B L U E T O O T H R E C E I V E R DESIGN BASED ON L A U R E N T ' S DECOMPOSITION by NOHA IBRAHIM B . E n g , American University of Beirut, 2003 A T H E S I S S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F THE REQUIREMENTS FOR T H E DEGREE OF MASTER O FAPPLIED SCIENCE in T H E F A C U L T Y O F G R A D U A T E STUDIES (Electrical and Computer Engineering) December, 2005 © Noha Ibrahim, 2005 Abstract B l u e t o o t h is a w i d e l y used c o m m u n i c a t i o n s t a n d a r d for wireless p e r s o n a l area networks ( W P A N ) . T h e B l u e t o o t h t r a n s m i t signal is G a u s s i a n frequency shift k e y i n g ( G F S K ) m o d u l a t e d . G F S K belongs to the f a m i l y of continuous-phase m o d u l a t i o n ( C P M ) signals, w h i c h achieve a g o o d trade-off between power a n d b a n d w i d t h efficiency a n d , due to constant envelope m o d u l a t i o n , allow for l o w - c o m p l e x i t y t r a n s m i t t e r implementa- t i o n . B l u e t o o t h devices often e m p l o y a s i m p l e d i s c r i m i n a t o r receiver, w h i c h is h i g h l y s u b o p t i m u m i n terms of power efficiency c o m p a r e d to the o p t i m u m receiver. O t h e r approaches p r o p o s e d i n the l i t e r a t u r e consider trellis-based d e t e c t i o n u s i n g the V i t e r b i or f o r w a r d - b a c k w a r d a l g o r i t h m . These schemes achieve significant performance i m p r o v e ments over d i s c r i m i n a t o r detectors w h i l e e n t a i l i n g a c o n s i d e r a b l y higher c o m p u t a t i o n a l c o m p l e x i t y . T h e m a i n challenges faced w h e n designing a B l u e t o o t h sequence detector is the v a r y i n g m o d u l a t i o n index, w h i c h results i n a v a r y i n g trellis structure, a n d the t i m e - v a r i a n t c h a n n e l phase, m a k i n g coherent detection w h i c h assumes perfect c h a n n e l phase e s t i m a t i o n a n almost i m p o s s i b l e task. In t h i s research w o r k , we present a receiver design for B l u e t o o t h t r a n s m i s s i o n based o n L a u r e n t ' s d e c o m p o s i t i o n of the B l u e t o o t h t r a n s m i t s i g n a l . T h e m a i n features of t h i s receiver are its l o w - c o m p l e x i t y c o m p a r e d to a l t e r n a t i v e solutions, its excellent performance close t o the t h e o r e t i c a l l i m i t , a n d its h i g h robustness against frequency offsets, phase noise, a n d m o d u l a t i o n i n d e x v a r i a t i o n s , w h i c h are characteristic for lowcost B l u e t o o t h devices. In p a r t i c u l a r , we show t h a t the devised noncoherent decision- ii m feedback e q u a l i z a t i o n receiver achieves a s i m i l a r performance as a recently p r o p o s e d 2-state noncoherent sequence detector, w h i l e it is advantageous i n terms of c o m p l e x i t y . T h e new receiver design is therefore h i g h l y a t t r a c t i v e for a p r a c t i c a l i m p l e m e n t a t i o n . Contents Abstract ii Contents iv List of Tables vii List of Figures viii Glossary xii Acknowledgements xvii 1 Introduction 2 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 Background 9 2.1 9 The Bluetooth System 2.1.1 Radio Front End 10 2.1.2 Physical Channel and Packet Definition 11 2.1.3 Physical Links 12 iv Contents 2.2 2.3 v 2.1.4 Modulation Scheme 13 2.1.5 Transmission Model 16 2.1.6 Error Correction 20 Bluetooth Receivers Literature Review 21 2.2.1 Optimum Receiver 22 2.2.2 Suboptimum Receivers 27 M L S D Lower Bound 31 3 Noncoherent Decision Feedback Equalizer Receiver Structure 3.1 33 Laurent's Representation of C P M 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 43 3.3 Discrete-time Transmission Model 47 3.4 Reduced-State Sequence Estimation 48 3.5 Decision-Feedback Equalizer Structure 50 3.5.1 Minimum Phase Channel 51 3.5.2 D F E Decision Rule 53 3.6 3.7 Noncoherent Detection 54 3.6.1 N D F E Decision Rule 57 3.6.2 Frequency Offset Estimation 58 3.6.3 Adaptive N D F E . 58 Summary of Proposed Receiver Structure 4 Performance Results and Discussion 4.1 60 62 Coherent Detection 63 4.1.1 Filter Selection 63 4.1.2 State Reduction 65 Contents 4.2 5 vi Noncoherent Detection 66 4.2.1 Performance with Constant Channel Phase 67 4.2.2 Performance Using the Modified Phase Reference Under Constant and Varying Channel Conditions 75 4.2.3 Adaptive N D F E 80 4.2.4 Performance in the Presence of Interference Conclusions Bibliography ' . : 83 86 89 List of Tables 2.1 Interference performance [51] 18 2.2 T h e values of r e l a t i v e l y - p r i m e integers m a n d p c o r r e s p o n d i n g to different m o d u l a t i o n indices h w i t h i n the range specified b y the B l u e t o o t h 3.1 standard 25 D u r a t i o n of L a u r e n t components.Cp(t) 38 vii List of Figures 1.1 The required 101ogi (iViVo) ^ 0 o rB ^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 C P M 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 E C B 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 3.1 31 Laurent pulses C (t) and Ci(t), and the main pulse P(t) for Bluetooth 0 GFSK signals with L = 2 and h = 1/3 3.2 Block diagram of optimum coherent receiver based on Laurent's decomposition 3.3 39 40 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 3.4 ix Impulse response of SRC, h Rc{t), before Hamming window (a), and S after Hamming window (b) 3.5 45 Magnitude frequency response for W M F , H (f), WMF and h = 0.35, and for SRC filter, H {f), SRC 3.6 assuming h = 0.28 with p = 0.3 46 The CIR of the (a) ISI channel h [k] compared to (b) its minimum phase c equivalent ho[k] 48 3.7 Block diagram of decision-feedback equalizer 51 3.8 The roots of H (z) compared to the roots of its minimum phase equivc alent H (z) 53 3.9 Structure of the proposed Bluetooth receiver 60 4.1 Performance comparison of the SRC and W M F as receiver filters. Co- 0 herent detection with h = 1/3 is assumed 4.2 Evaluation of the state reduction in the proposed receiver. Coherent detection with h = 1/3 is assumed 4.3 67 Performance of the N D F E using the /V-metric compared to MLSD, LDI, and coherent D F E , and h = 1/3 for all cases 4.6 66 Evaluation of the performance of the proposed receiver with state reduction in case of noncoherent detection 4.5 65 Performance of N D F E with a = 0.8 with F B F and F F F fixed at the receiver, or varying according to the value of h at the transmitter. . . . 4.4 64 69 Performance of the N D F E using the a-metric compared to MLSD, LDI, and coherent D F E , and h = 1/3 for all cases 69 4.7. Performance of the N D F E using the iV-metric h = 1/3 in the presence of phase jitter 4.8 70 Performance of N D F E using the a-metric h = 1/3 in the presence of phase jitter 70 List of Figures 4.9 x Performance of the N D F E with a = 0.8 for different values of the modulation index compared to the M L S D bound, L D I , and M L M - L D I . . . . 72 4.10 Performance of the N D F E for varying h with a = 0.4, 0.6, 0.8, and 0.9. 73 4.11 Performance of N D F E for varying h with a = 0.6 and 0.8 compared to that of N S D for a = 0.6 73 4.12 The required 10\og {E /N ) w b for B E R = 10~ 0 for N D F E with a = 0.8 3 with varying modulation index 4.13 lOlogio (E /N ) b 75 required for B E R = 1 0 0 for different combinations of - 3 a and (3 as a function of the modulation index with constant channel phase and h known 76 4.14 The required l0\og (E /N ) w b f o r B E R 0 = 1 0 " 3 f o r N D F E W l t h a = 0.6,(3 = 0.9 for varying h 78 4.15 The required mog (E /N ) 10 b f o r B E R Q = 1 0 " 3 f o r N D F E with a = 0.8, f3 = 0.9 for varying h 78 4.16 The required 10\og (E /N ) 10 b 0 for BER = 10 - 3 in the presence of fre- quency offset and phase jitter 79 4.17 The required 101ogi (^b/^Vo) for B E R = 10" 0 3 in the presence of fre- quency offset and phase jitter 79 4.18 1 0 1 o g i o ( £ y i V o ) required for B E R = 10~ for A N D F E with phase refer3 ence (3.46). The channel phase is constant and the notation A N D F E ( J V , H) e applies. 81 4.19 101bgio(£&//V ) required for B E R = 10~ for A N D F E with various com3 0 binations of a and /3 with phase reference (3.53). The channel phase is constant and the notation A N D F E ( A , H) applies r e 4.20 Mog (E /N ) 10 b 0 82 required for B E R = l O " for A N D F E and A N S D with 3 the favorable pair (a = 0.6, (3 = 0.9) and with constant channel phase. 82 List of Figures xi 4.21 Performance of N D F E in the presence of interference for the W M F and the SRC filter. The channel phase is constant and the notation IF(SIR,A/ ) applies 84 Cii 4.22 10\og (E /N ) 10 b 0 required for B E R = 1 ( T for N D F E and A N D F E in the 3 presence of interference. The channel phase is constant and the notation IF(SIR,A/ ) applies C|i 84 Glossary L i s t o f A b b r e v i a t i o n s (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 xiii Glossary 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 xiv Glossary 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 E Signal energy per modulation interval f[k] (N)DFE feedforward filter f 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 h [k] Minimum phase equivalent of h [k], the (N)DFE feedback filter h [k] Discrete-time channel impulse response (ISI channel) hsRc{t) Impulse response of the SRC filter hwMF(t) Impulse response of the W M F HsRc{f) Frequency response of the SRC filter HWMF(I) Frequency response of the W M F H Number of modulation index hypotheses used in A N D F E 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 s c 0 c c Glossary xv K Rician factor log Base-10 logarithm In Natural logarithm L C P M 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 N Length of estimation period used in A N D F E p the relatively prime denominator integer such that h = m/p P(t) Laurent main pulse q(t) Normalized phase pulse q f 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 e Te rect(i/T) rectangular pulse s(t) Equivalent complex baseband representation of transmitted signal S (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 n A/ j frequency offset of the desired signal compared to the interfering signal # 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) C) 0 xvi Glossary ifj(t,a) Information-carrying phase ty(t) G e n e r a l i z e d 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 R A from Dr. Lutz Lampe and Dr. Robert Schober through NSERC grants STPGP 257684 and RGPIN 283152-04, and T A 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 digital assistants (PDAs), and the evident success of cellular phones called for a wireless 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 industrybased 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 T h e E v o l u t i o n of the B l u e t o o t h T e c h n o l o g y In 1994, as the sales of cellular phones were increasing, Ericsson, one of leading telecommunications manufacturers, was investigating ways to add value to its phones in the crowded market. Consequently, Ericsson mobile research lab in Lund, Sweden, initiated 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". Bluetooth is the English derivative of the Viking word Blatand, and was named after the 10 th 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 technology 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 conducted 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 U W B offers, the Bluetooth SIG announced in May, 2005 the collaboration of the Bluetooth and UWB developers. The goal is to employ U W B in the next-generation Bluetooth products, while maintaining backward compatibility with the existing products [1]. 4 1.2 C h a l l e n g e s a n d M o t i v a t i o n 1.2 Challenges and Motivation T h e objective of the present w o r k is t o design a cost-effective, power-efficient, and s t r u c t u r a l l y - s i m p l e B l u e t o o t h receiver. T h e o p t i m u m B l u e t o o t h receiver has very h i g h s t r u c t u r a l a n d c o m p u t a t i o n a l c o m p l e x i t y , for it consists of a b a n k of m a t c h e d filters followed b y a c o h e r e n t 1 m a x i m u m l i k e l i h o o d sequence detector ( M L S D ) w i t h a n u m b e r of states v a r y i n g a c c o r d i n g to the m o d u l a t i o n index. Since the m o d u l a t i o n i n d e x i n B l u e t o o t h systems is allowed t o v a r y i n a r e l a t i v e l y w i d e range (0.28 < h < 0.35), the c o r r e s p o n d i n g n u m b e r of states i n the o p t i m u m receiver b r o a d l y varies w i t h a m i n i m u m of 12 (as w i l l be detailed i n the next chapter), m a k i n g it unfeasible. The p r a c t i c a l a n d s i m p l e alternative, namely, the l i m i t e r - d i s c r i m i n a t o r integrator ( L D I ) receiver, c u r r e n t l y used for B l u e t o o t h devices is a simple, low-cost receiver. However, it is h i g h l y s u b o p t i m a l , for it suffers a performance loss of m o r e t h a n 6 d B over the o p t i m u m receiver, as shown i n F i g . 1.1. T h e 6 d B gap suggests t h a t the L D I receiver consumes four times as m u c h s i g n a l power as the o p t i m u m receiver to achieve the same bit error rate ( B E R ) . In the present w o r k , we investigate the feasibility of a n a l t e r n a t i v e B l u e t o o t h receiver design, m o t i v a t e d b y the large possible power efficiency gains t h a t m a y be achieved over the c o n v e n t i o n a l L D I receiver. T h i s requires o v e r c o m i n g the following challenges faced w h e n designing a B l u e t o o t h receiver. T h e v a r y i n g m o d u l a t i o n i n d e x results i n a v a r y i n g trellis s t r u c t u r e w i t h a large n u m b e r of states, w h i c h poses a serious challenge w h e n c o n s i d e r i n g trellis-based detection. In a d d i t i o n , the m e t r i c s required for the detection of the m a x i m u m l i k e l i h o o d s y m b o l sequence are also dependent o n h. However, the trellis-based receivers achieve significant performance i m p r o v e m e n t s over the c o n v e n t i o n a l L D I receiver. Hence, we investigate 1 Coherent detection assumes perfect channel phase estimate at the receiver. 5 1.2 Challenges and Motivation 18 • T CO o 1— LDI receiver - * - • Optimum receiver] 17 • t 16 : k II CC 15 UJ CD O T3 14 >6dB a '5 CT13 <U ! 12 LU O O 11 10 9 0.29 0.3 0.31 0.32 0.33 Figure 1.1: The required 101ogi (£ 6/A o) for B E R = 1 0 , 0 / 0.34 -3 0.35 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 ± 1 0 0 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 literature. 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 outof-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. 6 1.3 Contributions 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) channel, we develop reduced-state trellis-based equalizers using the concepts of reducedstate 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 D F E receivers, which we refer to as noncoherent RSSE (NRSSE) and noncoherent D F E (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 impulse response, we show that decision-feedback equalization (DFE) achieves a performance close to the lower bound of maximum-likelihood sequence estimation (MLSE) 2 [17]. This is a remarkable result, since complete state reduction incurred considerable performance degradation in case of an alternative noncoherent 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]. 2 I n the context of the present work, "sequence estimation" a n d "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 N D F E 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 N D F E achieves almost the same performance as 2-state NSD. Due to its complexity advantage over NSD, the N D F E 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 M L S E 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 D F E . The minimum phase channel needed to achieve the maximum performance in reduced-state detectors is also discussed. The following 8 1.4 Thesis Outline section applies noncoherent detection to RSSE and D F E , 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 characteristics of the proposed receiver. Chapter 2 Background T h i s chapter provides the b a c k g r o u n d i n f o r m a t i o n necessary for the design, analysis, a n d e v a l u a t i o n of the B l u e t o o t h receiver presented thereafter. B l u e t o o t h was designed t o be a universal wireless technology, w h i c h operates at l o w power a n d c a n be i m p l e m e n t e d at low cost. A c c o r d i n g l y , the B l u e t o o t h specifications [51] described i n the first section of this chapter were carefully chosen so as to satisfy these requirements. Since the present w o r k involves a receiver design, the second section provides a l i t e r a t u r e survey o n the o p t i m u m a n d s u b o p t i m u m B l u e t o o t h receivers. T h e s e , i n a d d i t i o n to the M L S E lower b o u n d presented i n S e c t i o n 2.3 w i l l serve as b e n c h m a r k s w h e n e v a l u a t i n g the proposed s o l u t i o n . 2.1 The Bluetooth System T h i s section elaborates on the B l u e t o o t h specifications relevant for the present work. T h e r a d i o specifications w i l l be described first, followed b y the p h y s i c a l channel a n d packet definition. T h e types of p h y s i c a l l i n k s available for d a t a t r a n s m i s s i o n as well as the m o d u l a t i o n scheme used w i l l t h e n be presented. W e conclude t h i s section w i t h a n i l l u s t r a t i o n of the channel m o d e l a n d a n overview of the error c o r r e c t i o n schemes offered b y B l u e t o o t h . 9 2.1 The Bluetooth System 2.1.1 10 Radio Front E n d Bluetooth operates in the 2.4 GHz Industrial Scientific Medical (ISM) band, a frequency band that is globally available, license-free, and open to any radio system. The Bluetooth standard specifies a 1 MHz bandwidth for each R F 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 R F 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 interference 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 R F channels following a pseudo-random sequence [9], [13], [51]. Federal communications 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, respectively. 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 requirements) 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]. 11 2.1 The Bluetooth System 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 accommodate 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 occupy 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 specification 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) 12 2.1 The Bluetooth System ACCESS CODE GFSK HEADER GUARD • SYNC ~* ENHANCED DATA RATE PAYLOAD TRAILER • 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 transmitted. Further, a device can alternate between the two types of links during transmission, as long as an A C L link is established before using an SCO link. A C L 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 13 2.1 The Bluetooth System 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 transparent 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 modulation (CPM), is the modulation scheme used in the basic data rate mode of Bluetooth (cf. Section 2.1.2). C P M 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, C P M provides better spectral utilization than frequency shift keying (FSK) by introducing a continuous phase to smooth variations between symbols. The block diagram of a C P M modulator is shown in Fig. 2.3, where the frequency modulator consists of a voltage controlled oscillator (VCO). The resulting C P M passband signal is represented by [2] SRF(t) = C os (27r/ t + ib(t, a) + d ) , c (2.1) 0 where E denotes the signal energy per modulation interval T, f s c is the carrier fre- quency, #o i s 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 } i s the random binary data, h i s 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. \ 14 2.1 The Bluetooth System a[k] e { ± 1 } g(t) vco * ) SRF{t) FM Modulator 2nh Figure 2.3: Block diagram of a C P M 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 0 t J g{r)dT o 9(t) 1/2 t <0 0 < t < LT (2.3) t > LT When L > 1, the frequency pulse exceeds the symbol interval T, leading to intersymbol interference (ISI). However, this approach, known as partial response signaling, is desirable in C P M 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 C P M 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 transitions, 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) * r e c t ( * / T ) , where * denotes the convolution and h(t) is given by h(t) = 1 2ixaT ' / exp -t 2 2a T 2 2 (2.4) 15 2.1 The Bluetooth System 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 1/T for |t| < T/2 0 otherwise rect I — I = (2.5) is the frequency pulse shape given by 9(t) = ± Q(f(t + T/2J)-Q(£(t-T/2J) (2.6) where Q(t) = (l/\/27r) J e~ / d r is often referred to as the Gaussian Q-function. The t T 2 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 16 2.1 The Bluetooth System the restrictions imposed by the F C C 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 M o d e l 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 flatfading 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 dominant signal, 2cr is the variance of the diffuse path, and IQ(X) — l/2ir J exp(xcosc/!>)d(/> 2 o is the modified Bessel function of first kind and zeroth order. The Rician distribution is described by a Rician factor K — A /2a , 2 2 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 cr —> 0, the result is 2 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 17 2.1 The Bluetooth System 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) where <f> 0 = g^*°s(t) + n(t) , (2.8) 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 as 1 s(i)=exp(jV(t,a)) , (2.9) and n(£) is additive white Gaussian noise (AWGN) with two-sided power spectral density A^o/2. We note that the passband signal (prior to amplitude normalization) as s (t) SRp(t) RF (2.1) can be written in terms of = 3f?{s(£)e j27r/c s(i) *}, 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 jitter, 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 fluctuations 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) + 2 T T A / T + A(f>{t, T) . The constant phase term 0Q has been ignored. (2.10) 18 2.1 The Bluetooth System Table 2.1: Interference performance [51]. Frequency of Interference (A/ ) SIR 0 MHz (Co-channel interference) 11 dB 1 MHz (adjacent interference) 0 dB 2 MHz (adjacent interference) -30 dB > 3 MHz (adjacent interference) -40 dB C)i Consequently, the received signal in the presence of frequency offset and phase jitter is expressed as r(t) = e ' ' * ( S (t)+ * s ( t ) + n(t) . o) (2.11) Due to operation in the license-free ISM band and frequency reuse, Bluetooth can suffer from interference from other Bluetooth and W L A N devices. Co-channel interference 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 determined 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 f + A / c C>i for all the SIR ratios shown in Table 2.1 [51]. A / C i i represents the frequency offset of the interfering signal compared to the desired signal. After considering the interfering signal, the E C B representation of the received signal may be given by r(t) = geKM+^slt) + n(t) + i(t) , (2.12) where i(t) denotes the E C B 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- 19 2.1 The Bluetooth System Mt) e a[k] ai[k] {±1} CPM Modulator e {±1}- CPM Modulator G s(t) n(t) -0 r(t) i(t) Figure 2.5: Block diagram representing the E C B system model. dently, channel fading is irrelavant for receiver design. Consequently, in our design and after ignoring channel fading, the E C B representation of the received signal is r(t) = e>*W (t) + n{t) + i{t) . S (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 C P M 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 I E E E standard 802.15.1 based on Bluetooth and approved by the I E E E in 2001 [26]. The path loss is represented as a function of the separation distance d between the transmitter and receiver as follows { where L p 40.2 + 201og(d) d < 8 m 58.5 + 331og(d/8) d > 8 m (2-14) is the path loss in decibels (dB). The corresponding received power P R is then obtained by subtracting the path loss from the transmitted power. The receiver sensitivity threshold in Bluetooth is -70 d B m with a nominal 0 d B m transmit power. 2.1 The Bluetooth System 2.1.6 20 Error Correction Bluetooth packets are checked for errors or wrong delivery using the channel access code, the H E C in the header, and the CRC in the payload. In addition, three errorcorrection schemes are offered: • 1/3 rate F E C • 2/3 rate F E C • Automatic retransmission query (ARQ) The above schemes are used to transmit different parts of the packet. Bluetooth devices 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 F E C 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 F E C 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 information. For this reason, the Bluetooth developers kept the 1/3 F E C code optional. The presence or absence of the code depends on the packet type, defined in the 4-bit ' T Y P E ' 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 F E C 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 21 2.2 Bluetooth Receivers Literature Review Ol 02 o a\ as, 2 02 03 03 03 Figure 2.6: Illustration of Repetition code. the generator polynomial g(D) = (D © D © D © 1). If a packet contains less than 5 4 2 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 F E C code [51]. The ARQ scheme is used only on the payload of packets that have CRC. These include DV, D M , DH, and E V 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 C R C packet. If the H E C 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 C P M 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 C P M attractive also present difficulties for receiver designs. The next sections discuss several approaches for Bluetooth 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. 22 2.2 Bluetooth Receivers Literature Review 2.2.1 O p t i m u m Receiver The optimum receiver discussed here assumes coherent detection and an AWGN channel. 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 C P M receiver exploits the general state trellis structure of C P M to perform MLSD. Due to the memory inherent in a C P M 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 C P M 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 C P M using the VA, as will be described later in the section. Introducing the C P M memory, L , into Eq. (2.2), the information-carrying phase can be appropriately represented as n—L n a[k]q(t - kT) nT <t<{n+\)T . (2.15) k=n-L+l 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 (2.16) (2.17) 23 2.2 Bluetooth Receivers Literature Review 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 C P M signal at time t = nT can be uniquely defined by the combination of the phase state and the state vector as S = {9[n — L],a[n — L + 1],a[n — L], ...,a[n — 1]}. n Consequently, the number of states N in the trellis is s { pM ~ L (even m) Y (2.18) 2pM ~ L (odd m) 1 We have shown that a C P M 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 maximizes 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)\ d^ 2 . (2.19) Consequently, the MLSD decision rule is given by +oo a = argmax^- [ \r(t) - s(t)\ dt \ a (2.20) 2 • —oo +oo +oo argmax < - Jf \r \r(t)\ dt2 +O0 J \s{t)\ dt+ J 2M{r{t) • §*{t)} dt 2 2 \ (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 A FSM is a model of computation consisting of a set of states, having an input alphabet and a 2 transition function which maps the input symbols and current states to a next state. 24 2.2 B l u e t o o t h Receivers L i t e r a t u r e R e v i e w independent of | r ( £ ) | , w h i l e \s(t)\ is a constant t e r m since i t is expressed i n terms of 2 2 a n e x p o n e n t i a l (2.9). Hence, we c a n e l i m i n a t e these t e r m s a n d d i s r e g a r d t h e factor of 2 i n t h e r e m a i n i n g t e r m since i t has n o effect o n t h e decision, r e s u l t i n g i n t h e following MLSD decision rule a = argmax j J fH{r(t) • s * ( t ) } d t | . (2.22) T h e above e q u a t i o n shows t h a t the c o m p l e x i t y of t h e MLSD increases e x p o n e n t i a l l y w i t h t h e l e n g t h of t h e s y m b o l sequence a. T h i s c a n be avoided b y u s i n g t h e VA, w h i c h i n t r o d u c e s a recursively c a l c u l a t e d m e t r i c A (d) as follows n (n+l)T J A (a)= n M{r(t)-?(t)}dt —oo (n+l)T = A _ (d)+ n (2.23) 1 J $l{r(t)-s*(t)}dt (2.24) nT where A _i(d) is the a c c u m u l a t e d m e t r i c of the s u r v i v i n g sequence u p t o t i m e t = nT n a n d t h e second t e r m o n t h e r i g h t - h a n d side t h e e q u a t i o n , k n o w n as t h e b r a n c h m e t r i c A (d), is given b y n (n+l)T A (d) = n J 3f? J (t) • e x p r nT I -j d[n -L} + 2nh ]T a[k]q(t - kT) dt k=n-L+l (2.25) W e note t h a t t h e b r a n c h m e t r i c is interpreted as filtering t h e received s i g n a l r(t) t h r o u g h a b a n k of 2M L m a t c h e d filters a n d s a m p l i n g t o o b t a i n o u t p u t s w h i c h f o r m sufficient statistics for detection. W e notice t h a t t h e c o m p l e x i t y o f t h e MLSD u s i n g the VA increases o n l y l i n e a r l y w i t h the l e n g t h of t h e s y m b o l sequence. T h e VA c a n be best described w i t h t h e i l l u s t r a t i o n of i t s key steps: a d d , c o m p a r e , a n d select (ACS). T h e decision rule as given b y E q . (2.22) shows t h a t we have a m a x i m u m p r o b l e m . Therefore, the ACS consists of t h e following steps [8]. 25 2.2 Bluetooth Receivers Literature Review 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 symbol, 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 pM L metrics A (d) corresponding to all M L x L possible sequences and p (or 2p) possible phase L n states. This results in pM ~ (or 2pM ) (or 2pM ~ ) L surviving sequences at each time instant of l 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 (= 2M ) filters prior to sampling. The state vector L becomes S = {9[n — 2],a[n — 1]}. The variables needed to construct the trellis depend n 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 26 2.2 Bluetooth Receivers Literature Review Input symbol = -1 Input symbol = 1 [5TT/3,-1] •' i=0 p ^ s t a t e s {*,3,*,5*/3} i=T {0,2^/3,4^/3} i = 2T {TT/3, 7r/57r/3} i = 3T {02 , 7r/34 , 7r/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 V A ; 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}. A t each state, two trial symbols d[n] = +1 and a[n] = -1 are considered and the V A decides on the corresponding metric that maximizes (2.22). Hence, at each time instant of the V A , 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 V A paths at only three phase states at i = 0. These phase states were carefully chosen, for they lead to three 27 2.2 B l u e t o o t h Receivers L i t e r a t u r e R e v i e w different phase states at the next s y m b o l i n t e r v a l . F r o m T a b l e 2.2, we notice t h a t even a slight v a r i a t i o n of h c a n l e a d t o a t o t a l l y different trellis structure. Therefore, a s s u m i n g a fixed n o m i n a l h is not a n o p t i o n i n t h i s design. M o r e o v e r , the c o m p l e x i t y of the o p t i m u m receiver, defined b y the n u m b e r of states pM ~ L l (or 2pM ~ ), L varies w i t h h. 1 increases e x p o n e n t i a l l y as M a n d L increase, a n d t r e m e n d o u s l y It also assumes perfect c h a n n e l phase e s t i m a t i o n , w h i c h is not v a l i d i n p r a c t i c e (cf. S e c t i o n 2.1.5). W e conclude t h a t the o p t i m u m receiver performance m a y be considered as a b e n c h m a r k , b u t it is definitely not a feasible s o l u t i o n for the B l u e t o o t h receiver design. 2.2.2 Suboptimum Receivers Since the o p t i m u m receiver has a n e x t r e m e l y h i g h c o m p l e x i t y , B l u e t o o t h devices often e m p l o y a s i m p l e d i s c r i m i n a t o r receiver to recover the G F S K m o d u l a t e d d a t a . receiver uses a l i m i t e r - d i s c r i m i n a t o r integrator, a n d , therefore, is referred to as the L D I receiver. This hereafter It is a d o p t e d i n B l u e t o o t h due t o its l o w c o m p u t a t i o n a l a n d s t r u c t u r a l c o m p l e x i t y , m a k i n g it s i m p l e to i m p l e m e n t at a l o w cost. However, the L D I suffers f r o m a considerable performance loss ( > 6 d B ) over the o p t i m u m receiver, as i l l u s t r a t e d i n S e c t i o n 1.3. Since the i n t r o d u c t i o n of B l u e t o o t h devices i n the m a r k e t , several receiver designs have been p r o p o s e d i n literature. T h e L D I - b a s e d designs i n [47], [48], have o n l y a s l i g h t l y increased c o m p l e x i t y ; however, they offer o n l y a slight increase i n performance ( < 1 d B ) . These i n c l u d e a receiver based o n zero-crossing d e m o d u l a t i o n w i t h a d e c o r r e l a t i n g m a t c h e d filter [48], t e r m e d B T - Z X M F . A n o t h e r design p r o p o s e d b y the same a u t h o r s uses least squares-based post-integration filtering [47]. T h i s a p p r o a c h is further ex- tended i n [49] b y e m p l o y i n g a m a x - l o g - m a x i m u m l i k e l i h o o d ( M L M ) s y m b o l d e t e c t i o n w h i c h involves a f o r w a r d - b a c k w a r d a l g o r i t h m on a 4-state trellis, a n d the r e s u l t i n g detector is t e r m e d M L M - L D I [49], [50]. A l t h o u g h a significant performance i m p r o v e m e n t 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 C P M is properly taken into account. Consequently, many authors have considered trellis-based detection using the forward-backward algorithm [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, followed 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 coherent 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 M A P 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 29 2.2 Bluetooth Receivers Literature Review Pre-detetion filter T {t) F Li miterDiscriminator Integrate and Dump A* (f d Hard 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 impulse frequency response [54] where B is the 3-dB bandwidth of the filter, with an optimum value of 1.1 MHz in r 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 decomposition approach to C P M [41], [25]. It is shown in [41] that a C P M modulator can be 30 2.2 Bluetooth Receivers Literature Review 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 frequency f and modified unipolar information symbols a [fc], a modified phase state is r m obtained as [25] n—L Om[fc]]mod(p)e{0,l,..,p-1}. V [n-L]=[m m (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 p M and pM L L _ 1 = 2p states = 4p possible trajectories in each time interval, resulting in 4p time-limited, constant-envelope signal elements Pm(a [k])(t)> where a [k] = {a [k], a [k — 1],ty [n— m m m L]} is a unique address vector associated with each signal element. m m Applying this decomposition, the E C B transmit signal can be expressed as oo s(t) = Z)Pm(ai])(t-ir). m (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 improvement 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 31 2.3 M L S D Lower Bound Signal mapper with 4p m mod(p) T a [k] m a [k m Figure 2.9: - e L 1] •ZT] * [n-2] T m f » s(t) signal elements Block diagram of the G F S K modulator using the Rimoldi/Huber&Liu decomposition approach. a 2-state trellis search. W i t h complete state reduction, the N S D 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 N S D and the proposed receiver inspite of the proposed receiver's lower complexity. 2.3 M L S D L o w e r B o u n d A lower bound for the performance of maximum likelihood sequence detectors for C P M 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 M L S D in an A W G N channel is given by [2] (2.28) where d mm is the minimum normalized Euclidean distance between two sequences of information symbols a, and Ef, is the received energy per bit. For C P M signals, the squared and normalized Euclidean distance measure is given by (2.29) 32 2.3 MLSD Lower Bound oo where <p(t, 7 ) = 2nh ^ l[i]q(t - iT) and — a[i] — a[i] is the difference between the 2=0 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, d \ , is obtained as a function of h. It represents the distance between the pair m n 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 N m symbol intervals, then the upper bound on the minimum squared Euclidean distance reads d (h) = log (M) \ 2 mm 2 k i / 0 [l-cos<K£, n)]dt (2.30) 7mi ; Using the above equation, a lower performance bound of MLSD is obtained by substituting (2.30) into (2.28) as (2-31) Since the proposed receiver employs sequence estimation, this bound is used for comparison purposes in Chapter 4. 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 C P M signal presents various challenges for the receiver design, we adopt the linear representation of C P M , 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 G F S K 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 discretetime 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 M L S D and Maximum A Posteriori ( M A P ) are proven to be optimum. The M L S D is implemented using the Viterbi algorithm which determines the most likely symbol 33 34 3.1 Laurent's Representation of C P M Signals sequence as was detailed in Section 2.2.1. The M A P is implemented using the Bahl, Cocke, Jelinek, and Raviv (BCJR) algorithm [4] which determines the most probable 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. Furthermore, 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 D F E , a minimum 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 minimum phase channel, which serves as the feedback filter in the proposed D F E . 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 D F E . 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 C P M Signals There are two alternative representations of C P M in literature. One representation given by Rimoldi in [41] decomposes the C P M 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 C P M Signals 35 complete state reduction (1-state trellis). Another decomposition approach, referred to as Laurent's decomposition, was proposed by Pierre Laurent in 1986 [32]. It represents the C P M 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 C P M , including developing noncoherent sequence detection [11] and capacity bounds [56] for C P M . Moreover, using Laurent's representation, the challenges faced in the C P M 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] C P M 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 sections describe the derivation which gives an exact representation of the C P M signal. Furthermore, we illustrate its application to the Bluetooth GFSK signal, discuss the corresponding optimum receiver, and present an approximate representation of the C P M signal. 3.1.1 Laurent's Decomposition and Its Application to Bluetooth Laurent developed an alternative representation for binary C P M signals, which expresses the C P M signal as a sum of (2 ~ L 1 — 1) pulse amplitude modulated (PAM) 3.1 Laurent's Representation of C P M Signals 36 components . The complete derivation process presented in [32] will be described in 1 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 n t h interval (2.15) can now be written as n—L iP{t,a) = nh ^ fc=—oo n a[k]+ ^ a[k]ip{t - kT) , nT < t < (n + 1)T . (3.1) k=n—L+l Recalling the E C B 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 n t h interval as n—L s(t) = exp [jirh Yl n a W ) ' fc=—oo II e x P (M^Mt ~fcT)). (3.2) 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 m t h symbol can be defined as a sum of two terms exp (ja[m]ip(t — mT)^ =sin[-7r/i — •</?(£ — mT)\ sin(7r/i) /. , sin[^(* - mr)] + exp(jnh a m ) . , r n 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 A M P representation of a C P M signal with integer modulation index. We continue the derivation process by defining a generalized phase pulse function as ( ip(t) t < LT (3.4) 1 I n this work, "Laurent pulses", and "Laurent components", irh-<p(t-LT) t>and, LTi n some cases, " A M P (or P A M ) pulses (or components)", are used synonymously. 37 3.1 Laurent's Representation of C P M Signals In order to express Eq. (3.3) in terms of one function, we introduce the functions S (t) n sin (7m) Using these functions, Eq. (3.3) can be rewritten as exp (^ja[m]<f(t - mT)) = S - (t) L m + ex.p(jirh a[m])S- (t) m . (3.6) Introducing the above result in Eq. (3.2) leads to n—L n s(t) = exp [jivh J2 E[ k L-1 n-L ^2 S k=n—L+l fc=—oo = exp (jirh l L-k(t) + exp (jirh a[k]) S. (t)] i ]) a k k——oo H [S L-n{t) i+ + exp (jirh a[n - i\) Si- (t)} n . (3.7) i=0 The product term on the right-hand side of Eq. (3.7) can be expanded into a sum of 2 L terms. However, a close examination of the result reveals that only 2 L _ 1 of these terms are distinct functions of time. These form the A M P 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 =Y1 2 i _ 1 ' IK ' 0 < P < 21 1 - 1 (3.8) i=l where 7 p ; i 6 {0,1}. The A M P pulses are given by L-1 (t) . The Laurent pulses Cp(t) are nonzero in the interval [0, m i n (3.9) i = l i 2 ) ...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 C P M signal in terms of only one impulse as will be shown in Section 3.1.3. Finally, Laurent's representation of a C P M signal in terms of the A M P pulses reads +00 2 ~ -l L 1 s(t)= bp[k]C {t - kT) , P fc=-oo P=0 (3.10) 38 3.1 Laurent's Representation of C P M Signals Table 3.1: Duration of Laurent components Cp{t). Laurent Component Component duration C (t) (L + 1)T Ci(t) (L-l)T 0 C (t),C (t) 2 (L - 2)T 3 C (t),C (t),C (t),C (t) 4 5 6 (L - 3)T 7 C( i-l)/2, T C( i-l_i) 2 2 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 A [k]) P . (3.11) Eq. (3.10) is an exact representation of Eq. (2.9). However, as mentioned earlier Laurent's decomposition as derived above is restricted to binary C P M signals. The decomposition 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}C {t - kT) + b^C^t 0 - kT)) (3.12) fc=—oo where C (t) = S (t)Si{t) 0 0 and C (t) = S {t)S {t) x 0 3 are illustrated in Fig. 3.1 for h = 1/3. Since L = 2 for Bluetooth, we notice that C (t) has a duration of 3T, while C\(t) has 0 a duration of T and significantly less energy. The corresponding optimum receiver is described in the next section. 39 3.1 Laurent's Representation of C P M Signals 0.9; t/T-> Figure 3.1: Laurent pulses Co(t) and C\(t), and the main pulse P(t) for Bluetooth GFSK signals with L = 2 and h = 1/3. 3.1.2 O p t i m u m Coherent Receiver Based on Laurent's Decomposition Using the linear representation of C P M given in Eq. (3.10), a new optimum coherent detector for binary C P M was developed in [28]. This receiver requires 2 filters and a Viterbi sequence detector with p2 ~ L l L - 1 matched states, as shown in Fig. 3.2. To obtain the VA state vector, we conveniently express the complex symbols (3.16) as follows. When P = 0, b [k] = b [k - 1] exp (jirh a[k}), while for P = 1,2, ...2 ~ - 1, L 0 l 0 the complex symbols are represented in terms of &o [k — L] as b [k -L] = b [k -L)Y[ exp (jirh a[k - i\) ieip P (3.13) Q where Ip is a nonempty subset of the set { 0 , 1 , L — 1}. Hence, the state of the C P M signal can be uniquely represented at every time instant by the vector {b [k — L], a[k — 0 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 p2 ~ L l states are required for the 40 3.1 Laurent's Representation of C P M Signals Viterbi decoder CJ(-t) with r(i) states Figure 3.2: Block diagram of optimum coherent receiver based on Laurent's decomposition. 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 z [k] = (r(t) * Cp(-t)) P \ kT (for P = 0,1). t= 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] (3.14) A[fc] = K{z [fe]bS[fc] + ^ i [ ^ i W } 0 where z [k] and zi[k] are the sampled outputs of the matched filters to C (t) and C\(t), 0 Q respectively, and b [k] and b\[k] are the trial complex symbols corresponding to the trial 0 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 41 3.1 Laurent's Representation of C P M Signals 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 C P M signal using fewer Laurent pulses, as shown in [28] for binary C P M and in [10] for multilevel C P M . 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 M a i n Pulse For reduced-complexity receivers, it is desirable to develop a linear representation of a C P M 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 l=-oo (3.16) 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 optimization 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 42 3.1 Laurent's Representation of C P M Signals Real components 10 10.5 11 11.5 12 12.5 t/T 13 13.5 14 14.5 15 Imaginary components I -1 1 I I ^ — • — i — " " - - ^ ^ ^ ^ Exact CPM Signal ,1 10 Approximate CPM Signal Using Main Pulse . 1 10.5 11 I I I 11.5 12 12.5 t/T i 1 ' 13 13.5 i 1 14 i1 1 14.5 15 Figure 3.3: T h e 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 + mT) = P (r) with [32] m j P (r)= m P{t). —r^^ m—2 J] ' i=—oo oo c o s ^ r - H T ) ) J] cos (rm - ip(r - iT)) i=m+l (3.18) .[ sin (v?(r + m T ) ) . cos ( r + (m - 1)T) — cos(7r/i). sin (<p(r + (m — 1)T)). cos (irh — <p(r + mT)) ] . Observing E q . (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 C P M as given in (3.15) may be modelled as a 43 3.2 Filter Design 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 C P M 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). 3.2 F i l t e r D e s i g n 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 (3.19) The optimum filter that satisfies this property is the whitened matched filter (WMF) [17]. The W M F 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 # W M F ( / ) of the W M F corresponding to the main pulse P(t) (3.18) is given 44 3.2 F i l t e r D e s i g n by P*(-f) #WMF(/) = , 1 + p J o ) — J E |P(/-n/r)| • (3-20) 2 y r»=—oo T h e W M F depends o n the m a i n pulse P(t), w h i c h is f o r m u l a t e d i n terms of the m o d u - l a t i o n i n d e x h (cf. S e c t i o n 3.1.3). Hence, the W M F w o u l d i d e a l l y need to be adjusted a c c o r d i n g to the m o d u l a t i o n i n d e x of the t r a n s m i t t e d signal. T o a v o i d this, a n d to facilitate a p r a c t i c a l i m p l e m e n t a t i o n , we propose r e p l a c i n g the W M F w i t h the w i d e l y used square root-raised cosine ( S R C ) filter. T h e root-raised cosine filter is a v'Nyquist filter o b t a i n e d b y s p l i t t i n g the raised cosine filter i n t o two parts, w i t h the response HsRc{f) frequency of each p a r t b e i n g the square-root of the o r i g i n a l as [37] for | / | < = < - s f HSRCU) ( | / | - ^ ) f o r ^ < | / | < < ^ > . 0 for | / | > (3-21) ^ T h e c o r r e s p o n d i n g i m p u l s e response is given b y 4 p ( t / T ) cos (TT(1 + p)t/T) h s R c { t ) = + sin (TT(1 - p)t/T) 7Tt(i-(4 t/m/T P ( 3 - 2 2 ) where p denotes the roll-off factor of the raised cosine filter. T h e roll-off factor (0 < p < 1) determines the excess b a n d w i d t h o c c u p i e d b y the s i g n a l b e y o n d the N y q u i s t frequency 1/(2T). In a d d i t i o n , we consider an i m p o r t a n t issue i n the i m p l e m e n t a t i o n of a finite i m p u l s e response i n the following. T o o b t a i n a n i d e a l frequency response, the i m p u l s e response has t o be infinitely long. P r a c t i c a l l y , a n infinite i m p u l s e response cannot be i m p l e m e n t e d , a n d , thus, needs to be t r u n c a t e d . However, t r u n c a t i n g i n the t i m e d o m a i n leads to overshoots a n d ripples i n the frequency d o m a i n , w h i c h degrade the perfor- mance, p a r t i c u l a r l y i n the presence of interference. T o reduce these undesirable effects, the i d e a l i m p u l s e response m a y be m u l t i p l i e d b y a s u i t a b l e w i n d o w f u n c t i o n w h i c h 45 3.2 Filter Design 1 ^Hamming window 0.8 T 0.6; 0.4 5 0.2 I 0 -°- 6 2 0 (a) Figure 3.4: Impulse response of SRC, h (t), before Hamming window (a), and after SRC 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 w{t) = <( 0.54 + 0.46cos(27rf/ATjj) . - X (N ) L 0 where X (N ) L H H < t < Xu{N ) H (3.23) elsewhere and X (N ) L represent the upper limit and lower limit of the Ham- H ming window, respectively, and the length of the Hamming window is denoted by NH. X {N ) L H = ^ and Xu(N ) H = ^ if N H is odd, and X (N ) L H = X (N ) V H = 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 W M F and the SRC. This requires the multiplication of the W M F and SRC filter in the time domain by the Hamming window given by (3.23). Fig. 3.5 depicts 46 3.2 Filter Design 10 -WMF, h = 0.28| -WMF, h = 0.35 - SRC, p = 0.3 5 0 1 - 5 5f =| -10 6oT 8 -15 -20 -25 -30 -0.5 0 0.5 rr- Figure 3.5: Magnitude frequency response for W M F , #WMF(/), assuming h = 0.28 and h = 0.35, and for SRC filter, # SRC ( / ) , with p = 0.3. the magnitude frequency response of the W M F (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 W M F for different h. Moreover, while closely resembling the W M F in the passband, the SRC also accomplishes strong outof-band interference suppression. Since the SRC filter is widely used in practice with hardware implementations available, we adopt /ISRC(^) receiver. a s input filter for the proposed 47 3.3 Discrete-time Transmission Model 3.3 Discrete-time Transmission Model The discrete-time received signal after filtering with /ISRC(0 a n d symbol-rate sampling can be well-approximated by r{k}=r(t)*h (t)\t=kT (3.24) SRC Qh = e> m h [l}b[k c - I] + i[k] + n[k] (3.25) with the discrete-time channel impulse response (CIR) h [k] = P(t) * h c S R C (t)\ (3.26) t = k T of order q . n[k] is the sampled noise signal, which is still AWGN, and i[k] represents h 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)\ =kTt Through Laurent's representation (3.15), the memory of the C P M manifests itself as intersymbol interference (ISI), and h [k] is referred to hereafter as the ISI channel, c 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 /i|7]6[fc — I] can assume p qh+l c 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' r[k] jm - J2 hc[l]b[k 1=0 [ k ] - m \ - I] (3.27) 48 3.4 Reduced-State Sequence Estimation 0.6 T g 0.4 0.2 1 (a) T £ °- t 4 o 0.2 (b) Figure 3.6: The CIR of the (a) ISI channel h [k] compared to (b) its minimum phase c equivalent h [k]. 0 which may be easily derived from (2.20) and represents the Euclidean distance between the received signal e~ ^r[k] and the hypothetical transmitted signal y[k\. Observing j(t> Fig. 3.6 (a) which depicts the significant taps of the overall channel h [k] (3.26), we c can see that h [k] is of order q = 2. Taking into account that the appropriate p for c h 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 technique used to perform sequence estimation with a reduced-complexity VA. It provides a good tradeoff between performance and complexity by combining the high perfor- 49 3.4 Reduced-State Sequence Estimation mance of MLSD with the low complexity of D F E . 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 n that controls the number of states in the VA, and, as a result, the s 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 n of hypothetical data symbols a[k] as s a[k] = [a[k] ...a[k-n + 1]] . a (3.28) Therefore, the number of states required for the VA is reduced to 2 , which is indens pendent of p. A vector b[k] = [b[k — n ] . . . b[k — qn + 1]] of estimated symbols b[k] s is associated with each state (b[k] = [b[k — n ]] if n s > qh)- The symbols b[k] are s 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 n , such that one part is associated with the hypothetical data symbols a[k] s and the other retrieved from the survivor sequence as J2 [k] = ]T h [l]b[k - 1} + y c 1=0 c[l)b[k - I] h (3.29) l=n +l s where b[k -l]= b[k - n - 1] exp (jirh a[n] ) . s \ n=k—n s (3.30) / In other words, a state at time k does not carry all the necessary symbols, and the remaining (q^ — n ) required symbols b[k] are determined from the surviving sequence. s 50 3.5 Decision-Feedback Equalizer Structure 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 = d[k] X[k} - (3.31) h [l]b[k - I) c 1=0 where .Qh d[k] = e- * r[k] j [k] - c[l}b[k - 1} . (3.32) h (=n +l s The RSSE branch metric (3.31) is controlled by the design parameter n and the state s trellis (3.28) is independent of p . It allows the equalizer to vary from a full VA (n > qh) s to the simplest case of a DFE for n = 0. The DFE structure will be illustrated in the s following section. 3.5 The Decision-Feedback Equalizer Structure 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 D F E 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) f F = [/F[0], /F[1], /F[<?F]] of order q which suppresses the F contribution of the precursor ISI. To cancel the ISI in the present symbol, the feedback filter (FBF) f B and = [/B[0], /S[1], the output of the FBF /B[<7B]] of order q uses previously-detected symbols, B is then subtracted from the output of the FFF. Assuming correct previous decisions, the interference from previously detected symbols is removed 51 3.5 Decision-Feedback Equalizer Structure Decision device FFF a[k — ko] FBF Figure 3.7: Block diagram of decision-feedback equalizer. through the F B F , and the constant k , known as the decision delay, specifies the number 0 of future measurements processed before any decision is made on the present symbol. The input z[k] to the decision device may be expressed as QF QB = J2 fF[l]r[k 1=0 2 f [l]I[k B - Z - fed] - f [0]I[k B fco] (3.33) 1=1 where / is the vector of previously estimated symbols. Since the D F E operates under the assumption that the past decisions are correct, which is almost always violated, the D F E is evidently a suboptimum equalization scheme. The coefficients of the F F F and F B F filters obviously have a great impact on the D F E performance. Several techniques for computing the F F F and F B F coefficients were proposed in literature, of which a comprehensive review was given in [7]. In the present work, we adopt the simple D F E structure considering both coherent and noncoherent detection and compute the coefficients of the corresponding filters as outlined in the next section. 3.5.1 M i n i m u m 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, D F E . The finite impulse Usually a threshold decision is made, where the input to the decision device does not include the 2 last term on the right hand side of (3.33). 52 3.5 D e c i s i o n - F e e d b a c k E q u a l i z e r S t r u c t u r e response ( F I R ) transfer f u n c t i o n H (z) of a m i n i m u m phase c h a n n e l is such t h a t min H m{ ) z n a m s roots o n l y inside the u n i t circle. A m i n i m u m phase c h a n n e l /i i [fc] has the m n fastest decay, e n s u r i n g t h a t the energy c o n c e n t r a t i o n i n the first samples is m a x i m i z e d (3.34) where the /i j [fc] is the m i n i m u m phase equivalent of h[k] a n d lh is the l e n g t h of m n the C I R . Since the d e c i s i o n . i n a D F E l a r g e l y depends o n the first t a p (3.33), it is o b v i o u s t h a t a m i n i m u m - p h a s e c h a n n e l is desirable. Ideally, a m i n i m u m phase c h a n n e l is p r o d u c e d b y a p p l y i n g a discrete-time prefilter f[k] w i t h a n allpass characteristic to the s a m p l e d received s i g n a l r[k]. T h e m i n i m u m phase equivalent of the B l u e t o o t h ISI c h a n n e l h [k] (3.26) is o b t a i n e d b y prefiltering it b y f[k] c as (3.35) h [k] * f[k] c where the m i n i m u m phase equivalent, r e s u l t i n g i m p u l s e response. h [k], 0 represents the last qh + 1 taps of the T h e allpass filter f[k] the o v e r a l l m a g n i t u d e response, p r o d u c i n g a n alters the phase w i t h o u t A W G N affecting noise sequence at the o u t p u t . I n order not to change the m a g n i t u d e frequency response, the all-pass filter moves the r o o t s of the F I R transfer f u n c t i o n H (z) c of h [k] w h i c h are outside the u n i t circle s i m p l y c b y reflecting t h e m a b o u t the u n i t circle, r e s u l t i n g i n the r o o t s of its m i n i m u m phase HQ(Z), corresponding as i l l u s t r a t e d i n F i g . 3.8. F o r a p r a c t i c a l i m p l e m e n t a t i o n , a finite-impulse response ( F I R ) a p p r o x i m a t i o n of the allpass filter is of interest. A comprehensive review of the state-of-the-art prefiltering strategies is given i n [20]. In this research w o r k , the c o m p u t a t i o n of the prefilter is based o n linear p r e d i c t i o n , as was presented i n [20]. T h i s prefilter consists of the cascade of a m a t c h e d filter t o the c h a n n e l a n d a p r e d i c t i o n error filter, w h i c h is c a l c u l a t e d v i a the L e v i n s o n - D u r b i n a l g o r i t h m . It was f o u n d to be a n efficient prefiltering m e t h o d for the ISI c h a n n e l h [k] c i n a d d i t i o n to h a v i n g l o w c o m p u t a t i o n a l c o m p l e x i t y . T h e prefilter 53 3.5 D e c i s i o n - F e e d b a c k E q u a l i z e r S t r u c t u r e 1.O 0 Roots of H (z) x Roots of H (z) c 1 Q \ 0.5 0 i •O--: : ! ®* -0.5 -1 H C -5 -4 -3 - 2 - 1 Real Part 0 1 F i g u r e 3.8: T h e roots of H (z) c o m p a r e d t o t h e roots of i t s m i n i m u m phase equivalent c H (z). 0 is designed t o be of order qp = 6, a n d was found t o s l i g h t l y v a r y w i t h h, b u t o n l y insignificant performance differences i n c u r i f i t is designed for a fixed h. T h e overall c h a n n e l h [k] is of order q 0 B = 2 since i t is of t h e same l e n g t h as h [k], as was m e n t i o n e d c earlier, a n d is s h o w n for a fixed h = 1/3 i n F i g . 3.6 (b). 3.5.2 D F E Decision Rule F r o m the p r e v i o u s discussion, it is clear t h a t i n a D F E s t r u c t u r e , f[k] w o u l d serve as a n effective F F F , w h i l e h [k] w o u l d be its c o r r e s p o n d i n g F B F , a n d , hence, i n t h e present 0 work, we e m p l o y f[k] as t h e F F F a n d h [k] as t h e F B F i n t h e p r o p o s e d D F E structure. 0 T o further i l l u s t r a t e t h e role of t h e F B F i n r e d u c i n g t h e effect o f t h e ISI caused b y p r e v i o u s s y m b o l s , we m a y r e w r i t e r[k] * f[k] as r[k] * f[k] = e * l % [ 0 ] (b[k] + - p where t h e t e r m h [l]b[k - l]j 0 + i[k] * f[k] + n[k] * f[k] (3.36) QB J2h [l]b[k 0 represents t h e i n t e r s y m b o l interference. - I] A s s u m i n g coherent d e t e c t i o n , the D F E deci- 54 3.6 Noncoherent Detection sion rule reads ko}}} (3.37) ko-l]. (3.38) a[k — k ] = argmax < §R{d [A;']/io[0]6[/c 0 DFE a\k—ho] where QB d [k) DFE = e-^r[k] * f[k] -J2h [l]b[k 0 We may notice that the F B F subtracts out the ISI term from the decision. This decision rule corresponds to the branch metric (3.31) with n — 0. We have to note here that s for the proposed receiver the ISI channel is prefiltered to obtain its minimum phase equivalent only in case of D F E although, theoretically, a minimum phase channel is essential for RSSE as well. However, since the ISI channel h [k] only has 3 significant c 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 D F E . 3.6 Noncoherent Detection So far, we have successfully employed coherent detection using RSSE and D F E , 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] k k \[k] = ]T ( | r M | + \y[v}\ ) ~ 2 £ I W M I 2 2 • (3.39) 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, 55 3.6 N o n c o h e r e n t D e t e c t i o n C o l a v o l p e a n d R a h e l i [11] proposed r e d u c i n g t h e b r a n c h m e t r i c m e m o r y b y t r u n c a t i n g the w i n d o w of observations t o a finite value N, N > 2. Hence, (3.39) becomes k k X[k}= I M - 2 E 2 v=k-N+l The l « > ] l (3-40) - v=k-N+l t e r m |r[t>]| was e l i m i n a t e d i n (3.40) since i t is independent of t h e decision. A 2 noncoherent D F E ( N D F E ) was further derived from (3.40) i n [45], w i t h t h e b r a n c h m e t r i c , n o w s u b o p t i m u m , given b y k ANDFE[&] k = E I 2 / D F E H | 2 - 2 v=k-N+l where roFE[fe] (3-41) I ^ D F E H J / D F E M I v=k-N+l is the o u t p u t of the F F F a n d T / D F E ^ ] is t h e o u t p u t of t h e F B F . U s i n g the F B F a n d F F F r e c o m m e n d e d i n S e c t i o n 3.5, rcFE[fe] and j/DFE[fe] at t i m e k c a n be f o r m u l a t e d as QF rDFE[k] = ^f[l]r[k-l] (3.42) 1=0 and QB y E[fc] = 5 > ° [ » - f c o - Z ] (3.43) DF (=0 where t h e c o m p l e x s y m b o l s b[k — ko] are c a l c u l a t e d as (3.17). Hence, (3.41) becomes fe-i A N D F E W = E QB I ^ D F E M I 2 h [l]b[k -ko-l] + v=k-N+l 0 + ho[0]b[k - k ] 0 (3.44) i=i -2\C[k-l] + r [k}y* [k}\ DFE DPE where &-!]= A:-l E (3.45) ^ F E M ^ P E M . v=k-N+l QB Here, _ yr)FE[k] = Y^/ h [l}b[k — ko — l] + ho[0]b[k — k ], b[k — k — 1} o o 0 represent t h e p r e v i o u s l y - i=i detected s y m b o l s , w h i l e b[k — k ] is t h e h y p o t h e t i c a l c o m p l e x s y m b o l . T h e phase of 0 q^, [k — 1] { constitutes t h e estimate of the phase d i s t o r t i o n caused b y t h e channel, where the n u m b e r of phase reference c o n t r i b u t i o n s is c o n t r o l l e d b y t h e finite value N, a n d , 56 3.6 Noncoherent Detection 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 q^\k - 1] = aq^[k - 2] + r DFE (3.46) [/c - l ] & [ f c - 1] F E where the design parameter a, 0 < a < 1, acts as a forgetting factor, and q^ [k — 1] is ef 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? [k — 1] is referred to as the A^-metric, ei while that associated with q" [k — 1] is referred to as the a-metric. The performance of eS the N- and the a-metric was found to be approximately equivalent, precisely according to the following equation [46] (3.47) 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 involves estimating the channel phase is applied. However, in contrast to the N D F E , the 57 3.6 Noncoherent Detection 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\q? [k - 1] + r[k]y*[k]\ (3.48) q^[k - 1] - aq^[k - 2] + r[k - l]y*[k - 1] (3.49) 2 ef where and y[k] is calculated as for coherent RSSE (3.29). Therefore, similarly, the NRSSE decision rule is controlled by the design parameter n . Also similar to coherent RSSE, s prefiltering is not employed, as was explained in Section 3.5.2. 3.6.1 N D F E 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 N D F E decision rule expressed as b[k - k ] = argmin{|d 0 [/c] - h [0)b[k - k }\ } (3.50) 2 NDFE 0 0 b[k-k ] 0 = argmax m{d*i [k]h [0}b[k-k }}\ DFE Q (3.51) 0 6[fc-fco) where QB W W r [k] - J2h [l}b[k -k -l]. = DFE l9ref[fc-l]| Qref[k — 1] m a 0 i = (3.52) 0 1 y 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 N D F E 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 n = 0 (since this is a DFE), and with e~ Wr[k] replaced j,p s 1» WEf 58 3.6 Noncoherent Detection 3.6.2 Frequency Offset Estimation The N D F E 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 q ([k — 1] in re (3.46), the modified update equation reads q [k] = (aq [k - 1] + r Te( re{ DFE where the frequency offset estimate e> ^ \ 2wA Pref[fc] = PPretik ~ 1] + k [fc]y [fc]) e ^ ^ " ' , 1 DFE = p f[fc]/|Pref[k]| re '"DFE[k]2/DFEM('"DFE\k - l]y DFE (3.53) follows from [fc ~ 1])* • (3-54) Similar t o a , / 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 N D F E 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 59 3.6 N o n c o h e r e n t D e t e c t i o n as the F B F a n d F F F filters (cf. S e c t i o n 3.5) t u r n e d out t o be almost unaffected b y v a r y i n g h, considerable d e v i a t i o n s of the decision m e t r i c s w i l l o c c u r even for s m a l l m i s m a t c h e s between the m o d u l a t i o n i n d e x h assumed at the receiver a n d the a c t u a l h at the t r a n s m i t t e r . T h i s is evident f r o m the f o r m u l a t i o n of the c o m p l e x s y m b o l s given b y (3.30)'. Hence, even i n case of a correct e s t i m a t e d b i n a r y s y m b o l a[k], due to the m i s m a t c h i n h, the e s t i m a t e d c o m p l e x s y m b o l w i l l be incorrect. W e m a y assume a n o m i n a l value h at the receiver regardless of the m o d u l a t i o n i n d e x ' s a c t u a l value h. However, results presented i n C h a p t e r 4, show significant performance d e g r a d a t i o n since h at the t r a n s m i t t e r is allowed to v a r y i n a r e l a t i v e l y large i n t e r v a l (0.28 < h < 0.35). T h i s is especially the case for large values of a a n d (3 since the errors a c c u m u l a t e d f r o m previous estimates are of greater weight. T o account for this p r o b l e m , we propose to test a set 7i of m o d u l a t i o n i n d e x h y p o t h e ses h, a n d , after a n e s t i m a t i o n p e r i o d , N , e to select the "best" one. W e c a l c u l a t e \H\ N D F E a c c u m u l a t e d p a t h metrics, each for a t o t a l of N symbols. e A p r a g m a t i c ap- p r o a c h to d e t e r m i n e the best a l t e r n a t i v e is to decide for the h t h a t yields the smallest a c c u m u l a t e d p a t h m e t r i c . Therefore, the performance as well as the c o m p l e x i t y of this h a d a p t a t i o n depends on the n u m b e r of hypothesis a n d the l e n g t h of the e s t i m a t i o n period. W e e m p l o y t h i s a d a p t a t i o n technique t o the noncoherent decision feedback equalizer presented i n the previous sections, a n d the r e s u l t i n g receiver is referred t o as a d a p t i v e N D F E ( A N D F E ) . T h i s a d a p t a t i o n process is a p p l i e d t o the first packet received, a n d the e s t i m a t e d m o d u l a t i o n i n d e x used to detect the rest of the s y m b o l sequences, h is o b t a i n e d b y the following decision rule (3.55) C l e a r l y , the greater the set H a n d the e s t i m a t i o n p e r i o d N , the better the performance e of A N D F E . F r o m the results given i n C h a p t e r 4, we m a y deduce t h a t four m o d u l a t i o n i n d e x hypotheses a n d e s t i m a t i o n over 50 s y m b o l intervals is q u i t e sufficient to o b t a i n 60 3.7 S u m m a r y of P r o p o s e d Receiver S t r u c t u r e r(t)- Hammingwindowed SRC k] ~T=^kT [r[0],...,r[N -l\ e h- Estimator UJ LL Q F i g u r e 3.9: S t r u c t u r e of t h e p r o p o s e d B l u e t o o t h receiver. a performance close t o t h e case of h b e i n g perfectly k n o w n t o t h e receiver. W i t h these parameters, t h e c o m p l e x i t y increase of A N D F E c o m p a r e d t o N D F E is negligible, b u t results show significant performance i m p r o v e m e n t s under v a r y i n g h c o n d i t i o n s . 3.7 Summary of Proposed Receiver Structure T h e structure of the B l u e t o o t h receiver designed i n t h e present w o r k is s h o w n i n F i g . 3.9. T h e p r o p o s e d receiver is based o n L a u r e n t ' s d e c o m p o s i t i o n w h i c h presents the C P M s i g n a l as a s u p e r p o s i t i o n of P A M pulses (3.10). W e adopt t h e a p p r o x i m a t i o n w h i c h expresses t h e B l u e t o o t h G F S K s i g n a l i n terms of one ' m a i n pulse'. T h e r e c o m m e n d e d receiver i n p u t filter is t h e S R C filter (3.21), w h i c h is a p r a c t i c a l choice t h a t h i g h l y resembles t h e o p t i m a l W M F (3.20). T o increase interference suppression, a H a m m i n g w i n d o w (3.23) is a p p l i e d t o t h e S R C filter. T h e o u t p u t of t h e filter is s y m b o l p e r i o d s a m p l e d a n d fed i n t o a s i m p l e noncoherent decision-feedback equalizer, t h r o u g h w h i c h phase e s t i m a t i o n is e m p l o y e d as given i n (3.51). F u r t h e r m o r e , for r o b u s t detect i o n i n t h e presence of h i g h frequency offsets, frequency offset e s t i m a t i o n is e m p l o y e d u s i n g (3.53). M o r e o v e r , d u e t o t h e v a r y i n g m o d u l a t i o n i n d e x (0.28 < h < 0.35) i n B l u e t o o t h systems, we propose u s i n g A N D F E (3.55), where a n estimate of t h e m o d u l a t i o n i n d e x of t h e t r a n s m i t t e d s i g n a l is given. T h e chosen m o d u l a t i o n i n d e x is t h e one w h i c h m a x i m i z e s t h e p a t h m e t r i c over a l i m i t e d n u m b e r of samples N , (25 < N < 50), a n d e e is selected f r o m a predefined set of hypotheses, H, (\H\ = 2 , 4 ) . T h i s e s t i m a t e d h is t h e n used t o detect t h e rest of t h e received s i g n a l , a n d t h e final o u t p u t is t h e sequence 3.7 Summary of Proposed Receiver Structure of estimated binary symbols d. 61 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 B E R of 1 0 -3 since this is the required B E R 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 sampling start point to get the best performance possible using the adopted SRC filter. We 62 63 4.1 Coherent Detection 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 D F E . In case of D F E , the F F F and F B F are of orders q F = 6 and q B =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/N 0 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 B E R = 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 W M F and SRC filters was shown in Chapter 3. Furthermore, in this section, to support the decision of selecting 64 4.1 Coherent Detection 6 8 10 12 14 1 0 l o g ( E / N ) [dB]-> 10 b 18 20 0 Figure 4.1: Performance comparison of the SRC and W M F 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 W M F [Eq. (3.20)]. In Fig. 4.1, we show the BER as a function of the SNR for RSSE with 4 states and for D F E , using, in both cases, the W M F and the SRC, respectively. For comparison, the B E R curves for the MLSD bound and the LDI receiver are also included. We notice that for the 4-state RSSE, as well as the D F E , the curves for the SRC filter almost coincide with those of the W M F . 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 D F E , there is ~ 0.01 dB performance improvement when the W M F is used. Ideally, of course, the optimal W M F 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. 65 4.1 Coherent Detection 10" ! y i !• 10 10 ' DC LU m 10" 10" • x MLSD bound LDI receiver one-state NSD -a—DFE -e— 2-state RSSE -e>— 4-state RSSE 10 6 8 10 12 14 16 10log (E /N ) [dB]-> 10 b 18 20 22 24 0 Figure 4.2: Evaluation of the state reduction in the proposed receiver. Coherent detection 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 B E R curves in Fig. 4.2 for 2-state RSSE [Eqs. (3.31)-(3.32)], 4-state RSSE and D F E [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 D F E incurs a loss of ~ 0.7 dB compared to the 4-state RSSE, and less than 0.6 dB compared to the 2state 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. 66 4.2 N o n c o h e r e n t D e t e c t i o n 12.2 a = 0.8, FBF and FFF set according to h at transmitter 10.4 10.2 0.28 F i g u r e 4.3: 0.29 0.3 0.31 0.32 h at transmitter -> 0.33 0.34 0.35 P e r f o r m a n c e of N D F E w i t h a = 0.8 w i t h F B F a n d F F F fixed at the receiver, or v a r y i n g a c c o r d i n g t o the value of h at the t r a n s m i t t e r . S i m i l a r results are e x p e c t e d i n case of noncoherent D F E , w h i c h w i l l be considered i n the following sections. 4.2 Noncoherent Detection In t h i s section, we evaluate the performance of the p r o p o s e d receiver i n case of n o n coherent d e t e c t i o n , where the c h a n n e l phase is e s t i m a t e d u s i n g the phase reference. W e briefly discuss a n d d e m o n s t r a t e the performance of N R S S E [Eqs. (3.48)-(3.49)], a n d focus o n the performance of N D F E [Eqs. (3.51)-(3.52)], w h i c h constitutes the final receiver design (cf. S e c t i o n 3.7). T h e results for M L S D , L D I , a n d coherent D F E are i n c l u d e d for c o m p a r i s o n purposes. T h e m o d u l a t i o n i n d e x h = 1/3 a n d is k n o w n at the receiver unless stated otherwise. 67 4.2 N o n c o h e r e n t D e t e c t i o n 10 LU m •-*-• MLSD • bound LDI — e — 4-state N R S S E , a = 0.8 —±— 2 - s t a t e N R S S E , a = 0.8 — e — N D F E , a = 0.8 - A - 2-state N S D , a = 10 6 0.8 8 10 12 1 0 l o g ( E / N ) [dB]-> 10 b 14 16 18 20 0 F i g u r e 4.4: E v a l u a t i o n of the performance of the p r o p o s e d receiver w i t h state r e d u c t i o n i n case of noncoherent detection. 4.2.1 Performance with Constant Channel Phase In t h i s section, the channel phase is assumed to be t i m e - i n v a r i a n t . T h e performance under this c o n d i t i o n is evaluated w i t h state r e d u c t i o n , a n d for different values of N a n d a . M o r e o v e r , we discuss the effect of the v a r y i n g m o d u l a t i o n i n d e x at the t r a n s m i t t e r c o m p a r e d to the results o b t a i n e d w h e n h is k n o w n at the receiver. W e r e m i n d the reader t h a t the results presented i n this w o r k assume the F B F a n d F F F filters are fixed a c c o r d i n g t o h = 1/3. O n l y negligible, i f any, performance loss i n c u r s c o m p a r e d t o the case where h is v a r i e d a c c o r d i n g to the a c t u a l value at the t r a n s m i t t e r , as i l l u s t r a t e d i n F i g . 4.3. State Reduction W e p r e v i o u s l y found t h a t the D F E suffers fairly acceptable performance losses w i t h respect to the 2-state a n d 4-state R S S E . In this section, we i l l u s t r a t e t h a t a s i m i l a r result is o b t a i n e d i n case of noncoherent d e t e c t i o n a n d c o m p a r e the performance of the 68 4.2 N o n c o h e r e n t D e t e c t i o n p r o p o s e d receiver to the L D I a n d the N S D . F i g . 4.4 shows t h a t N R S S E a n d N S D y i e l d s i m i l a r performance. I n fact, the B E R curve of the 2-state N R S S E a l m o s t coincides w i t h t h a t of the 2-state N S D . M o r e o v e r , we notice t h a t N D F E suffers a performance loss of a b o u t 0.3 d B a n d 0.4 d B c o m p a r e d t o 2-state N R S S E a n d 2-state N S D , respectively, a n d achieves a performance g a i n of a b o u t 5.5 d B over the L D I . A s further state r e d u c t i o n t u r n e d out i m p o s s i b l e for the N S D a p p r o a c h of [27] based on R i m o l d i d e c o m p o s i t i o n , we c a n deduce t h a t L a u r e n t ' s d e c o m p o s i t i o n is preferable for the design of l o w - c o m p l e x i t y B l u e t o o t h receivers. F i n a l l y , we conclude t h a t N D F E provides a n excellent tradeoff between performance a n d c o m p l e x i t y , m a k i n g it the i d e a l equalizat i o n scheme for the p r o p o s e d receiver. H e n c e , the following discussion w i l l focus o n the performance e v a l u a t i o n of N D F E . Performance w i t h Different Values of N a n d a S t a r t i n g w i t h the N - m e t r i c (3.45), the B E R curves for N D F E w i t h different values of N are s h o w n i n F i g . 4.5. A s expected, w i t h o u t c h a n n e l phase v a r i a t i o n s , as the value of N increases, the performance closely approaches t h a t of coherent D F E . W e also observe t h a t even for the m i n i m u m value of N = 2, we have a performance i m p r o v e m e n t of m o r e t h a n 2.5 d B over the L D I receiver. F i g . 4.6 shows the performance u s i n g the a - m e t r i c . W i t h a = 0.95, the performance difference between coherent D F E a n d N D F E is a negligible 0.06 d B . S i m i l a r to N, as a increases, we get better power efficiency M o r e o v e r , r e c a l l i n g E q . (3.47), w h i c h compares the performance of the N D F E u s i n g the N- a n d a - m e t r i c , it m a y be verified b y a careful o b s e r v a t i o n of F i g s . 4.5-4.6. Specifically, we see t h a t the performance for N = 10 is equivalent to t h a t of a = 0.8, a n d a s i m i l a r statement is t r u e for N = 2 , 5 a n d a = 0,0.6, respectively. However, since the a - m e t r i c is c o m p u t a t i o n a l l y less c o m p l e x (cf. S e c t i o n 3.6), the following discussion w i l l be solely based o n the a - m e t r i c . 69 4.2 N o n c o h e r e n t D e t e c t i o n 6 8 10 12 14 10log (E /N )[dB]^ 10 b 0 F i g u r e 4.5: P e r f o r m a n c e of the N D F E u s i n g the A^-metric c o m p a r e d to M L S D , L D I , a n d coherent D F E , a n d h = 1/3 for a l l cases. 6 8 10 12 14 10log (E /N )[dB]^ 10 b 0 F i g u r e 4.6: P e r f o r m a n c e of the N D F E u s i n g the a - m e t r i c c o m p a r e d t o M L S D , L D I , a n d coherent D F E , a n d h = 1/3 for a l l cases. 70 4.2 N o n c o h e r e n t D e t e c t i o n 10 f 10" m •o ~ 1 0 " ^ LU CD O 10" o rx LU m 10 "~' ~.frr7 ~'.".T'.fri'. ~ \/~'.'fr:':~r~.'. ~. :frr. . "fr : MLSD bound 4 o 5 A 10 - F i g u r e 4.7: P e r f o r m a n c e of the N D F E u s i n g the TV-metric h — 1/3 i n the presence of phase j i t t e r . F i g u r e 4.8: P e r f o r m a n c e of N D F E u s i n g the cv-metric h = 1/3 i n the presence of phase jitter. 71 4.2 N o n c o h e r e n t D e t e c t i o n A s m e n t i o n e d above, increasing the value of a a n d N achieves better performance w h e n the c h a n n e l phase is constant. However, this is not the case i n the presence of phase j i t t e r , as c a n be seen i n F i g . 4.7 w h i c h p l o t s the B E R w i t h respect t o the s t a n d a r d d e v i a t i o n of the phase j i t t e r a&. In t h i s case, the performance of the p r o p o s e d receiver deteriorates as N increases since the a s s u m p t i o n of constant c h a n n e l phase is no longer v a l i d . A s i m i l a r o b s e r v a t i o n c a n be m a d e w h e n u s i n g the a - m e t r i c , as s h o w n i n F i g . 4.8. Therefore, the o p t i m a l value of N or a depends o n the c h a n n e l c o n d i t i o n s , a n d we m a y c o n c l u d e t h a t N D F E w i t h N between 5 a n d 10 or a between 0.6 a n d 0.8 is power efficient a n d gives satisfactory performance i n the presence of phase j i t t e r . Performance with Different Modulation Indices In the p r e v i o u s sections, we have assumed h = 1/3 at the t r a n s m i t t e r a n d receiver. However, since the G F S K s i g n a l depends o n h [Eqs. (2.1)-(2.2)], even w i t h the ass u m p t i o n t h a t h is k n o w n at the receiver, different performance results are o b t a i n e d for different m o d u l a t i o n indices. Specifically, the performance deteriorates as h de- creases since fd (cf. S e c t i o n 2.1.4) decreases. F i g . 4.9 shows the performance of N D F E w i t h a = 0.8 for h = 0.28, 1/3, a n d 0.35, a l o n g w i t h the c o r r e s p o n d i n g M L S D b o u n d s , the L D I , a n d the M L M - L D I . T h e performance of N D F E , w h i c h is equivalent to a 1 one-state t r e l l i s gives r e m a r k a b l e results w h e n c o m p a r e d t o the M L S D b o u n d w i t h a performance loss of o n l y a b o u t 1.2 d B . R e c a l l t h a t the o p t i m u m receiver (cf. S e c t i o n 2.2.1) requires 100 trellis states for h = 0.28 a n d 80 t r e l l i s states for h = 0.35, w h i l e the o p t i m a l receiver based o n L a u r e n t ' s d e c o m p o s i t i o n (cf. S e c t i o n 3.1.2) requires 50 a n d 40 states for h = 0.28 a n d h = 0.35, respectively. T h e performance gains c o m p a r e d to the L D I a n d M L M - L D I are more t h a n 5 d B a n d 1 d B , respectively. J T h e results for the M L M - L D I are taken from [49]. 72 4.2 N o n c o h e r e n t D e t e c t i o n 10 CE UJ CD MLSD bound NDFE with a = 0.8| LDI MLM-LDI h = 0.28 h = 1/3 h = 0.35 * • • 10 2 4 6 8 10 10log (E /N ) 10 b 0 12 [dB]-» 14 16 18 20 F i g u r e 4.9: P e r f o r m a n c e of the N D F E w i t h a = 0.8 for different values of the m o d u l a t i o n i n d e x c o m p a r e d to the M L S D b o u n d , L D I , a n d M L M - L D I . Performance with Unknown Modulation Index So far, we have assumed the m o d u l a t i o n i n d e x h to be k n o w n at the receiver. I n B l u e t o o t h devices, the v a r y i n g m o d u l a t i o n i n d e x (0.28 < h < 0.35) poses a serious challenge for the receiver design. T o s u p p o r t this statement, we show the B E R o b t a i n e d for a n S N R of 11 d B for different values of a = 0.4, 0.6, 0.8, a n d 0.9 w h e n a fixed m o d u l a t i o n i n d e x h = 0.28, 0.30, 1/3, a n d 0.35 is assumed at the receiver. T h e s i m u l a t i o n s are performed for different values of the a c t u a l m o d u l a t i o n i n d e x of the t r a n s m i t t e d s i g n a l , a n d the o b t a i n e d results are shown i n F i g . 4.10. It is clear t h a t as h at the receiver deviates from the a c t u a l h of the t r a n s m i t t e d s i g n a l , the performance degrades. We also notice t h a t the performance loss w i t h the h d e v i a t i o n is more severe for greater values of a. T h i s is due t o the fact t h a t a m i s m a t c h of h at the receiver takes longer t o forget i n case of a greater forgetting factor a. Hence, a lower value of a provides more robustness against v a r i a t i o n s i n the m o d u l a t i o n index. C o n s i d e r , for e x a m p l e , a = 0.4, a n o m i n a l value of h at the receiver w o u l d result i n v e r y slight changes from the case 73 4.2 N o n c o h e r e n t D e t e c t i o n • h = 0.30 • h = 0.28 - h = 1/3 - h known at receiver • h = 0.35 a = 0.4 0.28 0.29 0.3 0.31 0.32 0.33 h at transmitter -> a 0.28 0.29 0.34 0.35 0.28 0.29 0.3 0.31 0.32 0.33 h at transmitter - » a = 0.8 0.3 0.31 0.32 0.33 h at transmitter -> 0.34 0.35 0.28 0.29 0.34 0.35 =0.9 0.3 0.31 0.32 0.33 h at transmitter - » 0.34 0.35 F i g u r e 4.10: P e r f o r m a n c e of the N D F E for v a r y i n g h w i t h a = 0.4, 0.6, 0.8, a n d 0.9. h = 0.30 at receiver h = 0.28 at receiver T m ^ 10 V - NSD, a = 0.6 •V NDFE, a = 0.6 - V — NDFE, a = 0.8 m 10 C 0.28 T 0.29 0.3 0.31 0.32 0.33 h at transmitter - » 0.34 0.35 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.34 0.35 h at transmitter -> h = 0.35 at receiver : 1/3 at receiver 10° m •o £ 10" 10" .O JJ "~o ra o 10" 6 10" o rr LU m 0.28 10" a - •a s— 0.29 NSD, a = 0.6 N D F E , a = 0.6 NDFE, a = 0.8 0.3 0.31 0.32 0.33 h at transmitter -> 0.34 0.35 0.28 0.29 0.3 0.31 0.32 0.33 h at transmitter - » F i g u r e 4.11: P e r f o r m a n c e of N D F E for v a r y i n g h w i t h a = 0.6 a n d 0.8 c o m p a r e d t o t h a t of N S D for a = 0.6. 74 4.2 N o n c o h e r e n t D e t e c t i o n w h e n h is k n o w n at the receiver b u t s t i l l achieve more power efficiency t h a n the L D I . T o evaluate the p r o p o s e d receiver's robustness against h v a r i a t i o n s w i t h respect to the a l t e r n a t i v e noncoherent sequence detector i n l i t e r a t u r e , namely, N S D , we show a relative c o m p a r i s o n i n F i g . 4.11. T h e B E R curves o b t a i n e d for S N R = 11 d B are s h o w n for various m o d u l a t i o n indices h = 0.28, 0.3, 1/3, a n d 0.35 at the receiver, a n d are p l o t t e d for a v a r y i n g h at the t r a n s m i t t e r . W e choose the curves for N S D w i t h a = 0.6 since it was found to be the o p t i m a l value of a for N S D [27]. T h e N S D curves are c o m p a r e d t o of N D F E w i t h a = 0.6 a n d 0.8. T h e presented results show t h a t as h varies, a m u c h greater d e v i a t i o n is observed i n the B E R curves for N S D c o m p a r e d to those of N D F E , b o t h w i t h a = 0.6. C o n s i d e r i n g the case w h e n h = 0.28 at the receiver, as h varies at the t r a n s m i t t e r , the c o r r e s p o n d i n g B E R is almost constant i n case of N D F E , w h i c h is not the case for N S D . Specifically, the r e s u l t i n g percentage of error varies between 0.25% a n d 0.5% i n N D F E , a n d between 0.4% a n d 12% i n N S D . M o r e o v e r , we even observe less change i n the B E R of N D F E for a = 0.8 c o m p a r e d t o N D F E w i t h a = 0.6. W e conclude t h a t N D F E is more r o b u s t against h v a r i a t i o n s c o m p a r e d to N S D . A m o r e representative e v a l u a t i o n of the effect of v a r y i n g m o d u l a t i o n i n d e x is o b t a i n e d by c o n s i d e r i n g the r e q u i r e d S N R for a B E R of 1 0 ~ . 3 as before. W e a d o p t the same scenario F i g . 4.12 shows the required S N R for a B E R of 1 0 ~ as a function of the 3 m o d u l a t i o n i n d e x h for N D F E w i t h a = 0.8 a n d for the M L S D b o u n d a n d the L D I for c o m p a r i s o n purposes. T h e figure also i l l u s t r a t e s the i d e a l case of h k n o w n at the receiver. A s c a n be observed, severe power efficiency losses result w h e n h is u n k n o w n at the receiver. T h e v a r i a t i o n s i n the m o d u l a t i o n i n d e x m a y be m i n i m i z e d b y e s t i m a t i n g h at the t r a n s m i t t e r u s i n g A N D F E described i n S e c t i o n 3.6.3, a n d the c o r r e s p o n d i n g results w i l l be presented later i n this chapter. 75 4.2 N o n c o h e r e n t D e t e c t i o n i 0.29 i 1 i 0.3 i i J 0.31 0.32 0.33 0.34 0.35 h at transmitter —> F i g u r e 4.12: T h e r e q u i r e d 101og (E /N ) for B E R = 1 0 " for N D F E w i t h a = 0.8 3 10 b 0 w i t h v a r y i n g m o d u l a t i o n index. 4.2.2 Performance Using the Modified Phase Reference U n der Constant and Varying Channel Conditions T h e more realistic a n d thus m o r e relevant t r a n s m i s s i o n scenario considers a t i m e v a r y i n g c h a n n e l phase <p(t). I n the present w o r k , we account for c h a n n e l phase v a r i a t i o n s t h r o u g h the m o d i f i e d phase reference [see E q s . (3.53)-(3.54)] f o r m u l a t e d i n terms of t w o forgetting factors a a n d (3. T h r o u g h o u t this chapter, i f q f[k] f r o m (3.46) is ie used, o n l y t h e value of a is specified, a n d i f c7 f[fc] f r o m (3.53) is a p p l i e d , the values of re b o t h a a n d (3 are given. Performance With Constant Channel Phase Before c o n s i d e r i n g a t i m e - v a r i a n t channel phase, we first show the performance of t h e m o d i f i e d phase reference w h e n the c h a n n e l phase is constant. F i g . 4.13 illustrates the performance for different values of a a n d /3 as a f u n c t i o n of the m o d u l a t i o n i n d e x , 76 4.2 N o n c o h e r e n t D e t e c t i o n 0.28 F i g u r e 4.13: 1 0 i o g i 0.29 0 (E /N ) b 0 0.3 0.31 0.32 h at transmitter -> r e q u i r e d for B E R = 1 0 0.33 - 3 0.34 0.35 for different c o m b i n a t i o n s of a a n d (3 as a f u n c t i o n of t h e m o d u l a t i o n i n d e x w i t h constant c h a n n e l phase a n d h k n o w n . w h i c h is assumed t o be k n o w n at the receiver. W e notice t h a t there is a loss i n power efficiency w h e n (3.53) is used c o m p a r e d t o w h e n (3.46) is a p p l i e d d u e t o t h e phase noise i n t r o d u c e d b y t h e frequency e s t i m a t i o n . M o r e specifically, t h e gap between a = 0.6 a n d t h e p a i r (a = 0.6, j3 = 0.9) is a p p r o x i m a t e l y 0.4 d B . T h i s loss for t h e i d e a l case of a constant c h a n n e l phase has t o be accepted t o achieve a h i g h performance i n t h e more realistic scenario of a t i m e - v a r y i n g phase. M o r e o v e r , we observe t h a t as (3 increases t h e performance of q f[k] f r o m (3.53) closely approaches t h a t of q f[k] f r o m (3.46). It was ie Te found t h a t for various scenarios, t h e p a i r ( a = 0.6, (3 = 0.9) a p p e a r e d t o be a favorable choice. W e further observe t h a t t h e robust N D F E w i t h ( a = 0.6, (3 = 0.9) outperforms the L D I detector b y m o r e t h a n 4 d B , a n d i t is s t i l l s o m e w h a t m o r e power efficient t h a n t h e M L M - L D I detector, w h i c h requires a four-state f o r w a r d - b a c k w a r d a l g o r i t h m . F i n a l l y , we note t h a t N D F E w i t h frequency-offset e s t i m a t i o n performs very s i m i l a r t o the more c o m p l e x N S D w i t h t h e same values of a a n d (3. 77 4.2 N o n c o h e r e n t D e t e c t i o n Performance with Varying Modulation Index A s for the effect of the (3 factor o n the v a r y i n g m o d u l a t i o n i n d e x , the r e s u l t i n g performances of N D F E w i t h (a = 0.6,(3 = 0.9) a n d (a = 0.8,(3 = 0.9) are s h o w n i n F i g s . 4.14 a n d 4.15, respectively. T h e curves represent the required S N R for B E R = 10~ 3 for fixed values of the m o d u l a t i o n i n d e x at the receiver as f u n c t i o n of a v a r y i n g m o d u l a t i o n i n d e x at the t r a n s m i t t e r . W e notice t h a t the performance d e g r a d a t i o n res u l t i n g from a m i s m a t c h i n h encountered u s i n g (3.46) persists w h e n u s i n g (3.53). A s expected, w h e n a decreases the S N R differences between u n k n o w n h at the receiver a n d k n o w n h decreases. Regardless, u s i n g the m o d u l a t i o n i n d e x e s t i m a t o r (cf. S e c t i o n 3.6.3) is advisable, a n d w i l l be presented i n the S e c t i o n 4.2.3. Performance W i t h Varying Channel Phase The m o d i f i e d phase reference (3.53) was developed to account for the extreme (yet admissible) phase v a r i a t i o n s a n d frequency offsets of A / T = 0.1 i n B l u e t o o t h devices. W e n o w evaluate the performance of N D F E using this m o d i f i e d phase reference i n the presence of frequency offset a n d phase j i t t e r . for B E R = 1 0 ~ 3 F i g . 4.16 shows the required S N R as a f u n c t i o n of the n o r m a l i z e d frequency offset. T h e curves are p a r a m e t e r i z e d w i t h a s t a n d a r d d e v i a t i o n of a A = 0° (no j i t t e r ) a n d a A = 2 ° , 5° for the phase j i t t e r . T h e L D I detector fails for offsets A / T > 0.03 . . . 0.05, b u t we note t h a t D C offset c a n c e l l a t i o n m e t h o d s c o u l d be a p p l i e d t o m i t i g a t e the effect of frequency offset. W e further observe t h a t N D F E w i t h phase reference (3.46) cannot cope w i t h large frequency offsets of A / T > 0.02. However, c o n s i d e r i n g the p a i r ( a = 0.6,3 = 0.9), the N D F E w i t h phase reference (3.53) allows power-efficient t r a n s m i s s i o n even for extreme offsets. In a d d i t i o n , the m a x i m u m performance d e g r a d a t i o n w h e n a s s u m i n g a n a d d i t i o n a l phase j i t t e r w i t h a A = 5° is not more t h a n 1.5 d B . For the same scenario, F i g . 4.17 compares the performances of N D F E for the two pairs (a = 0.6,(3 = 0.9) a n d ( a = 0.8,(3 = 0.9) a n d N S D w i t h the pair (a = 0.6,(3 = 0.9), 78 4.2 N o n c o h e r e n t D e t e c t i o n h0.31at trans0.32 mtier ->0.33 F i g u r e 4.14: T h e required 1 0 1 o g i o ( £ / i V o ) for B E R = 1 0 " 3 6 0.35 for N D F E w i t h a = 0.6, B 0.9 for v a r y i n g h. 18i t 1 1 a = 0.8, = 0.9, f ixed h -e— h known at recevier — 17 CO NDFE, p V I O Ii or 16 LU ™ 15 o "O 2 14 •- _ _ _ \ h = 0.32 \h = 0.33 \h = 0.34 )l h0.31at trans0.32 mtier -»0.33 0.35 F i g u r e 4.15: T h e required 101ogi (£?6./JVo)'for B E R = 1 01~- 3 for N D F E w i t h a = 3 0 0.9 for v a r y i n g h. 0.8,8 79 4.2 N o n c o h e r e n t D e t e c t i o n n 1 1 i r a = 0.6, p = 0.9, o =0° _ a = 0.6, P = 0.9, O A 2° = 0 A a = 0.6, p = 0.9, o = 5° & •- a = 0.6,o = 2° - • - * - MLSD bound - * 8 "0 0.01 0.02 0.03 0.04 0.05 AfT —> 0.06 *. 0.07 0.08 0.09 0.1 F i g u r e 4.16: T h e r e q u i r e d 101ogi (j5fe/A o) for B E R = 1 0 ~ i n the presence of frequency r 3 0 offset a n d phase j i t t e r . NSD, a = 0.6, P = 0.9 N D F E , a = 0.8, P = 0.9 - N D F E , a = 0.6, (3 = 0.9 a =0" 16 15 14h 13 12, 11 0 0.01 0.02 0.03 0.04 0.05 AfT-» 0.06 0.07 0.08 0.09 0.04 0.05 AfT -> 0.06 0.07 0.08 0.09 F i g u r e 4.17: T h e r e q u i r e d 1 0 1 o g ( £ ' b / ^ o ) for B E R = I C T i n the presence of frequency v 10 offset a n d phase j i t t e r . 3 80 4.2 N o n c o h e r e n t D e t e c t i o n w h i c h was found most favorable i n [27]. W e observe t h a t the performance deteriorates w i t h i n c r e a s i n g phase j i t t e r as a increases ( b o t h w i t h 8 = 0.9). T h e N D F E w i t h the p a i r (a = 0.6,3 = 0.9) was found to have c o m p a r a b l e performance w i t h the 2-state N S D . T h e preferable performance for (a = 0.6,8 = 0.9) i n the presence of extreme phase v a r i a t i o n s is one of the reasons w h y this p a i r is considered to be the favorable choice for N D F E . 4.2.3 The Adaptive N D F E performance of N D F E severely deteriorates w h e n there is a m i s m a t c h between the value of h assumed at the receiver a n d the a c t u a l h at the t r a n s m i t t e r , as was i l l u s t r a t e d i n the previous sections (cf. F i g s . 4.12, 4.14, 4.15). W e now evaluate the performance of A N D F E (cf. S e c t i o n 3.6.3), where several N D F E s are executed for a n u m b e r of h y p o t h e t i c a l m o d u l a t i o n indices, a n d the decision m e t r i c s are a c c u m u l a t e d . W e consider different c o m b i n a t i o n s of a n e s t i m a t i o n p e r i o d N e set of 2 or 4 hypothesis H. = 25 or 50, a n d a W h e n \H\ = 2, the h y p o t h e t i c a l m o d u l a t i o n i n d e x set H = { 0 . 3 , 1 / 3 } , w h i l e H = { 0 . 2 8 , 0 . 3 0 , 0 . 3 2 , 0 . 3 4 } w i t h \H\ = 4. I n the following, we adopt the n o t a t i o n A N D F E ( A , \H\) to convey the e s t i m a t i o n p e r i o d a n d h y p o t h e t i c a l r e m o d u l a t i o n set used. F i g . 4.18 plots the S N R required for B E R = 1 0 - 3 vs. the m o d u l a t i o n i n d e x used at the t r a n s m i t t e r for N D F E w i t h h k n o w n at the receiver a n d A N D F E w i t h h u n k n o w n at the receiver. T h e phase reference (3.46), w h i c h does not account for frequency offset v a r i a t i o n s , is used for each of the values a = 0.6 a n d a = 0.8. O b s e r v i n g the results o b t a i n e d for a — 0.8, we notice t h a t the curves for N D F E a n d A N D F E ( 5 0 , 4 ) are v e r y close, a n d the A N D F E shows a r e m a r k a b l e performance i m p r o v e m e n t c o m p a r e d to the case where h is u n k n o w n at the receiver a n d e q u a l t o 0.32, r e a c h i n g 3 d B w h e n h at the t r a n s m i t t e r is 0.29. F o r a = 0.6 it is i l l u s t r a t e d t h a t even for a s m a l l e s t i m a t i o n p e r i o d a n d m o d u l a t i o n i n d e x hypothesis of N e = 25 a n d \7i\ = 2, the A N D F E is s t i l l 81 4.2 N o n c o h e r e n t D e t e c t i o n 0.31 0.32 h at transmitter • 0.35 F i g u r e 4.18: 101ogi (£(,/iVo) required for B E R = 1 0 " for A N D F E w i t h phase reference 3 0 (3.46). T h e c h a n n e l phase is constant a n d the n o t a t i o n A N D F E ( A , H) applies. r e robust. A g a i n , the results for A N D F E a n d a d a p t i v e N S D ( A N S D ) 2 w i t h a = 0.6 are a l m o s t i d e n t i c a l , as has been observed for the various aforementioned scenarios. In F i g . 4.19, we i l l u s t r a t e the performance of A N D F E for different pairs of a a n d 8 w h e n the phase reference (3.53) w i t h frequency offset e s t i m a t i o n is used. For comparison, the results of A N D F E ( 5 0 , 4 ) w i t h phase reference (3.46) a n d a = 0.6 are also s h o w n . W e m a k e the following observations. F i r s t , it is evident t h a t the a d a p t i v e version of N D F E ( A N D F E ) w i t h an estimation period N e = 50 c o m b i n e d w i t h \7i\ = 4 a n d h u n k n o w n at the receiver achieves a result a l m o s t i d e n t i c a l t o N D F E w i t h h k n o w n , as i l l u s t r a t e d for the p a i r (a = 0.6,/? = 0.8). Second, the results o b t a i n e d for a l m o s t a l l c o m b i n a t i o n s of a a n d 8 f o r m almost straight lines, i n d i c a t i n g t h a t t h e y are close to the h k n o w n case. T h i r d , as 8 approaches 1, the performance of A N D F E u s i n g the phase reference (3.53) approaches t h a t of (3.46), w h i c h is consistent w i t h the results 2 T h e A N S D is the N S D receiver w i t h a n adaptive ^.-estimator. 82 4.2 N o n c o h e r e n t D e t e c t i o n - • - LDI, h = 1/3 - * - MLSD bound h = 0.32 at receiver o=0,6,p=0,9 . ANDFE(50,4) a = 0.6, P = 0.6 15h ~ANDFE(50,4) a=0.6, p = 0.9 ,^ NDFE, h known at receiver : ANDFE(50,4) / a = 0.6, P=0.99 : ANDFE(50,4) a = 0.6 i ' - •* • t'r^-;-^^;;^^^^^ u ^ 0 6, P = 0.9 : »< ANDFE(50,4) t 10 a = 0.8, :P = 0.9 -*-~4 0.3 0.29 0.31 0.32 h at transmitter - 0.33 F i g u r e 4.19: 1 0 1 o g i ( S b / A ) required for B E R = 1 0 r 0 - 3 0 0.34 0.35 for A N D F E w i t h various c o m - b i n a t i o n s of a a n d 3 w i t h phase reference (3.53). T h e channel phase is constant a n d the n o t a t i o n A N D F E ( A T , H) applies. e 1 18i t 1 1 1 1 1 LDI detector 17 CO o 16 ANDFE(50,2) o •- 0.6, p. = 0,9 15 -e— NDFE, a=0.6, p=0.9, h known at receiver -e— ANDFE(50,4), o=0.6, p=0.9 1 3 05 -0 - NSD,a=0.6, p=0.9, h known at receiver - -a - ANSD(50,4), o=0.6, p=0.9 1 12 LJ o Hi o 10 9 0.28 0.31 0.32 h at transmitter - F i g u r e 4.20: 10\og (E /N ) w b 0 0.35 required for B E R = 1 0 ~ for A N D F E a n d A N S D w i t h the 3 favorable p a i r (a = 0.6, 3 = 0.9) a n d w i t h constant channel phase. 83 4.2 N o n c o h e r e n t D e t e c t i o n p r e v i o u s l y s h o w n for N D F E . F i g . 4.20 compares the performance of the A N D F E w i t h the a d a p t i v e N S D ( A N S D ) , w h i c h e m p l o y s a n a d a p t i v e scheme s i m i l a r to t h a t of A N D F E . T h e results show very close performance between the a d a p t i v e schemes of N D F E a n d N S D w h e n the same values of a a n d 3 are used (the m a x i m u m difference is a b o u t 0.4 d B ) , even w i t h a s m a l l e r set of \H\ = 2. W e c a n thus c o n c l u d e t h a t N D F E allows h i g h l y power-efficient d e t e c t i o n also for a priori u n k n o w n m o d u l a t i o n i n d e x . 4.2.4 Performance in the Presence of Interference D u e t o the o p e r a t i o n of B l u e t o o t h i n the I S M b a n d , e v a l u a t i n g the p r o p o s e d B l u e t o o t h receiver i n the presence of interference is necessary. W e consider the performance w i t h interference, such t h a t the signal-to-interference power r a t i o ( S I R ) a n d the carrier freq u e n c y difference A / j are chosen as specified i n the B l u e t o o t h s t a n d a r d a n d presented C i i n T a b l e 2.1. T h e n o t a t i o n I F ( S I R , A / ) is used t o convey the S I R i n d B a n d Af c i c>i in M H z . F i g . 4.21 shows the performance results for the o p t i m a l W M F a n d the p r o p o s e d S R C filter, where h = 1/3 is k n o w n at the receiver. T h e effective interference suppression p r o v i d e d b y b o t h filters is verified, w i t h the B E R for adjacent c h a n n e l interference almost equal to the curve o b t a i n e d w h e n no interference is considered. T h e results for b o t h filters are quite s i m i l a r , w i t h the S R C o u t p e r f o r m i n g the W M F i n case of adjacent c h a n n e l interference, a n d the reverse is observed w i t h co-channel interference. As a m a t t e r of fact, w h e n the S R C filter is a p p l i e d , there is no loss due t o adjacent c h a n n e l interference for the cases I F ( - 3 0 , 2 ) , a n d I F ( - 4 0 , 3 ) . M o r e o v e r , the B E R curve for I F ( 40,3) is s h o w n w h e n the S R C is a p p l i e d w i t h o u t the H a m m i n g w i n d o w , a n d a loss of ~ 0.5 d B is observed c o m p a r e d to the case w h e n the H a m m i n g w i n d o w is a p p l i e d . F i g . 4.22 i l l u s t r a t e s the performance of the A N D F E i n the presence of interference as a f u n c t i o n of the m o d u l a t i o n index. W e observe t h a t the results of A N D F E are almost i d e n t i c a l to those for N D F E i n the presence of adjacent c h a n n e l interference. A s for 84 4.2 N o n c o h e r e n t D e t e c t i o n 10 IF(11,0) IF(0,1) * IF(-30,2) 0 IF(-40,3) -with SRC — with WMF V • 10 12 14 10log (E /N )[dB]^ 10 b 0 F i g u r e 4 21: P e r f o r m a n c e of N D F E i n the presence of interference for the W M F a n d the S R C filter. T h e c h a n n e l phase is constant a n d the n o t a t i o n I F ( S I R , A / C i i ) applies. 26 r ANDFE, IF(11,0) 24 \ \ tr 22, N * LU m o : 13 - - N _ ^-^^ • i LDI, 1 F(11,0) — N ~ -. • •1: 20 NDFE, IF(11,0) 0> A . ~ - 1i - 1 _ _ CT 18 CD LDI, no interference i- - _ ~ -< _ ANDFE, IF(0,1) LU NDFE, IF(0,1) o o 12 10 NDFE, IF(-30,2), IF(-40,3) and no;interference ANDFE, IF(-30,2), IF(-40,3) and no interference 0.29 0.3 0.33 0.31 0.32 h at transmitter -> F i g u r e 4 .22: l O l o g i o l ^ / i V o ) required for B E R = l f T 3 0.34 0.35 for N D F E a n d A N D F E i n the presence of interference. T h e c h a n n e l phase is constant a n d the n o t a t i o n I F ( S I R , A / j ) C i applies. 4.2 N o n c o h e r e n t D e t e c t i o n 85 co-channel interference, a n accountable performance loss of u p to 3 d B over the N D F E is recorded for h < 0.3. A l s o , A N D F E performs worse t h a n the L D I for h < 0.295. However, i n p r a c t i c e , co-channel interference is a rare occurrence, a n d , adjacent c h a n n e l interference is of greater interest i n general. Chapter 5 Conclusions T h e c o m p l e x i t y constraints coupled w i t h the v a r y i n g m o d u l a t i o n i n d e x a n d the h i g h frequency offset v a r i a t i o n s i n B l u e t o o t h devices creates a h i g h l y c h a l l e n g i n g environment for the design of power-efficient B l u e t o o t h receivers. T h e c o n v e n t i o n a l L D I detector is a s i m p l e receiver, b u t is h i g h l y s u b o p t i m u m i n t e r m s of power efficiency, a n d c a n not cope w i t h the h i g h frequency offsets of up to ± 1 0 0 K H z admissible i n B l u e t o o t h systems. Several sequence detectors proposed i n l i t e r a t u r e have proven to be e x t r e m e l y power efficient c o m p a r e d to the L D I detector since t h e y p r o p e r l y take the C P M m e m ory into account. These i n c l u d e the M L M - L D I a n d the M A P receivers. However, the M L M - L D I has v e r y h i g h c o m p l e x i t y a n d the M A P receiver assumes perfect channel phase e s t i m a t i o n a n d knowledge of the m o d u l a t i o n i n d e x at the receiver, a n d , therefore, t h e y are i m p r a c t i c a l . T h e N S D receiver recently proposed i n [31] tackles these p r o b l e m s b y designing a noncoherent sequence detector w i t h a m o d u l a t i o n i n d e x estimator. A l t h o u g h the c o m p l e x i t y of the N S D is l o w c o m p a r e d to the M L M - L D I and the M A P receivers, it s t i l l requires a 2-state V i t e r b i decoder. In t h i s research w o r k , we have presented a s i m p l e noncoherent power-efficient receiver design for B l u e t o o t h t r a n s m i s s i o n based o n L a u r e n t ' s d e c o m p o s i t i o n , w h i c h transforms the a c t u a l n o n l i n e a r m o d u l a t i o n scheme i n t o a linear m o d u l a t i o n over a n i n t e r s y m b o l 86 87 interference channel. It was s h o w n t h a t the c o m b i n a t i o n of a n S R C filter, s y m b o l - rate s a m p l i n g , a n d a decision-feedback equalizer achieves a performance close t o the theoretical M L S E limit. W e have p r o p o s e d a noncoherent version, n a m e l y NDFE, w h i c h offers h i g h robustness t o l o c a l - o s c i l l a t o r d y n a m i c s a n d phase j i t t e r present i n B l u e t o o t h devices. N D F E was also extended to allow a d a p t a t i o n of the decision m e t r i c t o a n a priori u n k n o w n m o d u l a t i o n i n d e x h. T h e presented s i m u l a t i o n results showed t h a t N D F E performs v e r y s i m i l a r t o 2-state N S D p r o p o s e d i n [31]. T h e key advantages of the p r o p o s e d design are o u t l i n e d i n the following a n d a s u m m a r y of the c o r r e s p o n d i n g s i m u l a t i o n results presented i n C h a p t e r 4 is given. • U s i n g L a u r e n t ' s d e c o m p o s i t i o n , the o p t i m a l receiver i n p u t filter for the r e s u l t i n g P A M s i g n a l is the w h i t e n e d m a t c h e d filter ( W M F ) [17]. However, since the W M F is dependent o n the m o d u l a t i o n i n d e x h, we propose r e p l a c i n g it b y the p r a c t i c a l S R C filter w h i c h has s i m i l a r characteristics. T h e s i m u l a t i o n s (cf. F i g . 4.1) prove t h a t the r e s u l t i n g performance u s i n g the W M F a n d S R C are almost i d e n t i c a l . M o r e o v e r , h i g h adjacent c h a n n e l interference suppression is achieved t h r o u g h the S R C , as i l l u s t r a t e d i n F i g . 4.22. • F o l l o w i n g the S R C filter, a noncoherent decision feedback equalizer is e m p l o y e d , where the phase is i m p l i c i t l y e s t i m a t e d u s i n g the developed phase reference (3.46). T h i s phase reference is f o r m u l a t e d i n terms of a forgetting factor a a n d shows r e m a r k a b l e performance i m p r o v e m e n t c o m p a r e d to the L D I , closely a p p r o a c h i n g the M L S D lower b o u n d as a increases (cf. F i g . 4.6). However, since this phase reference fails i n the presence of extreme frequency offset v a r i a t i o n s a n d phase j i t t e r , a m o d i f i e d phase reference (3.53) is devised. A g a i n , the m o d i f i e d phase reference is c o n t r o l l e d b y a forgetting factor 8, a n d the r e s u l t i n g decision rule was found t o be e x t r e m e l y robust against c h a n n e l phase v a r i a t i o n s , as s h o w n i n F i g . 4.16. T h e p a i r (a = 0.6, 8 = 0.9) w h i c h was f o u n d t o be favorable i n N D F E for various scenarios, has almost the same power efficiency as the 2-state N S D 88 using the same values of a a n d 3. • T h e N D F E is the simplest sequence detector possible, c o n s i s t i n g of a one-state trellis, thus, p r o v i d i n g an e x t r e m e l y h i g h r e d u c t i o n i n c o m p l e x i t y c o m p a r e d t o the 2p-state trellis required b y the o p t i m a l detector. O n l y a slight performance loss was recorded b y decreasing the trellis f r o m a full-state to a one-state trellis (cf. F i g . 4.4). T h i s is a r e m a r k a b l e result since the one-state trellis i n the N S D receiver was s h o w n to have worse performance t h a n the l o w - c o m p l e x i t y L D I detector [27]. • O n e of the p r i m a r y disadvantages of the sequence detectors proposed i n l i t e r a t u r e is a s s u m i n g a n o m i n a l m o d u l a t i o n i n d e x h at the receiver. I n the new receiver design, we propose e m p l o y i n g an a d a p t i v e ^.-estimation scheme w i t h a n a d a p t a t i o n p e r i o d of 7V s y m b o l s , a n d a set Ti of h y p o t h e t i c a l m o d u l a t i o n indices. T h i s e a d a p t i v e version of N D F E , w h i c h does not require knowledge of the m o d u l a t i o n i n d e x h, was found h i g h l y effective w i t h i V = 50 a n d \Ti\ = 2 or 4, o n l y a d d i n g a e slight increase i n c o m p l e x i t y . T h e s i m u l a t i o n results presented i n F i g s . 4.19-4.20 show t h a t it performs almost i d e n t i c a l l y t o the case w h e n h is k n o w n , a n d very close to the a d a p t i v e version of N S D w i t h the same values of N e and \H\. In c o n c l u s i o n , the new receiver design is h i g h l y a t t r a c t i v e for a p r a c t i c a l i m p l e m e n t a t i o n , a c h i e v i n g a performance g a i n of m o r e t h a n 4 d B over the L D I detector w i t h o n l y a slight increase i n c o m p l e x i t y , a n d a performance loss of o n l y a b o u t 2 d B c o m p a r e d to the M L S D lower b o u n d w i t h a n extreme decrease i n c o m p l e x i t y . In a d d i t i o n , it achieves performance results s i m i l a r t o the N S D i n various scenarios, inspite of its c o m p l e x i t y advantage over N S D . Consequently, the N D F E receiver is favorable over a l l state-of-the-art B l u e t o o t h receivers available i n literature. 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Bluetooth receiver design based on Laurent’s decomposition Ibrahim, Noha 2005
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Title | Bluetooth receiver design based on Laurent’s decomposition |
Creator |
Ibrahim, Noha |
Date Issued | 2005 |
Description | 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) signals, which achieve a good trade-off between power and bandwidth efficiency and, due to constant envelope modulation, allow for low-complexity transmitter implementation. Bluetooth devices often employ a simple discriminator receiver, which is highly suboptimum in terms of power efficiency compared to the optimum receiver. Other approaches proposed in the literature consider trellis-based detection using the Viterbi or forward-backward algorithm. These schemes achieve significant performance improvements 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 lowcost Bluetooth devices. In particular, we show that the devised noncoherent decision 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. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-01-06 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0065532 |
URI | http://hdl.handle.net/2429/17636 |
Degree |
Master of Applied Science - MASc |
Program |
Electrical and Computer Engineering |
Affiliation |
Applied Science, Faculty of Electrical and Computer Engineering, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 2006-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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