 or 4 hypothesis H. When \H\ = 2, the hypothetical modulation index set H = {0.3,1/3}, while H = {0.28,0.30,0.32,0.34} with \H\ = 4. In the following, we adopt the notation ANDFE(Are, \H\) to convey the estimation period and hypothetical modulation set used. Fig. 4.18 plots the SNR required for BER = 10-3 vs. the modulation index used at the transmitter for NDFE with h known at the receiver and ANDFE with h unknown at the receiver. The phase reference (3.46), which does not account for frequency offset variations, is used for each of the values a = 0.6 and a = 0.8. Observing the results obtained for a — 0.8, we notice that the curves for NDFE and ANDFE(50,4) are very close, and the ANDFE shows a remarkable performance improvement compared to the case where h is unknown at the receiver and equal to 0.32, reaching 3 dB when h at the transmitter is 0.29. For a = 0.6 it is illustrated that even for a small estimation period and modulation index hypothesis of Ne = 25 and \7i\ = 2, the ANDFE is still 4.2 Noncoherent Detection 81 0.31 0.32 h at transmitter • 0.35 Figure 4.18: 101ogi0(£(,/iVo) required for BER = 10"3 for ANDFE with phase reference (3.46). The channel phase is constant and the notation ANDFE(Are, H) applies. robust. Again, the results for ANDFE and adaptive NSD (ANSD)2 with a = 0.6 are almost identical, as has been observed for the various aforementioned scenarios. In Fig. 4.19, we illustrate the performance of ANDFE for different pairs of a and 8 when the phase reference (3.53) with frequency offset estimation is used. For comparison, the results of ANDFE(50,4) with phase reference (3.46) and a = 0.6 are also shown. We make the following observations. First, it is evident that the adaptive version of NDFE (ANDFE) with an estimation period Ne = 50 combined with \7i\ = 4 and h unknown at the receiver achieves a result almost identical to NDFE with h known, as illustrated for the pair (a = 0.6,/? = 0.8). Second, the results obtained for almost all combinations of a and 8 form almost straight lines, indicating that they are close to the h known case. Third, as 8 approaches 1, the performance of ANDFE using the phase reference (3.53) approaches that of (3.46), which is consistent with the results 2The ANSD is the NSD receiver with an adaptive ^.-estimator. 4.2 Noncoherent Detection 82 h = 0.32 at receiver o=0,6,p=0,9 15h 10 . ANDFE(50,4) a = 0.6, P = 0.6 - • - LDI, h = 1/3 -*- MLSD bound ~ANDFE(50,4) a=0.6, p = 0.9 ,^ • NDFE, h known at receiver t'r^-;-^^;;^^^^^ : u ^ 0 6, P = 0.9 : ANDFE(50,4) a = 0.6 i ANDFE(50,4) / : »< ' - •* a = 0.6, P=0.99 t ANDFE(50,4) a = 0.8, :P = 0.9 -*-~4 0.29 0.3 0.31 0.32 h at transmitter -0.33 0.34 0.35 Figure 4.19: 101ogi0(Sb/Ar0) required for BER = 10-3 for ANDFE with various com binations of a and 3 with phase reference (3.53). The channel phase is constant and the notation ANDFE(ATe, H) applies. 18i 1 1 1 1 1 1 t CO o 3 05 LJ o o 17 16 15 1 1 12 Hi 10 9 0.28 LDI detector ANDFE(50,2) o •- 0.6, p. = 0,9 -e— NDFE, a=0.6, p=0.9, h known at receiver -e— ANDFE(50,4), o=0.6, p=0.9 -0 - NSD,a=0.6, p=0.9, h known at receiver - -a - ANSD(50,4), o=0.6, p=0.9 0.31 0.32 h at transmitter -0.35 Figure 4.20: 10\ogw(Eb/N0) required for BER = 10~3 for ANDFE and ANSD with the favorable pair (a = 0.6, 3 = 0.9) and with constant channel phase. 4.2 Noncoherent Detection 83 previously shown for NDFE. Fig. 4.20 compares the performance of the ANDFE with the adaptive NSD (ANSD), which employs an adaptive scheme similar to that of ANDFE. The results show very close performance between the adaptive schemes of NDFE and NSD when the same values of a and 3 are used (the maximum difference is about 0.4 dB), even with a smaller set of \H\ = 2. We can thus conclude that NDFE allows highly power-efficient detection also for a priori unknown modulation index. 4.2.4 Performance in the Presence of Interference Due to the operation of Bluetooth in the ISM band, evaluating the proposed Bluetooth receiver in the presence of interference is necessary. We consider the performance with interference, such that the signal-to-interference power ratio (SIR) and the carrier fre quency difference A/Cij are chosen as specified in the Bluetooth standard and presented in Table 2.1. The notation IF(SIR,A/ci) is used to convey the SIR in dB and Afc>i in MHz. Fig. 4.21 shows the performance results for the optimal WMF and the proposed SRC filter, where h = 1/3 is known at the receiver. The effective interference suppres sion provided by both filters is verified, with the BER for adjacent channel interference almost equal to the curve obtained when no interference is considered. The results for both filters are quite similar, with the SRC outperforming the WMF in case of adjacent channel interference, and the reverse is observed with co-channel interference. As a matter of fact, when the SRC filter is applied, there is no loss due to adjacent channel interference for the cases IF(-30,2), and IF(-40,3). Moreover, the BER curve for IF(-40,3) is shown when the SRC is applied without the Hamming window, and a loss of ~ 0.5 dB is observed compared to the case when the Hamming window is applied. Fig. 4.22 illustrates the performance of the ANDFE in the presence of interference as a function of the modulation index. We observe that the results of ANDFE are almost identical to those for NDFE in the presence of adjacent channel interference. As for 4.2 Noncoherent Detection 84 10 V IF(11,0) • IF(0,1) * IF(-30,2) 0 IF(-40,3) -with SRC — with WMF Figure 4 the SRC 10 12 14 10log10(Eb/N0)[dB]^ 21: Performance of NDFE in the presence of interference for the WMF and filter. The channel phase is constant and the notation IF(SIR,A/Cii) applies. 26 r 24 22, tr LU m o 20 0> CT 18 CD LU o o 12 10 ANDFE, IF(11,0) \ \ : 13 N * - -N_ ^-^^ • N i — ~ -. • • LDI, 1 1: . F(11,0) A NDFE, IF(11,0) ~ - 1 i-1 _ _ LDI, no interference i- - _ ~ -< _ ANDFE, IF(0,1) NDFE, IF(-30,2), IF(-40,3) and no;interference NDFE, IF(0,1) ANDFE, IF(-30,2), IF(-40,3) and no interference 0.29 0.3 0.31 0.32 h at transmitter -> 0.33 0.34 0.35 Figure 4 presence applies. .22: lOlogiol^/iVo) required for BER = lfT3 for NDFE and ANDFE in the of interference. The channel phase is constant and the notation IF(SIR,A/Cij) 4.2 Noncoherent Detection 85 co-channel interference, an accountable performance loss of up to 3 dB over the NDFE is recorded for h < 0.3. Also, ANDFE performs worse than the LDI for h < 0.295. However, in practice, co-channel interference is a rare occurrence, and, adjacent channel interference is of greater interest in general. Chapter 5 Conclusions The complexity constraints coupled with the varying modulation index and the high fre quency offset variations in Bluetooth devices creates a highly challenging environment for the design of power-efficient Bluetooth receivers. The conventional LDI detector is a simple receiver, but is highly suboptimum in terms of power efficiency, and can not cope with the high frequency offsets of up to ±100KHz admissible in Bluetooth systems. Several sequence detectors proposed in literature have proven to be extremely power efficient compared to the LDI detector since they properly take the CPM mem ory into account. These include the MLM-LDI and the MAP receivers. However, the MLM-LDI has very high complexity and the MAP receiver assumes perfect channel phase estimation and knowledge of the modulation index at the receiver, and, there fore, they are impractical. The NSD receiver recently proposed in [31] tackles these problems by designing a noncoherent sequence detector with a modulation index esti mator. Although the complexity of the NSD is low compared to the MLM-LDI and the MAP receivers, it still requires a 2-state Viterbi decoder. In this research work, we have presented a simple noncoherent power-efficient receiver design for Bluetooth transmission based on Laurent's decomposition, which transforms the actual nonlinear modulation scheme into a linear modulation over an intersymbol 86 87 interference channel. It was shown that the combination of an SRC filter, symbol-rate sampling, and a decision-feedback equalizer achieves a performance close to the theoretical MLSE limit. We have proposed a noncoherent version, namely NDFE, which offers high robustness to local-oscillator dynamics and phase jitter present in Bluetooth devices. NDFE was also extended to allow adaptation of the decision metric to an a priori unknown modulation index h. The presented simulation results showed that NDFE performs very similar to 2-state NSD proposed in [31]. The key advantages of the proposed design are outlined in the following and a summary of the corresponding simulation results presented in Chapter 4 is given. • Using Laurent's decomposition, the optimal receiver input filter for the resulting PAM signal is the whitened matched filter (WMF) [17]. However, since the WMF is dependent on the modulation index h, we propose replacing it by the practical SRC filter which has similar characteristics. The simulations (cf. Fig. 4.1) prove that the resulting performance using the WMF and SRC are almost identical. Moreover, high adjacent channel interference suppression is achieved through the SRC, as illustrated in Fig. 4.22. • Following the SRC filter, a noncoherent decision feedback equalizer is employed, where the phase is implicitly estimated using the developed phase reference (3.46). This phase reference is formulated in terms of a forgetting factor a and shows remarkable performance improvement compared to the LDI, closely approaching the MLSD lower bound as a increases (cf. Fig. 4.6). However, since this phase reference fails in the presence of extreme frequency offset variations and phase jitter, a modified phase reference (3.53) is devised. Again, the modified phase reference is controlled by a forgetting factor 8, and the resulting decision rule was found to be extremely robust against channel phase variations, as shown in Fig. 4.16. The pair (a = 0.6, 8 = 0.9) which was found to be favorable in NDFE for various scenarios, has almost the same power efficiency as the 2-state NSD 88 using the same values of a and 3. • The NDFE is the simplest sequence detector possible, consisting of a one-state trellis, thus, providing an extremely high reduction in complexity compared to the 2p-state trellis required by the optimal detector. Only a slight performance loss was recorded by decreasing the trellis from a full-state to a one-state trellis (cf. Fig. 4.4). This is a remarkable result since the one-state trellis in the NSD receiver was shown to have worse performance than the low-complexity LDI de tector [27]. • One of the primary disadvantages of the sequence detectors proposed in literature is assuming a nominal modulation index h at the receiver. In the new receiver design, we propose employing an adaptive ^.-estimation scheme with an adapta tion period of 7Ve symbols, and a set Ti of hypothetical modulation indices. This adaptive version of NDFE, which does not require knowledge of the modulation index h, was found highly effective with iVe = 50 and \Ti\ = 2 or 4, only adding a slight increase in complexity. The simulation results presented in Figs. 4.19-4.20 show that it performs almost identically to the case when h is known, and very close to the adaptive version of NSD with the same values of Ne and \H\. 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