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Bluetooth receiver design based on Laurent’s decomposition 2005
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Title | Bluetooth receiver design based on Laurent’s decomposition |
Creator |
Ibrahim, Noha |
Date Created | 2010-01-06 |
Date Issued | 2010-01-06 |
Date | 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 |
Collection |
Retrospective Theses and Dissertations, 1919-2007 |
Series | UBC Retrospective Theses Digitization Project |
Date Available | 2010-01-06 |
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 |
Degree |
Master of Applied Science - MASc |
Program |
Electrical and Computer Engineering |
Affiliation |
Applied Science, Faculty of |
Degree Grantor | University of British Columbia |
Graduation Date | 2006-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
URI | http://hdl.handle.net/2429/17636 |
Aggregated Source Repository | DSpace |
Digital Resource Original Record | https://open.library.ubc.ca/collections/831/items/1.0065532/source |
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B L U E T O O T H RECEIVER DESIGN BASED ON LAURENT'S DECOMPOSITION by N O H A I B R A H I M B.Eng, American University of Beirut, 2003 A THESIS 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 T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F A P P L I E D S C I E N C E in T H E F A C U L T Y O F G R A D U A T E S T U D I E S (Electrical and Computer Engineering) December, 2005 © Noha Ibrahim, 2005 Abstract B l u e t o o t h is a wide ly used communica t ion s tandard for wireless personal area networks ( W P A N ) . T h e B lue too th t ransmit signal is Gauss ian frequency shift keying ( G F S K ) modula ted . G F S K belongs to the fami ly of continuous-phase modu la t i on ( C P M ) sig- nals, wh ich achieve a good trade-off between power and b a n d w i d t h efficiency and, due to constant envelope modu la t ion , al low for low-complexi ty t ransmit ter implementa- t ion . B l u e t o o t h devices often employ a s imple d iscr imina tor receiver, which is h igh ly s u b o p t i m u m i n terms of power efficiency compared to the o p t i m u m receiver. Other ap- proaches proposed i n the l i terature consider trellis-based detect ion using the V i t e r b i or forward-backward a lgor i thm. These schemes achieve significant performance improve- ments over d i sc r imina tor detectors while entai l ing a considerably higher computa t iona l complexi ty. T h e m a i n challenges faced when designing a B l u e t o o t h sequence detector is the vary ing modu la t i on index, which results i n a va ry ing trell is structure, and the t ime-variant channel phase, mak ing coherent detection which assumes perfect channel phase es t imat ion an almost impossible task. In this research work, we present a receiver design for B l u e t o o t h t ransmiss ion based on Laurent ' s decomposi t ion of the B l u e t o o t h t ransmi t signal . T h e m a i n features of this receiver are its low-complexi ty compared to al ternative solutions, i ts excellent performance close to the theoret ical l imi t , and its h igh robustness against frequency offsets, phase noise, and modu la t i on index variat ions, wh ich are characterist ic for low- cost B l u e t o o t h devices. In par t icular , we show that the devised noncoherent decision- i i m feedback equal izat ion receiver achieves a s imi lar performance as a recently proposed 2-state noncoherent sequence detector, whi le i t is advantageous i n terms of complexity. T h e new receiver design is therefore h igh ly at t ract ive for a p rac t ica l implementa t ion . Contents Abstract ii Contents iv List of Tables vii List of Figures viii Glossary xii Acknowledgements xvii 1 Introduction 1 1.1 The Evolution of the Bluetooth Technology 2 1.2 Challenges and Motivation 4 1.3 Contributions 6 1.4 Thesis Outline 7 2 Background 9 2.1 The Bluetooth System 9 2.1.1 Radio Front End 10 2.1.2 Physical Channel and Packet Definition 11 2.1.3 Physical Links 12 iv Contents v 2.1.4 Modulation Scheme 13 2.1.5 Transmission Model 16 2.1.6 Error Correction 20 2.2 Bluetooth Receivers Literature Review 21 2.2.1 Optimum Receiver 22 2.2.2 Suboptimum Receivers 27 2.3 M L S D Lower Bound 31 3 Noncoherent Decision Feedback Equalizer Receiver Structure 33 3.1 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 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 3.7 Summary of Proposed Receiver Structure 60 4 Performance Results and Discussion 62 4.1 Coherent Detection 63 4.1.1 Filter Selection 63 4.1.2 State Reduction 65 Contents vi 4.2 Noncoherent Detection 66 4.2.1 Performance with Constant Channel Phase 67 4.2.2 Performance Using the Modified Phase Reference Under Con- stant and Varying Channel Conditions 75 4.2.3 Adaptive NDFE ' 80 4.2.4 Performance in the Presence of Interference . : 83 5 Conclusions 86 Bibliography 89 List of Tables 2.1 Interference performance [51] 18 2.2 T h e values of re la t ively-pr ime integers m and p corresponding to differ- ent modu la t i on indices h w i t h i n the range specified by the B l u e t o o t h s tandard 25 3.1 D u r a t i o n of Lauren t components.Cp(t) 38 v i i List of Figures 1.1 The required 101ogi0(iViVo) ^ o r B ^ R = ^ 3 as a function of the mod- . ulation index h 5 2.1 General basic rate packet format [51] 11 2.2 General enhanced data rate packet format [51] 12 2.3 Block diagram of a CPM modulator 14 2.4 Frequency pulse shape g(t) and phase pulse q(t) for Gaussian frequency shift keying with time-bandwidth product BT = 0.5 15 2.5 Block diagram representing the ECB system model 19 2.6 Illustration of Repetition code 21 2.7 Phase state trellis structure for Bluetooth with h = 1/3 26 2.8 LDI detector block diagram 29 2.9 Block diagram of the GFSK modulator using the Rimoldi/Huber&Liu decomposition approach 31 3.1 Laurent pulses C0(t) and Ci(t), and the main pulse P(t) for Bluetooth GFSK signals with L = 2 and h = 1/3 39 3.2 Block diagram of optimum coherent receiver based on Laurent's decom- position 40 3.3 The imaginary and real components of an exact Bluetooth signal s(t) and an approximate signal s(t) generated using only the main pulse P(i). 42 viii List of Figures ix 3.4 Impulse response of SRC, hSRc{t), before Hamming window (a), and after Hamming window (b) 45 3.5 Magnitude frequency response for WMF, HWMF(f), assuming h = 0.28 and h = 0.35, and for SRC filter, HSRC{f), with p = 0.3 46 3.6 The CIR of the (a) ISI channel hc[k] compared to (b) its minimum phase equivalent ho[k] 48 3.7 Block diagram of decision-feedback equalizer 51 3.8 The roots of Hc(z) compared to the roots of its minimum phase equiv- alent H0(z) 53 3.9 Structure of the proposed Bluetooth receiver 60 4.1 Performance comparison of the SRC and W M F as receiver filters. Co- herent detection with h = 1/3 is assumed 64 4.2 Evaluation of the state reduction in the proposed receiver. Coherent detection with h = 1/3 is assumed 65 4.3 Performance of NDFE with a = 0.8 with FBF and F F F fixed at the receiver, or varying according to the value of h at the transmitter. . . . 66 4.4 Evaluation of the performance of the proposed receiver with state re- duction in case of noncoherent detection 67 4.5 Performance of the NDFE using the /V-metric compared to MLSD, LDI, and coherent DFE, and h = 1/3 for all cases 69 4.6 Performance of the NDFE using the a-metric compared to MLSD, LDI, and coherent DFE, and h = 1/3 for all cases 69 4.7. Performance of the NDFE using the iV-metric h = 1/3 in the presence of phase jitter 70 4.8 Performance of NDFE using the a-metric h = 1/3 in the presence of phase jitter 70 List of Figures x 4.9 Performance of the N D F E with a = 0.8 for different values of the mod- ulation index compared to the M L S D bound, LDI, 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 NSD for a = 0.6 73 4.12 The required 10\ogw{Eb/N0) for B E R = 10~3 for N D F E with a = 0.8 with varying modulation index 75 4.13 lOlogio (Eb/N0) required for B E R = 1 0 - 3 for different combinations of a and (3 as a function of the modulation index with constant channel phase and h known 76 4.14 The required l0\ogw(Eb/N0) f o r B E R = 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 mog10(Eb/NQ) f o r B E R = 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\og10(Eb/N0) for B E R = 1 0 - 3 in the presence of fre- quency offset and phase jitter 79 4.17 The required 101ogi0(^b/^Vo) for B E R = 10"3 in the presence of fre- quency offset and phase jitter 79 4.18 101ogio(£y iVo) required for B E R = 10~3 for A N D F E with phase refer- ence (3.46). The channel phase is constant and the notation A N D F E ( J V e , H) applies. 81 4.19 101bgio(£&//V 0) required for B E R = 10~3 for A N D F E with various com- binations of a and /3 with phase reference (3.53). The channel phase is constant and the notation A N D F E ( A r e , H) applies 82 4.20 Mog10(Eb/N0) required for B E R = l O " 3 for A N D F E and A N S D with the favorable pair (a = 0.6, (3 = 0.9) and with constant channel phase. 82 List of Figures xi 4.21 Performance of NDFE in the presence of interference for the W M F and the SRC filter. The channel phase is constant and the notation IF(SIR,A/C i i) applies 84 4.22 10\og10(Eb/N0) required for BER = 1(T 3 for NDFE and A N D F E in the presence of interference. The channel phase is constant and the notation IF(SIR,A/C | i) applies 84 Glossary L i s t o f A b b r e v i a t i o n s (In alphabetical order) A C L Asynchronous connectionless link A N D F E Adaptive noncoherent decision-feedback equaliz* A M P Amplitude modulated pulses A R Q Automatic retransmission query A W G N Additive white Gaussian noise B E R Bit error rate C P M Continuous phase modulation C R C Cyclic redundancy check D F E Decision-feedback equalizer D M Medium data rate D V Data voice E C B Equivalent complex baseband E D R Enhanced data rate E V Extended voice eSCO extended synchronous connection-oriented F B F Feedback filter F C C Federal Communications Commission F E C Forward error correction xii Glossary xiii F H S Frequency hop synchronization FIR Finite impulse response F F F Feedforward filter G F S K Gaussian frequency shift keying H E C Header error check H V High-quality voice ISI Intersymbol interference ISM Industrial, scientific, medical LDI Limiter-discriminator integrator M L S ( D / E ) Maximum-likelihood sequence (detector/estimator) N D F E Noncoherent decision-feedback equalizer N R S S E Noncoherent reduced-state sequence estimation NSD Noncoherent sequence detector P S K Phase shift keying PSP Per-survivor processing R S S E Reduced-state sequence estimation S C O Synchronous connection-oriented SIG Special interest group SIR Signal-to-interference power ratio SD Sequence detector SRC Square-root raised cosine U W B Ultra-wide band V A Viterbi algorithm W L A N Wireless local area network W M F Whitened matched filter Glossary xiv Operators and Notation t continuous time [k] Discrete time a Input binary data vector of a[fc] € { ± 1 } b[k] Zeroth complex symbols bp[k] Complex symbols B 3-dB bandwidth Cp(t) Laurent pulses exp Exponential function Ef, Received energy per bit Es Signal energy per modulation interval f[k] (N)DFE feedforward filter fc Carrier frequency fd Frequency deviation with respect to the carrier frequency g(t) Frequency shaping pulse h Modulation index h(t) Gaussian low-pass filter h0[k] Minimum phase equivalent of hc[k], the (N)DFE feedback filter hc[k] Discrete-time channel impulse response (ISI channel) hsRc{t) Impulse response of the SRC filter hwMF(t) Impulse response of the WMF HsRc{f) Frequency response of the SRC filter HWMF(I) Frequency response of the WMF H Number of modulation index hypotheses used in 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 Glossary xv K Rician factor log Base-10 logarithm In Natural logarithm L CPM memory m the relatively prime numerator integer such that h = m/p n{t) Additive white Gaussian noise No One-sided noise power spectral density of the passband noise process Ne Length of estimation period used in 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 qTef Phase reference used in noncoherent detection Q(t) Gaussian Q-function r(t) Equivalent complex baseband representation of received signal 7"DFE [k] output of feedforward filter rect(i/T) rectangular pulse s(t) Equivalent complex baseband representation of transmitted signal Sn(t) Special sin functions used in Laurent's derivation process T Symbol period a Phase estimation forgetting factor (3 Frequency offset estimation forgetting factor A / Frequency offset between the transmitter and receiver oscillator A/ C ) j frequency offset of the desired signal compared to the interfering signal #0 Constant phase p Roll-off factor of the SRC filter 0o Constant phase rotation as a result of fading 4>(t) Time-varying phase tp(t) Phase function <p(t) = 2irhq(t) Glossary x v i ifj(t,a) Informat ion-carrying phase ty(t) Genera l ized phase pulse function Acknowledgements I would like to express my gratitude to my research supervisors, Dr. Lutz Lampe and Dr. Robert Schober, for their guidance and encouragement, and their valuable contribution to this research work. Their productive supervision throughout the past two years not only helped me broaden my scope of knowledge, but also increased my appreciation and dedication to this challenging research project, which has proven to be highly rewarding. I also gratefully acknowledge the financial support in the forms of RA from Dr. Lutz Lampe and Dr. Robert Schober through NSERC grants STPGP 257684 and RGPIN 283152-04, and TA from the Electrical and Computer Engineering Department. I would like to thank my colleagues in the communication theory group for sustaining a friendly work environment. Finally, I would like to acknowledge the love and consistent support my family has provided me throughout my graduate studies in UBC. I would also like to thank my friends, especially Axel Davidian, for always being there, while being physically present at the other side of the globe. xvii Chapter 1 Introduction The pervasive use of mobile computing devices such as laptops and personal digi- tal assistants (PDAs), and the evident success of cellular phones called for a wire- less technology to connect these devices together. This technological vision became a reality with the introduction of the Bluetooth standard for wireless personal area networks (WPANs) which enables wireless communication among various electronic devices. Bluetooth has revolutionized the wireless world, for it provides low-power, low-cost, and short-range radio links with secure and reliable transmissions and global compatibility. Bluetooth is an open standard, which enables manufacturers to take full advantage of the capabilities of the technology and build products according to its specifications, thus expanding the Bluetooth applications to diverse market areas. With seamless voice and data connections to virtually all mobile devices, the human imagination remains the only limit to application options. The Bluetooth technology specification, currently in its fourth version of the core specification, is developed by an industry- based association, the Bluetooth Special Interest Group (SIG), and will be outlined in detail in the next chapter. The following section gives a brief history of the Bluetooth technology, its current 1 1.1 The Evolution of the Bluetooth Technology 2 status, and its future path. Section 1.2 states the motivation behind the present work and the challenges faced, and a summary of the various contributions is provided in Section 1.3. Finally, a brief description of the contents of the thesis concludes this chapter. 1.1 T h e E v o l u t i o n o f t h e 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 telecom- munications manufacturers, was investigating ways to add value to its phones in the crowded market. Consequently, Ericsson mobile research lab in Lund, Sweden, initi- ated a study to investigate the feasibility of a low-power, low-cost wireless technology to connect mobile phones and their accessories without the need of cumbersome cables. The study rapidly progressed, and as researchers realized the developed technology can be used to exchange data among numerous wireless and wired devices, the study quickly turned into a large project, which was given an internal code name,"Bluetooth". Blue- tooth is the English derivative of the Viking word Blatand, and was named after the 10th century Danish Viking King Harald Blatand who united Denmark and Norway during a time of fighting. Bluetooth developers considered the name appropriate for the technology since they anticipated it will unite the telecommunications world, just as Blatand united his world [5]. To allow Bluetooth to be an accepted industry standard, five major companies from three diverse business areas formed the Bluetooth Special Interest Group (SIG). The group was formally announced on May 20, 1998, and included two leading companies in mobile telephony, Ericsson and Nokia, two leading companies in laptop computing, IBM and Toshiba, and one leading company in digital signal processing, Intel. Today, the Bluetooth SIG has over 3,400 member companies all over the world [21]. In its mission statement, the Bluetooth SIG affirms that it "will support a collaborative 1.1 The Evolution of the Bluetooth Technology 3 environment and drive programs to develop and advance Bluetooth wireless technology in order to exceed personal connectivity expectations and meet the needs of a changing world". Versions 1.0, 1.0b, and 1.1 of the Bluetooth specifications were released in 1999, 2000, 2001, respectively. Bluetooth received additional support when the IEEE Standards Association approved the IEEE Std. 802.15.1, derived from the Bluetooth Specification, in March, 2002. The standard was published three months later, and introduced minor changes to the physical and medium access control (MAC) layers. Version 1.2 of the Bluetooth specification was formally ratified on November 2003, while the latest Bluetooth Core Specification Version 2.0+ Enhanced Data Rate (EDR) was released a year later, and provided increased data rates and lower power consumption. The market for Bluetooth devices has been rapidly growing, and, as a result, the tech- nology has met the significant milestone of five million Bluetooth units shipped per week. The number of Bluetooth-enabled devices doubled from 2003 to 2004, and is expected to reach 500 million units by the end of 2005. Moreover, a recent study con- ducted in Japan, United states, and United Kingdom has shown an increased customer awareness of Bluetooth. Another emerging technology, ultra-wide band (UWB) appears to have great potential for the wireless applications which currently use Bluetooth. UWB transmits with very low power at extremely large bahdwidths, but there are still some challenges before this technology can be realized. To meet the future market demands, and take advantage of the high data rates that UWB offers, the Bluetooth SIG announced in May, 2005 the collaboration of the Bluetooth and UWB developers. The goal is to employ UWB in the next-generation Bluetooth products, while maintaining backward compatibility with the existing products [1]. 1.2 Challenges and M o t i v a t i o n 4 1.2 Challenges and Motivation T h e objective of the present work is to design a cost-effective, power-efficient, and s t ruc tura l ly-s imple 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 high s t ruc tura l and computa t iona l complexi ty, for it consists of a bank of matched filters followed by a coherent 1 m a x i m u m l ike l ihood sequence detector ( M L S D ) w i t h a number of states vary ing according to the modu la t ion index. Since the modu la t i on index in B l u e t o o t h systems is allowed to vary i n a relat ively wide range (0.28 < h < 0.35), the corresponding number of states i n the o p t i m u m receiver broadly varies w i t h a m i n i m u m of 12 (as w i l l be detai led in the next chapter), mak ing it unfeasible. T h e prac t ica l and simple alternative, namely, the l imi te r -d i sc r imina tor integrator ( L D I ) receiver, current ly used for B lue too th devices is a simple, low-cost receiver. However, i t is h igh ly subop t imal , for i t suffers a performance loss of more than 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 that the L D I receiver consumes four t imes as much s ignal power as the o p t i m u m receiver to achieve the same bit error rate ( B E R ) . In the present work, we investigate the feasibili ty of an al ternat ive B l u e t o o t h receiver design, mot iva ted by the large possible power efficiency gains that may be achieved over the conventional L D I receiver. T h i s requires overcoming the fol lowing challenges faced when designing a B l u e t o o t h receiver. T h e vary ing modu la t i on index results i n a vary ing trel l is s tructure w i t h a large number of states, which poses a serious challenge when considering trell is-based detection. In add i t ion , the metrics required for the detection of the m a x i m u m l ike l ihood symbol se- quence are also dependent on h. However, the trell is-based receivers achieve significant performance improvements over the conventional L D I receiver. Hence, we investigate 1 Coherent detection assumes perfect channel phase estimate at the receiver. 1.2 Challenges and Motivation 5 18 17 T CO o 16 1— II CC UJ 15 CD O T3 14 a '5 CT <U 13 12 LU 11 O O 10 9 • LDI receiver - * - • Optimum receiver] t • k : >6dB ! 0.29 0.3 0.31 0.32 0.33 0.34 0.35 Figure 1.1: The required 101ogi0(£,6/A/o) for BER = 10 - 3 as a function of the modula- tion index h. the design of a reduced-state sequence detector independent of the modulation index. A carrier frequency deviation of up to ±100 KHz is allowed in Bluetooth systems [51], resulting in fast channel phase variations. Several coherent detectors which achieve significantly higher power efficiency compared to the LDI are available in the litera- ture. However, perfect channel phase estimation is assumed at the receiver, which is impractical in Bluetooth systems. Therefore, we consider the suboptimum noncoherent detection schemes which perform implicit channel phase estimation. Since Bluetooth operates in the license-free ISM band, it is vulnerable to interference from other Bluetooth and wireless local area network (WLAN) devices operating in the same frequency band. The design of the optimal receiver filter is dependent on the modulation indices, and, moreover, it is not guaranteed to accomplish strong out- of-band interference suppression. The present work studies the design of a practical input receiver filter independent of h and capable of accomplishing strong adjacent and co-channel interference suppression. 1.3 Contributions 6 1 .3 C o n t r i b u t i o n s In the present work, we consider noncoherent sequence detection for Bluetooth signals based on Laurent's decomposition which models the GFSK signal as a superposition of amplitude modulation pulses, as will be described in Chapter 3 [32]. We then make the following contributions. • As with Laurent's decomposition the actual nonlinear modulation scheme is transformed into a linear modulation over an intersymbol-interference (ISI) chan- nel, we develop reduced-state trellis-based equalizers using the concepts of reduced- state sequence estimation (RSSE) [16] and per-survivor processing (PSP) [39], which will be presented in Chapter 3. We also devise noncoherent versions of the RSSE and 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 im- pulse response, we show that decision-feedback equalization (DFE) achieves a performance close to the lower bound of maximum-likelihood sequence estima- tion (MLSE) 2 [17]. This is a remarkable result, since complete state reduction incurred considerable performance degradation in case of an alternative nonco- herent sequence detector (NSD) in the literature [31], [27]. We also note that a similar approach based on Laurent's decomposition required a 4-state trellis decoder to achieve a similar performance [43]. 2 I n the context of the present work, "sequence est imat ion" and "sequence detection" are used synonymously. 1.4 Thesis Outline 7 • We develop a simple frequency estimation method which provides high robustness to (extreme) frequency offsets and phase jitter. • NRSSE and NDFE are extended to adapt the decision metric to the modulation index h used at the transmitter side. This adaptive algorithm offers a simple solution to the problem of varying modulation index, and provides acceptable performance for all values of h. • By means of simulation results for various transmission scenarios, we show that NDFE achieves almost the same performance as 2-state NSD. Due to its com- plexity advantage over NSD, the NDFE receiver is an attractive solution for a practical implementation. 1.4 T h e s i s O u t l i n e This thesis is organized as follows. Chapter 2 introduces the background information required for the implementation and evaluation of a Bluetooth receiver. Moreover, a literature review on the optimum and suboptimum Bluetooth receivers, as well as the MLSE lower bound are presented in order to provide solid grounds for comparison. Chapter 3 describes in detail the structure of the proposed receiver. It starts by introducing Laurent's decomposition, which forms the basis of the present work. It points out the decomposition's major strengths and presents the optimum Bluetooth receiver based on the decomposition. After discussing the input receiver filter design, the corresponding discrete-time transmission model is introduced. The decision metric for coherent RSSE is then given, leading to the description of the structure of the extreme case of RSSE, the DFE. The minimum phase channel needed to achieve the maximum performance in reduced-state detectors is also discussed. The following 1.4 Thesis Outline 8 section applies noncoherent detection to RSSE and 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 character- istics of the proposed receiver. Chapter 2 Background T h i s chapter provides the background informat ion necessary for the design, analysis, and evaluat ion 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 to be a universal wireless technology, which operates at low power and can be imple- mented at low cost. Accord ing ly , 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 work involves a receiver design, the second section provides a l i terature sur- vey on the o p t i m u m and subop t imum Blue too th receivers. These, i n add i t ion to the M L S E lower bound presented i n Sect ion 2.3 w i l l serve as benchmarks when evaluat ing the proposed solut ion. 2.1 T h e B l u e t o o t h S y s t e m 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 radio specifications w i l l be described first, followed by the physical channel and packet definit ion. T h e types of physical l inks available for da t a t ransmission as wel l as the modu la t i on scheme used w i l l then be presented. W e conclude this section w i t h an i l lus t ra t ion of the channel model and an overview of the error correct ion schemes offered by B lue too th . 9 2.1 The Bluetooth System 10 2.1.1 Radio Front End Bluetooth operates in the 2.4 GHz Industrial Scientific Medical (ISM) band, a fre- quency band that is globally available, license-free, and open to any radio system. The Bluetooth standard specifies a 1 MHz bandwidth for each RF channel. However, there are variations in the range of frequencies available in the ISM band, and the Bluetooth specifications were defined to accommodate these differences. In most countries, where at least 80 MHz of bandwidth is allocated to the ISM band, Bluetooth defines 79 RF channels located at (2402 + k) MHz, k = 0,1,... 78. In other countries, such as France, Spain, and Japan, which allow the usage of only a reduced spectrum for the ISM band, 23 Bluetooth channels are defined, located at (2454 + k) MHz, k = 0,1,..., 22. Since Bluetooth devices do not have exclusive use of the frequency band, possible in- terference from other devices (garage door openers, baby monitors, microwave ovens, cordless phones, etc.) can be experienced. To minimize the effect of interference, a frequency-hopping spread spectrum approach is used. The radio hops through all the RF channels following a pseudo-random sequence [9], [13], [51]. Federal communica- tions commission (FCC) regulations part 15.247 require that a device operates on a given channel for a maximum of 0.4 seconds within a 30-second interval [34]. The Bluetooth core specification defines three transmit-power classes: 100 mW (20 dBm), 2.5 mW (4 dBm), and 1 mW (0 dBm), corresponding to Class 1, 2, and 3 devices, re- spectively. With Class 1 devices, the communication range may increase up to 100 m. Power-save modes including Sniff, Hold, and Park (in decreasing order of power require- ments) modes are used to reduce power consumption. Power control is used for devices with transmitter power higher than 4 dBm, and thus, necessary for Class 1 devices. For devices with lower transmitter power, power control is optional for optimizing the power consumption and the interference level [51]. 2.1 The Bluetooth System 11 ACCESS CODE HEADER PAYLOAD Figure 2.1: General basic rate packet format [51]. 2.1.2 Physical Channel and Packet Definition Bluetooth devices communicate with each other by forming piconets, which can ac- commodate up to eight devices. Several piconets can establish interconnections using bridge devices to form a larger network known as a scatternet. A piconet channel is divided into slots, each having a duration of 625 /usee. Users sharing the same channel are each assigned a time slot, and are time- and hop-synchronized to the channel to ensure reliable communication. Information between Bluetooth devices is exchanged using packets. A packet may oc- cupy one, three, or five time slots, and a frequency hop occurs only once the packet is complete. The hopping rate varies based on the number of slots occupied per packet, and reaches a maximum rate of 1600 hops/sec. The Bluetooth Standard defines sixteen different types of packets depending upon the application. In the Bluetooth specifi- cation Version 2.0 + EDR, higher transmission rates (relative to the older Bluetooth specification versions) and corresponding packet formats were introduced. In addition to the basic rate of 1 Mbps, which uses the Gaussian frequency shift keying (GFSK) modulation scheme, an enhanced data rate with two new modulation modes was also defined. The primary modulation mode, 7r/4-DQPSK (differential encoded quaternary phase shift keying), provides a data rate of 2 Mbps, while the secondary mode, 8-DPSK (differential encoded 8-ary phase shift keying), increases the data rate to up to 3 Mbps. The general basic rate packet format is shown in Fig. 2.1. Each packet begins with an access code, unique for the channel. The access code consists of 4 Preamble bits, a 64-bit sync word and an optional 4-bit trailer appended to the sync word only if a header packet follows. The access code is used for synchronization, direct current (DC) 2.1 The Bluetooth System 12 ACCESS CODE HEADER GUARD SYNC ENHANCED DATA RATE PAYLOAD TRAILER GFSK • ~* DPSK • Figure 2.2: General enhanced data rate packet format [51]. offset compensation, and identification. A 72-bit header may follow the access code. The header contains important control information such as packet type, flow control, and header error control (HEC), and is encoded with a 1/3 forward error correction (FEC) code. The payload field has a length varying between 0 and 2745 bits [51]. The general enhanced date rate packet format is shown in Fig. 2.2. Each packet consists of an access code, a header, a guard period, a synchronization sequence, an enhanced data rate payload, and a trailer. As noted in the figure, the access code and the header use GFSK, the same modulation scheme used in basic rate packets. The guard time then allows for transition into the higher data rate modulation scheme. Since this dissertation only involves the basic data rate modulation scheme, only GFSK will be elaborated in the following section. 2.1.3 Physical Links Bluetooth uses two types of links for voice and data transmission, the synchronous, connection-oriented (SCO) links and the asynchronous connectionless links (ACL). Bluetooth devices can use either link, depending on the type of packets being trans- mitted. Further, a device can alternate between the two types of links during trans- mission, as long as an ACL link is established before using an SCO link. ACL links support point-to-multipoint connections with a maximum data rate of 732.2 Kbps in an asymmetrical mode or 433.9 Kbps in a symmetrical mode and are primarily used for file and data transfers. SCO links support point-to-point connections with a data rate of 64 Kbps and are typically reserved for voice traffic. The SCO link was further improved in Version 1.2 of the Bluetooth specification, by defining a new extended 2.1 The Bluetooth System 13 SCO (eSCO) link. New packet types were defined for this link as extended voice (EV) packets, which are the original high quality voice (HV) packets with cyclic redundancy check (CRC) enabled to achieve higher reliability. The eSCO may be used for trans- parent synchronous user data and audio transmissions, and data rates of 384 Kbps and 564 Kbps can be achieved. These data rates are the same as those enabled by the third generation (3G) Universal Mobile Telecommunications Service (UMTS) systems, making the eSCO the most suitable link for audio and video streaming applications over 3G networks [6], [22]. 2.1.4 Modulation Scheme Gaussian frequency shift keying (GFSK), a special case of continuous phase modula- tion (CPM), is the modulation scheme used in the basic data rate mode of Bluetooth (cf. Section 2.1.2). CPM has been widely used in communications due to its power and bandwidth efficiency as well as its constant envelope. Constant envelope mod- ulation schemes are known to allow for efficient power amplification. Further, CPM provides better spectral utilization than frequency shift keying (FSK) by introducing a continuous phase to smooth variations between symbols. The block diagram of a CPM modulator is shown in Fig. 2.3, where the frequency modulator consists of a voltage controlled oscillator (VCO). The resulting CPM passband signal is represented by [2] SRF(t) = Cos (27r/ct + ib(t, a) + d0) , (2.1) where Es denotes the signal energy per modulation interval T, fc is the carrier fre- quency, #o 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. \ 2.1 The Bluetooth System 14 a[k] e { ± 1 } g(t) vco ) * SRF{t) FM Modulator 2nh Figure 2.3: Block diagram of a CPM modulator. The frequency shaping pulse g(t) determines the smoothness of ip(t, a) and is assumed to be of finite duration LT, where L is known as the CPM memory. A time-limited g(t) leads to an appropriate representation of q(t) as 9(t) 0 t < 0 t J g{r)dT 0 < t < LT o 1/2 t > LT (2.3) When L > 1, the frequency pulse exceeds the symbol interval T, leading to inter- symbol interference (ISI). However, this approach, known as partial response signaling, is desirable in CPM since it introduces additional memory and allows for a narrower power spectrum with low spectral sidelobes. Varying the parameter h, and choosing different frequency shaping pulses, g(t), results in various CPM schemes. Continuous phase frequency shift keying (CPFSK) signals, for example, use a rectangular pulse shape of duration T, but this causes sudden frequency transitions, resulting in a large bandwidth [42]. To achieve a more compact spectrum and smoother frequency transi- tions, the frequency shaping pulse in GFSK consists of a rectangular pulse pre-filtered by a Gaussian low-pass filter h(t) as [40] g(t) = h(t) * r ec t (* /T) , where * denotes the convolution and h(t) is given by 1 ' / -t2 h(t) = 2ixaT exp 2a2T2 (2.4) 2.1 The Bluetooth System 15 Figure 2.4: Frequency pulse shape g(t) and phase pulse q(t) for Gaussian frequency shift keying with time-bandwidth product BT = 0.5. with a = yj\n{2)/ (2vr) BT and BT is the 3-dB bandwidth-time product (B: 3-dB bandwidth of the Gaussian filter). The result of the convolution of the Gaussian filter with the rectangular pulse that is defined as rect I — I = is the frequency pulse shape given by 1/T for |t| < T/2 0 otherwise (2.5) 9(t) = ± Q(f(t + T/2J)-Q(£(t-T/2J) (2.6) where Q(t) = (l/\/27r) J e~T / 2 dr is often referred to as the Gaussian Q-function. The t bandwidth of the GFSK spectrum and the resulting bit error rate at the receiver are both affected by two parameters: BT and the modulation index h = 2faT, where fd is the frequency deviation (maximum frequency shift with respect to the carrier frequency). It is desirable to have a low BT product and a small modulation index, for it results in a narrower spectrum. However, a low BT product produces higher ISI. In Bluetooth systems, T — 1 psec, BT = 0.5, and fa may vary between 140 KHz and 175 KHz, resulting in 0.28 < h < 0.35. The low modulation index values are due to 2.1 The Bluetooth System 16 the restrictions imposed by the FCC section 15.247 rule governing frequency hopping spread spectrum in ISM devices [51]. The GFSK frequency pulse shape g(t) used in Bluetooth and the resulting phase pulse q(t) are shown in Fig. 2.4. 2.1.5 Transmission Model For short-range wireless systems, such as Bluetooth, communication often occurs in residential homes, office buildings, or commercial areas (factories, shopping centers, etc.). The signal propagation depends on the indoor environment and its topography, and varies with time due to the motion of people and equipment. In such settings, Doppler and delay spreads are minimal, resulting in a stationary or a slow-varying flat- fading channel. The envelope of such channels is comparable to a Rician distribution. The probability density function of the Rician distribution is defined as [40] [o for(0<O) where g is the envelope of the channel gain, A, (A > 0), is the amplitude of the domi- nant signal, 2cr2 is the variance of the diffuse path, and IQ(X) — l/2ir J exp(xcosc/!>)d(/> o is the modified Bessel function of first kind and zeroth order. The Rician distribution is described by a Rician factor K — A2/2a2, which is the ratio of the power of the dominant path to the power of the diffuse path. In the special case of a weak dominant signal (A —> 0), a Rayleigh distribution is obtained, while as cr2 —> 0, the result is an additive white Gaussian noise (AWGN) channel. The indoor propagation model in the 2.4 GHz unlicensed ISM band in which Bluetooth operates (cf. Section 2.1.1) was investigated in [29]. The root mean square (rms) of the delay spread was found to be below 70 nsec, with an average value of approximately 50 nsec. These values are very low compared to the symbol duration of 1 /isec in Bluetooth. The channel propagation model was classified into two major categories. The first category includes channels with a line-of-sight path, which may be approximated by a Rician distribution with 2.1 The Bluetooth System 17 K — 5 (~ 7 dB). The second category describes channels with an obstructed path, and was found to have the same distribution with K = 2 (~ 3 dB), which is very close to the Rayleigh distribution. In a slow-fading channel, and after being affected by noise, the equivalent complex baseband (ECB) representation of the received signal is given by r(t) = g^*°s(t) + n(t) , (2.8) where <f>0 is a phase rotation resulting from fading and is constant over time, s(t) denotes the complex envelope of the transmitted signal that is written with a normalized unit amplitude as1 s(i)=exp(jV(t,a)) , (2.9) and n(£) is additive white Gaussian noise (AWGN) with two-sided power spectral den- sity A^o/2. We note that the passband signal sRF(t) (2.1) can be written in terms of s(i) (prior to amplitude normalization) as SRp(t) = 3f?{s(£)e j 2 7 r / c*}, where 3?(.) represents the real part of the signal. Oscillators are used to frequency modulate the Bluetooth information-carrying signal (cf. Section 2.1.4). The frequency stability of an oscillator is defined as the measure of the degree to which an oscillator maintains the same value of frequency over a given time. Oscillator instability results in a phenomenon known as phase noise or timing jit- ter, which introduces an additional time-varying phase rotation to the signal, A0(£,r), where A(p(t,r) is a zero-mean Gaussian variable with a variance CF\(T) depending on the employed oscillator and linearly increasing with time. Further, frequency fluctua- tions of a signal occur due to the frequency offset ( A / ) between the transmitter and receiver oscillator, contributing to a greater phase variation [14], [33]. The resulting time-varying phase is represented as a function of the aforementioned parameters as 0(t + r) = <f>{t) + 2 T T A / T + A(f>{t, T) . (2.10) The constant phase term 0Q has been ignored. 2.1 The Bluetooth System 18 Table 2.1: Interference performance [51]. Frequency of Interference (A/C ) i) SIR 0 MHz (Co-channel interference) 1 MHz (adjacent interference) 2 MHz (adjacent interference) > 3 MHz (adjacent interference) 11 dB 0 dB -30 dB -40 dB Consequently, the received signal in the presence of frequency offset and phase jitter is expressed as r(t) = S e ' ' ( * ( t ) + * o ) s ( t ) + n(t) . (2.11) Due to operation in the license-free ISM band and frequency reuse, Bluetooth can suffer from interference from other Bluetooth and WLAN devices. Co-channel interfer- ence occurs when two or more piconets occupy the same frequency at the same time, while adjacent channel interference occurs when two piconets operate at neighboring frequencies in the same band. The performance in the presence of interference is de- termined by measuring the signal-to-interference (SIR) power ratio. The Bluetooth standard requires that the bit error rate be less than or equal to 0.1% in the presence of an interfering signal at fc + A / C > i for all the SIR ratios shown in Table 2.1 [51]. A / C i i represents the frequency offset of the interfering signal compared to the desired signal. After considering the interfering signal, the ECB representation of the received signal may be given by r(t) = geKM+^slt) + n(t) + i(t) , (2.12) where i(t) denotes the ECB representation of the interfering signal. In the present work, we investigate and implement a novel receiver design for Bluetooth systems, as mentioned in Section 1.3. Since the channel gain can be assumed constant for the duration of one packet, and since detection for different packets is performed indepen- 2.1 The Bluetooth System 19 a[k] G {±1} ai[k] e {±1}- CPM s(t) Modulator eMt) n(t) -0 CPM Modulator i(t) r(t) Figure 2.5: Block diagram representing the 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>*WS(t) + n{t) + i{t) . (2.13) A block diagram of the channel model is shown in Fig. 2.5, were ai[k] represents randomly-generated binary data symbols which are independent of a[k]. The 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 {40.2 + 201og(d) d < 8 m (2-14) 58.5 + 331og(d/8) d > 8 m where Lp is the path loss in decibels (dB). The corresponding received power P R is then obtained by subtracting the path loss from the transmitted power. The receiver sensitivity threshold in Bluetooth is -70 dBm with a nominal 0 dBm transmit power. 2.1 The Bluetooth System 20 2.1.6 Error Correction Bluetooth packets are checked for errors or wrong delivery using the channel access code, the HEC in the header, and the CRC in the payload. In addition, three error- correction schemes are offered: • 1/3 rate 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 de- vices may switch between these schemes as the signal level improves or degrades, thus maintaining performance at different noise levels. The three schemes are described in the following. The 1/3 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 infor- mation. 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 'TYP 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 2.2 Bluetooth Receivers Literature Review 21 as, a\ Ol 02 o2 02 03 03 03 Figure 2.6: Illustration of Repetition code. the generator polynomial g(D) = (D5 © D4 © D2 © 1). If a packet contains less than 10 bits, zeros are appended to the codeword before applying the code. All single and double adjacent errors can be corrected through the 2/3 F E C code [51]. The ARQ scheme is used only on the payload of packets that have CRC. These include DV, DM, 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 CRC packet. If the HEC or CRC of any packet fails, or if no access code is detected, the ARQN bit is again set to NACK. The data payload is retransmitted until a positive acknowledgement is received or a time-out is exceeded [51]. 2.2 B l u e t o o t h R e c e i v e r s L i t e r a t u r e R e v i e w CPM is considered an attractive modulation choice for wireless systems. This is partly due to the continuous phase and memory inherent in the signal, providing protection from channel errors. However, these properties that make CPM attractive also present difficulties for receiver designs. The next sections discuss several approaches for Blue- tooth receiver designs in literature. The optimum receiver is presented first to serve as a lower bound for the suboptimum receivers discussed in the following section. 2.2 Bluetooth Receivers Literature Review 22 2.2.1 Optimum Receiver The optimum receiver discussed here assumes coherent detection and an AWGN chan- nel. It uses the maximum-likelihood sequence detector (MLSD), which determines the most likely sequence of received symbols given all possible combinations of transmitted symbols. The optimum algorithm for performing (MLSD), the Viterbi algorithm (VA) [18], is a trellis search algorithm originally proposed in 1967 for decoding convolutional codes [55]. The optimum CPM receiver exploits the general state trellis structure of CPM to perform MLSD. Due to the memory inherent in a 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 CPM using the VA, as will be described later in the section. Introducing the CPM memory, L, into Eq. (2.2), the information-carrying phase can be appropriately represented as The first term of the right-hand side of Eq. (2.15) constitutes the accumulated phase up to time (n — L)T. It is referred to as the phase state and denoted hereafter by 9[n — L]. Using the fact that 9[n — L] and 9[n — L}mod(2ir) are physically indistinguishable, and conveniently representing the modulation index h in terms of two relatively prime integers m and p as h = m/p, the phase state becomes n—L n k=n-L+l a[k]q(t - kT) nT <t<{n+\)T . (2.15) (2.16) (2.17) 2.2 Bluetooth Receivers Literature Review 23 The second term in Eq. (2.15) is a sum of the phase contribution due to the most recent symbol a[n] and the state vector (a[n — 1], a[n — 2 ] , a [ n — L + 1]). Hence, the state of the CPM signal at time t = nT can be uniquely defined by the combination of the phase state and the state vector as Sn = {9[n — L],a[n — L + 1],a[n — L], ...,a[n — 1]}. Consequently, the number of states Ns in the trellis is { pML~Y (even m) (2.18) 2pML~1 (odd m) We have shown that a CPM signal can be represented by a trellis consisting of a finite number of states, which form what is referred to as a finite state machine (FSM) 2. To perform detection through MLSD, the VA decides on the path metric which maxi- mizes the log-likelihood function log[p(r(£)|s(£))], where p(r(t)\s(t)) is the conditional probability density function (pdf) defined for AWGN channels by - p.(r(t)\s(t))cxexp^J \r(t)-s(t)\2d^ . (2.19) Consequently, the MLSD decision rule is given by +oo a = argmax^- [ \r(t) - s(t)\2dt \ (2.20) a • —oo argmax < +oo +oo +O0 - f \r J \r(t)\2dt- J \s{t)\2dt+ J 2M{r{t) • §*{t)}2dt \ (2.21) -oo where a denotes the estimated symbol sequence, d represents the trial symbol sequence, and s(t) represents the hypothetical transmitted signal sequence associated to the hypothetical symbol sequence d. We notice that in Eq. (2.21) the decision rule is 2A FSM is a model of computation consisting of a set of states, having an input alphabet and a transition function which maps the input symbols and current states to a next state. 2.2 B l u e t o o t h Receivers Li te ra ture Rev iew 24 independent of | r ( £ ) | 2 , whi le \s(t)\2 is a constant t e rm since i t is expressed i n terms of an exponent ia l (2.9). Hence, we can el iminate these terms and disregard the factor of 2 i n the remain ing t e rm since i t has no effect on the decision, resul t ing i n the following MLSD decision rule a = argmax j J fH{r(t) • s * ( t ) } d t | . (2.22) T h e above equat ion shows that the complexi ty of the MLSD increases exponent ia l ly w i t h the length of the symbol sequence a. T h i s can be avoided by using the VA, which introduces a recursively calculated metr ic An(d) as follows (n+l)T An(a)= J M{r(t)-?(t)}dt (2.23) —oo (n+l)T = A n_ 1(d)+ J $l{r(t)-s*(t)}dt (2.24) nT where An_i(d) is the accumulated metr ic of the su rv iv ing sequence up to t ime t = nT and the second t e rm on the r ight -hand side the equation, known as the branch metr ic An(d), is given by (n+l)T An(d) = J 3f? J r(t) • exp I -j nT d[n -L} + 2nh ]T a[k]q(t - kT) k=n-L+l dt (2.25) W e note that the branch metr ic is interpreted as fi l tering the received signal r(t) th rough a bank of 2ML matched filters and sampl ing to ob ta in outputs wh ich form sufficient statist ics for detection. W e notice that the complex i ty of the MLSD using the VA increases only l inear ly w i t h the length of the symbo l sequence. T h e VA can be best described w i t h the i l lus t ra t ion of its key steps: add, compare, and select (ACS). T h e decision rule as given by E q . (2.22) shows that we have a m a x i m u m problem. Therefore, the ACS consists of the following steps [8]. 2.2 Bluetooth Receivers Literature Review 25 Table 2.2: The values of relatively-prime integers m and p corresponding to different modulation indices h within the range specified by the Bluetooth standard. h 0.28 0.29 0.30 0.31 0.32 0.33 0.34 0.35 m 7 29 3 31 8 1 17 7 P 25 100 10 100 25 3 50 20 1. Add: At time t — (n + 1)T, we examine the two branches which lead to a common state. For each of these branches, we add the calculated branch metric to the corresponding accumulated metric. 2. Compare: The result of the sums generated in step 1 are compared. 3. Select: The branch (and corresponding data symbol) associated with the maximum path metric is selected. After performing the ACS step, the branch metric corresponding to the selected sym- bol, also known as the surviving metric, is retained. The VA repeats the ACS step till the end of the symbol sequence, with one out of M branch metrics selected at each state. Hence, at each time instant, the VA computes all possible pML (or 2pML) metrics A n(d) corresponding to all ML possible sequences and p (or 2p) possible phase states. This results in pML~x (or 2pML~l) surviving sequences at each time instant of the VA. The value of the final accumulated path metric corresponds to the maximum path metric value for all possible sequences of symbols. The symbol sequence esti- mate corresponding to this maximum path metric d constitutes the output of the VA. Considering the special case of Bluetooth with binary GFSK and L = 2, the received vector is required to pass through 8 (= 2ML) filters prior to sampling. The state vector becomes Sn = {9[n — 2],a[n — 1]}. The variables needed to construct the trellis depend on the integers m and p, which are given in Table 2.2 for various values of h (in the range specified in the Bluetooth standard). Obviously, the minimum values for m and 2.2 Bluetooth Receivers Literature Review 26 Input symbol = -1 Input symbol = 1 [5TT/3,-1] •' i = 0 i = T p ^ s t a t e s {*,3,*,5*/3} {0,2^/3,4^/3} i = 2T i = 3T {TT/3, 7r/57r/3} {0,27r/3,47r/3} Figure 2.7: Phase state trellis structure for Bluetooth with h = 1/3. p result in the least complex receiver. To illustrate the complexity of the optimum receiver even in the simplest scenarios, we choose the case of h = 1/3. For Bluetooth with h = 1/3, there would be 12 states in the VA; each state is a unique combination of a[n — 1] G { ± 1 } and 8[n — 2] G {0,7r/3, 27r/3, 7T, 47r/3,57r/3}. At each state, two trial symbols d[n] = +1 and a[n] = - 1 are considered and the 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 2.2 B l u e t o o t h Receivers L i te ra ture Rev iew 27 different phase states at the next symbol interval . F r o m Table 2.2, we notice that even a slight var ia t ion of h can lead to a to ta l ly different trel l is structure. Therefore, assuming a fixed nomina l h is not an opt ion i n this design. Moreover , the complex i ty of the o p t i m u m receiver, defined by the number of states pML~l (or 2pML~1), increases exponent ia l ly as M and L increase, and tremendously varies w i t h h. It also assumes perfect channel phase es t imat ion, which is not va l id i n practice (cf. Sect ion 2.1.5). W e conclude that the o p t i m u m receiver performance may be considered as a benchmark, but it is definitely not a feasible so lu t ion 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 an extremely high complexi ty , B l u e t o o t h devices often employ a s imple d iscr iminator receiver to recover the G F S K modula ted data. T h i s receiver uses a l imi te r -d iscr imina tor integrator, and, therefore, is referred to hereafter as the L D I receiver. It is adopted i n B l u e t o o t h due to i ts low computa t iona l and s t ruc tura l complexi ty, mak ing i t s imple to implement at a low cost. However, the L D I suffers from a considerable performance loss (> 6 d B ) over the o p t i m u m receiver, as i l lus t ra ted i n Sect ion 1.3. Since the in t roduc t ion of B lue too th devices i n the market , several receiver designs have been proposed i n l i terature. T h e LDI -based designs i n [47], [48], have only a s l ight ly increased complexi ty ; however, they offer only a slight increase in performance ( < 1 d B ) . These include a receiver based on zero-crossing demodula t ion w i t h a decorrelat ing matched filter [48], termed B T - Z X M F . An o t h e r design proposed by the same authors uses least squares-based post- integrat ion filtering [47]. T h i s approach is further ex- tended i n [49] by employing a max- log -max imum l ike l ihood ( M L M ) symbol detect ion which involves a forward-backward a lgor i thm on a 4-state trel l is , and the resul t ing de- tector is termed M L M - L D I [49], [50]. A l t h o u g h a significant performance improvement 2.2 Bluetooth Receivers Literature Review 28 (~ 3.5 dB) is observed, the design complexity generated by the postprocessing and the 4-state trellis is not acceptable for the low-cost Bluetooth devices. As was observed in [49], sequence detection (SD) introduces tremendous performance gains since the memory of the CPM is properly taken into account. Consequently, many authors have considered trellis-based detection using the forward-backward al- gorithm [49] or the Viterbi algorithm [53], [43], [31]. A sequence detector based on Laurent's decomposition, termed Maximum aposteriori probability (MAP) receiver, was proposed in [43]. It consists of a filter matched to the first Laurent pulse, fol- lowed by a 4-state trellis decoder. It performs better than all the previously-described receivers, reaching performance gains of approximately 6 dB. However, it employs co- herent detection, where a perfect channel phase estimate is assumed, and, therefore, it is impractical. In another SD design, [53], the trellis states in the Viterbi decoder were reduced to two by eliminating the effect of additional phase states. The main disadvantage of these trellis-based designs is that they assume a certain nominal value for the modulation index h. Since in Bluetooth h is allowed to vary in a relatively wide range (cf. Section 2.1.4), assuming an incorrect h at the receiver tremendously degrades the performance. Recently, a noncoherent sequence detector (NSD) which uses an adaptive algorithm that estimates the modulation index h has been proposed [31]. It achieves a performance gain similar to the MAP receiver with a 2-state trellis. However, none of the sequence detectors proposed in literature offer performance gains (relative to the LDI) with complete state reduction. Specifically, a one-state trellis in the NSD receiver was shown to lead to a performance loss of more than 4 dB over the LDI receiver [27]. The two receivers relevant for comparison purposes are the LDI and NSD receivers, and, therefore, will be described in detail in the following. The LDI is considered a benchmark for all designed Bluetooth receivers since it is employed in state-of-the-art Bluetooth devices. The NSD receiver is relevant to the present work since, similar to 2.2 Bluetooth Receivers Literature Review 29 Pre-detetion TF{t) Li miter- Integrate and A * d ( f Hard filter Discriminator Dump Decision a[k] Figure 2.8: LDI detector block diagram. the proposed receiver, it uses noncoherent sequence detection and a modulation index estimator. L D I detector The conventional limiter-discriminator integrator (LDI) detector is illustrated in Fig. 2.8. The received signal is first filtered by a pre-detection bandpass Gaussian filter with im- pulse frequency response [54] where Br is the 3-dB bandwidth of the filter, with an optimum value of 1.1 MHz in an AWGN channel [52]. The output of the Gaussian filter, rp(t), consists of a signal envelope, a distorted signal phase and a filtered noise term. The limiter-discriminator then outputs the derivative of the phase of rp(t), ip'jj,). The integrate and dump filter reintegrates the differentiated phase, producing a phase difference A$<f(£) , which represents the change over a symbol time of the signal phase plus the change in phase noise contributed by the AWGN. Hard decision is then performed on A$d(i), where a bit decision of '1' results if A$d(t) > 0, and '0' otherwise. The LDI detector was simulated in [27], and the obtained results will be used in this work for comparison purposes. N S D This receiver, described in [27], [31], is based on the Rimoldi/Huber&Liu decomposi- tion approach to CPM [41], [25]. It is shown in [41] that a CPM modulator can be 2.2 Bluetooth Receivers Literature Review 30 decomposed into a trellis encoder and a signal mapper. The trellis encoder consists of shift registers with L delay elements and a recursive phase integrator. A time-invariant trellis is achieved by introducing a slope function. Further, using a new reference fre- quency fr and modified unipolar information symbols am[fc], a modified phase state is obtained as [25] n—L Vm[n-L]=[m Om[fc]]mod(p)e{0,l,..,p-1}. (2.26) fc=—oo From Eq. (2.26), we can deduce that the number of phase states is equal to p. Hence, the phase trellis for binary GFSK with L = 2 would consists of p M L _ 1 = 2p states and pML = 4p possible trajectories in each time interval, resulting in 4p time-limited, constant-envelope signal elements Pm(am[k])(t)> where am[k] = {am[k], am[k — 1], tym[n — L]} is a unique address vector associated with each signal element. Applying this decomposition, the ECB transmit signal can be expressed as oo s(t) = Z)Pm(ami])(t-ir). (2.27) i=0 The resulting block diagram of the GFSK modulator using the Rimoldi decomposition approach is illustrated in Fig. 2.9. The filter design for this receiver uses time-limited exponential functions as described in [25], [30]. In case of GFSK signals in Bluetooth, it was found in [27] that the use of only one filter, hi(t) yields excellent performance. The final filter consisted of a cascade of hi(t) and a Gaussian pre-filter. In contrast to the previous designs, the concept of noncoherent detection [11] is adopted, eliminating the need for channel phase estimation. A decision metric which accounts for the time-varying phase and frequency offset was also developed. In addition, an adaptive sequence detector was designed to cope with the varying modulation index h. It was found that a 2-state Viterbi decoder provides significant performance improve- ment over the conventional LDI detector. NSD is shown to be robust against phase variations and frequency offset. The main disadvantage is the complexity required for 2.3 M L S D Lower Bound 31 Signal mapper T m mod(p) e L •ZT] *m[n-2] T f » with 4p signal elements s(t) am[k] am[k - 1] Figure 2.9: Block diagram of the G F S K modulator using the Rimoldi/Huber&Liu decomposition approach. a 2-state trellis search. With complete state reduction, the NSD performs poorly in terms of power efficiency [27]. The results presented in [27] will be used in the present work to illustrate the similarity in performance between NSD and the proposed receiver inspite of the proposed receiver's lower complexity. A lower bound for the performance of maximum likelihood sequence detectors for 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] where dmm 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 . 3 M L S D L o w e r B o u n d (2.28) (2.29) 2.3 MLSD Lower Bound 32 oo where <p(t, 7 ) = 2nh ̂ l[i]q(t - iT) and — a[i] — a[i] is the difference between the actual transmitted symbol a[i] and the hypothetical symbol a[i]. Therefore, for binary symbols G {0, ± 2 } . We notice that this distance depends on the modulation index h. It is clear from Eq. (2.29) that the Euclidean distance is a nondecreasing function of iV, the number of symbol intervals. In [2, Chapter 3], an upper bound for the minimum d, dm\n, is obtained as a function of h. It represents the distance between the pair of sequences of infinite length that merges the earliest for any h. A 'merger' occurs once the phases of two sequences merge at a certain point in time, and coincide for all subsequent time intervals. If the merger between these two sequences occurs after Nm symbol intervals, then the upper bound on the minimum squared Euclidean distance reads k 0 ; Using the above equation, a lower performance bound of MLSD is obtained by substi- tuting (2.30) into (2.28) as Since the proposed receiver employs sequence estimation, this bound is used for com- parison purposes in Chapter 4. 2=0 d2mm(h) = log2(M) \ i / [l-cos<K£,7min)]dt (2.30) (2-31) Chapter 3 Noncoherent Decision Feedback Equalizer Receiver Structure This chapter provides a description and analysis of the proposed receiver structure. The proposed receiver is designed to be structurally simple, yet power efficient and compliant with the Bluetooth specifications outlined in Chapter 2. As the nonlinear structure of the 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 discrete- time model is presented, resulting in an overall intersymbol interference channel. To perform reliable detection, the proposed receiver must then employ an equalization scheme. Several equalization techniques have been proposed in literature, of which the M L S D and Maximum A Posteriori (MAP) are proven to be optimum. The M L S D is implemented using the Viterbi algorithm which determines the most likely symbol 33 3.1 Laurent's Representation of CPM Signals 34 sequence as was detailed in Section 2.2.1. The MAP is implemented using the Bahl, Cocke, Jelinek, and Raviv (BCJR) algorithm [4] which determines the most proba- ble symbol at a given time. However, due to the high computational complexity of these optimum methods, alternative low-complexity suboptimum schemes have been developed. In the present work, we employ a well-known suboptimum equalization scheme, reduced-state sequence estimation (RSSE), described in Section 3.4. Further- more, we adopt a special case of RSSE, decision feedback equalization (DFE), of which an illustration is given in Section 3.5. To obtain a high performance with DFE, a min- imum phase channel with maximum energy concentration in the first taps, is essential. Therefore, we employ an all-pass prefilter that transforms the ISI channel into a min- imum phase channel, which serves as the feedback filter in the proposed DFE. For a practical implementation which accounts for the channel phase variations in Bluetooth signals, noncoherent detection, where the phase is implicitly estimated, is employed to the proposed RSSE and DFE. Section 3.6 describes the techniques used for the channel phase and frequency offset estimation. Moreover, since the modulation index varies in a relatively wide range in Bluetooth, an adaptive /i-estimator is presented to tackle this problem. Finally, a brief summary of the proposed receiver structure is given in Section 3.7. 3.1 L a u r e n t ' s R e p r e s e n t a t i o n o f C P M S igna l s There are two alternative representations of CPM in literature. One representation given by Rimoldi in [41] decomposes the CPM into a trellis encoder and a signal mapper (cf. Section 2.2.2). This approach was adopted for the NSD receiver design described in [27]. As mentioned in Section 2.2.2, the NSD receiver achieves high performance gains with a 2-state trellis, but performs worse than the conventional LDI in case of 3.1 Laurent's Representation of CPM Signals 35 complete state reduction (1-state trellis). Another decomposition approach, referred to as Laurent's decomposition, was pro- posed by Pierre Laurent in 1986 [32]. It represents the CPM signal either exactly or approximately as a linear superposition of a finite number of amplitude modulated pulses (AMP). Laurent's decomposition has been the basis of a considerable amount of research on CPM, including developing noncoherent sequence detection [11] and capacity bounds [56] for CPM. Moreover, using Laurent's representation, the chal- lenges faced in the CPM receiver design due to the nonlinear structure of the signal are eliminated. As a result, it has been shown to be an effective tool for constructing reduced-complexity coherent [28], [10] and noncoherent [24], [11] CPM receivers. More specifically, a Bluetooth receiver based on Laurent's decomposition was proposed in [43], and achieved considerable performance gains over the conventional LDI. However, this receiver employed coherent detection, and the varying modulation index, channel phase variations, and frequency offsets were not addressed (cf. Section 2.2.2). In the present work, we adopt Laurent's decomposition approach. The following sec- tions describe the derivation which gives an exact representation of the CPM signal. Furthermore, we illustrate its application to the Bluetooth GFSK signal, discuss the corresponding optimum receiver, and present an approximate representation of the C P M signal. 3.1.1 Laurent's Decomposition and Its Application to Blue- tooth Laurent developed an alternative representation for binary CPM signals, which ex- presses the CPM signal as a sum of (2L~1 — 1) pulse amplitude modulated (PAM) 3.1 Laurent's Representation of CPM Signals 36 components1. The complete derivation process presented in [32] will be described in the following. For convenience, we define a new phase function <p(t) = 2irhq(t), where q(t) was given in (2.3), resulting in ip(t) = irh for t > LT. The expression of ip(t, a) in the n t h interval (2.15) can now be written as n—L n iP{t,a) = nh ^ a[k]+ ^ a[k]ip{t - kT) , nT < t < (n + 1)T . (3.1) fc=—oo k=n—L+l Recalling the ECB representation of the transmitted signal formulated in (2.9), and observing one signaling interval only, we may use Eq. (3.1) to express s(t) in the n t h interval as n—L n s(t) = exp [jirh Yl a W ) ' II e x P (M^Mt ~ fcT)) . (3.2) fc=—oo k=n—L+l The next step in the derivation is referred to by Laurent as the 'most important step'. It involves the usage of Euler's formula in addition to exploiting the fact that cos(a[k]ip(t— kT)) = cos{<p(t - kT)) and sm(a[k]<p(t - kT)) = a[k] sin(v?(£ - kT)). After several mathematical manipulations, the complex exponential associated to the m t h symbol can be defined as a sum of two terms sin[-7r/i — •</?(£ — mT)\ exp (ja[m]ip(t — mT)^ = sin(7r/i) / . , r n sin[^(* - mr)] { } + exp(jnh a m ) . , , sin(7m) n - L + 1 < m < n , nT < t < (n + 1)T . We may notice that this formulation is not valid for integer values of h since it would result in sin(7r/i) = 0. An alternative solution, proposed in [23], derives the AMP repre- sentation of a CPM signal with integer modulation index. We continue the derivation process by defining a generalized phase pulse function as ( ip(t) t < LT (3.4) irh-<p(t-LT) t> LT 1 I n this work, "Laurent pulses", and "Laurent components", and, in some cases, " A M P (or P A M ) pulses (or components)" , are used synonymously. 3.1 Laurent's Representation of CPM Signals 37 In order to express Eq. (3.3) in terms of one function, we introduce the functions Sn(t) sin (7m) Using these functions, Eq. (3.3) can be rewritten as exp (^ja[m]<f(t - mT)) = SL-m(t) + ex.p(jirh a[m])S-m(t) . (3.6) Introducing the above result in Eq. (3.2) leads to n—L n s(t) = exp [jivh J2 E[ lSL-k(t) + exp (jirh a[k]) S.k(t)] fc=—oo k=n—L+l n-L L-1 = exp (jirh ^2 aik]) H [Si+L-n{t) + exp (jirh a[n - i\) Si-n(t)} . (3.7) k——oo i=0 The product term on the right-hand side of Eq. (3.7) can be expanded into a sum of 2L terms. However, a close examination of the result reveals that only 2 L _ 1 of these terms are distinct functions of time. These form the AMP components that constitute Laurent's representation. To obtain a general form of these impulses, we define the radix-2 representation of the index P L-1 P = Y 1 2 i _ 1 ' IK ' 0 < P < 21-1 - 1 (3.8) i = l where 7 p ; i 6 {0,1}. The AMP pulses are given by L-1 (t) . (3.9) The Laurent pulses Cp(t) are nonzero in the interval [0, min 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 CPM signal in terms of only one impulse as will be shown in Section 3.1.3. Finally, Laurent's representation of a CPM signal in terms of the AMP pulses reads +00 2L~1-l s(t)= bp[k]CP{t - kT) , (3.10) fc=-oo P=0 3.1 Laurent's Representation of CPM Signals 38 Table 3.1: Duration of Laurent components Cp{t). Laurent Component Component duration C0(t) (L + 1)T Ci(t) ( L - l ) T C2(t),C3(t) (L - 2)T C4(t),C5(t),C6(t),C7(t) (L - 3)T C( 2i-l)/2, C( 2 i - l_ i ) T where the complex symbols bp[fc], referred to in [35] as pseudo-symbols, are expressed in terms of the binary data symbols as bp[k] = exp ^jnh ^ ^ a[l] — ̂ a[k — i] • Jp,i^j ̂ — exp (jirh AP[k]) . (3.11) Eq. (3.10) is an exact representation of Eq. (2.9). However, as mentioned earlier Lau- rent's decomposition as derived above is restricted to binary C P M signals. The decom- position approach was further extended in [35] to M-ary signals. Considering the special case of Bluetooth GFSK signals with L = 2 and applying Laurent's decomposition, the resulting linear representation is given by +oo s(t)= (bo[k}C0{t - kT) + b^C^t - kT)) (3.12) fc=—oo where C0(t) = S0(t)Si{t) and Cx(t) = S0{t)S3{t) are illustrated in Fig. 3.1 for h = 1/3. Since L = 2 for Bluetooth, we notice that C0(t) has a duration of 3T, while C\(t) has a duration of T and significantly less energy. The corresponding optimum receiver is described in the next section. 3.1 Laurent's Representation of CPM Signals 39 0.9; t/T-> Figure 3.1: Laurent pulses Co(t) and C\(t), and the main pulse P(t) for Bluetooth GFSK signals with L = 2 and h = 1/3. 3.1.2 Optimum Coherent Receiver Based on Laurent's De- composition Using the linear representation of CPM given in Eq. (3.10), a new optimum coherent detector for binary CPM was developed in [28]. This receiver requires 2 L - 1 matched filters and a Viterbi sequence detector with p2L~l states, as shown in Fig. 3.2. To obtain the VA state vector, we conveniently express the complex symbols (3.16) as follows. When P = 0, b0[k] = b0[k - 1] exp (jirh a[k}), while for P = 1,2, ...2L~l - 1, the complex symbols are represented in terms of &o [k — L] as bP[k -L] = bQ[k -L)Y[ exp (jirh a[k - i\) (3.13) ieip where Ip is a nonempty subset of the set { 0 , 1 , L — 1}. Hence, the state of the CPM signal can be uniquely represented at every time instant by the vector {b0[k — L], a[k — L+l ] , ...,a[k — 2],a[k — 1]}. Recalling that the modulation index h may be represented as h = m/p, we deduce that bo[k — L] can take p values. Taking into account the remaining (L — 1) binary symbols, it is evident that p2L~l states are required for the 3.1 Laurent's Representation of CPM Signals 40 r(i) CJ(-t) Viterbi decoder with states Figure 3.2: Block diagram of optimum coherent receiver based on Laurent's decompo- sition. Viterbi algorithm. For Bluetooth GFSK signals (3.12), the optimum receiver is implemented as follows. The received signal r(t) is fed into a bank of 2 filters CQ( —t) and Cl(—t) matched to the Laurent pulses Co(t) and Ci(t), which provide sufficient statistics for the VA decision metric. The output of each of the matched filters is sampled at every symbol interval and is given at time t = kT by zP[k] = (r(t) * Cp(-t)) \t=kT (for P = 0,1). The VA consists of 2p states, and searches for the symbol sequence that maximizes the total path metric given that the branch metric is expressed as [28] A[fc] = K{z0[fe]bS[fc] + ^ i [ ^ i W } (3.14) where z0[k] and zi[k] are the sampled outputs of the matched filters to CQ(t) and C\(t), respectively, and b0[k] and b\[k] are the trial complex symbols corresponding to the trial sequence of binary symbols a. This metric is used to search the trellis by performing MLSD and outputing the maximum-likelihood symbol sequence as described in Section 2.2.1. Comparing this receiver to the optimum receiver described in Section 2.2.1, we notice that in case of Bluetooth GFSK signals, the number of filters is reduced from 8 to 3.1 Laurent's Representation of CPM Signals 41 2 and the number of states required in the VA is reduced in case of odd m from 4p to 2p. Moreover, the complexity of this receiver may be easily reduced by approximating the CPM signal using fewer Laurent pulses, as shown in [28] for binary CPM and in [10] for multilevel CPM. However, the resulting performance would of course be suboptimum. Regardless, the number of trellis states are still dependent on the modulation index h, and coherent detection is assumed. Hence, this approach is not a feasible solution for the Bluetooth receiver design, and, therefore, will not be pursued in the present work. 3.1.3 Main Pulse For reduced-complexity receivers, it is desirable to develop a linear representation of a CPM signal using only one pulse. Co(t) is the most important component in (3.10), for it has the longest duration and contains most of the energy of the signal (as was illustrated in Fig. 3.1). Laurent used this fact to derive a 'main pulse', P(t), which has the same duration as Cn(£), and presents by itself the best approximation of s(t) as s(t)= b[k]P(t-kT) (3.15) k——oo where b[k] = bo[k] = exp j jirh ^ a[l] J (3.16) \ l=-oo J may be represented as differentially-encoded complex symbols b[k] = b[k - 1] exp [jnh a[k}), (3.17) as previously mentioned. The desired main pulse P(t) is obtained through an opti- mization criterion which minimizes the average energy of the difference between the exact signal s(t) and its approximation s(t). P(T + mT), 0 < r < T, is given by 3.1 Laurent's Representation of C P M Signals 42 Real components 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15 t/T Imaginary components I - 1 1 I I ^ — • — i — " " - - ^ ^ ^ ^ Exact CPM Signal Approximate CPM Signal Using Main Pulse i i i ,1 1 . I I I ' 1 1 1 1 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15 t/T Figure 3.3: The imaginary and real components of an exact Bluetooth signal s(t) and an approximate signal s(t) generated using only the main pulse P{t). P(T + mT) = Pm(r) with [32] j m—2 oo Pm(r)= — r ^ - J] c o s ^ r - H T ) ) J] cos (rm - ip(r - iT)) ^ ' i=—oo 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 Eq. (3.18) closely, we may notice that P(t) is zero for t < 0 and t > (L + 1)T, which proves our previous statement that P(t) is the same duration, as Co(t). Laurent gives an alternative representation for the main pulse in [32] where it is shown to be a weighted superposition of time-shifted versions of the Laurent pulses Cp(t). In both representations, we notice that the main pulse depends on the modulation index h. Hence, varying h affects both the complex symbols b[k] as well as the main pulse P(i). Also note that the representation of C P M as given in (3.15) may be modelled as a 3.2 Filter Design 43 phase shift keying (PSK) signal constellation of p points, and the main pulse may be considered as the pulse shaping filter. Hence, the memory of the CPM manifests itself as intersymbol interference. Figure 3.3 shows the negligible difference in the imaginary and real components of an exact Bluetooth signal as formulated in (2.9) and an approximate signal using only the main pulse as given in (3.15), both simulated for the same set of binary data, a, and for h = 1/3. In the following sections, we assume that the Laurent's approximation given by Eq. (3.15) holds with equality, and, thus, will be denoted hereafter by s(t). The optimum demodulator for a linear system with a pulse shaping filter P(t) is known to be the matched filter P*(—t) [38]. The noise sequence at the output of the matched filter is zero-mean Gaussian distributed noise. Although the noise variance is reduced, the noise sequence is correlated since P(t) is not a VNyquist filter in general, and, therefore, the cascade of the main pulse and its matched filter does not satisfy the Nyquist criterion. To obtain uncorrelated additive white noise which is more convenient to deal with, a \/Nyquist receiver filter with the frequency response H(f) is required such that The optimum filter that satisfies this property is the whitened matched filter (WMF) [17]. The 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 #WMF ( / ) of the W M F corresponding to the main pulse P(t) (3.18) is given 3 . 2 F i l t e r D e s i g n (3.19) 3.2 F i l t e r Des ign 44 by P*(-f) # W M F ( / ) = , + p o 1 J ) — • (3-20) J E | P ( / - n / r ) | 2 y r»=—oo T h e W M F depends on the m a i n pulse P(t), which is formulated i n terms of the modu- la t ion index h (cf. Section 3.1.3). Hence, the W M F wou ld ideal ly need to be adjusted according to the modu la t ion index of the t ransmi t ted signal . T o avoid this, and to facil i tate a prac t ica l implementa t ion , we propose replacing the W M F w i t h the widely- used square root-raised cosine ( S R C ) filter. T h e root-raised cosine filter is a v'Nyquist filter obta ined by sp l i t t ing the raised cosine filter into two parts, w i t h the frequency response HsRc{f) of each part being the square-root of the or ig ina l as [37] for | / | < HSRCU) = < - s f ( | / | - ^ ) f o r ^ < | / | < < ^ > . (3-21) 0 for | / | > ^ T h e corresponding impulse response is given by 4 p ( t / T ) cos (TT(1 + p)t/T) + sin (TT(1 - p)t/T) h s R c { t ) = 7Tt(i-(4Pt/m/T ( 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 bandwid th occupied by the signal beyond the Nyqu i s t frequency 1 / ( 2 T ) . In add i t ion , we consider an impor tan t issue in the implementa t ion of a finite impulse response i n the fol lowing. T o ob ta in an ideal frequency response, the impulse response has to be infini tely long. Prac t ica l ly , an infinite impulse response cannot be imple- mented, and, thus, needs to be t runcated. However, t runca t ing i n the t ime domain leads to overshoots and ripples i n the frequency domain , which degrade the perfor- mance, pa r t i cu la r ly i n the presence of interference. T o reduce these undesirable effects, the ideal impulse response may be mul t ip l i ed by a sui table window function which 3.2 Filter Design 45 1 0.8 T 0.6; 5 0.4 0.2 0 -°-26 Ĥamming window 0 (a) I Figure 3.4: Impulse response of SRC, hSRC(t), before Hamming window (a), and after Hamming window (b). allows the resulting impulse response to smoothly decay to zero. Therefore, to increase interference suppression, which is mandatory for Bluetooth devices (cf. Section 2.1.5), we apply a windowing method and choose the widely used Hamming window given by 0.54 + 0.46cos(27rf/ATjj) - XL(NH) < t < Xu{NH) w{t) = <( . (3.23) 0 elsewhere where XL(NH) and XL(NH) represent the upper limit and lower limit of the Ham- ming window, respectively, and the length of the Hamming window is denoted by NH. XL{NH) = ^ and Xu(NH) = ^ if NH is odd, and XL(NH) = XV(NH) = if NH is even. Fig. 3.4 depicts the root-raised cosine filter in the time domain (3.22) before and after the Hamming window. The Hamming window is illustrated by the dashed curve in Fig. 3.4 (a) for a predefined value of NH equal to 64, which is eight times the oversampling time, and is zero otherwise. In the following, we assume that a Hamming window with NH — 64 have been applied to both the 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 3.2 Filter Design 46 10 5 0 1 -5 5f =| -10 6T o 8 -15 -20 -25 -30 -WMF, h = 0.28| -WMF, h = 0.35 - SRC, p = 0.3 -0.5 0 r r - 0.5 Figure 3.5: Magnitude frequency response for WMF, #WMF(/), assuming h = 0.28 and h = 0.35, and for SRC filter, # S R C(/), 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 out- of-band interference suppression. Since the SRC filter is widely used in practice with hardware implementations available, we adopt /ISRC(^) a s input filter for the proposed receiver. 3.3 Discrete-time Transmission Model 47 3.3 Di s c r e t e - t i m e T r a n s m i s s i o n M o d e l 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)*hSRC(t)\t=kT (3.24) Qh = e>m hc[l}b[k - I] + i[k] + n[k] (3.25) with the discrete-time channel impulse response (CIR) hc[k] = P(t) * h S R C ( t ) \ t = k T (3.26) of order qh. n[k] is the sampled noise signal, which is still AWGN, and i[k] represents the sampled interference signal. Furthermore, the time variance of the phase <f>(t) is assumed to be slow compared to the modulation interval T, and, thus, <p[k] = <p(t)\t=kT- Through Laurent's representation (3.15), the memory of the CPM manifests itself as intersymbol interference (ISI), and hc[k] is referred to hereafter as the ISI channel, which may be described by a finite state machine with a trellis diagram. Recalling that the complex symbols b[k] belong to a p-ary PSK constellation (3.16), and since the ISI channel was found to be of length qh +1, the trellis would consist of p^+i states Qh (since /ic|7]6[fc — I] can assume pqh+l different values) determined by the state vector 1=0 b[k] = ^b[k],b[k — 1], ...,b[k — qh] of hypothetical (or trial) symbols b[k]. Assuming the channel phase <f>[k] is known at the receiver, the VA searches for the symbol sequence which minimizes the path metric given that the branch metric at time k is expressed by m = e' jmr[k] - J2 hc[l]b[k - I] 1=0 [ k ] - m \ (3.27) 3.4 Reduced-State Sequence Estimation 48 0.6 T g 0.4 0.2 1 (a) T £ °-4t o 0.2 (b) Figure 3.6: The CIR of the (a) ISI channel hc[k] compared to (b) its minimum phase equivalent h0[k]. which may be easily derived from (2.20) and represents the Euclidean distance between the received signal e~j(t>^r[k] and the hypothetical transmitted signal y[k\. Observing Fig. 3.6 (a) which depicts the significant taps of the overall channel hc[k] (3.26), we can see that hc[k] is of order qh = 2. Taking into account that the appropriate p for modulation indices 0.28 < h < 0.35 can be significantly larger than 8 (cf. Table 2.2), the complexity of MLSD becomes prohibitive. Hence, state reduction is mandatory, and will be employed in the present work using the concepts of reduced-state sequence estimation (RSSE) [16] and per-survivor processing [39]. 3.4 Red u c e d - S t a t e Sequence E s t i m a t i o n Reduced-state sequence estimation (RSSE) [15], [16] is an increasingly popular tech- nique used to perform sequence estimation with a reduced-complexity VA. It provides a good tradeoff between performance and complexity by combining the high perfor- 3.4 Reduced-State Sequence Estimation 49 mance of MLSD with the low complexity of DFE. It truncates the number of states searched by the VA by determining the additional information no longer present in the current state from previous decisions. Moreover, it provides flexibility through a design parameter ns that controls the number of states in the VA, and, as a result, the complexity. In Bluetooth, and due to the varying modulation index, it is essential to achieve state reduction independent of the modulation index h = m/p with potentially large p (cf. Table 2.2). As observed in the previous section, the state vector of the trellis is defined in terms of the complex symbols b[k], which are expressed as (3.16), and are therefore dependent on p. To tackle this problem, we define a trellis state vector reduced to a certain number ns of hypothetical data symbols a[k] as a[k] = [a[k] ...a[k-na + 1]] . (3.28) Therefore, the number of states required for the VA is reduced to 2ns, which is inde- pendent of p. A vector b[k] = [b[k — ns] . . . b[k — qn + 1]] of estimated symbols b[k] is associated with each state (b[k] = [b[k — ns]] if ns > qh)- The symbols b[k] are determined by applying per-survivor processing [39], and are calculated as (3.17). The RSSE simply involves partitioning the ISI channel at each time k into two using the parameter ns, such that one part is associated with the hypothetical data symbols a[k] and the other retrieved from the survivor sequence as y[k] = ]T hc[l]b[k - 1} + J2 hc[l)b[k - I] (3.29) 1=0 l=ns+l where b[k -l]= b[k - ns - 1] exp ( j irh a[n] ) . (3.30) \ n=k—ns / In other words, a state at time k does not carry all the necessary symbols, and the remaining (q^ — ns) required symbols b[k] are determined from the surviving sequence. 3.5 Decision-Feedback Equalizer Structure 50 A constant channel phase will be assumed here, but this constraint will be relaxed in the following sections. The VA branch metric at time k may be developed by applying RSSE in (3.27), resulting in X[k} = where d[k] - hc[l]b[k - I) 1=0 (3.31) .Qh d[k] = e-j*[k]r[k] - hc[l}b[k - 1} . (3.32) (=ns+l The RSSE branch metric (3.31) is controlled by the design parameter ns and the state trellis (3.28) is independent of p . It allows the equalizer to vary from a full VA (ns > qh) to the simplest case of a D F E for ns = 0. The D F E structure will be illustrated in the following section. 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 The decision-feedback equalizer (DFE), which may be interpreted as a one-state VA, is the simplest nonlinear equalizer that constitutes an attractive compromise between complexity and performance. The main idea behind the 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) fF = [/F[0], /F[1], /F[<?F]] of order qF which suppresses the contribution of the precursor ISI. To cancel the ISI in the present symbol, the feedback filter (FBF) fB = [/B[0], /S[1], /B[<7B]] of order qB uses previously-detected symbols, and the output of the F B F is then subtracted from the output of the F F F . Assuming correct previous decisions, the interference from previously detected symbols is removed 3.5 Decision-Feedback Equalizer Structure 51 FFF Decision device FBF a[k — ko] Figure 3.7: Block diagram of decision-feedback equalizer. through the FBF, and the constant k0, known as the decision delay, specifies the number of future measurements processed before any decision is made on the present symbol. The input z[k] to the decision device may be expressed as2 QF QB = J2 fF[l]r[k fB[l]I[k - Z - fed] - fB[0]I[k - fco] (3.33) 1=0 1=1 where / is the vector of previously estimated symbols. Since the 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 FBF filters obviously have a great impact on the D F E performance. Several techniques for computing the F F F and FBF 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 Minimum Phase Channel A minimum-phase overall impulse response is essential to obtain a high performance with RSSE [16], especially in the extreme case of RSSE, DFE. The finite impulse 2Usually a threshold decision is made, where the input to the decision device does not include the last term on the right hand side of (3.33). 3.5 Decision-Feedback Equa l ize r St ructure 52 response ( F I R ) transfer function Hmin(z) of a m i n i m u m phase channel is such that Hmm{z) n a s roots only inside the uni t circle. A m i n i m u m phase channel /imi n[fc] has the fastest decay, ensuring that the energy concentrat ion i n the first samples is max imized where the /imj n[fc] is the m i n i m u m phase equivalent of h[k] and lh is the length of the C I R . Since the dec i s ion . in a D F E largely depends on the first tap (3.33), i t is obvious that a minimum-phase channel is desirable. Ideally, a m i n i m u m phase channel is produced by app ly ing a discrete-time prefilter f[k] w i t h an allpass characterist ic to the sampled received signal 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 channel hc[k] (3.26) is obtained by prefil tering i t by f[k] as where the m i n i m u m phase equivalent, h0[k], represents the last qh + 1 taps of the resul t ing impulse response. T h e allpass filter f[k] alters the phase wi thou t affecting the overal l magni tude response, p roduc ing an A W G N noise sequence at the output . In order not to change the magni tude frequency response, the all-pass filter moves the roots of the F I R transfer funct ion Hc(z) of hc[k] wh ich are outside the uni t circle s imply by reflecting them about the uni t circle, resul t ing i n the roots of i ts corresponding m i n i m u m phase HQ(Z), as i l lus t ra ted i n F i g . 3.8. For a p rac t ica l implementa t ion , a finite-impulse response ( F I R ) approx imat ion of the allpass filter is of interest. A comprehensive review of the state-of-the-art prefil tering strategies is given i n [20]. In this research work, the computa t ion of the prefilter is based on linear predic t ion , as was presented i n [20]. T h i s prefilter consists of the cascade of a matched filter to the channel and a predic t ion error filter, wh ich is calculated v i a the L e v i n s o n - D u r b i n a lgor i thm. It was found to be an efficient prefil tering method for the ISI channel hc[k] i n add i t ion to having low computa t iona l complexi ty. T h e prefilter (3.34) hc[k] * f[k] (3.35) 3.5 Decision-Feedback Equa l ize r St ructure 53 0 Roots of Hc(z) x Roots of HQ(z) \ • O - - : i : ! ® * -5 -4 -3 - 2 - 1 0 1 Real Part Figure 3.8: T h e roots of Hc(z) compared to the roots of its m i n i m u m phase equivalent H0(z). is designed to be of order qp = 6, and was found to s l ight ly vary w i t h h, but only insignificant performance differences incur i f i t is designed for a fixed h. T h e overall channel h0[k] is of order qB = 2 since i t is of the same length as hc[k], as was mentioned earlier, and is shown for a fixed h = 1/3 i n F i g . 3.6 (b). 1.O 1 0.5 0 -0.5 -1 H C 3.5.2 DFE Decision Rule F r o m the previous discussion, it is clear that i n a D F E structure, f[k] wou ld serve as an effective F F F , while h0[k] would be its corresponding F B F , and , hence, i n the present work, we employ f[k] as the F F F and h0[k] as the F B F i n the proposed D F E structure. T o further i l lus t ra te the role of the F B F i n reducing the effect of the ISI caused by previous symbols , we may rewrite r[k] * f[k] as r[k] * f[k] = e * l % [ 0 ] (b[k] + - p h0[l]b[k - l]j + i[k] * f[k] + n[k] * f[k] (3.36) where the t e rm QB J2h0[l]b[k - I] represents the in tersymbol interference. A s s u m i n g coherent detection, the D F E deci- 3.6 Noncoherent Detection 54 sion rule reads a[k — k0] = argmax < §R{dDFE[A;']/io[0]6[/c a\k—ho] ko}}} (3.37) where QB dDFE[k) = e-^r[k] * f[k] -J2h0[l]b[k ko-l]. (3.38) We may notice that the FBF subtracts out the ISI term from the decision. This decision rule corresponds to the branch metric (3.31) with ns — 0. We have to note here that for the proposed receiver the ISI channel is prefiltered to obtain its minimum phase equivalent only in case of D F E although, theoretically, a minimum phase channel is essential for RSSE as well. However, since the ISI channel hc[k] only has 3 significant taps, a 4-state RSSE is interpreted as full state VA, and a 2-state RSSE is not severely affected if the channel is not minimum phase. Consequently, the all-pass filter f[k] is only employed in case of DFE. 3.6 N o n c o h e r e n t D e t e c t i o n So far, we have successfully employed coherent detection using RSSE and DFE, and the results presented in Chapter 4 show remarkable performance results. However, Bluetooth signals encounter channel phase variations due to oscillator phase noise and frequency instabilities of Bluetooth transceivers (cf. Section 2.1.5), and, as a result, coherent detection which assumes reliable phase estimates <j)[k] of <p\k\ is not possible in practice. Noncoherent equalization algorithms based on sequence detection were proposed in [19] and [11]. These schemes are power efficient and highly robust against channel phase variations, for the channel phase is implicitly estimated. The optimum noncoherent MLSD branch metric is given at time k by [11] Evidently, this branch metric has unlimited memory, for at each time k it depends on the entire previous sequence of symbols. To limit the computational complexity, k k \[k] = ]T ( | r M | 2 + \y[v}\2) ~ 2 £ I W M I • (3.39) 3.6 Noncoherent Detec t ion 55 Colavolpe and R a h e l i [11] proposed reducing the branch metr ic memory by t runca t ing the window of observations to a finite value N, N > 2. Hence, (3.39) becomes k k X[k}= I M 2 - 2 E l « > ] l - (3-40) v=k-N+l v=k-N+l T h e t e rm |r[t>]|2 was e l iminated in (3.40) since i t is independent of the decision. A noncoherent D F E ( N D F E ) was further derived from (3.40) i n [45], w i t h the branch metr ic , now subop t imum, given by k k A N D F E [ & ] = E I 2 / D F E H | 2 - 2 I ^ D F E H J / D F E M I (3-41) v=k-N+l v=k-N+l where roFE[fe] is the output of the F F F and T / D F E ^ ] is the output of the F B F . U s i n g the F B F and F F F recommended i n Sect ion 3.5, rcFE[fe] and j/DFE[fe] at t ime k can be formulated as QF rDFE[k] = ^f[l]r[k-l] (3.42) 1=0 and QB yDFE[fc] = 5 > ° [ » - f c o - Z ] (3.43) (=0 where the complex symbols b[k — ko] are calculated as (3.17). Hence, (3.41) becomes fe-i QB h0[l]b[k -ko-l] + ho[0]b[k - k0] i=i (3.44) A N D F E W = E I ^ D F E M I 2 + v=k-N+l -2\C[k-l] + rDFE[k}y*DPE[k}\ where A:- l &-!]= E ^ F E M ^ P E M . (3.45) v=k-N+l QB _ Here, yr)FE[k] = Y^/ ho[l}b[k — ko — l] + ho[0]b[k — ko], b[k — k0 — 1} represent the previously- i=i detected symbols , while b[k — k0] is the hypothe t ica l complex symbol . T h e phase of q^,{[k — 1] consti tutes the estimate of the phase d i s to r t ion caused by the channel, where the number of phase reference contr ibut ions is control led by the finite value N, and, 3.6 Noncoherent Detection 56 therefore, [k — 1] is referred to as the N-ph&se reference. For a time-invariant channel phase, the greater the N, the less noise variance in the phase estimate, and, as a result, the better the performance. However, the computational complexity increases as well due to the increasing number of terms in (3.45). Schober and Gerstacker [44], [45], suggest avoiding this problem by recursively calculating the phase reference as where the design parameter a, 0 < a < 1, acts as a forgetting factor, and q^ef[k — 1] is referred to as the a-phase reference. Similar to JV, in case of constant channel phase, as a increases, the performance improves. However, phase jitters and frequency offsets contribute to a time-varying phase which does not represent the actual channel phase (cf. Section 2.1.5). In this case, the noise variance of the phase estimate will be high, and will accumulate as N or a become larger, degrading the performance. Consequently, the optimum values of N and a vary and depend on the channel parameters, and the phase jitter and frequency offset variations. Assuming constant channel phase, the performance approaches that of coherent detection as a —> 1 and N —• oo. This is an expected result since these values lead to the optimal branch metric (3.39) with no memory truncation. In the following, the metric associated with q?ei[k — 1] is referred to as the A^-metric, while that associated with q"eS[k — 1] is referred to as the a-metric. The performance of the N- and the a-metric was found to be approximately equivalent, precisely according to the following equation [46] However, since the a-metric requires less number of arithmetic operations [46], we adopt the a-metric in the present work. To perform noncoherent RSSE (NRSSE) using the VA, the same concept which in- volves estimating the channel phase is applied. However, in contrast to the NDFE, the q^\k - 1] = aq^[k - 2] + rD F E[/c - l ] & F E [ f c - 1] (3.46) (3.47) 3.6 Noncoherent Detection 57 previously estimated symbols are dependent upon the surviving sequence of the state being considered. The NRSSE decision rule involves minimizing the following branch metric X[k] = \y[k)\2 - 2\q?ef[k - 1] + r[k]y*[k]\ (3.48) where q^[k - 1] - aq^[k - 2] + r[k - l]y*[k - 1] (3.49) and y[k] is calculated as for coherent RSSE (3.29). Therefore, similarly, the NRSSE decision rule is controlled by the design parameter ns. Also similar to coherent RSSE, prefiltering is not employed, as was explained in Section 3.5.2. 3.6.1 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 NDFE decision rule expressed as b[k - k0] = argmin{|dNDFE[/c] - h0[0)b[k - k0}\2} (3.50) b[k-k0] = argmax m{d*iDFE[k]hQ[0}b[k-k0}}\ (3.51) 6[fc-fco) where W W = rDFE[k] - J2h0[l}b[k -k0-l]. (3.52) QB l 9 r e f [ f c - l ] | i = 1 Qref[k — 1] m a 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 NDFE receiver is the final adopted receiver design which refers to the receiver employing the decision rule given by (3.50). We may notice that this decision is similar to the coherent decision metric (see (3.32) and (3.31)) with ns = 0 (since this is a DFE), and with e~j,pWr[k] replaced 1» WEf 3.6 Noncoherent Detection 58 3.6.2 Frequency Offset Estimation The NDFE decision rule presented in the previous section has proven to be extremely power efficient and robust against channel phase variations (cf. Chapter 4). However, for Bluetooth-standard compliant devices, deviations from the center frequency as large as A / T = 0.1 are admissible [51], and these contribute to additional phase variations which are not accounted for in (3.50). Variations of the phase reference which explicitly account for a frequency offset have been proposed in [12],[31]. Applied to qre([k — 1] in (3.46), the modified update equation reads qTe([k] = (aqre{[k - 1] + rD F E[fc]yD F E[fc]) e ^ ^ " 1 ' , (3.53) where the frequency offset estimate e>2wA^k\ = p r ef[fc]/|Pref[k]| follows from Pref[fc] = PPretik ~ 1] + '"DFE[k]2/DFEM('"DFE\k - l]yD F E[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 3.6 Noncoherent Detec t ion 59 as the F B F and F F F filters (cf. Section 3.5) turned out to be almost unaffected by va ry ing h, considerable deviat ions of the decision metrics w i l l occur even for smal l mismatches between the modu la t i on index h assumed at the receiver and the actual h at the t ransmit ter . T h i s is evident from the formulat ion of the complex symbols given by (3.30)'. Hence, even i n case of a correct est imated b ina ry symbol a[k], due to the mismatch i n h, the est imated complex symbol w i l l be incorrect. W e may assume a nomina l value h at the receiver regardless of the modu la t i on index 's ac tual value h. However, results presented i n Chap te r 4, show significant performance degradat ion since h at the t ransmit ter is allowed to vary i n a relat ively large interval (0.28 < h < 0.35). T h i s is especially the case for large values of a and (3 since the errors accumulated from previous estimates are of greater weight. T o account for this problem, we propose to test a set 7i of modu la t i on index hypothe- ses h, and, after an es t imat ion per iod, Ne, to select the "best" one. W e calculate \H\ N D F E accumulated pa th metrics, each for a to ta l of Ne symbols . A pragmat ic ap- proach to determine the best al ternative is to decide for the h that yields the smallest accumulated pa th metr ic . Therefore, the performance as wel l as the complex i ty of this h adapta t ion depends on the number of hypothesis and the length of the es t imat ion per iod . W e employ this adapta t ion technique to the noncoherent decision feedback equalizer presented i n the previous sections, and the resul t ing receiver is referred to as adapt ive N D F E ( A N D F E ) . T h i s adapta t ion process is appl ied to the first packet received, and the est imated modu la t i on index used to detect the rest of the symbol sequences, h is obta ined by the fol lowing decision rule Clear ly , the greater the set H and the es t imat ion per iod Ne, the better the performance of A N D F E . F r o m the results given i n Chap te r 4, we may deduce that four modu la t i on index hypotheses and es t imat ion over 50 symbol intervals is quite sufficient to ob ta in (3.55) 3.7 S u m m a r y of Proposed Receiver St ructure 60 r(t)- Hamming- windowed SRC ~T=^kT k] [r[0],...,r[Ne-l\ h- Estimator UJ LL Q Figure 3.9: St ructure of the proposed B l u e t o o t h receiver. a performance close to the case of h being perfectly known to the receiver. W i t h these parameters, the complex i ty increase of A N D F E compared to N D F E is negligible, but results show significant performance improvements under va ry ing h condi t ions. 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 the present work is shown i n F i g . 3.9. T h e proposed receiver is based on Laurent ' s decomposi t ion which presents the C P M signal as a superposi t ion of P A M pulses (3.10). W e adopt the approxima- t ion wh ich expresses the B l u e t o o t h G F S K signal i n terms of one ' m a i n pulse'. T h e recommended receiver input filter is the S R C filter (3.21), which is a prac t ica l choice tha t h igh ly resembles the 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 window (3.23) is appl ied to the S R C filter. T h e output of the filter is symbol per iod sampled and fed into a s imple noncoherent decision-feedback equalizer, th rough which phase es t imat ion is employed as given i n (3.51). Fur thermore , for robust detec- t ion i n the presence of h igh frequency offsets, frequency offset es t imat ion is employed using (3.53). Moreover , due to the vary ing modu la t i on index (0.28 < h < 0.35) i n B lue - too th systems, we propose using A N D F E (3.55), where an estimate of the modu la t i on index of the t ransmi t ted s ignal is given. T h e chosen m o d u l a t i o n index is the one which maximizes the pa th metr ic over a l imi t ed number of samples Ne, (25 < Ne < 50), and is selected from a predefined set of hypotheses, H, (\H\ = 2 ,4) . T h i s est imated h is then used to detect the rest of the received signal , and the final output is the sequence 3.7 Summary of Proposed Receiver Structure 61 of estimated binary symbols d. Chapter 4 Performance Results and Discussion This chapter presents and analyzes the performance results of the designed Bluetooth receiver described in Chapter 3, which was simulated using C++ and Matlab. We assume the channel model is as given by (2.13), and perform detection for a block of 1500 data points at one time. These blocks are generated until a minimum of 1000 errors is obtained, giving a high level of accuracy. To put the results into context, the proposed receiver is compared to that of the NSD and the LDI, which are considered relevant benchmarks, as discussed in Section 2.2.2. Furthermore, we consider the MLSD bound (cf. Section 2.3) as the theoretical performance limit [2]. The results for the LDI and NSD are obtained from [27], and the MLSD bound is calculated in the present work for different values of h. When comparing the performance of different receivers, we measure the signal-to-noise ratio required to achieve a BER of 10 - 3 since this is the required BER for Bluetooth receivers [51]. Different performance results were observed when the sampling start point was changed. Therefore, when obtaining the discrete-time channel model (3.25), we optimize our sam- pling start point to get the best performance possible using the adopted SRC filter. We 62 4.1 Coherent Detection 63 use an oversampling time of 8, and the total length of the employed Hamming window, and as a result, the receiver input filter, is given by 8T. The first section of this chapter shows the coherent detection performance results for RSSE and DFE. In case of DFE, the F F F and FBF are of orders qF = 6 and qB = 2 (cf. Section 3.5.1), and are computed for a fixed modulation index h = 1/3 throughout the chapter. Moreover, the modulation index of the transmitted signal is also assumed to be 1/3 and known at the receiver and the receiver input filter is the SRC filter, unless stated otherwise. The second section is devoted to noncoherent detection were the channel phase is implicitly estimated in various scenarios. Most of the presented figures investigate the bit error rate (BER) performance as a function of the signal-to-noise ratio (SNR) Eb/N0 in decibels (dB), where Eb is the received energy per bit and No is the one-sided noise power spectral density of the underlying passband noise process. In other cases, the required SNR for BER = 10~3, which is representative in case of Bluetooth devices [51], is recorded for different values of the modulation index h. 4.1 Coherent Detection In this section, we present the results obtained for coherent detection; i.e, assuming a perfectly estimated channel phase at the receiver. These serve as the basis for our filter selection and our evaluation of the performance loss which the proposed receiver suffers as a result of state reduction. 4.1.1 Filter Selection We have selected the SRC filter as the input receiver filter for the proposed receiver. The similarity in the magnitude frequency responses for the W M F and SRC filters was shown in Chapter 3. Furthermore, in this section, to support the decision of selecting 4.1 Coherent Detection 64 6 8 10 12 14 10 log 1 0 (E b /N 0 ) [dB]-> 18 20 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 DFE, using, in both cases, the W M F and the SRC, respectively. For comparison, the BER curves for the MLSD bound and the LDI receiver are also included. We notice that for the 4-state RSSE, as well as the DFE, the curves for the SRC filter almost coincide with those of the WMF. As a matter of fact, a closer observation would reveal that in case of a 4-state RSSE, there is ~ 0.05 dB performance improvement when the SRC is used, while in case of a DFE, there is ~ 0.01 dB performance improvement when the 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. 4.1 Coherent Detection 65 10" 10 10 ' DC LU m 10" 10" 10 ! y i !• MLSD bound • LDI receiver x one-state NSD -a—DFE -e— 2-state RSSE -e>— 4-state RSSE 6 8 10 12 14 16 18 20 22 24 10log 1 0(Eb/N 0) [dB]-> Figure 4.2: Evaluation of the state reduction in the proposed receiver. Coherent de- tection with h — 1/3 is assumed. . • Therefore, the proposed SRC filter is a practical and ideal choice for the input receiver filter, and will be used for the following simulations unless states otherwise. 4.1.2 State Reduction Observing the BER curves in Fig. 4.2 for 2-state RSSE [Eqs. (3.31)-(3.32)], 4-state RSSE and 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 2- state RSSE, which is highly acceptable considering the reduction in complexity. Third, comparing the performance of the proposed receiver and NSD using a one-state trellis (or DFE), we notice tremendous differences with the NSD suffering a performance loss of more than 10 dB. The proposed receiver, on the other hand, outperforms the LDI by more than 5 dB, which is evidently a remarkable result using a one-state trellis. 4.2 Noncoherent Detec t ion 66 12.2 a = 0.8, FBF and FFF set according to h at transmitter 10.4 10.2 0.31 0.32 h at transmitter -> 0.33 0.34 0.35 0.28 0.29 0.3 Figure 4.3: Performance of N D F E w i t h a = 0.8 w i t h F B F and F F F fixed at the receiver, or vary ing according to the value of h at the t ransmit ter . S imi l a r results are expected i n case of noncoherent D F E , which w i l l be considered i n the fol lowing sections. 4.2 Noncoherent Detection In this section, we evaluate the performance of the proposed receiver in case of non- coherent detection, where the channel phase is est imated using the phase reference. W e briefly discuss and demonstrate the performance of N R S S E [Eqs. (3.48)-(3.49)], and focus on the performance of N D F E [Eqs. (3.51)-(3.52)], wh ich consti tutes the final receiver design (cf. Section 3.7). T h e results for M L S D , L D I , and coherent D F E are included for compar ison purposes. T h e modu la t i on index h = 1/3 and is known at the receiver unless stated otherwise. 4.2 Noncoherent Detec t ion 67 10 LU m 10 • - * - • M L S D bound • LDI — e — 4 -state N R S S E , a = 0.8 —±— 2 -state N R S S E , a = 0.8 — e — N D F E , a = 0.8 - A - 2 -state N S D , a = 0.8 6 8 10 12 14 16 18 20 10 log 1 0 (E b /N 0 ) [dB]-> Figu re 4.4: E v a l u a t i o n of the performance of the proposed receiver w i t h state reduct ion i n case of noncoherent detection. 4.2.1 Performance with Constant Channel Phase In this section, the channel phase is assumed to be t ime-invariant . T h e performance under this condi t ion is evaluated w i t h state reduct ion, and for different values of N and a . Moreover , we discuss the effect of the vary ing m o d u l a t i o n index at the t ransmit ter compared to the results obta ined when h is k n o w n at the receiver. W e remind the reader that the results presented i n this work assume the F B F and F F F filters are fixed according to h = 1/3. O n l y negligible, i f any, performance loss incurs compared to the case where h is varied according to the ac tual value at the t ransmit ter , as i l lus t ra ted i n F i g . 4.3. State Reduction W e previously found that the D F E suffers fair ly acceptable performance losses w i t h respect to the 2-state and 4-state R S S E . In this section, we i l lustrate that a s imi lar result is obta ined in case of noncoherent detect ion and compare the performance of the 4.2 Noncoherent Detec t ion 68 proposed receiver to the L D I and the N S D . F i g . 4.4 shows that N R S S E and N S D yie ld s imi la r performance. In fact, the B E R curve of the 2-state N R S S E almost coincides w i t h that of the 2-state N S D . Moreover , we notice that N D F E suffers a performance loss of about 0.3 d B and 0.4 d B compared to 2-state N R S S E and 2-state N S D , re- spectively, and achieves a performance gain of about 5.5 d B over the L D I . A s further state reduct ion turned out impossible for the N S D approach of [27] based on R i m o l d i decomposi t ion, we can deduce that Laurent ' s decomposi t ion is preferable for the design of low-complexi ty B l u e t o o t h receivers. F i n a l l y , we conclude that N D F E provides an excellent tradeoff between performance and complexi ty, m a k i n g it the ideal equaliza- t ion scheme for the proposed receiver. Hence, the fol lowing discussion w i l l focus on the performance evaluat ion of N D F E . P e r f o r m a n c e w i t h D i f f e r e n t V a l u e s o f N a n d a S ta r t ing 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 shown i n F i g . 4.5. A s expected, wi thou t channel phase variat ions, as the value of N increases, the performance closely approaches that of coherent D F E . W e also observe that even for the m i n i m u m value of N = 2, we have a performance improvement of more than 2.5 d B over the L D I receiver. F i g . 4.6 shows the performance using the a-metr ic . W i t h a = 0.95, the performance difference between coherent D F E and N D F E is a negligible 0.06 d B . S imi l a r to N, as a increases, we get better power efficiency Moreover , recal l ing E q . (3.47), wh ich compares the performance of the N D F E using the N- and a-metr ic , i t may be verified by a careful observation of F igs . 4.5-4.6. Specifically, we see tha t the performance for N = 10 is equivalent to that of a = 0.8, and a s imi lar statement is true for N = 2,5 and a = 0,0.6, respectively. However, since the a-metr ic is computa t iona l ly less complex (cf. Sect ion 3.6), the fol lowing discussion w i l l be solely based on the a-metr ic . 4.2 Noncoherent Detec t ion 69 6 8 10 12 14 10log 1 0(E b /N 0) [dB]^ Figure 4.5: Performance of the N D F E using the A^-metric compared to M L S D , L D I , and coherent D F E , and h = 1/3 for a l l cases. 6 8 10 12 14 10log 1 0(E b/N 0)[dB]^ Figure 4.6: Performance of the N D F E using the a -met r ic compared to M L S D , L D I , and coherent D F E , and h = 1/3 for a l l cases. 4.2 Noncoherent Detec t ion 70 1 0 f 1 0 " m •o ~ 1 0 " ^ LU CD O o rx LU m 1 0 " 1 0 "~' ~. frr 7 ~'.".T'.fri'. ~ \/~'.'fr:':~r~.'. ~. : fr r. . " f r : MLSD bound 4 5 o A - 1 0 Figure 4.7: Performance of the N D F E using the TV-metric h — 1/3 in the presence of phase j i t ter . F igure 4.8: Performance of N D F E using the cv-metric h = 1/3 i n the presence of phase j i t ter . 4.2 Noncoherent Detec t ion 71 A s mentioned above, increasing the value of a and N achieves better performance when the channel phase is constant. However, this is not the case i n the presence of phase j i t ter , as can be seen i n F i g . 4.7 wh ich plots the B E R w i t h respect to the s tandard devia t ion of the phase j i t te r a&. In this case, the performance of the proposed receiver deteriorates as N increases since the assumption of constant channel phase is no longer va l id . A s imi lar observation can be made when using the a-metr ic , as shown i n F i g . 4.8. Therefore, the op t ima l value of N or a depends on the channel condi t ions, and we may conclude that N D F E w i t h N between 5 and 10 or a between 0.6 and 0.8 is power efficient and gives satisfactory performance i n the presence of phase j i t ter . Performance with Different Modulation Indices In the previous sections, we have assumed h = 1/3 at the t ransmi t te r and receiver. However, since the G F S K signal depends on h [Eqs. (2.1)-(2.2)], even w i t h the as- sumpt ion that h is known at the receiver, different performance results are obta ined for different modu la t i on indices. Specifically, the performance deteriorates as h de- creases since fd (cf. Sect ion 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, and 0.35, along w i t h the corresponding M L S D bounds, the L D I , and the M L M - L D I 1 . T h e performance of N D F E , wh ich is equivalent to a one-state trel l is gives remarkable results when compared to the M L S D bound w i t h a performance loss of only about 1.2 d B . Reca l l that the o p t i m u m receiver (cf. Sect ion 2.2.1) requires 100 trell is states for h = 0.28 and 80 trel l is states for h = 0.35, whi le the o p t i m a l receiver based on Laurent ' s decomposi t ion (cf. Sect ion 3.1.2) requires 50 and 40 states for h = 0.28 and h = 0.35, respectively. T h e performance gains compared to the L D I and M L M - L D I are more than 5 d B and 1 d B , respectively. J The results for the M L M - L D I are taken from [49]. 4.2 Noncoherent Detec t ion 72 10 CE UJ CD 10 MLSD bound NDFE with a = 0.8| LDI MLM-LDI * h = 0.28 • h = 1/3 • h = 0.35 16 18 20 2 4 6 8 10 12 14 10log 1 0(E b/N 0) [ d B ] - » Figure 4.9: Performance of the N D F E w i t h a = 0.8 for different values of the modula - t ion index compared to the M L S D bound , L D I , and M L M - L D I . Performance with Unknown Modulation Index So far, we have assumed the modu la t i on index h to be known at the receiver. In B l u e - too th devices, the va ry ing modu la t i on index (0.28 < h < 0.35) poses a serious challenge for the receiver design. T o support this statement, we show the B E R obtained for an S N R of 11 d B for different values of a = 0.4, 0.6, 0.8, and 0.9 when a fixed modu la t i on index h = 0.28, 0.30, 1/3, and 0.35 is assumed at the receiver. T h e s imulat ions are performed for different values of the ac tual modu la t ion index of the t ransmi t ted s ignal , and the obta ined results are shown i n F i g . 4.10. It is clear that as h at the receiver deviates from the ac tual h of the t ransmi t ted s ignal , the performance degrades. W e also notice that the performance loss w i t h the h dev ia t ion is more severe for greater values of a. T h i s is due to the fact that a mismatch of h at the receiver takes longer to forget i n case of a greater forgetting factor a. Hence, a lower value of a provides more robustness against variat ions i n the modu la t i on index. Consider , for example, a = 0.4, a nomina l value of h at the receiver wou ld result i n very slight changes from the case 4.2 Noncoherent Detec t ion 73 • h = 0.28 • h = 0.30 - h = 1/3 • h = 0.35 - h known at receiver a = 0.4 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 h at transmitter -> a = 0.8 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 h at transmitter - » a =0.9 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 h at transmitter -> 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 h at transmitter - » Figure 4.10: Performance of the N D F E for vary ing h w i t h a = 0.4, 0.6, 0.8, and 0.9. h = 0.28 at receiver h = 0.30 at receiver T m ^ 10 m 10 C T 10° m •o £ 10" 10" .O JJ "~o ra 10" o o 6 10" rr LU m 10" V - NSD, a = 0.6 • V NDFE, a = 0.6 - V — NDFE, a = 0.8 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 h at transmitter - » : 1/3 at receiver 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 h at transmitter -> h = 0.35 at receiver a - NSD, a = 0.6 •a NDFE,a = 0.6 s— NDFE, a = 0.8 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 h at transmitter -> 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 h at transmitter - » Figure 4.11: Performance of N D F E for vary ing h w i t h a = 0.6 and 0.8 compared to that of N S D for a = 0.6. 4.2 Noncoherent Detec t ion 74 when h is known at the receiver but s t i l l achieve more power efficiency than the L D I . T o evaluate the proposed receiver's robustness against h variat ions w i t h respect to the al ternative noncoherent sequence detector i n l i terature, namely, N S D , we show a relative compar ison i n F i g . 4.11. T h e B E R curves obta ined for S N R = 11 d B are shown for various modu la t i on indices h = 0.28, 0.3, 1/3, and 0.35 at the receiver, and are p lo t ted for a vary ing h at the t ransmit ter . W e choose the curves for N S D w i t h a = 0.6 since i t 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 compared to of N D F E w i t h a = 0.6 and 0.8. T h e presented results show that as h varies, a much greater devia t ion is observed i n the B E R curves for N S D compared to those of N D F E , b o t h w i t h a = 0.6. Cons ider ing the case when h = 0.28 at the receiver, as h varies at the t ransmit ter , the corresponding B E R is almost constant i n case of N D F E , wh ich is not the case for N S D . Specifically, the resul t ing percentage of error varies between 0.25% and 0.5% i n N D F E , and between 0.4% and 12% i n N S D . Moreover , we even observe less change i n the B E R of N D F E for a = 0.8 compared to N D F E w i t h a = 0.6. W e conclude that N D F E is more robust against h variat ions compared to N S D . A more representative evaluat ion of the effect of va ry ing m o d u l a t i o n index is obta ined by considering the required S N R for a B E R of 10~ 3 . W e adopt the same scenario as before. F i g . 4.12 shows the required S N R for a B E R of 10~ 3 as a function of the modu la t i on index h for N D F E w i t h a = 0.8 and for the M L S D bound and the L D I for compar ison purposes. T h e figure also i l lustrates the ideal case of h known at the receiver. A s can be observed, severe power efficiency losses result when h is unknown at the receiver. T h e variat ions i n the modu la t i on index may be m i n i m i z e d by es t imat ing h at the t ransmit ter using A N D F E described i n Section 3.6.3, and the corresponding results w i l l be presented later i n this chapter. 4.2 Noncoherent Detec t ion 75 i i i 1 i i J 0.29 0.3 0.31 0.32 0.33 0.34 0.35 h at transmitter —> Figure 4.12: T h e required 101og10(Eb/N0) for B E R = 1 0 " 3 for N D F E w i t h a = 0.8 w i t h va ry ing modu la t i on index. 4.2.2 Performance Using the Modified Phase Reference U n - der Constant and Varying Channel Conditions T h e more realist ic and thus more relevant t ransmiss ion scenario considers a t ime- vary ing channel phase <p(t). In the present work, we account for channel phase var ia- t ions th rough the modif ied phase reference [see Eqs . (3.53)-(3.54)] formulated i n terms of two forgetting factors a and (3. Th roughou t this chapter, i f qief[k] from (3.46) is used, only the value of a is specified, and i f c7ref[fc] from (3.53) is appl ied, the values of b o t h a and (3 are given. Performance With Constant Channel Phase Before considering a t ime-variant channel phase, we first show the performance of the modif ied phase reference when the channel phase is constant. F i g . 4.13 il lustrates the performance for different values of a and /3 as a funct ion of the modu la t i on index, 4.2 Noncoherent Detec t ion 76 0.28 0.29 0.3 0.33 0.34 0.35 0.31 0.32 h at transmitter -> Figure 4.13: 10 iog i 0 (Eb/N0) required for B E R = 1 0 - 3 for different combinat ions of a and (3 as a funct ion of the modu la t ion index w i t h constant channel phase and h known. wh ich is assumed to be known at the receiver. W e notice that there is a loss in power efficiency when (3.53) is used compared to when (3.46) is appl ied due to the phase noise in t roduced by the frequency est imat ion. M o r e specifically, the gap between a = 0.6 and the pair (a = 0.6, j3 = 0.9) is approximate ly 0.4 d B . T h i s loss for the ideal case of a constant channel phase has to be accepted to achieve a h igh performance i n the more realist ic scenario of a t ime-vary ing phase. Moreover , we observe that as (3 increases the performance of qief[k] f rom (3.53) closely approaches that of qTef[k] from (3.46). It was found that for various scenarios, the pair ( a = 0.6, (3 = 0.9) appeared to be a favorable choice. W e further observe that the robust N D F E w i t h ( a = 0.6, (3 = 0.9) outperforms the L D I detector by more than 4 d B , and i t is s t i l l somewhat more power efficient t han the M L M - L D I detector, which requires a four-state forward-backward a lgor i thm. F ina l ly , we note that N D F E w i t h frequency-offset es t imat ion performs very s imi lar to the more complex N S D w i t h the same values of a and (3. 4.2 Noncoherent Detec t ion 77 P e r f o r m a n c e w i t h V a r y i n g M o d u l a t i o n I n d e x A s for the effect of the (3 factor on the vary ing modu la t i on index, the result ing per- formances of N D F E w i t h (a = 0.6,(3 = 0.9) and (a = 0.8,(3 = 0.9) are shown i n F igs . 4.14 and 4.15, respectively. T h e curves represent the required S N R for B E R = 10~ 3 for fixed values of the modu la t i on index at the receiver as funct ion of a vary ing modu la t i on index at the t ransmit ter . W e notice that the performance degradat ion re- sul t ing from a mismatch i n h encountered using (3.46) persists when using (3.53). A s expected, when a decreases the S N R differences between unknown h at the receiver and known h decreases. Regardless, using the m o d u l a t i o n index est imator (cf. Sect ion 3.6.3) is advisable, and w i l l be presented i n the Sect ion 4.2.3. P e r f o r m a n c e W i t h V a r y i n g C h a n n e l P h a s e T h e modif ied phase reference (3.53) was developed to account for the extreme (yet admissible) phase variat ions and frequency offsets of A / T = 0.1 i n B l u e t o o t h devices. W e now evaluate the performance of N D F E using this modif ied phase reference i n the presence of frequency offset and phase j i t ter . F i g . 4.16 shows the required S N R for B E R = 10~ 3 as a function of the normal ized frequency offset. T h e curves are parameterized w i t h a s tandard devia t ion of a A = 0° (no j i t ter) and a A = 2° , 5° for the phase j i t ter . T h e L D I detector fails for offsets A / T > 0.03 . . . 0.05, but we note that D C offset cancel la t ion methods could be appl ied to mi t igate the effect of frequency offset. W e further observe that 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, considering the pair ( a = 0.6,3 = 0.9), the N D F E w i t h phase reference (3.53) allows power-efficient t ransmiss ion even for extreme offsets. In add i t ion , the m a x i m u m performance degradat ion when assuming an add i t iona l phase j i t te r w i t h a A = 5° is not more than 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) and ( a = 0.8,(3 = 0.9) and N S D w i t h the pair (a = 0.6,(3 = 0.9), 4.2 Noncoherent Detec t ion 78 0.31 0.32 0.33 h at transmiter -> 0.35 Figure 4.14: T h e required 101ogio(£ 6 / iVo) for B E R = 1 0 " 3 for N D F E w i t h a = 0.6, B 0.9 for va ry ing h. 18i 1 1 t 17 CO I O Ii 16 or LU ™ 15 o "O 2 14 — N D F E , a = 0.8, p = 0.9, fixed h -e— h known at receiver \ h = 0.32 V • - _ _ _ \h = 0.33 \h = 0.34 )l 0.31 0.32 0.33 h at transmiter -» 1 - 3 0.35 Figure 4.15: T h e required 101ogi 0(£?6 . /JVo)'for B E R = 10~ 3 for N D F E w i t h a = 0.8,8 0.9 for va ry ing h. 4.2 Noncoherent Detec t ion 79 8 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 - - * *. "0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 AfT —> Figure 4.16: T h e required 101ogi0(j5fe/A ro) for B E R = 10~ 3 i n the presence of frequency offset and phase j i t ter . - NDFE, a = 0.6, (3 = 0.9 NDFE, a = 0.8, P = 0.9 NSD, a = 0.6, P = 0.9 a =0" 16 15 14h 13 12, 11 0.04 0.05 A f T - » 0.06 0.07 0.08 0.09 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 AfT -> Figure 4.17: T h e required 101og 1 0 (£ 'b /^ v o) for B E R = I C T 3 i n the presence of frequency offset and phase j i t ter . 4.2 Noncoherent Detec t ion 80 which was found most favorable i n [27]. W e observe that the performance deteriorates w i t h increasing phase j i t te r as a increases (both w i t h 8 = 0.9). T h e N D F E w i t h the pair (a = 0.6,3 = 0.9) was found to have comparable 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 var iat ions is one of the reasons why this pai r is considered to be the favorable choice for N D F E . 4.2.3 Adaptive N D F E T h e performance of N D F E severely deteriorates when there is a mismatch between the value of h assumed at the receiver and the ac tual h at the t ransmit ter , as was i l lus t ra ted i n the previous sections (cf. F igs . 4.12, 4.14, 4.15). W e now evaluate the performance of A N D F E (cf. Sect ion 3.6.3), where several N D F E s are executed for a number of hypothe t ica l modu la t i on indices, and the decision metrics are accumulated. W e consider different combinat ions of an es t imat ion per iod Ne = 25 or 50, and a set of 2 or 4 hypothesis H. W h e n \H\ = 2, the hypothe t ica l m o d u l a t i o n index set H = { 0 . 3 , 1 / 3 } , while H = {0.28,0 .30,0 .32,0 .34} w i t h \H\ = 4. In the fol lowing, we adopt the no ta t ion A N D F E ( A r e , \H\) to convey the es t imat ion per iod and hypothe t ica l modu la t i on 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 index used at the t ransmit ter for N D F E w i t h h known at the receiver and A N D F E w i t h h unknown at the receiver. T h e phase reference (3.46), wh ich does not account for frequency offset variat ions, is used for each of the values a = 0.6 and a = 0.8. Observ ing the results obta ined for a — 0.8, we notice that the curves for N D F E and A N D F E ( 5 0 , 4 ) are very close, and the A N D F E shows a remarkable performance improvement compared to the case where h is unknown at the receiver and equal to 0.32, reaching 3 d B when h at the t ransmit ter is 0.29. For a = 0.6 it is i l lus t ra ted that even for a smal l es t imat ion per iod and modu la t i on index hypothesis of Ne = 25 and \7i\ = 2, the A N D F E is s t i l l 4.2 Noncoherent Detec t ion 81 0.31 0.32 h at transmitter • 0.35 Figure 4.18: 101ogi 0 (£( , / iVo) required for B E R = 1 0 " 3 for A N D F E w i t h phase reference (3.46). T h e channel phase is constant and the no ta t ion A N D F E ( A r e , H) applies. robust. A g a i n , the results for A N D F E and adaptive N S D ( A N S D ) 2 w i t h a = 0.6 are almost ident ical , as has been observed for the various aforementioned scenarios. In F i g . 4.19, we i l lustrate the performance of A N D F E for different pairs of a and 8 when the phase reference (3.53) w i t h frequency offset es t imat ion is used. For comparison, the results of A N D F E ( 5 0 , 4 ) w i t h phase reference (3.46) and a = 0.6 are also shown. W e make the fol lowing observations. F i r s t , i t is evident that the adapt ive version of N D F E ( A N D F E ) w i t h an es t imat ion per iod Ne = 50 combined w i t h \7i\ = 4 and h unknown at the receiver achieves a result almost ident ica l to N D F E w i t h h known, as i l lus t ra ted for the pair (a = 0.6,/? = 0.8). Second, the results obta ined for almost a l l combinat ions of a and 8 form almost straight lines, ind ica t ing that they are close to the h known case. T h i r d , as 8 approaches 1, the performance of A N D F E using the phase reference (3.53) approaches that of (3.46), wh ich is consistent w i t h the results 2 T h e A N S D is the N S D receiver wi th an adaptive ^.-estimator. 4.2 Noncoherent Detec t ion 82 h = 0.32 at receiver o=0,6,p=0,9 15h 10 . ANDFE(50,4) a = 0.6, P = 0.6 - • - LDI, h = 1/3 - * - MLSD bound ~ANDFE(50,4) a=0.6, p = 0.9 ,̂ • NDFE, h known at receiver t'r^-;-^^;;^^^^^ : u ̂ 0 6, P = 0.9 : ANDFE(50,4) a = 0.6 i ANDFE(50,4) / : »< ' - •* a = 0.6, P=0.99 t ANDFE(50,4) a = 0.8, :P = 0.9 - * - ~ 4 0.29 0.3 0.31 0.32 h at transmitter - 0.33 0.34 0.35 Figure 4.19: 101og i 0 (Sb /A r 0 ) required for B E R = 1 0 - 3 for A N D F E w i t h various com- binat ions of a and 3 w i t h phase reference (3.53). T h e channel phase is constant and the no ta t ion A N D F E ( A T e , H) applies. 18i 1 1 1 1 1 1 t CO o 3 05 L J o o 17 16 15 1 1 12 H i 10 9 0.28 LDI detector ANDFE(50,2) o •- 0.6, p. = 0,9 -e— NDFE, a=0.6, p=0.9, h known at receiver -e— ANDFE(50,4), o=0.6, p=0.9 -0 - NSD,a=0.6, p=0.9, h known at receiver - -a - ANSD(50,4), o=0.6, p=0.9 0.31 0.32 h at transmitter - 0.35 Figure 4.20: 10\ogw(Eb/N0) required for B E R = 10~ 3 for A N D F E and A N S D w i t h the favorable pair (a = 0.6, 3 = 0.9) and w i t h constant channel phase. 4.2 Noncoherent Detec t ion 83 previously shown 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 adapt ive N S D ( A N S D ) , which employs an adaptive scheme s imi lar to that of A N D F E . T h e results show very close performance between the adaptive schemes of N D F E and N S D when the same values of a and 3 are used (the m a x i m u m difference is about 0.4 d B ) , even w i t h a smaller set of \H\ = 2. W e can thus conclude that N D F E allows h ighly power-efficient detect ion also for a priori unknown modu la t i on index. 4.2.4 Performance in the Presence of Interference D u e to the operat ion of B l u e t o o t h i n the I S M band, evaluat ing the proposed 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 that the signal-to-interference power ra t io (SIR) and the carrier fre- quency difference A / C i j are chosen as specified i n the B l u e t o o t h s tandard and presented i n Table 2.1. T h e no ta t ion I F ( S I R , A / c i ) is used to convey the S I R i n d B and Afc>i i n M H z . F i g . 4.21 shows the performance results for the o p t i m a l W M F and the proposed S R C filter, where h = 1/3 is known at the receiver. T h e effective interference suppres- sion provided by bo th filters is verified, w i t h the B E R for adjacent channel interference almost equal to the curve obta ined when no interference is considered. T h e results for b o t h filters are quite s imi lar , w i t h the S R C outperforming the W M F i n case of adjacent channel interference, and the reverse is observed w i t h co-channel interference. A s a mat ter of fact, when the S R C filter is appl ied, there is no loss due to adjacent channel interference for the cases IF(-30,2) , and IF(-40,3) . Moreover , the B E R curve for I F ( - 40,3) is shown when the S R C is appl ied wi thou t the H a m m i n g window, and a loss of ~ 0.5 d B is observed compared to the case when the H a m m i n g window is appl ied. F i g . 4.22 i l lustrates the performance of the A N D F E i n the presence of interference as a funct ion of the modu la t i on index. W e observe that the results of A N D F E are almost ident ical to those for N D F E i n the presence of adjacent channel interference. A s for 4.2 Noncoherent Detec t ion 84 10 V IF(11,0) • IF(0,1) * IF(-30,2) 0 IF(-40,3) -with SRC — with WMF Figure 4 the S R C 10 12 14 10log1 0(Eb/N0)[dB]^ 21: Performance of N D F E i n the presence of interference for the W M F and filter. T h e channel phase is constant and the no ta t ion I F ( S I R , A / C i i ) applies. 26 r 24 22, tr LU m o 20 0> CT 18 CD LU o o 12 10 ANDFE, IF(11,0) \ \ : 13 N * - - N _ ^ - ^ ^ • N i — ~ - . • • LDI, 1 1: . F(11,0) A NDFE, IF(11,0) ~ - 1 i - 1 _ _ LDI, no interference i - - _ ~ -< _ ANDFE, IF(0,1) NDFE, IF(-30,2), IF(-40,3) and no;interference NDFE, IF(0,1) ANDFE, IF(-30,2), IF(-40,3) and no interference 0.29 0.3 0.31 0.32 h at transmitter -> 0.33 0.34 0.35 Figure 4 presence applies. .22: l O l o g i o l ^ / i V o ) required for B E R = l f T 3 for N D F E and A N D F E i n the of interference. T h e channel phase is constant and the no ta t ion I F ( S I R , A / C i j ) 4.2 Noncoherent Detec t ion 85 co-channel interference, an accountable performance loss of up 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 han the L D I for h < 0.295. However, i n practice, co-channel interference is a rare occurrence, and, adjacent channel interference is of greater interest i n general. Chapter 5 Conclusions T h e complexi ty constraints coupled w i t h the vary ing modu la t i on index and the high fre- quency offset variat ions i n B lue too th devices creates a h igh ly chal lenging environment for the design of power-efficient B l u e t o o t h receivers. T h e conventional L D I detector is a s imple receiver, but is h ighly s u b o p t i m u m i n terms of power efficiency, and can not cope w i t h the high 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 terature have proven to be extremely power efficient compared to the L D I detector since they proper ly take the C P M mem- ory into account. These include the M L M - L D I and the M A P receivers. However, the M L M - L D I has very high complexi ty and the M A P receiver assumes perfect channel phase es t imat ion and knowledge of the modu la t i on index at the receiver, and, there- fore, they are imprac t i ca l . T h e N S D receiver recently proposed i n [31] tackles these problems by designing a noncoherent sequence detector w i t h a m o d u l a t i o n index esti- mator . A l t h o u g h the complexi ty of the N S D is low compared to the M L M - L D I and the M A P receivers, i t s t i l l requires a 2-state V i t e r b i decoder. In this research work, we have presented a s imple noncoherent power-efficient receiver design for B l u e t o o t h t ransmission based on Laurent ' s decomposi t ion, wh ich transforms the ac tual nonlinear modu la t i on scheme into a l inear modu la t i on over an in tersymbol 86 87 interference channel. It was shown that the combina t ion of an S R C filter, symbol - rate sampl ing , and a decision-feedback equalizer achieves a performance close to the theoret ical M L S E l i m i t . W e have proposed a noncoherent version, namely N D F E , wh ich offers h igh robustness to local-osci l lator dynamics and phase j i t te r present i n B l u e t o o t h devices. N D F E was also extended to al low adapta t ion of the decision metr ic to an a priori unknown modu la t i on index h. T h e presented s imula t ion results showed that N D F E performs very s imi lar to 2-state N S D proposed i n [31]. T h e key advantages of the proposed design are out l ined i n the fol lowing and a summary of the corresponding s imula t ion results presented i n Chap te r 4 is given. • U s i n g Laurent ' s decomposi t ion, the op t ima l receiver input filter for the result ing P A M signal is the whi tened matched filter ( W M F ) [17]. However, since the W M F is dependent on the modu la t i on index h, we propose replacing i t by the prac t ica l S R C filter which has s imi lar characteristics. T h e s imulat ions (cf. F i g . 4.1) prove that the resul t ing performance using the W M F and S R C are almost ident ical . Moreover , h igh adjacent channel interference suppression is achieved through the S R C , as i l lus t ra ted i n F i g . 4.22. • Fo l lowing the S R C filter, a noncoherent decision feedback equalizer is employed, where the phase is i m p l i c i t l y est imated using the developed phase reference (3.46). T h i s phase reference is formulated i n terms of a forgett ing factor a and shows remarkable performance improvement compared to the L D I , closely approaching the M L S D lower bound as a increases (cf. F i g . 4.6). However, since this phase reference fails i n the presence of extreme frequency offset variat ions and phase j i t ter , a modif ied phase reference (3.53) is devised. A g a i n , the modif ied phase reference is control led by a forgetting factor 8, and the resul t ing decision rule was found to be extremely robust against channel phase variat ions, as shown i n F i g . 4.16. T h e pair (a = 0.6, 8 = 0.9) which was found to 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 and 3. • T h e N D F E is the simplest sequence detector possible, consist ing of a one-state trell is , thus, p rov id ing an extremely high reduct ion i n complexi ty compared to the 2p-state trel l is required by the op t ima l detector. O n l y a slight performance loss was recorded by decreasing the trell is f rom a full-state to a one-state trell is (cf. F i g . 4.4). T h i s is a remarkable result since the one-state trel l is i n the N S D receiver was shown to have worse performance than the low-complexi ty L D I de- tector [27]. • One of the p r imary disadvantages of the sequence detectors proposed i n l i terature is assuming a nomina l modu la t i on index h at the receiver. In the new receiver design, we propose employing an adaptive ^.-estimation scheme w i t h an adapta- t ion per iod of 7Ve symbols , and a set Ti of hypothe t ica l modu la t i on indices. T h i s adaptive version of N D F E , which does not require knowledge of the modu la t ion index h, was found h ighly effective w i t h i V e = 50 and \Ti\ = 2 or 4, only adding a slight increase i n complexi ty. T h e s imula t ion results presented i n F igs . 4.19-4.20 show that i t performs almost ident ical ly to the case when h is known, and very close to the adapt ive version of N S D w i t h the same values of Ne and \H\. In conclusion, the new receiver design is h igh ly a t t ract ive for a prac t ica l implemen- ta t ion , achieving a performance gain of more than 4 d B over the L D I detector w i t h only a slight increase i n complexi ty, and a performance loss of on ly about 2 d B com- pared to the M L S D lower bound w i t h an extreme decrease i n complexi ty. 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