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Noncoherent sequence detection receiver for bluetooth systems Jain, Mani 2004

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NONCOHERENT SEQUENCE DETECTION RECEIVER FOR BLUETOOTH SYSTEMS by Mani Jain B.Tech., Regional Engineering College Kurukshetra, India, 1995 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of Applied Science in THE FACULTY OF GRADUATE STUDIES (Department of Electrical and Computer Engineering) We accept this thesis as conforming to the required standard The University of British Columbia September 2004 © Mani Jain, 2004 lUBCl THE UNIVERSITY OF BRITISH C Q H . M P I A FACULTY OF GRADUATE STUDIES Library Authorization for scholarly purposes m a y b e g r a n t e d b y l h „ h e a d „ , _ . . °y ' ™ 8 9 r , e , h a l " e ™ » « > " e*tonsi»e copying „ , , h i s l h e s i s p - » c a l i o „ o n h i S 1 , s l s f o r , i r o n C i a , g a : s : : : : ; : : ~ Name of Author (please print) Date (dd/mm/yyyy) Title of Thesis Degree: M A C ^ ' L Z U L ^ . Year: _ _ ^ ^ l _ _ L _ _ _ _ _ _ _ _ _ _ _ Department of jxQ * * n The University of B r i t i ^ I u n ^ ~ ^ ^ C^k^^*-i=£a_^ Vancouver, BC Canada L ) ~ gfad.ubc.ca/forms/?formlD=THS page 1 of 1 /as/ t/pcfe Abstract Bluetooth is an increasingly popular and widely deployed standard for wireless personal area networks (WPAN). The Bluetooth physical layer employs Gaussian frequency shift keying (GFSK), which is a particular form of continuous phase modulation (CPM). GFSK provides a favorable trade-off between power and bandwidth efficiency, and allows for low-complexity transmitter and receiver implementations. A simple discriminator detector is used to recover the GFSK modulated data in Bluetooth devices. Though structurally and computationally simple, discriminator detectors are very power inefficient. Coherent sequence detectors are significantly more power efficient for modula-tion schemes with memory, since the memory introduced by CPM is properly taken into account. However, realization of sequence detection (SD) for Bluetooth systems is very difficult because the modulation index h is allowed to vary in a wide interval. This varying modulation index leads to a varying trellis structure for SD with possibly a large number of states. The design of the optimal receiver filter for a sufficient statistic for SD after sampling is also dependent on h. The receiver filter design is further complicated by the operation of Bluetooth systems in a license-free band, thereby, requiring the designed receiver filter to be robust against interference from other devices operating in the same band. Moreover, coherent SD requires phase synchronization, which is a difficult task as well because of the frequency hopping radio of Bluetooth and the allowed local oscillator dynamics. Several approaches to a simple and power-efficient receiver design for Bluetooth have ii iii been discussed in the literature. The drawbacks of these approaches are that either the achieved power efficiency is insufficient or perfect channel phase estimation at the receiver has been assumed. These designs are restricted to a particular value of h and the effects of high frequency offsets at the receiver oscillator on the performance have not been accounted for. Therefore, the practical applicability of these receivers is limited. In the present research work, a novel noncoherent SD (NSD) receiver for Bluetooth sys-tems is proposed. The receiver design is based on the decomposition approach to CPM and the concept of noncoherent sequence detection of CPM. A low-complexity implementation of the receiver is presented with only one receiver filter and NSD on a two-state trellis, which accomplishes significant performance gains of more than 4 dB over the discriminator-based detector. The proposed receiver caters to the requirements of Bluetooth systems compre-hensively in that (a) the entire range of possible h is considered and an adaptive solution to account for varying h is provided, (b) a frequency offset compensator is incorporated into NSD to cope with the large local oscillator frequency deviations allowed in Bluetooth sys-tems, and (c) improved decoding methods for the forward error correction (FEC) schemes employed in Bluetooth are devised. Simulation and analytical results verify that the pre-sented NSD receiver operates close to the theoretical limits. The proposed receiver is robust and simple, and therefore, is an attractive solution for Bluetooth devices. Contents Abstract ii List of Figures x List of Tables xi Acknowledgments xii 1 Introduction 1 1.1 The Bluetooth System 3 1.2 Challenges and Motivation 5 1.3 Contributions 7 1.4 Thesis Organization 7 2 Background and State-of-the-Art 9 2.1 Bluetooth System Model 9 2.1.1 GFSK Modulation 10 2.1.2 Channel and Interference Model 12 2.1.3 Error Correction 15 2.2 Optimum Receiver 16 2.2.1 Optimum Receiver in the Presence of Random Phase in the Channel 19 2.2.2 Lower Bound for Performance of MLSD 20 iv CONTENTS v 2.3 Benchmark Receivers for Bluetooth 22 2.3.1 LDI Detector 23 2.3.2 MLM-LDI Detector 23 3 Noncoherent Sequence Detection Receiver for Bluetooth 25 3.1 Rimoldi/Huber&Liu Representation of GFSK 26 3.1.1 The Decomposition Approach to CPM 27 3.1.2 Receiver Structure 29 3.1.3 Application of the Decomposition Approach to Bluetooth 33 3.2 Filter Design 34 3.3 Noncoherent Sequence Detection 36 3.3.1 Rectangular Windowing 38 3.3.2 Exponential Windowing 38 3.3.3 Frequency Offset Estimation 39 3.4 State Reduction 41 3.5 Adaptive Noncoherent Sequence Detection 47 3.6 Channel Coding 49 3.6.1 Error Analysis 49 3.6.2 Joint NSD and Decoding for Rate 1/3 RC 53 3.6.3 Modified Decoding for Rate 2/3 HC 56 3.6.4 LDI Detector with Modified Decoding 58 3.7 Summary of the Proposed Receiver Structure 59 4 Results and Discussion 61 4.1 Performance in an AWGN Channel for Uncoded Transmission 62 4.1.1 Filter Selection 62 4.1.2 NSD with Implicit Phase Estimation 64 CONTENTS vi 4.1.3 State Reduction 67 4.1.4 Conclusions 68 4.2 Performance for Uncoded Transmission 69 4.2.1 Adaptive NSD Performance 69 4.2.2 Performance in the Presence of Channel Phase Variations 73 4.2.3 Performance in the Presence of Interference 79 4.3 Performance with Channel Coding 81 4.4 Packet Transmission over Fading Channel 84 4.5 Summary 85 5 Conclusions and Future Work 88 5.1 Conclusions 88 5.2 Recommendations for Future Work 91 Glossary 92 Bibliography 98 List of Figures 1.1 General packet format [BTS03] 3 1.2 Performance comparison of LDI and optimum receiver 6 2.1 Bluetooth baseband system model 14 2.2 Time-variant trellis of phase state for h = 1/3, M = 2, and L — 2 17 2.3 Block diagram for LDI receiver 23 3.1 Time-invariant trellis of phase state for h = 1/3, M = 2, and L = 2 28 3.2 Equivalent representation of CPM modulator using decomposition approach. 30 3.3 Matched filter receiver for CPM using decomposition approach 32 3.4 Equivalent GFSK modulator using decomposition approach 33 3.5 (a) Magnitude frequency responses for different filters, (b) Noise autocorre-lation function (/?„„[«] after filter he(t) and sampling 35 3.6 Two-state trellis structure 42 3.7 One-state trellis structure 43 3.8 An example for the path trellis for h = 1/3 to illustrate state reduction. . . . 43 3.9 Full-state path trellis for two error event example 44 3.10 Two-state path trellis for two error event example 45 3.11 One-state path trellis for one error event example 47 vn LIST OF FIGURES viii 3.12 Error pattern frequency for (a) NSD with two-states at 11 dB with a = 0.6, B = 0.9, and (b) LDI detector at 16 dB. (Error patterns in a block of 10 symbols: 1-'010'; 2-'0110'; 3-'01010',4-l010010', 5-'0100010', 6-others, where '1' represents an error.) 50 3.13 BER for uncoded transmission and WER for transmission with 1/3 rate repetition code and (15,10) Hamming code over AWGN channel. Ana-lytical results according to Eq. (2.25) (BER), Eq. (3.40)(WER-RC) and Eq. (3.41)(WER-HC) 5 1 3.14 Error-gap distribution (EGD) Pr{0m|l} for AWGN channel. E G D L D I : LDI detector output at 16 dB, E G D F S , N S D : Full-state NSD with a = 0.6, (3 = 0.9 at 11 dB, E G D 2 S , N S D : Two-state NSD with a = 0.6, Q = 0.9 at 11 dB, and E G D B S C : ideal BSC with approximately same BER as LDI and NSD 5 2 3.15 The most likely error events for the considered channel 5 3 3.16 (a) Joint NSD and decoding of repetition code with original two-state trellis (b) modified joint NSD and decoding using modified trellis 5 5 3.17 Normalized minimum Euclidean distance <S^mlRc as a function of modulation index h for different coding and decoding schemes 56 3.18 Proposed receiver structure 5 9 4.1 Performance comparison of different receiver filters: Coherent detection, full-state trellis, h = 1/3, AWGN channel with constant phase 63 4.2 Performance of NSD using ./V-metric with filter he(t) and h — 1/3 in an AWGN channel with constant phase 65 4.3 Performance of NSD using iV-metric in the presence of phase jitter with filter he(t) and h = 1/3 in an AWGN channel 65 4.4 Performance of N S D using a-metric with filter he(t) and h = 1/3 in an AWGN channel with constant phase 66 LIST OF FIGURES ix 4.5 Performance of NSD using a-metric in the presence of phase jitter with filter he(t) and h = 1/3 in an AWGN channel 66 4.6 Performance with reduced number of states in the Viterbi processor: h = 1/3, AWGN channel with constant phase 67 4.7 Performance comparison of NSD, LDI, and MLM-LDI detectors for different values of h 68 4.8 Performance of the proposed receiver for varying h with a = 0.8 in an AWGN channel with constant phase 71 4.9 Performance of the proposed receiver for varying h with a = 0.6 in an AWGN channel with constant phase 71 4.10 Performance of the proposed receiver for varying h with a = 0.4 in an AWGN channel with constant phase 72 4.11 The required 101og 1 0(£ s//vo) for BER = IO - 3 for the proposed receiver for varying h with a = 0.6 in an AWGN channel with constant phase 72 4.12 Performance comparison of the proposed receiver with ANSD for unknown h with a = 0.6 in an AWGN channel with constant phase for different combination of A^ and \H\ represented by ANSD(Are,|7i|) 74 4.13 Performance of proposed NSD receiver in the presence of frequency offset with h= 1/3 76 4.14 Performance of proposed NSD receiver in the presence of frequency offset and phase jitter with variance o_(T) 76 4.15 Performance of proposed NSD receiver for different values of (3 with a — 0.6 and h = 1/3 in an AWGN channel with constant channel phase 78 4.16 Performance the proposed NSD receiver for unknown h in an AWGN channel with constant phase. Eq. (3.32) for phase estimation has been used with a = 0.6 and (3 = 0.9 if not stated otherwise 78 LIST OF FIGURES x 4.17 Performance of ANSD with \7i\ = 4, Ne = 50 as a function of h in an AWGN channel with constant channel phase 79 4.18 Performance of the proposed ANSD with \H\ = 4, Ne = 50, and R = 0.9 in the presence of interference using the representation IF(A/ c i in MHz,CIR in dB). he(t) is used if not stated otherwise 80 4.19 Performance of the proposed NSD receiver with a = 0.6, B = 0.9, h = 1/3 and rate 1/3 RC for an AWGN channel with constant phase 83 4.20 Performance of the proposed NSD receiver with a = 0.6, B — 0.9, h = 1/3 and (15,10) HC for an AWGN channel with constant phase 83 4.21 Packet error rate in Rayleigh fading channel for DM3 and DH3 packet types [BTS03]. h = 1/3, NSD with phase reference (Eq. (3.32)) with a = 0.6 and 8 = 0.9 85 4.22 Packet error rate in Ricean fading channel with K = 3 for DM3 and DH3 packet types [BTS03]. h = 1/3, NSD with phase reference (Eq. (3.32)) with a = 0.6 and B = 0.9 86 4.23 Packet error rate in Ricean fading channel with K = 10 for DM3 and DH3 packet types [BTS03]. h = 1/3, NSD with phase reference (Eq. (3.32)) with a - 0.6 and B = 0.9 86 List of Tables 2.1 Interference performance [BTS03] 14 3.1 Branch metric table for h = 1/3 46 3.2 The modified syndrome table for (15,10) Hamming code 58 4.1 Representation of modulation index h for NSD 73 xi Acknowledgments It is a pleasure to convey my gratitude to my research supervisor Dr. Robert Schober and co-supervisor Dr. Lutz Lampe for guiding me throughout this research work. Their constant encouragement and critical assessment of the thesis work helped me finish a chal-lenging project. The various discussions we had from time to time were fruitful and the team work was instrumental in allowing me to acquire the required skills and achieve the proposed research objectives. The courses offered by Dr. Robert Schober on 'Detection and Estimation of Signals in Noise' and by Dr. Cyril Leung on 'Communication and Informa-tion Theory' helped me build strong fundamentals in digital communications and were of immense value in allowing me to think independently. I thank National Sciences and Engineering Research Council (NSERC) and Dr. Robert Schober for funding this project. I would also like to thank the colleagues at the department of Electrical and Computer Engineering, UBC, for creating a stimulating and a friendly environment at work. xii Chapter 1 Introduction The growing dependence of modern man on portable electronic devices created the need to allow these devices to communicate with each other. The sophisticated wireless technology of today provides the solution for convenient wireless connection of electronic devices, giv-ing rise to wireless personal area networks (WPAN). Bluetooth™ 1 is a WPAN technology that enables devices like cellular phones, personal digital assistants, laptops, desktop com-puters and printers, to communicate with each other at a raw bit rate of 1 Mega bits per second (Mbps) over a short range without using cables [BTS03]. There is a multitude of other useful applications of Bluetooth enabled devices like home and industrial automation, entertainment devices, toys, digital cameras, headsets etc., that can benefit people in many ways [SheOl]. The important differentiating features of Bluetooth devices from other short range wireless technologies are their low cost, robustness, low complexity, and low power consumption [SheOl]. The need for an improved global solution for short range wireless communications led Ericsson, IBM, Intel, Nokia, and Toshiba to guide research efforts towards developing the Bluetooth industry standard in 1998 [BisOl]. A Special Interest Group (SIG) was formed by these five companies aimed at creating a user-friendly, flexible, and efficient standard, : The B L U E T O O T H trademark in owned by Bluetooth SIG, Inc., USA [SheOl]. 1 2 which was later joined by over 3000 other companies [BT.org]. The standard is named after the 10th century Danish king Harald Blatand (Bluetooth in English), who had been instrumental in uniting warring Scandinavian people just as Bluetooth wireless technology allows collaboration between computing, mobile phone, and automotive industries [BT.org]. Since then Bluetooth has grown into a well accepted technology in the telecommunications industry. Bluetooth technology is complementary to the 802.11 wireless local area network (WLAN) technology as the latter caters to higher data rates in bigger areas at higher cost and power consumption. The emerging technology for high data rate short range wireless communications, known as "ultra-wideband", targets similar applications as Bluetooth, but the standard is still in the developmental stage. Bluetooth devices already have a big market and the number of Bluetooth enabled devices entering the market is growing every year. According to the studies made by a market research firm called In-Stat/MDR [BT.com] [Zdnet], the number of Bluetooth chip shipments reached 69 million in 2003, up from 35.8 million in 2002. Shipments are expected to exceed 100 million in 2004 and with the continuing trend are likely to surpass 510 million by the year 2006. The promising market for Bluetooth devices is the motivating factor to explore the pos-sibility of an improved receiver design for Bluetooth systems. The state-of-the-art receiver for Bluetooth systems uses the noncoherent limiter-discriminator with integrate and dump filter (LDI) [SVOla], which although being very simple and robust, is highly power ineffi-cient. The aim of the present research work is to design a more power efficient receiver, complying with the requirements of the Bluetooth standard. The following section provides an overview of the Bluetooth system and specification as defined in the standard [BTS03]. The second section of this chapter states the objective of the thesis work more precisely including its motivation and the associated challenges. The contributions of this work are briefly summarized in the third section, and the concluding 1.1 The Bluetooth System 3 section outlines the organization of this thesis. 1.1 The Bluetooth System Bluetooth devices can communicate with each other using point-to-point or point-to -multipoint radio links within a range of 10 m extendable to 100 m [SVOlb]. An inter-connection of up to eight such devices forms a piconet. The Bluetooth network may be expanded by interlinking several such piconets together to form a scatternet. For communi-cation of voice and data within piconets and scatternets, Bluetooth provides asynchronous connectionless (ACL) links using packet switching and synchronous connection oriented (SCO) links which use circuit switching [BisOl]. For voice traffic, SCO links are used, which can support voice traffic at 64 Kbps. Time slots at regular intervals are reserved for SCO traffic. For asynchronous data transmission, the ACL provides best-effort links. The information between the Bluetooth devices is exchanged using packets. A packet can occupy up to five time slots with each time slot having a duration of 625 us. Each packet is transmitted over a different hop frequency. The general format of a packet is shown in Fig. 1.1. LSB 68/72 bits 54 bits 0-2745 bits MSB ACCESS CODE HEADER PAYLOAD Figure 1.1: General packet format [BTS03]. The packet consists of a 68 or 72 bit long access code, a 54 bit long header, and up to 2745 payload bits. The access code consists of 4 preamble bits, a 64 bit sync word and an optional 4 bit trailer. The generation of the sync word is based on the device's address. Therefore, the sync word is known to the communicating devices, and is used for synchronization and timing acquisition. The header contains relevant link information. Depending upon the 1.1 The Bluetooth System 4 requirements of the communication process, 15 different packet types have been defined. Error correction schemes are also defined for each packet type. 1/3 rate FEC, 2/3 rate FEC or an automatic repeat request (ARQ) scheme for data is used for reliable error free transmission of packets. To guarantee the compatibility between different radios used in the system and to define the quality of the system, the Bluetooth specifications define the following requirements for Bluetooth radios [BTS03]: • Bluetooth devices operate in the license free industrial, scientific and medical (ISM) 2.4 GHz band. • The radio employs frequency hopping spread spectrum (FHSS). The radio hops on a packet by packet basis in a pre-determined sequence at a frequency of 1600 hops/second on 79 channels of 1 MHz bandwidth, located in the 2.400-2.4835 GHz frequency range. The specifications permit a reduced channel hop over only 23 chan-nels for countries that have restrictions in their ISM band [BTS03]. • Gaussian frequency shift keying (GFSK) modulation is used to map binary data symbols to modulated symbols. • Time division duplexing is employed for full duplex transmission. • Three classes of Bluetooth devices are defined depending on the maximum transmitter power. Class 1, 2, and 3 devices have maximum transmit powers of 20 dBm (100 mW), 4 dBm (2.5 mW), and 0 dBm (1 mW), respectively [SheOl]. • The maximum allowed raw bit error rate is specified as 0.1%. 1.2 Challenges and Motivation 5 1.2 Challenges and Motivation The objective of this research work is to design a power efficient, low cost, and robust receiver for Bluetooth devices. The optimum receiver for GFSK modulated signals consists of a correlator followed by a maximum likelihood sequence detector (MLSD) [ProOl, Ch. 5]. However, the trellis structure for sequence detection (SD) is dependent on the modulation index h, which is allowed to vary in a broad interval of 0.28 to 0.35 in the Bluetooth standard [BTS03]. This relaxation results in a varying trellis structure for SD possibly with a large number of states that differs with h, and the complexity of SD increases with the number of states (cf. Section 2.2). Hence, the realization of MLSD for Bluetooth systems involves a complex receiver structure. On the other hand, for practical implementation it is necessary to keep the receiver structure simple and cost effective. Acquisition of a phase reference for coherent MLSD is also a difficult task as Bluetooth systems allow up to 100 KHz carrier frequency deviation [BTS03]. The permitted variation of the modulation index and the frequency hopping make it even more difficult to estimate the phase reference explicitly. Therefore, in this work, noncoherent sub-optimum sequence detection schemes are investigated for the detection of Bluetooth signals. The design of the optimal receiver filters which are necessary to obtain a sufficient statis-tic after sampling, is also dependent on h [ARS81]. Moreover, the operation of Bluetooth in the ISM band makes it vulnerable to interference from other users of Bluetooth, WLANs, and cordless phones, operating in the same frequency band. Therefore, a careful receiver filter design is needed to suppress co-channel and adjacent channel interference as far as possible. As has been mentioned earlier, the popular receiver for Bluetooth systems uses the LDI detector which offers a very cost effective, structurally simple, and robust solution for 1.2 Challenges and Motivation 6 Bluetooth devices. However, the performance of the LDI detector is highly sub-optimum which is verified by the bit error rate (BER) vs. 10\ogw(Es/N0) graph of the LDI detector and optimum coherent MLSD performance shown in Fig. 1.2, where Es denotes the average symbol energy of the transmitted signal and /V0 is the single-sided power spectral density of the noise. The comparison of the LDI detector and optimum coherent MLSD performance reveals that to achieve a BER of 0.1% the LDI detector requires a 6 dB higher signal power than the optimum receiver. This observation indicates the possibility of a reduction in power consumption by a factor of four, thereby achieving increased throughput or better coverage for Bluetooth systems at the same signal power. Inspite of the aforementioned challenges, the large possible power efficiency gains motivate an investigation of an alternative approach for receiver design for Bluetooth systems. 10° 10' .-6 0 2 4 6 8 10 12 14 16 18 lOlog 1 0 ( E / N 0 ) Figure 1.2: Performance comparison of LDI and optimum receiver. 1.3 Contributions 7 1.3 Contributions The main contributions of the present research work are as follows: • A novel receiver design is proposed that shows performance gains of 3 to 5 dB over the conventional LDI detector. The receiver design is based on the Rimoldi/Huber&Liu decomposition of GFSK [Rim88] [HL89] and the concept of non-coherent SD (NSD) [CR99]. A simple receiver structure is obtained consisting of a single filter and a subsequent two-state sequence detector. The reduction to two states for any value of h is achieved using per survivor processing [RPT95] which solves the varying trellis structure problem associated with MLSD. • A new filter design for the novel receiver is proposed to effectively combat interference. • A simple frequency estimation method is devised for robustness against high frequency offsets. • To obtain acceptable performance for all values of h, an adaptive NSD strategy is proposed, which allows for fast adaptation of NSD to the actually used value of h. The proposed strategy offers a low complexity solution to estimate h thereby improving the performance of the detector. • The capability of the (15,10) expurgated Hamming code to correct double errors, is used to modify the decoder to achieve improvement in the performance of NSD for coded transmission. The conventional LDI receiver also benefits from the modification with negligible added complexity. 1.4 Thesis Organization The thesis is organized as follows. 1.4 Thesis Organization 8 In Chapter 2, the required theoretical background for this research work is reviewed. There, the structure of the optimum receiver for GFSK is presented followed by a discussion of the state-of-the-art receivers for Bluetooth systems. Chapter 3 provides the theory and the structure of the proposed receiver. The decompo-sition approach given by Rimoldi/Huber&Liu [Rim88] [HL89] is discussed. Furthermore, the application of NSD to GFSK and the phase estimation techniques are presented. The fre-quency estimation technique devised to provide robustness against frequency offsets is also discussed. Subsequently, a filter design is presented that is extremely effective in combating interference. The varying modulation index h affects the receiver performance adversely. Therefore, a simple estimation algorithm for h is devised which significantly improves the performance. Thereafter, the proposed approach to achieve state reduction is presented followed by the modified FEC decoding algorithm. Chapter 4 discusses the performance of the proposed receiver for different design pa-rameters. A comparison of performance for the available filter designs, phase estimation techniques, and numbers of states directs an appropriate selection of the receiver param-eters. The performance of the designed receiver is then evaluated under a set of different channel conditions to test the effectiveness of the proposed h and frequency estimation tech-niques, along with the modified FEC decoding strategies. A comparison of the performance of the new receiver with that of state-of-the-art receivers is also presented. Finally, the research work is summarized in Chapter 5, and possible future work is suggested. Chapter 2 Background and State-of-the-Art In this chapter, the background information required for the present thesis work is reviewed and the benchmark receivers are introduced. The Bluetooth standard specifies the require-ments for a Bluetooth radio and the expected receiver performance [BTS03]. This research work precisely follows the widely accepted Bluetooth radio specifications to develop a sim-ple and power efficient receiver for short range wireless communications. In Section 2.1 of this chapter the Bluetooth system model is described in detail including the modulation scheme and the channel. Section 2.2 discusses optimum detection and disadvantages of the optimum detector. A theoretical lower bound for the receiver performance is also obtained. The subsequent section introduces the popular LDI detector [SVOlb], and other state-of-the-art receivers for Bluetooth such as the max-log-maximum-likelihood LDI (MLM-LDI) [SWJ04] and the maximum-a-posteriori probability (MAP) receiver [SHS03]. 2.1 Bluetooth System Model This section outlines the specifications required for the receiver design and presents the sys-tem model based on the channel characteristics defined in the Bluetooth standard [BTS03]. 9 2.1 Bluetooth System Model 10 2.1.1 G F S K Modulation As has been mentioned earlier (cf. Section 1.1), Bluetooth employs frequency hopping spread spectrum (FHSS). The implemented modulation scheme is binary partial response Gaussian frequency shift keying (GFSK) which provides a favorable trade-off between power and bandwidth efficiency, and allows for low-complexity transmitter and receiver implemen-tations. GFSK belongs to the family of continuous phase modulation (CPM), which is a non-linear modulation scheme with memory. The high spectral efficiency of CPM is at-tributed to the following characteristics of the modulated signal: • The transmitted signal has a constant envelope, i.e., non-linear amplifiers do not cause out-of-band power. • The phase of the modulated signal is constrained to be continuous which avoids large spectral side-lobes outside of the main spectral band of the signal [ProOl, Ch. 4], This also introduces memory in the modulated signal. The modulated CPM signal can be represented as [AAS86] SRF(t,a) = -^-cos(27Tfct + 4>(t,a)) (2.1) where E denotes the signal energy per modulation interval T, fc is the carrier frequency, a is the random input stream of binary data bits a[i], a[i] £ {±1}- a) is the information carrying phase, given by oo <j)(t, a) = 2nh J _ a[i]q(t - iT) (2.2) i=—oo where h is the modulation index and q[t) = g(r)dT is the phase pulse. For GFSK, the frequency pulse g(t) is obtained by filtering a rectangular pulse with a Gaussian filter h(t) [LVNKM03], i.e., g(t) = h(t) * rect 2.1 Bluetooth System Model 11 where * denotes convoloution and h(t) is a Gaussian low pass filter given by , , , cB ( c\Btf with B as the 3 dB bandwidth of the Gaussian low pass filter, and constant c = 2u /^/ln(2). The rectangular pulse is defined as «* (l) = { llT for l(i < T ' 2 . (2.3) \T) 10 otherwise Hence, the frequency pulse can be expressed as [LVNKM03] 9(t) = ^ [Q(c • B(t - 772)) - Q(c • B(t + 772))] (2.4) where Q(t) is the standard Gaussian Q function, Q(t) — l/y/zlr JT°° e_T2/2<ir. The phase pulse q(t) is normalized such that it is given by { 0 t < 0 f*g(T)dT 0<t<LT , (2.5) 1/2 t > LT where L is the span of frequency pulse g(t). In the Bluetooth standard [BTS03], the time-bandwidth product of g(t) is specified as BT — 0.5 with T = 10 - 6 s. Since q{LT) ~ 1/2 is true for L — 2 [ProOl, Ch. 4], GFSK can be approximated as a partial response CPM with a truncated frequency pulse of duration 2T, i.e., L = 2. Hence, the data symbol is spread over two symbol durations which introduces additional memory and reduces the spectral occupancy of the modulated signal. Introducing L in Eq. (2.2), the phase deviation in the interval nT < t < (n + 1)T can be written as (f>(t, a) = 2irh J _ a[i]q(t - IT) + uh J _ a[i\. (2.6). i=n—L+l i=—oo For Bluetooth devices the modulation index h is allowed to vary in the range of 0.28 to 0.35 [BTS03], which is lower than the classical minimum shift keying (MSK) requirement of 0.5. The sub-MSK modulation index affects the system performance adversely [ProOl]. How-ever, this restriction is imposed to satisfy federal communications commission (FCC) section 2.1 Bluetooth System Model 12 15.247 rule governing frequency hopping spread spectrum ISM devices [FCC]. Hence, as the modulation index value affects the peak frequency deviation, the maximum frequency deviation is between 140 and 175 KHz. For the design of the sequence detector the modu-lation index is appropriately quantized such that h can be expressed as a rational number h = k/p with relatively prime integers k and p [ProOl, Ch. 4]. 2.1.2 Channel and Interference Model Bluetooth systems are designed for short range wireless applications. For such applications the root mean square (rms) delay introduced by the time-spreading mechanism of the channel is below 70 ns for indoor radio at 2.4 GHz [ZW99][JSP96][SV01a], which is very low when compared with the symbol duration of 1 ps for Bluetooth systems. Moreover, at high carrier frequencies, the channel is very slowly varying [SVOla] and the Doppler frequency shift can be ignored. This renders the channel frequency non-selective with a static fading gain for the duration of one packet. Furthermore, a line-of-sight (LOS) path is assumed to be available, therefore, the envelope of the faded signal closely follows the Ricean distribution [JNG02]. The Ricean distribution is given by [Rap96] where g is the envelope of the channel gain, 2cr2 is the variance of the multipath , Pi o s is the peak amplitude of the LOS signal, and I0(.) is the 0 th order Bessel modified function of the first kind. The average channel power gain is assumed to be one, i.e., P\os + 2a2 = 1. The ratio P\os/2a2 is defined as the Ricean factor K. Rayleigh fading is obtained as a special case for P\os —> 0 and the static AWGN channel results for a2 —> 0. The equivalent complex baseband (ECB) received signal in a slow fading channel can be expressed as r(t) = ge?*» • s(t,a) + n(t), 0<t<Tp, (2.8) 2.1 Bluetooth System Model 13 where g is constant over one packet length Tp and is independent from packet to packet. s(t, a) is the ECB transmit signal, n(t) is complex-baseband AWGN and 0o is the time-invariant phase introduced due to fading. s(t, a) is related to the pass-band signal SRF(£, a) by SRF(£, a) = 9ft{s(i, a)ej2nfct} where denotes real part of the complex number x. The Bluetooth specifications allow a ±100 kHz frequency offset between transmitter and receiver oscillator, which is fairly relaxed compared to the 1 MHz bandwidth of the signal. This undesirable high frequency offset A / deteriorates the receiver performance to a great extent as shown in the performance results in Chapter 4. Besides introducing frequency offset, the receiver oscillator also introduces phase jitter, A^ c (£ , r ) . For a given t and r, Acf)c(t, T) can be modelled as a white Gaussian process with variance cr_(r) = (2TT fc)2kr, where k is a constant depending on the employed oscillator [DMR00]. The variance increases linearly with time. The time varying phase introduced by the channel can be expressed as <t>c(t + r) = <f>c{t) + 27rA/r + A<f>c(t, r). (2.9) The received signal in the presence of phase noise and frequency offset becomes r(t) = ge>(Mt)+*o)s(tj _) + n ( i ) ) o < t < Tp. (2.10) As Bluetooth operates in the license-free ISM band, there is no control over the usage of the band. Therefore, other piconets operating independently in the vicinity may cause undesirable interference. The Bluetooth standard specifies the co-channel and adjacent channel interference performance of the receiver [BTS03]. Apart from other Bluetooth devices, interference may also be caused by other users of the ISM band e.g. WLANs. In the present work, only the effect of interference from other Bluetooth piconets, synchronized to the packet boundaries, is considered. The signal-to-interferer power ratio (CIR) for an interferer at fc + A / C ; i , where A / C i i is the frequency offset of interferer compared to the desired signal, is given in Table 2.1. Taking into account this unwanted interference i(t), 2.1 Bluetooth System Model 14 the Bluetooth ECB system model can be mathematically expressed as r(t) = g^^+^sit, a) + i(t) + n(t), 0<t<Tp. (2.11) Table 2.1: Interference performance [BTS03]. Type of Interference A 4 i CIR Co-channel interference 0 MHz 11 dB Adjacent channel (1 MHz) interference 1 MHz OdB Adjacent channel (2 MHz) interference 2 MHz -30 dB Adjacent channel (>3 MHz) interference 3 MHz -40 dB To summarize, the Bluetooth ECB system model can be graphically represented by the block diagram shown in Fig. 2.1, where a[i] is the signal estimated by the receiver. Though the channel is the same for the desired signal and the interferer, the Ricean distributed fading gain g' and phase shift (f>'Q observed by the interfering signal may be different from that of the desired signal as the paths between the transmitter and receiver may not be the same for the two. Transmitter 27th a[i] Gaussian filter with BT = 0.5 Phase Modulator Interfering signal (Bluetooth) Channel gexp(j<t>o) exp(j<t>c(t)) i>—(XV-g'exp(j0o ) Figure 2.1: Bluetooth baseband system model. r(D Receiver 2.1 Bluetooth System Model 15 2.1.3 Error Correction The Bluetooth specifications define different packet types depending upon the application and its requirements of quality and data rate. For error free reception of the packets three error correction schemes are defined, which are described as follows: 1. An automatic repeat request (ARQ) scheme is defined to acknowledge the correct reception of the payload of a packet. A cyclic redundancy check (CRC) code is added to the packet to determine if the received packet is error free or not. If the packet is not correctly received, it is retransmitted until it is received correctly. ARQ is used for payload protection of all the data packets and the special control packet for frequency hop synchronization (FHS). 2. A simple 1/3 rate FEC scheme is defined for the protection of header and high quality voice (HV1) packets. Each bit is repeated three times to implement this repetition code. Though the overhead is high, it provides better reliability as compared to uncoded or (15,10) Hamming coded transmission. Therefore, the header is always 1/3 rate FEC protected as it carries important link information. Since it is undesirable to retransmitted the voice packets, HV1 packets are also repetition coded to maintain good quality for the voice traffic. 3. A 2/3 rate (15,10) expurgated1 Hamming code, is defined to protect medium rate data packets (DM1, DM3, DM5), data voice (DV) packets, the voice packet type HV2, and the FHS control packets. This code is used to correct single errors [BTS03] in a codeword of 15 bits. Five parity bits are appended to a block of 10 information bits. The calculation of parity bits is as follows: if 10 binary data symbols are represented by m = [mo, m i , m g ] and associated with the message polynomial 1In the specifications, the (15,10) Hamming code is referred to as 'shortened' Hamming code. 2.2 Optimum Receiver 16 m(x) = mo + m\X + ...ITIQX9, then the parity bits are given by the remainder obtained from dividing m(x)x5 by the generator polynomial g(x) = 1 + x2 + x4 + x5 [BTS03]. 2.2 Opt imum Receiver In this section, the structure of an optimum receiver for GFSK in a pure AWGN channel is discussed. Considering the inherent memory in CPM signals an optimum detector takes decisions based on the sequence of received signals, minimizing the probability of error. The interdependence of successive symbols can be represented by a trellis diagram, the minimum Euclidean distance path through which gives the maximum-likelihood sequence [ProOl, Ch. 5]. As mentioned in Section 2.1.1, the phase of the GFSK modulated signal, for L = 2 can be rewritten as (cf. Eq. (2.6)) where nT < t < (n + 1)T. In the time interval nT < t < (n + 1)T, the phase deviation of the modulated signal is dependent on (a) the current symbol and the previous (L — 1) input symbols represented by the first term in Eq. (2.12) and (b) the accumulated phase till time {n — L)T, represented by the second term. The phase transitions are controlled by the current symbol and the (L — 1) last symbols only, as the phase pulse q(t) is constant for t > LT (cf. Eq. (2.5)). Hence, the vector (9[n — L],a[n — L + 1 ] ,a[n — 1], a[n\) uniquely specifies the signal element during n t h interval, where 9[n — L] represents the accumulated phase referred to as phase state. Since the phase is unique only in the interval from 0 to 27r, the phase state can be defined as [Liu90] n n - 2 (2.12) V 7T n—L k ^ a[i] mod(2p) (2.13) For even k, there are p phase states and 9[n — L] belongs to 2.2 Optimum Receiver 17 and for odd k, there are 2p phase states which belong to the set j Trk 2nk (2p- l)irk I V V V For odd k, the term in the bracket in Eq. (2.13) is alternating between even and odd resulting in twice the number of phase states compared to even k. This alternation between even and odd phase states for an odd k introduces time variance in the trellis as the possible set of attainable states in one time interval of duration T differs from the next time interval. Nevertheless, the number of attainable states per time interval is equal to p. This is shown in the phase state trellis diagram in Fig. 2.2 which is drawn for binary GFSK with h — 1/3 and L = 2. *- Input symbol-1 Input symbol+1 [6[i-2],a[i-l]] i'=0 i = T i = 2T ' = 3T Attainable . { j t / 3 _ n > 5 j t / 3 } { 0 > 2 j [ / 3 > 4 m } { 7 [ / 3 ] ^ 5 j [ / 3 } { 0 , 2n/3,4n/3} Phase states Figure 2.2: Time-variant trellis of phase state for /i = 1/3, M = 2, and L = 2. Thus, GFSK can be represented as a finite state time-variant Markov process with pML~l (or 2pML~1) states and the state vector Sn = [6[n - L],a[n - L + 1], ...a[n - 1]]. 2.2 Optimum Receiver 18 For binary GFSK with L = 2, the state vector Sn becomes [6[n — 2],a[n — 1]]. The number of possible states is 2p for even k and Ap for odd k. The state vector at t — (n + 1)T is obtained as Sn+l = [6[n-l],a[n]] (2.14) where d[n - 1] = 6[n - 2] + irha[n - 1]. (2.15) For optimum demodulation of GFSK, the Viterbi algorithm (VA) is an efficient method to perform MLSD [ARS81] which takes decisions based on the best path metrics of the survivor paths. The path metric is the sum of the branch metric and the corresponding survivor path metric for each branch. The preliminary decision is taken in favor of that path metric which maximizes the log-likelihood function. The log-likelihood function is proportional to the cross correlation metric [ProOl, Ch. 5] f(n+l)T CMn(a) = / ^{r(t)e-j^'a)}dt (2.16) J—oo r(n+l)T = CM n_ 1(a)+ / ^{r(t)e-^a)}dt (2.17) JnT where CMn(a) represents the path metric up to time nT for the surviving sequences, d are the possible hypothetical sequences, r(t) is the received signal, r(t) — s(t,a) + n(t). For each state there is one survivor path and M possible branches. Therefore, the number of possible branches or hypothetical sequences per symbol interval is pML (or 2pML ). Hence, in each symbol interval there are pML (or 2pML) cross correlation values to be calculated. To obtain the cross-correlation metric using a basic quadrature receiver the number of matched baseband filters required is 2ML with the impulse response [ARS81] * , ( , , * ) _ / c o s [ 2 r t E L - t + 1 * M ( i - s ) r - f ) ] ( 2 1 8 ) ] 0 for t outside [0,T], 2.2 Optimum Receiver 19 and hs(t,a) = l ^ [ ^ E l - L + i ^ M d - O r - t ) ] ( 2 . 1 9 ) y J | 0 for i outside [0,T]. V ' Thus, the optimum receiver consists of a bank of matched filters hc(t,a) and hs(t,d) fol-lowed by an MLSD processor. From the aforementioned discussion, it may be observed that the complexity of an optimum receiver increases exponentially with M and L. Also, the modulation index h is an important parameter that influences the trellis structure, the number of phase states, and the filter design. The variable h for Bluetooth systems complicates the optimum receiver design to a great extent. For h = 1/3, there are 12 possible states, therefore 24 computations per symbol interval have to be performed. For h = 0.29 = 29/100, there are 400 states and 800 computations, that are required to be performed in each symbol interval. Such a high number of computations increases the cost and compromises the robustness of the receiver, thereby, defeating the purpose of the design. 2.2.1 Optimum Receiver in the Presence of Random Phase in the Channel In the discussion so far, the channel is assumed to be a pure AWGN, hence no phase distortion is introduced by the channel. But for practical Blutooth devices, a random time-variant phase distortion 4>c(t) may be introduced in the received signal as described in Section 2.1.2. For an optimum coherent receiver, the phase 4>c{t) has to be estimated explicitly, e.g. by introducing known pilot symbols into the transmitted signal stream and using a phase-locked loop [ProOl, Ch.6], at the expense of increased complexity. For optimum NSD with the assumption of a constant phase rotation 0O, the received signal r(t) is expressed as r(i) = s{t,a)^° + n(t). The phase distortion <po is modeled as random variable with uniform distribution in the interval [0, 27r) [CR99]. The decoding 2.2 Optimum Receiver 2 0 strategy for optimum NSD for any modulation format is expressed as [CR99] d = argmax \s(t, a)\2dt + log/0 (Jj- jf r(i)s*(i, d)dt^ | ( 2 . 2 0 ) where d is the detected sequence, a is a hypothetical information sequence, and To is the observation interval. As the observation interval To increases linearly with time, the lengths of the transmitted sequence and the hypothetical sequence increase causing an exponential increase in the receiver complexity with time. Thus, the channel phase introduces infinite memory in the transmitted signal rendering the realization of a simple receiver based on optimum NSD infeasible. 2 . 2 . 2 Lower Bound for Performance of MLSD It is desirable to establish a theoretical bound for the best achievable performance of MLSD. For large average signal bit energy to noise ratios (SNRs), the probability of a bit error of coherent MLSD in an AWGN channel is given by [AAS86, Ch. 3] ^ e ~ Q h / ^ f H ( 2 . 2 1 ) where r i m i n is the minimum normalized Euclidean distance between two possible sequences of symbols a[i] and Rc denotes the code rate. The Euclidean distance between two sequences a^N and a2tN of length N for constant envelop signals depends on the phases of the two signals, and hence, depends on h [AAS86, Ch. 3]. The squared Euclidean distance between a^ jv and a2iN can be written as [AS81] D2(ahN,a2,N) = / \s(t,aitN) - s{t,a2,N)\2dt, ( 2 . 2 2 ) Jo which results in the following expression for the minimum Euclidean distance normalized by average bit energy Eb [AS81] C J V = log2(M) • min j j V - i cos[<f>(tnN)}dt} . ( 2 . 2 3 ) 2.2 Optimum Receiver 21 Here 4>(t,jN) = 2ixh YHiLo 7[*]<?(* — ^0 (Eq. (2.2)) with jN defined as the difference of the two sequences, i.e., ~fN = a^ jv - a>2,N (cf- [AS81]). The Euclidean distance increment over each symbol interval is nonnegative. Therefore, the Euclidean distance is a nondecreasing function of the length of the observation interval N. An upper bound for the minimum distance may be obtained as a function of h for any N by considering the infinitely long sequence pair that merges the earliest for any h. Two sequences are said to have merged at a certain time if their phases coincide for all subsequent time intervals. Therefore, for establishing the performance bounds for a particular h for different cases of coded and uncoded transmission, first the sequence pair of infinite length that merges the earliest is found. Then, using the difference sequence of the pair an upper bound for minimum Euclidean distance is obtained as in Eq. (2.23). If the pair merges after NB observation intervals, then the upper bound becomes Here, the notation d_in(/i) is used to indicate the dependence of the minimum Euclidean distance on h. Replacing d m i n in Eq. (2.21) by the upper bound dmin(h) of the minimum distance where d m i n < dmin(h), the probability of error as a function of h becomes As Q(x) is a decreasing function of x, a lower performance bound of MLSD is achieved which is a lower performance bound for NSD too. Therefore, BER is dependent on the normalized squared Euclidean distance c^in(h) which allows for a quantitative comparison of different coding and decoding schemes. For reference, d^ in for binary phase shift keying (BPSK) is 2 units which means the minimum difference in the energies of the two BPSK modulated sequences is 2Eb [AS81]. (2.24) (2.25) 2.3 Benchmark Receivers for Bluetooth 22 2.3 Benchmark Receivers for Bluetooth Since the idea of Bluetooth system originated in 1998, many researchers have been working on the development of low complexity, robust, and power efficient receivers [PYPJC01][HL01] [SJ03b][SWJ04][SHS03]. The popular receiver used in Bluetooth devices is the limiter-discriminator with integrate and dump filter [SVOla], which is a simple-to-implement, low cost, and robust noncoherent receiver. However, the LDI receiver is very power inefficient compared to the optimum receiver, as shown in Fig. 1.2. To improve the performance of the LDI receiver, several techniques have been devised such as channel estimation [PYPJC01], fractionally-spaced differential detection [HL01], and least squares based post-integration filtering [SJ03a]. To simplify the receiver structure, modified zero-crossing demodulation of intermediate frequency (IF) signals has also been proposed [SJ03b]. However, the per-formance gain achieved by these schemes over the conventional LDI receiver, at the target BER of IO - 3 , is in the range of 0.5 to 1 dB [SJ03b][SWJ04], which is not substantial. Re-cently, Scholand et al. proposed a receiver referred to as max-log-maximum-likelihood LDI (MLM-LDI) [SWJ04], which achieves significant performance gains over the conventional LDI detector but at the cost of increased complexity. Another receiver which achieves close-to-optimum performance is the maximum-a-posteriori probability (MAP) receiver, proposed by Schiphorst et al. [SHS03]. This receiver is based on Laurent's decomposition of GFSK. The MAP receiver has a similar performance as coherent MLSD and achieves close to 6 dB gain in performance as compared to the LDI detector, however, it requires an exact knowledge of the channel phase and the modulation index. In the following section the concepts of the LDI detector and the MLM-LDI detector are briefly introduced. 2.3 Benchmark Receivers for Bluetooth 23 2.3.1 LDI Detector The classic LDI receiver consists of a limiter-discriminator for data demodulation, followed by an integrate and dump (I&D) post-detection filter. The block diagram of the LDI detector is shown in Fig. 2.3. r(t) Band pass filter Limiter- <P'(t) Integrate A(p(t)^ Discriminator & Dump Hard limiter Figure 2.3: Block diagram for LDI receiver. A pre-detection band-pass Gaussian filter is applied, with equivalent low pass impulse response K(t) = y F R e - ( S D < " " ) ' , (2.26) where Br is the 3 dB bandwidth of the filter. The optimum bandwidth of the filter is given by Br = 1.1/(2T) [SVOla]. The limiter-discriminator outputs the derivative of the filtered signal phase <p(t) [Paw81]. The I&D filter is a rectangular filter of duration T and amplitude one. The I&D filter re-integrates the derivative over one symbol duration giving a phase difference which is the net increment or decrement in the phase over a symbol duration [Paw81]. The decision is taken at each interval T after sampling at the maximum eye opening instant [SVOla]. The hard limiter compares the sampled data with the set decision level which is zero for binary GFSK. If the phase difference is greater than zero, which means increasing phase of the received signal, the decision is taken in favor of +1. To get the results in various scenarios like varying h and interference, the LDI receiver has been simulated in the present research work. 2.3.2 M L M - L D I Detector The MLM-LDI detector employs a max-log-maximum likelihood (MLM) symbol estimation post processor followed by a least-squares based post-integration filter to the output of a 2.3 Benchmark Receivers for Bluetooth 24 digital LDI [SWJ04] [SJ03b]. The MLM involves a forward-backward (FB) algorithm on a four state trellis [KB90]. This post-processor improves the performance by approximately 3.5 dB, but an increased design complexity is introduced because of the post processing and the four state trellis involved in MLM. For comparison purposes in the present research work, the results presented in [SWJ04, Figure 1] have been used. Chapter 3 Noncoherent Sequence Detection Receiver for Bluetooth In this chapter, the proposed receiver design for Bluetooth devices is presented. As dis-cussed in the previous chapter, the Bluetooth receiver is desired to be cost-effective and power-efficient, at the same time meeting the Bluetooth radio specifications [BTS03]. The most challenging problem in the design of the receiver is to ensure good performance with a low-complexity receiver under the high permissible frequency offset and the varying mod-ulation index. For a simple yet power efficient receiver design, the Rimoldi/Huber&Liu representation of CPM [Rim88] [HL89] has been adopted as shall be explained in Section 3.1. In Section 3.2, a new receiver filter design is proposed which ensures good perfor-mance in the presence of interference. The novel receiver employs noncoherent sequence detection (NSD) where the phase is implicitly estimated. Methods for implicit phase and frequency offset estimation for NSD are described in Section 3.3. The variable modulation index results in a trellis with a variable number of states (cf. Section 2.2), consequently increasing the receiver complexity. This problem is solved by reducing the number of states to two as discussed in Section 3.4. Besides the complexity of the receiver, a deviation in h from its true value also affects the receiver performance adversely. To tackle this problem a 25 3.1 Rimoldi/Huber&Liu Representation of GFSK 26 simple h estimation scheme is presented in Section 3.5. In Section 3.6, channel coding and techniques to improve the decoding performance are discussed. Finally, the receiver design is summarized in the concluding section of this chapter. 3.1 Rimoldi/Huber&Liu Representation of GFSK The high complexity of the optimum CPM receivers due to the inherent memory of CPM motivated the search for reduced complexity receivers. Laurent (1986) [Lau86] proposed an alternative representation of CPM where CPM is represented as a superposition of finite amplitude modulated pulses. This results in significant complexity reduction be-cause for modulations with good spectral characteristics, the representation with only one pulse achieves high accuracy. However, for noncoherent sequence detection an additional whitening filter is necessary [CR98]. Laurent's approach to represent CPM has been ap-plied to Bluetooth receivers in [SHS03], but the varying modulation index problem has not been addressed in the receiver design. An alternative representation of CPM, addressed as the decomposition approach in the present work, was proposed independently by Rimoldi (1988) [Rim88] and Huber and Liu (1989) [HL89]. The decomposition approach shows that CPM can be represented as a linear time-invariant trellis encoder with memory and a time-invariant memoryless signal mapper. This results in the decoupling of filter design and trellis search. Complexity reduction is also achieved as four or six signal elements are suffi-cient to represent all signal elements of CPM with M < 4 while maintaining good accuracy [HL89]. In the present work, the decomposition approach given by Rimoldi/Huber&Liu has been adopted. The following section discusses the decomposition approach in brief. 3.1 Rimoldi/Huber&Liu Representation of GFSK 27 3.1.1 The Decomposition Approach to C P M In Section 2.2, it was shown that CPM has a time-variant phase state trellis. In the decomposition approach, first the time-variant phase state trellis is transformed into a time-invariant phase state trellis, thereby enabling the decomposition of nonlinear CPM with memory into a simple trellis encoder and a signal mapper. To achieve a time-invariant trellis, a slope function is introduced which is defined as [HL89] { 0 t < 0 0<t<LT (3.1) t > LT Adding a zero term to the phase of the CPM modulated signal SRF(£, a).(Eq. (2.1)), the phase of the modulated signal can be written as n n $(*, a) = 27r/ci - 2-uh c(t - iT) + <p(t, a) + 2irh ] T c{t - iT). (3.2) i——oo i=—oo For convenience, a new reference frequency fr is defined as M - 1 fr = fc - h-^r- (3-3) and a modified unipolar information symbol b[i] is introduced that is given by H i l - ^ ' Y " ' £{0,1,.. .M-1}. (3.4) With the new variables fr and b[i], the phase in Eq. (3.2) can be rewritten as [HL89] $(i, 6) = 27r/rt + (^ or-+ — *[n-L] + 27r/i V p(t - iT, b[i}), (3.5) ' i=n-L+l where b is the vector [6[0], 6[1]..., 6[n]], <p0r is the initial phase and ty[n — L] is the normalized modified phase state. The phase transition function p(t, b[i]) is defined as p(t, b[i]) = (26[z] - (M - l))q(t) + c(t). (3.6) 3.1 Rimoldi/Huber&Liu Representation of GFSK 28 The new modified information carrying phase becomes 9 " 4(t,b) = — V[n - L] + 2Kh J _ p(t-iT,b[i]) P i-n-L+l with modified normalized phase state definition as n—L (3.7) #[n -L] = k J _ 6[i] mod(p) G {0,1,...,p- 1}. (3.8) The phase state is updated recursively as *[n - L + 1] = (*[n - L] + fc • b[n - L + 1]) mod(p). (3.9) Hence, the number of modified phase states is equal to p for any k with respect to the new reference frequency, achieving the desired time-invariance in the phase trellis as shown in Fig. 3.1. Comparison of Fig. 2.2 with Fig. 3.1, which has been drawn for the same h [h = 1/3), reveals that the trajectories in the modified trellis are the same for any time interval unlike the original trellis of Fig. 2.2. Input symbol 0 *- Input symbol +1 m i-2]2rc/p,b[i-l]] [0.1 [4n3,0) i = 0 i = T i = 2T i = 3T Figure 3.1: Time-invariant trellis of phase state for h = 1/3, M = 2, and L = 2. There are pML~l possible states with M branches each, resulting in pML possible trajectories in each time interval. Hence, there are pML different modulated signals of 3.1 Rimoldi/Huber&Liu Representation of GFSK 29 duration T which are referred to as the signal elements p(t,d[i]). d[i] is a unique address vector associated with each signal element, Therefore, the CPM modulator can be decomposed into a trellis encoder which generates the unique address vector based on the recursively updated phase state and previous L data inputs, and a signal mapper which maps the encoded address to a signal element representing the modulated signal. Hence, the transmitted ECB signal can be represented as a sequence of time-limited signal elements p(t, d[i]) Here d represents the sequence of associated addresses, i.e., d — [d[0],d[l], ...,d[n]]. The decomposed equivalent CPM modulator is shown in Fig. 3.2. The intersymbol-interference (ISI) memory due to the previous (L — 1) symbols is represented by (L — 1) shift registers. The phase memory is recursively updated using a simple phase integrator as shown in the Fig. 3.2. The signal mapper stores a table of pML signal elements for the exact representation of CPM. Having represented CPM as a simple combination of a trellis encoder and a signal map-per, the corresponding reduced complexity receiver structure is presented in the following section. 3.1.2 Receiver Structure The optimum receiver requires a bank of 2ML matched filters [AAS86] for detecting the received signal. However, it has been shown that the required number of complex filters D to obtain sufficient statistics of the received signal is given by 2D — mm(2Dmax,2ML) d[n] = [V[n-L],b[n-L + l],...,b[n-l],b[n]]. (3.10) oo (3.11) 3.1 Rimoldi/Huber&Liu Representation of GFSK 30 Time invariant Jrellis encoder b[n] b[n- l ] , b[n] b[n-L+l] b[n-l] mod(p) b[n-L+l] T[n-L] Signal mapper with p M L signal elements s(t,d) Recursive phase integrator Figure 3.2: Equivalent representation of CPM modulator using decomposition approach. [HL89], where Dmax is given by 2 £ > m o i = 2 r i . l l / i ( M - l ) +2.221. (3.12) The factor of 2 present in Eq. (3.12) is because of complex baseband signal elements. The dimension D is the number of baseband time-limited functions which represent the signal space formed by the modulated signal with sufficient accuracy. Hence, for Bluetooth systems, with a maximum h = 0.35, the dimension D of the matched filter bank reduces to three. To find a proper set of basis functions with dimension D, which spans the whole bandwidth of all the pML signal elements p(t,d[i}), Huber and Liu [HL89] proposed a simple set of D pairs of sine and cosine passband functions of duration T. The proposed basis functions are a set of non-orthogonal exponential functions which are symmetrical to the carrier frequency and equally spaced on the frequency axis. The receiver low-pass filters with respect to the carrier frequency fc are given by [HL89]: with frequencies given by U *k(2d-l-D), d e { l , 2 , . . . , £ > } . (3.13) (3.14) 3.1 Rimoldi/Huber&Liu Representation of GFSK 31 A/a is the frequency spacing between the basis function frequencies, 0 < A/aT < 1. It has been shown that the performance degrades negligibly for 0.5 < A/^T < 0.7 with 0.25 < h < 0.5 for D = 3 and D = 2 [Liu90, Fig. 5.4] [HL89, Fig. 6]. Moreover, in Section 3.2 it shall be shown that D = 1 can be used for the proposed receiver for which fd = 0. Hence, in the present work it has not been attempted to optimize A / j . Therefore, A/aT == 0.5 has been used for D > 1 in all the simulation work. The matched filter bank necessary for demodulation is specified by a vector of D coor-dinates hrj(t) = [fi£)\t), ...h^\t)}. The D samples of the filtered received signal in the n t h time interval are given by x[n] = r(t) * h*D(-t)\t=nT = {xi[n},...,xD[n}} (3.15) where h*D(—t) represents complex conjugate of matrix hr>(—t). \x\\n\,xo[n]] are the suf-ficient statistic to estimate the input information sequence a using a Viterbi processor. The Viterbi processor stores D values for each signal element p(d[i]) = [pi(d[i]),pD(d[i])}, which are obtained by multiplying the correlation of D basis functions with signal ele-ments p(t, d[i]) corresponding to pML hypothetical sequences with address vector d[i], and a matrix C - 1 [Liu90], p(d[i])= / p{t,d\i])h*D(t)dt • C'1. (3.16) The matrix C is a D-hy-D array of cross-correlations of basis functions, introduced to take into account the effect of non-orthogonal basis functions hrj(t). It is given by C = ^ £ hTD(t)h*D(t)dt, (3.17) where h^\(t) represents transpose of matrix h£>(t). For coherent detection the Viterbi processor cross-correlates the received sample Xd[i] with pd(d[i}) for all hypothetical sequences to get the branch metric A[i] = H{x[z]-p"(#])}, (3-18) 3.1 Rimoldi/Huber&Liu Representation of GFSK 32 where pH(d[i]) represents Hermitian transposition of matrix p(d[i]). A survivor path traces the estimated symbol in each time interval for every state. The calculated branch metric is added to the corresponding survivor path to get the path metric of that branch. For each state the decision is taken in favor of the branch with maximum path metric. The corresponding symbol is added to the relevant survivor path and the path metric of the survivor path is updated. The estimated sequence is the survivor path with maximum path metric among all the states. The number of states in the Viterbi processor is pML~1 using the decomposition approach, which is only half the number of states required for an odd value of k if the conventional representation of CPM is used. It is worth observing that the number of states in the Viterbi processor is independent of the filters preceding it. The modified receiver structure is shown in Fig. 3.3. t r(t) r C(-t) t = kT = kT Viterbi processor with p M L - ' states Figure 3.3: Matched filter receiver for CPM using decomposition approach. Comparing the receiver structure obtained with the Rimoldi/Huber&Liu's decomposi-tion approach with the optimum receiver structure described in Section 2.2, it is observed that the bank of 2ML filters has been replaced by a bank of sub-optimum D-dimensional filters hr)(t). Unlike the optimal filters, the baseband filters ho(t) are independent of h and the number of states in the Viterbi processor. Also, as mentioned above, the number of states required in the Viterbi processor reduces to pML~l from 2pML~l for an odd k while it remains the same for an even k. 3.1 Rimoldi/Huber&Liu Representation of GFSK 33 In the discussion so far, it has been assumed that the channel does not introduce any phase distortion. The effect of phase distortion is discussed in Section 3.3 which describes NSD and the corresponding branch metric calculation. 3.1.3 Application of the Decomposition Approach to Bluetooth The equivalent transmitter structure for Bluetooth systems using the decomposition ap-proach is shown in Fig 3.4. As mentioned earlier in Section 2.1.1, for Bluetooth systems, L = 2, M = 2, and h can take any value from 0.28 to 0.35. Therefore, there is only one shift register with the number of signal elements in the signal mapper varying with h. b[n] b [ n - l ] b [n] T b [ n - l ] mod(p) Signal mapper with 4p signal elements s(t,d) Figure 3.4: Equivalent GFSK modulator using decomposition approach. The baseband receiver can be represented by a bank of 3 complex exponential filters (cf. Section 3.1.2), followed by a Viterbi processor with 2p states. However, for a low cost receiver it is desirable to reduce the number of filters even further. With this intent, a discussion on the required number of filters and on an improved filter design is presented in the following section. Moreover, the problems of varying trellis structure with varying p and the high computational complexity still exist, implying that, the number of states needs to be reduced for all values of h. A state reduction method will be presented in Section 3.4. 3.2 Filter Design 34 3.2 Filter Design Using the time-limited complex-exponential functions hp}(t), three complex matched filters are required as has been explained in the previous section. However, as GFSK is a very bandwidth efficient modulation scheme, a reduced number of basis function may represent the GFSK signal space with sufficient accuracy. Therefore, the performance of receivers with D = 2 and D = 1 was investigated. It was found that there is no degradation in performance for D = 2, while for D = 1 the required lOlog10(Es/No) increases by only 0.5 dB to achieve the target B E R of 10 - 3 (cf. Section 4.1.1). The number of filters required reduces to one with D = 1 as fd = 0. Hence, as reduction in complexity is achieved at the expense of very small loss in performance, it is proposed to use D = 1. As mentioned in Section 1.1, Bluetooth systems operate in the license-free ISM band. Therefore the receiver is required to be robust against co-channel interference (CCI) and adjacent channel interference (ACI) (cf. Section 2.1.2). For effective suppression of ACI, the receiver filter should be able to stop the frequencies outside the passband of 1 MHz of the desired signal. Fig. 3.5 (a) shows the magnitude frequency response of the filter h^\t) with D = 1 from which it may be inferred that the filters proposed in Eq. (3.13) can accomplish only limited ACI suppression. Therefore, the passband characteristics of the proposed receiver filter hf\t) need to be modified for interference suppression. The LDI detector is quite robust in the presence of interference [SVOla]. Fig. 3.5 (a) shows the magnitude of the frequency response of the Gaussian pre-filter applied in the LDI receiver with impulse response where Br is the 3 dB bandwidth of the filter, a typical value of which is BrT - 0.55 [SVOla], While the passband spans more than twice the bandwidth of the desired signal, the stop-(3.19) 3.2 Filter Design 35 17 0.4 ®" 0.2 1 ! ! ! ( ) , , i 1 i. j A & € ^ 6 © Figure 3.5: (a) Magnitude frequency responses for different filters, (b) Noise autocorrelation function </?„„[«;] after filter he(t) and sampling. band efficiently suppresses the interference from adjacent channels separated by more than 2 MHz. Therefore, a new filter design is proposed with an additional Gaussian pre-filter hr(t) cascaded with filter h^\t), with an effective impulse response he(t) = hr(t)*hiji\t), (3.20) that may be expressed as he(t) = ^ (Q(c • Br(t - T/2)) - Q(c • Br(t + T/2))) (3.21) with constant c = 27r/^ /ln(2) and Q(r) as the standard Gaussian Q-function. The combined filter characteristics he(t) are also shown in Fig. 3.5 (a). The passband of he(t) is narrower than that of hr(t) and the stop-band characteristics are significantly improved as compared to the originally proposed filter h^\t). Therefore, the combined filter is capable of improved interference suppression as demonstrated in the performance results in Section 4.1.1. 3.3 Noncoherent Sequence Detection 36 Since he(t) is not time-limited and he(t) * h*e(—t) is not a Nyquist pulse, ISI and colored noise are unavoidable after sampling. However, these effects are minor as can be seen from the rapidly decaying noise autocorrelation function depicted in Fig. 3.5 (b). For this reason and for the sake of simple implementation, ISI and noise coloring are neglected in the following NSD design. The performance results, however, include these effects which are not major as seen in the results presented in Section 4.1.1. 3.3 Noncoherent Sequence Detection As discussed in Section 2.2, coherent detection or optimum noncoherent detection are not feasible solutions for the desired low cost Bluetooth receiver design. For derivation of a sub-optimum NSD metric for GFSK from the optimum NSD metric discussed in Section 2.2.1, the channel phase is assumed to be constant. However, since practical Bluetooth devices are required to cope with time-variant channel phase (cf. Section 2.1.2), the assumption of constant channel phase will be relaxed later in this section. For linear modulation schemes with constant envelop it has been shown [CR99] that the maximum likelihood noncoherent sequence detector derived from Eq. (2.20) for optimum NSD, for a block of NT transmitted symbols maximizes the sequence metric {J V T - 1 n - 1 "| j>[n]c*[n]5y [/]£[*][ (3.22) 71=1 1=0 ) where c is the code sequence uniquely associated with the hypothetical information sequence a and r[i] are the received samples. The incremental metric X[i] = A[i + 1] — A[i] can be written as A[z] = K{r[z]c*[z]^r*[/]c[/]}. (3.23) 1=0 The receiver front end derived in Section 3.1.2, allows the application of the noncoherent detection method for linear modulation given by Eq. (3.22) directly to CPM [LSEH01]. Unlike other demodulation schemes for CPM, cf. e.g. [CR98], a whitening matched filter is 3.3 Noncoherent Sequence Detection 37 not required for the receiver design discussed in Section 3.1.2. Hence, the sequence metric for a block of NT symbols and the incremental branch metric for MLSD of CPM using the decomposition approach can be written respectively as [LSEH01] r j v T - i A[NT "1] = » |E ^}PH(d[n])q;ei[n - 1] j , (3.24) and A[z] = K{*[i]p"(d[i])& f[i-l]} (3.25) where qre[ [i — 1] is the reference phase for the survivor path corresponding to the hypothetical information sequence address vector d[i], 0 < i < NT — 2, and is defined as i - l qie{[t-l] = J2^}PH(d[l}). (3.26) (=0 The expression for branch metric given by Eq. (3.25) is similar to branch metric com-putation for coherent metric using Eq. (3.18) where channel phase is known at receiver. Since under low noise conditions x[l]pH(d[l]) « e^0 for correct hypothetical sequence, the phase reference gives the average of the estimated channel phase over all transmitted bits. Therefore, for constant channel phase the phase reference can be interpreted as the im-plicit estimation of the phase distortion introduced by the channel. However, in practice, the assumption of constant channel phase over the whole transmission period is not true. Moreover, the computation of the phase reference in Eq. (3.26) is done over all the trans-mitted symbols. The number of transmitted symbols grows linearly with time, thereby, introducing unlimited phase memory. Therefore, in order to limit the complexity of the receiver and to relax the constant channel phase assumption, the application of a rectan-gular and a exponential observation window has been proposed [CR99][SG99] as will be explained in the following two sections. 3.3 Noncoherent Sequence Detection 38 3.3.1 Rectangular Windowing Colavolpe and Raheli proposed the use of a rectangular window of length N, N <§C Nj and TV > 2, to limit the number of observations while calculating the phase reference, with equal weight given to each observation in the window. Hence, the phase memory is truncated to the N most recent symbols, the phase reference calculation thus becomes [CR99] < & - ! ] = E i>N-l, (3.27) l=i-N+l where q^{[i — 1] represents the modified phase reference. The channel phase is assumed to be constant only over the period of N symbol intervals. The new branch metric, referred to as ./V-metric is defined as A"[i] = n^)PHCm)& - !]}• (3-28) The phase reference is updated as <&] = ^ \PHm) + &[i - 1] - x[i -N+ l]pH{d[i -N+l\). (3.29) With the increase in the length of the window N, the number of data samples increases, resulting in a decrease in noise variance of the phase estimation. Therefore, the phase estimation becomes better with an increase in N for approximately constant channel phase. However, if the phase of the channel varies with time, for example in the presence of phase jitter or frequency offset, the noise variance may increase with increasing N, causing a deterioration in the performance. The optimum value of N depends on the SNR and the channel characteristics. Performance results for the proposed receiver with iV-metric for different values of iV and under different channel conditions are given in Section 4.1.2. 3.3.2 Exponential Windowing An exponentially decaying window was proposed in [SG99], where an unlimited size window is adopted but the weight given to each observation decays exponentially with time. The 3.3 Noncoherent Sequence Detection 39 most recent observation is given a maximum weight of one. The phase reference is updated recursively by [SG99] q^[i\ = x\i\pH(d\i}) + a • q«{[i - 1] (3.30) where a is the forgetting factor, 0 < a < 1. The corresponding branch metric is called a-metric. The iV-metric and a-metric are identical for the special cases N = 2, a = 0 and N —> oo, a —> 1. As for the Af-metric the optimum value of a depends on the SNR and the channel characteristics. As a approaches one, all the observations are given the same weight of 1, which is favorable if the channel phase is constant, but affects the performance adversely in a time-varying channel. In terms of the computational processing required for updating the phase reference, a comparison of Eqs. (3.29) and (3.30) reveals that the a-metric requires less arithmetic operations compared to the A -^metric. Hence, for a simple receiver design the a-metric is preferable. A performance comparison of the two metrics is given in Section 4.1.2. 3 . 3 . 3 Frequency Offset Estimation The NSD using the a-metric or the Af-metric is very power efficient and robust against channel phase variations. However, the frequency offset A / allowed in the Bluetooth sys-tems is of the order of ±100 kHz [BTS03], i.e., the normalized offset A / T can be as large as ±0.1. To cope with such high frequency deviations it is mandatory to estimate the fre-quency offset A / and incorporate the effect of A / into the definition of the phase reference. The maximum-likelihood function for sequence detection now has to maximize the joint function for data detection and frequency estimation [CR02]. The sequence metric equiva-lent to Eq. (3.22) for equal-energy, linearly modulated signals is obtained by introducing a 3.3 Noncoherent Sequence Detection 40 trial value of the frequency offset A / [CR02, Eq. 18] { NT-l n - l "| £ r[n]c*[n] £ r * [ / ] c [ / ] e - ^ " - ' ^ (3.31) n=l (=0 J in which the inner sum (index I) is the new estimate of the channel phase. Applying Eq. (3.31) to CPM and truncating the memory using exponential windowing, the modified reference symbol can now be defined as: QreS\ = - 1] + *[i]p"(d[i])) e ^ 1 - 1 ' (3.32) with the frequency offset estimate = (3.33) obtained using the adaptive estimator PrefW = fared* " 1] + x[i]P"[d\l])(p{d[l - l\)xH[i - 1]). (3.34) 6 is the forgetting factor for the calculation of pref[*]> 0 < (3 < 1. The second term in Eq. (3.34) can be interpreted as an estimate of the frequency drift over one symbol duration because for high SNR and low noise conditions x[i]pH(d[i])(p(d[i — l])xH[i — 1]) « ej(*c[i]-*c[i-i]) f o r d[i) = d{i). By using Eq. (3.32) in the branch metric Eq. (3.25), NSD is still robust against random phase noise and residual frequency offsets (A/ — A/) , but it now explicitly accounts for the most prominent and systematic contributor to the phase variations in Bluetooth systems, i.e., oscillator frequency offset. The chosen approach is similar to frequency estimation for NSD and linear modulation advocated by Colavolpe and Raheli in [CR02]. Moreover, at typically required SNRs the estimator in Eq. (3.34) resembles Kay's frequency estimator [Kay89], which was also found advantageous in [CR02]. Both Eq. (3.32) and Eq. (3.34) are different from [CR02] and [Kay89] respectively, in that recursive update equations are formulated, which (a) involves 3.4 State Reduction 41 less arithmetic operations than rectangular windowing and (b) are specially well-suited for per-state tracking in the trellis of NSD. It is also worth noting that the devised frequency offset estimation for NSD corresponds to the use of DC offset cancellation methods for the LDI receiver, which were found necessary to compensate the severe frequency offsets encountered in Bluetooth, cf. e.g. [LDF93]. 3 . 4 State Reduction So far, with the application of the decomposition approach and NSD to Bluetooth, a simple receiver structure has been obtained consisting of a single filter and a Viterbi processor with a simple phase estimation algorithm. However, the varying number of states in the Viterbi processor complicates the receiver structure. A reduction in the number of states to a value independent of h is desirable, and this is readily accomplished by following the per-survivor-processing (PSP) approach [RPT95]. As described in Section 3.1.1, the state vector of the modified trellis using a unipolar information sequence and normalized phase state for Bluetooth system is given by (using Eq. (3.10)) S * = [ t f [n-2] ,&[n- l ] ] (3.35) where ^[n — 2] is updated using (cf. Eq. (3.9)) *[n - 2] = (¥[n - 3] + kb[n - 2]) mod(p). (3.36) There are 2p states, i.e., if h = 1/3, the number of states is 6 and if h = 0.29 = 29/100, the number of states is 200. The large and variable number of states is mainly contributed by the phase state \I>[n], which can take values from 0 to (p — 1). Following the PSP approach [RPT95], the number of states can be reduced if the phase state is decided tentatively according to the survivor path terminating in the current state, thus reducing the number 3.4 State Reduction 42 of states to two. The new state vector is defined as S£ = [*[n-2],6[n-l]] (3.37) where ^[n — 2] is the phase state of the survivor path at n t h time interval for the current state. As the phase state is not allowed to vary anymore, the only variable in the definition of the state vector is b[n — 1], which can take two possible values {0,1}. Hence, the state is defined by b[n — 1] only and the associated phase state is used for branch metric calculation. Fig. 3.6 shows the two-state trellis structure. ^fm[i] is the phase state associated with state m, m € {0,1} at instant iT. i = 2 i =3 i = 4 Figure 3.6: Two-state trellis structure. State reduction to one can similarly be achieved by taking a tentative decision on the phase state as well as b[n — 1] for the survivor path. Hence, S1n = [*[n-2],b[n-l]] (3.38) where both ^[n — 2] and b[n — 1] are fixed and correspond to the phase state of the survivor path and the last added symbol to the survivor path. Hence, there is only one survivor path and the symbol is detected based on the branch metric instead of the path metric. The modified one-state trellis is shown in Fig. 3.7. The following example demonstrates the difference between the three trellis structures for h = 1/3. The phase state is updated using W[n - 2] = [*[n - 3] + b[n - 2]] mod(3). 3.4 State Reduction 43 Input symbol 0 Input symbol +1 i = 2 i = 3 i = 4 Figure 3.7: One-state trellis structure. The path trellis for the example is drawn in Fig. 3.8, where all the states are shown. However, the attainable states depend on the trellis structure that is explained below. The full-state trellis has six survivor paths and therefore, can attain all the six states {(0,0), (1,0), (2,0), (0,1), (1,1), (2,1)}, where the first term in the bracket is the phase state and the second term is the previous input according to the definition of state vector in Eq. (3.35). Similarly, the possible two states for the two-state trellis are either from {(0,0), (1,0), (2,0)} or from {(0,1), (1,1), (2,1)} and the one-state trellis has only one sur-vivor path which can attain any one state from (0,0), (1,0), (2, 0), (0,1), (1,1) and (2,1). In this example, the initial state at i = 0 is assumed to be (1,1). [W-2],b[i-l]] Figure 3.8: An example for the path trellis for h = 1/3 to illustrate state reduction. At instant i = 1, the two-state trellis has two survivor paths terminating at states (2,0) and (2,1), but for the one-state trellis only the path with the best branch metric out of 3.4 State Reduction 44 these two survives and the symbol added to the survivor path corresponds to 6[0] with the higher branch metric. At instant i = 2, the two-state trellis selects one out of the two survivor paths terminating at (0,1) and (2,1) for state '1' and one out of (0,0) and (2,0) for state '0' based on the respective path metric. However, in a full-state trellis none of these paths are terminated and all the four paths continue to trace the sequence. It is interesting to consider the minimum distance for all the three cases of full-state, two-state, and one-state trellis. As the possible trajectories and error events change with state reduction, the minimum distance changes too, thereby affecting the BER. The distance between the correct and the erroneous path is the difference of their path metrics, which is calculated by adding the branch metric of each trajectory in the path. The branch metric is calculated using Eq. (3.25), i.e., = ffi{x[i]p H (d[i])q?*{[i — 1]}, where q°el[i — 1] is given by Eq. (3.32) with a = 0.6 and 3 = 0.9, and x[i] represents noiseless received signal samples here. Table 3.1 gives all possible branch metrics for h = 1/3. Fig. 3.9 shows an example of a two adjacent error events path along with the corresponding correct path for the full-state trellis. For the path metric calculation of the erroneous path, the address vector for x[i] [vrti-2],b[i-i]] [0,0] [1,0] [2,0] 10,11 [i,i] [2,1] 000 "x 001 i = 2 i = 3 Correct path Erroneous path i = 0 i = l Figure 3.9: Full-state path trellis for two error event example. are the digits over the correct path and the address vector for p(d[i\) are the digits over the 3.4 State Reduction 45 erroneous path in Fig. 3.9. Therefore, the difference of the path metrics at instant i = 3 is Df = 5(000,000) + £(001,001) + 5(010,010) - 5(000,001) - 5(001,010) - 5(010,100) = 0.7593. where B(x,y) refers to the entry in the Table 3.1. Similarly, for the two-state trellis, Fig. 3.10 shows the trellis path for the same example of two adjacent error events. Comparing Figs. 3.9 and 3.10 it can be observed that the distance between the correct and erroneous path for the two-state trellis is the same as that for full-state trellis, i.e., D\ = 5(000,000) + 5(001,001) + 5(010,010) - 5(000,001) - 5(001,010) - 5(010,100) = 0.7593. [b[i-l]] *~ Correct path Erroneous part Figure 3.10: Two-state path trellis for two error event example. For the one-state trellis the distance between the two paths is given by the difference of their branch metrics. Fig. 3.11shows an example of a one error event path. The distance between the two paths is D\ = 5(000,000) - 5(000,001) = 0.01135. The comparison of these three examples illustrates that while for the full-state trellis and two-state trellis the distance in the correct and the erroneous path is the same, for the 3.4 State Reduction 46 in r» 00 r~ in ON ON O O NO cn CO ci d d — m NO d in !£} r~ 2 s ° 2 K s 8 o I — in —< 8 d m CO r- o 00 00 ^ - cn ON m oo oo ON ON CO in NO NO NO d I ? d 9 d -H — CO 8 S-l O o I 2 N§ NO O d d c- — NO d i 8 8 \o —• d o o l NO o d NO d NO d o l o c| PQ o © 8 in — o I o I in m 00 00 co m 00 NO ND o in d d d oo in CN m r -© CN ON o ON m NO 5 m cn d i 9 d 9 NO d i 8 d — in NO — d ©' d o r- cn ON cn ^ -00 ON 00 00 NO 00 ON O NO NO NO d d I 9 d NO d o I g >n cn 00 OO in cn 00 ON i n ON m NO cn <? d i d g o I o I o I m — o I s o I Q. 0.70978 -0.35489 -0.35489 —0.16077 -0.53958 0.70035 0.69843 -0.54385 -0.15458 —0.36018 -0.34957 0.70975 § § 8 CN o o 110 o CN 8 o o CN o CN 3.5 Adaptive Noncoherent Sequence Detection 47 000 V|/[i-2]=0, b [ i - l ] = 0 Correct path Erroneous pat! 001 i = 0 Figure 3.11: One-state path trellis for one error event example. one state trellis the distance drops by a factor of 65 approximately. Therefore, the probabil-ity of error in the presence of noise increases for the one-state trellis while for the two-state trellis it is expected to remain the same as that for the full-state trellis if double adjacent errors is the most prominent error event. This observation is verified by the results shown in Section 4.1.3. Therefore, a two-state trellis structure is a viable solution for the reduced complexity Bluetooth receiver. 3.5 Adaptive Noncoherent Sequence Detection In the previous section, the effect of a varying modulation index h on the number of states in the trellis has been considered. However, the demodulation frequency fr is also dependent on h as can be observed from Eq. (3.3), i.e., fr = fc — h H ^ } . The deviation of estimated modulation index h at the receiver from the true modulation index h of the transmitted signal introduces a frequency offset of Afh — , hence, deteriorating the performance of NSD. Secondly, h affects p(d[i]) because of its dependence on fr and d[i]. As d[i] — \^[i — 2], b[i — 1, b[i]], \J/[i — 2] G {0,1,(p — 1)}, the number of address vectors is dependent on h, irrespective of the number of states defined in the trellis. Lastly, h also affects the phase reference qret[i] as it is dependent on the received filtered samples x[i] and p(d[i\). Therefore, for an optimum performance of the receiver the modulation index of the transmitted signal is required to be known at the receiver. 3.5 Adaptive Noncoherent Sequence Detection 48 For a low-complexity receiver, the sequence detector could operate with an assumed that there is a considerable performance degradation as h at the transmitter is allowed to vary in a relatively large interval of 0.28 to 0.35. Therefore, for an acceptable performance under variable h conditions an h estimation scheme is suggested. For a simple receiver design, it is proposed to estimate h by performing NSD for a small number of hypotheses h G Ti, where H is the set of hypotheses, over an estimation period of Ne symbols. The hypothesis yielding the best metric (Eq. (3.24)) in the estimation period Ne is chosen as the estimated modulation index h and is used to detect the rest of the sequence, i.e., This adaptive NSD can be regarded as an approximate maximum likelihood joint detec-tion and estimation with quantization of the unknown parameter h. For a reasonably small number of tested hypotheses and short estimation periods, e.g. \H\ = 2,4 and Ne = 25,50, the complexity increase due to adaptation is almost negligible. However, considerable im-provement in the performance under varying h conditions is obtained. The performance results of adaptive NSD (ANSD) are discussed in Section 4.2.1. Under low noise conditions, the performance of ANSD remains the same whether the symbols during the estimation period are known or not. The complexity of ANSD can be reduced by using smaller values of Ne. Reduction of Ne from 50 to 25 introduces only 0.3 dB loss in performance (cf. Sec-tion 4.2.2). The performance improves if the number of hypotheses is increased, however, if frequency offset cancellation is also employed then even two hypotheses yield good results. The reason of better performance with frequency offset estimation is that the frequency drift introduced by the mismatched h is implicitly alleviated by the proposed frequency offset estimation method (cf. Section 3.3.3). nominal value h, regardless of its actual value. But the results given in Section 4.2.1 show (3.39) 3.6 Channel Coding 49 3.6 Channel Coding As described in Section 2.1.3, Bluetooth devices employ a rate 1/3 repetition code (RC) and a rate 2/3 expurgated Hamming code (HC) for error correction purpose at the receiver. A conventional FEC decoder uses hard-decision decoding subsequent to NSD to retrieve a[i]. Conventionally, both FEC schemes are utilized to correct only single errors incurred in the transmission for LDI detectors with hard decoding. However, an analysis of the error patterns of the decoded information sequence shows that there is a very high probability of two consecutive errors. For example, an information sequence '101' is received as 'Oil' with a high probability. In general, if a[i] denotes a complement of a[i], then the most likely error event is {a[z],a[z + 1] = a[i],a[i + 2]} received as {d[i],d[i + l],a[i + 2]}, i.e., double adjacent errors (DAE) are incurred, which cannot be corrected using the rate 1/3 RC or a single error correcting HC. Fig. 3.12 shows the frequency of five different error patterns occurring in a block of 10 symbols in the two-state NSD and the LDI detector output confirming high occurrence of DAE. The frequency of DAE is much higher in the proposed receiver as compared to the LDI detector. The following error analysis gives an in-depth investigation of the observed error patterns. 3.6.1 Error Analysis To study the behaviour of the channel for coded and uncoded transmission the analytical results for the binary symmetric channel (BSC) are compared with simulations results. Section 2.2.2 introduced the theoretical lower performance bound for MLSD. Fig. 3.13 shows the graph for the theoretical and the simulated BER vs. 101og(Es/A7o). It may be observed that the simulation results are in agreement with the analytical MLSD performance for uncoded BER. For coded transmission, the analytical and simulation results of performance can be compared for code-word error rate. For a BSC with error probability BER, the code-3.6 Channel Coding 50 Figure 3.12: Error pattern frequency for (a) NSD with two-states at 11 dB with a = 0.6, 6 = 0.9, and (b) LDI detector at 16 dB. (Error patterns in a block of 10 symbols: 1-'010'; 2-'0110'; 3-'01010',4-'010010', 5-'0100010', 6-others, where '1' represents an error.) word error rate (WER) for the rate 1/3 RC is obtained as [ProOl, Ch. 8] WER = 1 - (1 - BER) 3 - 3 • BER(1 - BER) 2. (3.40) The WER for (15,10) HC is given by [ProOl, Ch. 8] WER = 1 - (1 - BER) 1 5 - 15 • BER(1 - BER) 1 4. (3.41) The analytical and simulation results for the WER of RC and HC are also plotted in Fig. 3.13. It can be observed that for BER = 10~3, the 10log(Es/N0) corresponding to the analytical results for RC (Eq. (3.40)) is 5 dB while it is observed as 7 dB for the simulation results. Similarly for HC at BER = 10 - 3 the difference in 10 \og(Es/No) for analytical and simulated value is 3 dB. The considerable gap between analytical and simulation results for both the RC and the HC WER curves suggests that the memoryless channel model assumed for Eqs. (3.40) and (3.41) is inadequate. To investigate the channel memory in more detail, the error-gap-distribution (EGD) of the NSD receiver output and LDI detector output is observed. The EGD is obtained as 3.6 Channel Coding 51 oa £ 1 0 + BER, uncoded 0 WER, RC • WER, HC - Analytical result - Simulation result 2 4 6 101ogI0(E/N0) — Figure 3.13: BER for uncoded transmission and WER for transmission with 1/3 rate rep-etition code and (15,10) Hamming code over AWGN channel. Analytical results according to Eq. (2.25) (BER), Eq. (3.40)(WER-RC) and Eq. (3.4.1)(WER-HC). [AFK72] Pr{0m|l} = ^Pr(0 f c l | l ) (3.42) k=T) where a one in the sequence represents an error and 0fe 1 is the representation of a sequence of k 0's followed by a one. Thus, the EGD gives the probability of an error-free run of length m after the occurrence of an error. Therefore, m = 0 is the event of consecutive errors or DAEs, m = 1 is the error pattern of type '101' and error patterns for m > 1 can be similarly obtained. The EGDs of the outputs of full-state NSD, two-state NSD, LDI detector, and for a BSC are shown in Fig. 3.14. To obtain EGD of the BSC, an interleaver is implemented between encoder and modulator. A corresponding deinterleaver is employed at the receiver after the Viterbi decoder, thus, eliminating the effect of memory introduced by GFSK modulation. 3.6 Channel Coding 52 From Fig. 3.14 it may be observed that for the BSC, the probability of error is indepen-dent of the error gap, whereas, for the NSD and LDI detectors the probability of m — 0 is quite high as compared to non-zero values of m, implying that the probability of DAE is higher as compared to single error events. The inherent memory of partial response GFSK introduces DAE in the output of all the three detectors. Full-state NSD and two-state NSD employs sequence detection which properly takes into account the GFSK modulation memory and therefore, suffers from higher DAE. The two-state NSD further enhances this effect due to the application of PSP in the detection algorithm. o / ! i 1 1 \ 1 1 1 \ E G 1 11 II ii u u II II u E G P F S "n \\ \" -i u 2S.NSD ...... .. 1 1 03I I 1 1 1 ' 1 1 0 2 4 6 8 10 12 14 m * • Figure 3.14: Error-gap distribution (EGD) Pr{Om|l} for AWGN channel. EGD L D i : LDI detector output at 16 dB, EGDFS,NSD: Full-state NSD with a = 0.6, 0 = 0.9 at 11 dB, EGD2S,NSD: Two-state NSD with a = 0.6, f3 = 0.9 at 11 dB, and EGD B S c: ideal BSC with approximately same BER as LDI and NSD. As mentioned before, the most likely error event for NSD is {a[z],a[i + 1] = a[i],a[z + 2]} —> {a[i\, d[i + l),a[i + 2]} which is now supported by EGD analysis. The reason for this is that the first error introduces a phase slip resulting in a lith difference in the correct phase state. Due to PSP of the survivor path, this phase slip affects the next decision, 3.6 Channel Coding 53 causing another error event that compensates the phase slip, restoring the correct phase state of the survivor path. The next most likely event is of type {a[i], a[i + 1] = a[i]} being detected as {a[i],a[i + 1]}, i.e., a single erroneous decision, which is followed by another error event with high probability to restore the correct phase. Fig. 3.15 depicts DAE and an example for the second most likely error event. Correct path Erroneous path ' Error event [0] [1] i = 0 i = 5 Starting at i=l , Correct Path: 100, Erroneous Path: 010 (a) Double adjacent error event. i = 5 Starting at i=l , Correct Path: 1100, Erroneous Path: 0110 (b) A n example for the second most likely error event. Figure 3.15: The most likely error events for the considered channel. Motivated by the high probability of burst errors of length two, improved decoding schemes are suggested for RC and HC in the following sections. 3.6.2 Joint NSD and Decoding for Rate 1/3 R C The most likely error event introduces two errors in two blocks of three symbols at the NSD output, e.g. the coded symbol sequence '000111' may be erroneously received as '001011' 3.6 Channel Coding 54 at the NSD output. These two errors can be corrected using repetition hard decoding. However, the high probability of DAE may introduce two DAEs affecting a single block of 3 bits, e.g., a coded sequence of '000111000' is detected as '001010100' after NSD introducing 2 incorrigible errors. Therefore, to improve the receiver performance, it is proposed to perform NSD and decoding jointly, i.e., to decode while detecting the sequence. The joint NSD and decoding restricts the decoded sequence for a coded sequence '000111000' to be either correct '000111000' or incorrect '000000000', i.e., the information sequence {a[i], a[i + l],a[i + 2],a[i + 3]} can be erroneously estimated as {a[i],a[z + l],a[i + 2],a[i + 3]}, where due to encoding a[i] — a[i + 1] = a[i + 2] is enforced. The modified trellis in Fig. 3.16 shows the paths for the correct coded sequence i.e 'OHIO' and the most likely error event '00000'. For hard decoding, the decision about the survivor path for each state is taken at every instant and is decoded after the sequence detection is complete. In joint NSD and decoding of RC, instead of determining the survivor path at every instant for each state, it is determined once in every three symbol durations based on the accumulated path metric for each trajectory. There are only four possible trajectories from instant i = 3nT to instant i = (3n + 3)T, as shown in Fig. 3.16 (a). Joint NSD and decoding achieves considerable performance gains as will be shown in Section 4.3. The complexity can be reduced by using modified joint NSD and decoding in which the survivor path is determined at instant i = (3n + 1)T instead of determining it at i = (3n + 3)T as shown in Fig. 3.16 (b). At the subsequent two instants only the branch metric corresponding to the survivor path for the two states is calculated and added to the path metric. This introduces only a negligible performance loss (cf. Fig. 4.19), however, the number of required branch metric calculations per symbol interval reduces to two-third on average. The normalized Euclidean minimum distance for the modified joint NSD and decoding can be approximated by the difference sequence corresponding to the most likely error event, i.e., {2 2 2 0} (cf. Section 2.2.2). The corresponding curve for d^in(h)/.Rc is shown 3.6 Channel Coding 55 Correct path 3 , Erroneous path Decision Decision Decision Figure 3.16: (a) Joint NSD and decoding of repetition code with original two-state trellis (b) modified joint NSD and decoding using modified trellis. in Fig. 3.17. The normalized Euclidean distance squared is increased as compared to the uncoded transmission. For example, for h = 1/3 the distance dlin(h = 1/3)/Rc = 3.10 is obtained, which is 2 units higher than uncoded transmission. As <i^ in(/i) corresponds to energy normalized by energy per bit [AAS86], 4.5 dB gain is expected in terms of 101og(£'s/Aro) for h = 1/3 with modified joint NSD and decoding. The simulation results of the proposed decoding algorithm will be discussed in Section 4.3. 3.6 Channel Coding 56 4 3.5 r-3 h Repetition code and joint NSD and decoding 2.5 Hamming code and modified decoding Uncoded transmission 0.5 0.32 0.33 0.34 0.35 0.28 0.29 0.3 0.31 h Figure 3.17: Normalized minimum Euclidean distance c?^in/i?c as a function of modulation index h for different coding and decoding schemes. 3.6.3 Modified Decoding for Rate 2/3 H C For joint decoding of a (n, k) cyclic code over Galois field of order q GF(g), such as the rate 2/3 HC, an alternative and equivalent trellis representation associated with the code is given in [W6178]. The number of states differs with time with a maximum of g(n-fc) states. For the (15,10) expurgated Hamming code specified for Bluetooth systems, the maximum number of states is 32 and on average 34 branch metric calculations per symbol interval are required compared to only 4 in the hard decoding case. Therefore, joint decoding is prohibitively complex for a simple receiver design. From the error gap analysis given in Section 3.6.1, it is inferred that the channel between the FEC encoder output and decoder input is not memoryless but that burst errors of length two occur with high probability. Based on this observation an improved decoder for the rate 2/3 HC is devised that can correct double errors. This improved decoder is described 3.6 Channel Coding 57 in the following para. The (15,11) HC is strictly a single-error-correcting (SEC) code which can correct 15 error patterns, however, the (15,10) expurgated HC can correct additional 16 error-patterns which can be used to correct errors with Hamming weight larger than one [Bla80, Table 5.1]. This property of the (15,10) HC is utilized to correct highly likely burst-error of length 2. Referred to as a single error correcting-double adjacent error correcting (SEC-DAEC) decoder, it is used to correct error patterns of the type xl + xl+1, 0 < i < 13. In the simulations, the decoder has been implemented using a syndrome table [Wic95]. The syndrome S for the received vector of 15 symbols R is calculated as S = R • HT. The matrix H is given by H = [-^ xsl-Pioxs] where Js X5 is the identity matrix and Piox5 is the parity matrix given by [BTS03] Pioxs = [[HOlO], [01101], [11100], [OHIO], [00111], [11001], [10110], [01011], [11111], [10101]]T. To implement the modified decoding, 9 rows are added to the syndrome table corresponding to 9 DAE events in a code-word of 10 information bits. The modified syndrom table is given in Table 3.2. Since the last five bits in a systematic codeword of 15 bits are parity bits, the error correction is performed over the first 10 bits in a code-word of 15 bits. Therefore, the error patterns for the last five bits are not included in the syndrome table. Thus, the modified decoding for (15,10) expurgated HC allows to correct a single error and DAE in a block of 15 data symbols. As explained in Section 3.6.1, the next most likely error events are those where the second error event occurs within a block of 15 data symbols with a high probability after the first error followed by a correct data symbol have already been detected. The difference sequence for these incorrigible error events can be approx-imately given by {2 0}. The minimum distance corresponding to the modified Hamming decoding is shown in Fig. 3.17, which indicates an increase in the squared minimum distance from 1.1 to 1.3 for h = 1/3. The minimum ditance for the conventional decoder results 3.6 Channel Coding 58 Table 3.2: The modified syndrome table for (15,10) Hamming code. Error Pattern Syndrome 1 0 0 0 0 0 0 0 0 0 11010 0 1 0 0 0 0 0 0 0 0 01101 0 0 1 0 0 0 0 0 0 0 11100 0 0 0 1 0 0 0 0 0 0 OHIO 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 0 0 0 0 11001 0 0 0 0 0 0 1 0 0 0 10110 0 0 0 0 0 0 0 1 0 0 01011 0 0 0 0 0 0 0 0 1 0 11111 0 0 0 0 0 0 0 0 0 1 10101 110 0 000000 10111 0 1 1 0 0 0 0 0 0 0 10001 0 0 1 1 0 0 0 0 0 0 10010 0 0 0 1 1 0 0 0 0 0 01001 0 0 0 0 1 1 0 0 0 0 11110 0 0 0 0 0 1 1 0 0 0 01111 0 0 0 0 0 0 1 1 0 0 11101 0 0 0 0 0 0 0 1 1 0 10100 0 0 0 0 0 0 0 0 1 1 01010 in the same dmm(/i) as uncoded transmission because the most prominent error event is DAE for both the cases. Thus, in terms of 10log(E3/N0), a gain of 0.7 dB is expected with modified decoding for HC. The corresponding simulation results are presented in Section 4.3. 3.6.4 LDI Detector with Modified Decoding Though the LDI detector does not employ sequence detection, the memory of GFSK modu-lation introduces DAE as may be observed from the EGD shown in Fig. 3.14. It is expected that the proposed modified decoding may be useful for LDI detector as well. For the rep-etition code, the application of soft decoding does not improve the performance over hard 3.7 Summary of the Proposed Receiver Structure 59 decoding since the frequency of DAE in the LDI detector output is not as high as for the NSD detector (cf. Section 4.3). However, the application of SEC-DAEC decoding helps to improve the performance of the conventional LDI detectors for Bluetooth devices because of the presence of DAE at the LDI output. This has been confirmed by results shown in Section 4.3. Note that in the case of a memoryless channel no considerable performance gain is achieved by making use of this enhanced error-correcting capability of the (15,10) HC. This is probably the reason why this enhanced capability has not been exploited for the LDI detector in the Bluetooth literature so far. 3.7 Summary of the Proposed Receiver Structure Fig. 3.18 shows the structure of the receiver designed for Bluetooth devices in the present research work. The proposed receiver is based on the decomposition approach given by Viterbi processor with 2 states h*(-t) t = / *[n].  nT yes n'<N„? rate 1/3 RC? yes (n' = (n) mod T p ) (Phase and frequency estimation using Eq. (3.32)) Figure 3.18: Proposed receiver structure. rate 2/3 HC ? yes, SEC-DAEC decoder Rimoldi/Huber&Liu to represent CPM [Rim88] [HL89], resulting in a receiver consisting of one filter h*(—t) followed by a Viterbi processor. The novel combined filter he(t), given by Eq. (3.21), provides robustness against interference. The filtered received signal is sampled and fed into a two state Viterbi processor. The Viterbi processor employs NSD where the phase is implicitly estimated using the a-metric as described in Section 3.3.2. The frequency 3.7 Summary of the Proposed Receiver Structure 60 offset estimation using Eq. (3.32) is implemented to provide good performance under high frequency offset conditions. To ensure acceptable performance for all permitted values of h, 0.28 < h < 0.35, it is proposed to use adaptive NSD, in which h is estimated using a 25 to 50 symbol estimation period Ne and quantized hypotheses for h € {hi, / 1 2 , h \ H \ } . The hypothesis yielding the maximum metric over the estimation period is the estimated h which is used to detect the rest of the received signal in a packet. The value of h is estimated for each packet whenever a new packet arrives, i.e., for the packet duration Tp, h is estimated when time instant n satisfies the expression (n)mod(Tp) < Ne. For efficient decoding performance of the rate 1/3 RC, modified joint NSD and decoding is used, for which the Viterbi processor switches to the modified trellis (cf. Fig. 3.18) for calculation of the path metric as described in Section 3.6.2. For the rate 2/3 HC, a modified SEC-DAEC decoder is used after NSD. The performance results for the proposed receiver are discussed in the following chapter. Chapter 4 Results and Discussion The performance of the receiver design proposed for Bluetooth devices in the third chapter was simulated using C++ language. The simulation results of the proposed NSD are presented in this chapter, along with a discussion on the performance and a comparison with state-of-the-art techniques. The channel model described in Section 2.1.2 has been used for simulations. Performance under various channel conditions is presented. To get reliable performance results, over 2.5 million information bits are transmitted in each run in blocks of 10000 bits. A minimum of 500 bit errors are observed achieving average percentage deviation of upper and lower limits of 99% confidence interval from the mean of less that 0.1 %. Initially, in the first section of this chapter, h — 1/3 is assumed for the presented performance results as only 6 states are required in the Viterbi processor. The performance for other values of the modulation index is provided wherever found necessary. Section 4.1 gives the performance in the AWGN channel with uncoded transmission for different receiver filters, phase estimation metrics, and numbers of states. Thus in the first section the best combination of these parameters for the proposed receiver is obtained, which is then applied in the following section, where the performance under varying modulation 61 4.1 Performance in an AWGN Channel for Uncoded Transmission 62 index conditions and in the presence of interference and frequency offset is considered. In Section 4.3, the performance of coded transmission with different decoding strategies is presented, followed by the performance under Rayleigh and Ricean channels in Section 4.4. The performance results of the proposed receiver in various scenarios are summarized in the concluding section. 4.1 Performance in an AWGN Channel for Uncoded Transmission The BER performance as a function of 10 \ogw(Es/N0) is investigated in an AWGN channel for uncoded transmission with h = 1/3 unless stated otherwise. The performance of the proposed receiver is compared with that of the LDI and the MLM-LDI detectors. The LDI receiver is simulated as described in Section 2.3.1. The performance results of the MLM-LDI detector are cited from [SWJ04]. In the rest of this section, the performances for different receiver filters, phase estimation methods, and trellis structures are presented, thus making a selection of the most suitable values for these parameters for the Bluetooth receiver possible. 4.1.1 Filter Selection The BER performances of filter banks with 1, 2, and 3 matched filters (Eq. (3.13)) and that of the novel filter he(t) (Eq. (3.21)) are compared using the receiver structure described in Section 3.1.2 with a signal mapper with 12 signal elements corresponding to At = 1/3 and assuming that the channel phase is known at the receiver. The BER vs. 10\ogl0(Es/N0) curves for the four different cases are shown in Fig. 4.1. The curves for the LDI detector and the MLSD bound (cf. Eq. (2.25)) are also included for comparison purpose. The curves for D = 3 and D = 2 coincide, indicating that there is no performance loss 4.1 Performance in an AWGN Channel for Uncoded Transmission 63 lOlog 1 0 ( E s / N 0 ) Figure 4.1: Performance comparison of different receiver filters: Coherent detection, full-state trellis, h = 1/3, AWGN channel with constant phase. if the number of matched filters is reduced from 3 to 2. By reducing the dimension of the filter bank to one, D — 1, the observed performance loss at BER = 10 - 3 is only approxi-mately 0.5 dB. The high power efficiency with just one filter can be attributed to the high spectral efficiency of GFSK which allows only one basis function to represent the occupied bandwidth with sufficient accuracy. It is also worth noting that the curves closely follow the MLSD bound, indicating a negligible performance loss incurred by the application of alternative representation of CPM proposed by Rimoldi/Huber&Liu [Rim88] [HL89] with reduced complexity receiver filters. The performance improves with the new one dimen-sional filter he(t) by approximately 0.25 dB compared to the one dimensional matched filter h^\t). As the filter set ho(t) proposed by Huber and Liu [HL89] is only an approximate set of basis functions, the minor improvement obtained using the modified filter he[t) is not surprising. Therefore, it is proposed to use the new filter he{t) for the novel receiver for Bluetooth devices. 4.1 Performance in an AWGN Channel for Uncoded Transmission 64 4.1.2 NSD with Implicit Phase Estimation The performance of NSD using the a-metric and the A-metric is evaluated in this section. The receiver employs one filter with impulse response he(t). The performance of NSD using the A -^metric is shown in Fig. 4.2. The performance of NSD improves with increasing window size N, since the channel phase is assumed to be constant. Fig. 4.3 shows the performance in the presence of phase jitter for 10\og10(Es/No) = 11 dB. The BER is plotted against 0 A ( T ) where <JA(T) denotes the variance of phase jitter. The standard deviation 0A(T ) is denoted by a A in the following discussion. For longer window length N the performance deteriorates at higher phase jitter variance because the assumption of constant channel phase is no longer valid. The BER vs. 101og10(£' s/Aro) and BER vs. a A curves for the a-metric are shown in Figs. 4.4 and 4.5, respectively. As a increases the BER performance improves (cf. Fig. 4.4), closely approaching coherent detection performance at a = 0.95. However, the deterioration in performance with channel phase variations increases for high a values (cf. Fig. 4.5) as expected. The identical curves for N = 2 in Fig. 4.2 and a = 0.0 in Fig. 4.4 verifies the theory presented in Section 3.3.2. The performance of NSD using the Af-metric is similar to that of the a-metric as can be inferred by a comparison of Fig. 4.2 with Fig. 4.4 and Fig. 4.3 with Fig. 4.5. However, since the a-metric needs less arithmetic operations compared to the A -^metric (cf. Section 3.3), the former will be used in the following discussion on NSD receiver performance. For better power efficiency, a value of a close to one is desirable, however, for robustness against channel phase variations a value of a close to zero is more suitable. Therefore, the appropriate choice of a depends on the channel conditions. 4.1 Performance in an AWGN Channel for Uncoded Transmission 65 Figure 4.3: Performance of NSD using /V-metric in the presence of phase jitter with filter he(t) and h = 1/3 in an AWGN channel. 4.1 Performance in an AWGN Channel for Uncoded Transmission 66 10 0 1 2 3 4 S 6 7 8 9 10 0" A (in degrees) *• Figure 4.5: Performance of NSD using a-metric in the presence of phase jitter with filter he(t) and h = 1/3 in an AWGN channel. 4.1 Performance in an AWGN Channel for Uncoded Transmission 67 4.1.3 State Reduction The performance with a reduced number of states is compared using coherent detection and a matched filter bank with D = 3. Fig. 4.6 shows that the loss in performance with a two-state trellis is negligible compared to a full-state trellis at high ES/N0 ratios. This is evident from the fact that at high ES/NQ ratios DAEs are the most prominent error events. As explained in Section 3.4, the minimum distance for DAE events remains the same for full-state and two-state trellis which explains the negligible performance loss with the two-state trellis compared to the full-state trellis at high ES/NQ ratios. On the other hand, the one-state trellis performance is highly degraded because the minimum distance drops by a factor of 65 compared to the full-state trellis as illustrated by an example in Section 3.4. The performances of NSD with a two-state trellis using one filter he{t) and a = 0.4 and a = 0.8 closely follow the full-state trellis performance as can be observed from a comparison of Fig. 4.6 and Fig. 4.4. 101og 1 0 (E/N 0 ) " Figure 4.6: Performance with reduced number of states in the Viterbi processor: h = 1/3, AWGN channel with constant phase. 4.1 Performance in an AWGN Channel for Uncoded Transmission 68 The performance of the two-state trellis with the a metric for h = 0.28 and h = 0.35 is shown in Fig. 4.7. For h = 0.35 a full-state Viterbi processor requires 40 states, however, as shown in the graph (cf. Fig. 4.7) the Viterbi processor with only two states gives satisfactory performance. The performance deteriorates with decreasing modulation index since the peak frequency deviation from the carrier frequency decreases [ProOl, Ch. 4]. The corresponding performances of the LDI detector and the MLM-LDI detector are also shown. The proposed receiver is more power efficient than the LDI detector because at BER = IO - 3 the LDI detector needs 5 to 6 dB more power. The MLM-LDI detector is inferior to the proposed receiver by 1 to 2 dB at BER of 10 -3. 1 Olog 1 0 (EM o) • Figure 4.7: Performance comparison of NSD, LDI, and MLM-LDI detectors for different values of h. 4.1.4 Conclusions From the above discussion, it may be concluded that one receiver filter with impulse re-sponse he(t) (Eq. (3.21)) provides good performance. NSD with phase estimation using 4.2 Performance for Uncoded Transmission 69 the a-metric (Eq. (3.30)) with a between 0.4 to 0.8 is very efficient and gives satisfactory results in the presence of phase jitter. The two-state trellis results in a very simple re-ceiver structure irrespective of the value of h with a performance comparable to that of the full-state trellis. Hence, the proposed receiver structure which is used for the further study of the receiver performance in this chapter, consists of one filter he(t) followed by a two-state Viterbi processor and employs NSD with implicit phase estimation using the a metric unless mentioned otherwise. 4.2 Performance for Uncoded Transmission The performance of the proposed receiver is now evaluated under various channel conditions specified in the Bluetooth standard. So far, the performance has been studied only for fixed h in the absence of frequency offset and interference. In this section, these conditions are relaxed. However, only uncoded transmission is considered. 4.2.1 Adaptive NSD Performance The performance of the proposed receiver for varying modulation indices is now investigated. Fig. 4.8 shows the BER assuming h = 0.28, 0.30, 0.3333, and 0.35 at the receiver for a = 0.8 at 10\og10(Es/NQ) = 11 dB, as a function of the modulation index of the transmitted signal. Figs. 4.9 and 4.10 show the corresponding curves for a = 0.6 and 0.4. It may be observed that the performance deteriorates as h at receiver deviates from the h at the transmitter. As the value of a decreases, the performance degradation with deviation from the transmitter h decreases because the mismatched h at receiver introduces frequency variation, and as discussed in Section 3.3.2, the performance deteriorates with increasing a if the channel phase is not constant. Considering for example the results for receiver h = 0.30, the observed BER increases by a factor of 10 if the h at transmitter deviates by -0.007 or 4.2 Performance for Uncoded Transmission 70 +0.012 for a = 0.8 (cf. Fig. 4.8). For a = 0.6, the range of h broadens as the respective values are -0.02 and +0.035, while for a = 0.4 the repective values are -0.02 and +0.05, i.e., the range of h broadens further to 0.28 < h < 0.35. Therefore, a lower value of a provides better robustness in case of a varying modulation index. The optimum performance is achieved if h is exactly known at the receiver. The corresponding performance curve is shown for a = 0.6 in Fig. 4.9. The deviation from the optimum performance increases with the deviation from the true value of h. Fig. 4.11 shows the required SNR at BER = 10~3 as a function of the modulation index h for (a) the MLSD lower bound, (b) known h at the NSD receiver, (c) 8 cases of fixed h at the NSD receiver and (d) the LDI detector at h = 1/3. The number of bits transmitted in the simulations to obtain these curves was reduced to 1 million keeping the minimum observed errors the same as 500. The required Es/N0 ratio increases with the deviation from the true value of h as observed before. It can be noted that for h = 0.31 at receiver the required 10 log 1 0(£' s/A /o) increases by up to 1.0 dB compared to the case of known h at the receiver, when the h at transmitter varies from 0.30 to 0.32. As a simple solution to estimate h, adaptive NSD was introduced in Section 3.5. The performance of ANSD is shown in Fig. 4.12 for a = 0.6 and different numbers of hypotheses \H\ and different estimation periods in symbols Ne. The modulation index h has been quantized in steps of 0.01, except for h = 0.33 where h = 1/3 has been chosen. Table 4.1 gives the corresponding values of k and p where h = k/p. The performance with 8, 4, and 2 hypotheses with estimation period A^ of 50 symbols is shown in Fig. 4.12. To study the effect of decreasing A^, the performance of ANSD with 8 hypotheses and estimation period of 25 symbols is also shown. If \H\ = 4, the h is chosen from the set {0.28,0.30,0.32,0.34}, while for 2 hypotheses h is either one of {0.30,0.33}. Comparing the curves for Ne = 50, it may be observed that the performance with 8 4.2 Performance for Uncoded Transmission 71 1 4 £ | & ti% i I : : : : : ! : : : : : : : ; : ; : ; : : ;; :; ;;• ; ^ >h=0.35 ** 2 h-0,28 h=0.30 —jjjU 1 ' h=Q.3333 \ / : 3 \ - — •. \ ; ".T.'.-f*^,^ TV. ~ : nt,— „ MLSD. bound .... 3 ~ ~ *-4 5 i i : : : : : : : : : : : : : : : : ' : ? : ' : • * - ( ^ -0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 h at transmitter »-Figure 4.8: Performance of the proposed receiver for varying h with a = 0.8 in an AWGN channel with constant phase. h at transmitter Figure 4.9: Performance of the proposed receiver for varying h with a = 0.6 in an AWGN channel with constant phase. 4.2 Performance for Uncoded Transmission 72 h at transmitter Figure 4.10: Performance of the proposed receiver for varying h with a = 0.4 in an AWGN channel with constant phase. 181 1 1 1 1 1 r 9I i i i 1 1 1 1 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 h at transmitter Figure 4.11: The required l0log10(Es/N0) for BER = 10 3 for the proposed receiver for varying h with a — 0.6 in an AWGN channel with constant phase. 4.2 Performance for Uncoded Transmission 73 Table 4.1: Representation of modulation index h for NSD. h 0.28 0.29 0.30 0.31 0.32 0.33 0.34 0.35 k/p 7/25 29/100 3/10 31/100 8/25 1/3 17/50 7/20 hypotheses is the best with a maximum deviation of 0.5 dB in the required ES/NQ ratio from the optimum performance when h is known at the receiver. For 4 and 2 hypotheses the maximum deviation is 0.75 and 3.5 dB respectively. Since the deterioration in performance is quite high with the deviation of h from its true value, the observed degradation in performance with reduced \H\ is expected. As A7,, is reduced to 25 for \TL\ = 8, the maximum deviation in the required ES/NQ ratio from the optimum performance increases to 0.8 dB compared to 0.5 dB for NE = 50. For further study of the performance of ANSD, \TL\ = 4 and NE = 50 has been assumed which gives satisfactory performance under varying modulation index conditions. The performance of the LDI detector with /i=l/3 with variable h at the transmitter is 4 to 5 dB inferior to the proposed ANSD with \7i\ = 4 and NE = 50, while ANSD outperforms the MLM-LDI detector by approximately 0.8 dB at h = 0.28 and 0.6 dB at h = 0.35. 4.2.2 Performance in the Presence of Channel Phase Variations In practice, Bluetooth devices are required to cope with high frequency offsets A / and time-variant channel phases. The BER performance of NSD vs. A / T at 10 log 1 0(£' s/A ro) = 11 dB is shown in Fig. 4.13 for a = 0.9, 0.6, and 0.3 with h = 1/3. It may be observed that for a = 0.9 and a = 0.6, the BER rapidly increases as A / T approaches 0.02, while, the permitted normalized frequency offset A / T is 0.1. a = 0.3 provides some robustness against frequency offset, but cannot cope with offsets as large as A / = 100 kHz. The implementation of Eq. (3.32) with forgetting factor 3 for frequency estimation alleviates 4.2 Performance for Uncoded Transmission 74 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 h at transmitter — Figure 4.12: Performance comparison of the proposed receiver with ANSD for unknown h with a = 0.6 in an AWGN channel with constant phase for different combination of A7,, and \H\ represented by ANSD(A7e,|H|). the effect of frequency offset to a great extent as can be observed from Fig. 4.13. The performance for three values of 3 = 0.99, 0.9, and 0.6 are shown in combination with a = 0.6. As 3 approaches one, the performance deteriorates at high A / T , and if 3 is decreased to 0.6 the power efficiency decreases, however, the performance becomes independent of the frequency offset. For a better comparison, the required 10 log10(Es/No) for a BER of 10 - 3 vs. A / T is plotted in Fig. 4.14 for h = 1/3, where the effect of phase jitter is also included. Eq. (3.30) was used for phase estimation for the curves with a = 0.6 and 0.2, while, for the curves with a = 0.6 and 3 = 0.9 Eq. (3.32) was used for the phase and frequency estimation. The performance of NSD without frequency estimation deteriorates very fast as A / T increases. a = 0.2 provides better stability compared to a = 0.6 because of the presence of a time-varying phase. Furthermore, it is observed that the performance for the implicit phase 4.2 Performance for Uncoded Transmission 75 estimation according to Eq. (3.32) is almost constant for 0 < A / T < 0.1, i.e., it provides the desired stability of the performance of NSD in the presence of high frequency offsets and phase jitter. Increasing CTA from 2° to 5° does not affect the stability of NSD at high frequency offsets, though the required 101og 1 0(£ , s/A r 0) increases by 0.4 dB. Also, the performance is close to the MLSD bound which presumes perfect synchronization between transmitter and receiver oscillator, hence implying that the proposed frequency estimation technique is very effective in combating the permitted high frequency offsets in Bluetooth devices. There is a loss in the performance due to the phase noise introduced by frequency estimation according to Eq. (3.32) as can be observed from Fig. 4.15 which has been plotted for h = 1/3 and an AWGN with constant channel phase. At BER = 10~3, the application of Eq. (3.32) deteriorates the performance by 1.3 dB with 0 = 0.6 and by 0.5 dB for 0 — 0.9 compared to the case when Eq. (3.30) is applied for phase reference estimation. For 0 = 0.99, there is negligible loss in performance, however, as seen from Fig. 4.13, the performance deteriorates at high frequency offsets. Thus the acceptable stability in the presence of frequency offset is achieved at 0 = 0.9 which comes at the expense of 0.5 dB increase in the required Es/N0 ratio at BER = 10 -3. The new phase reference estimation using Eq. (3.32) also improves performance of NSD, if h is unknown as shown in Fig. 4.16, where the required 101og 1 0(£ , s/Af 0) for BER = 10 - 3 is plotted as a function of h for an AWGN channel with constant phase. As seen from the curves drawn for fixed h at the receiver, the deterioration in performance with deviation from the true h is lower than that in Fig. 4.11. For h = 0.31 at receiver the required 10\oglo(Es/No) increases only by up to 0.3 dB compared to the case of known h at the receiver, when the h at transmitter varies from 0.30 to 0.32. As was noted in Section 4.2.1, the corresponding increase is 1.0 dB without frequency estimation. The increase of 0.5 4.2 Performance for Uncoded Transmission 76 Figure 4.14: Performance of proposed NSD receiver in the presence of frequency offset and phase jitter with variance cr\(T). 4.2 Performance for Uncoded Transmission 77 dB in the required 10\og10(Es/No) with the application of frequency estimation technique when h is known at receiver is expected as discussed in the previous paragraph. The effect of 0 on ANSD performance in varying h conditions is shown in Fig. 4.17 where the required 10\og10(Es/No) for the target BER for ANSD is plotted as a function of h in an AWGN channel with constant phase for different combinations of a and 0. The loss in the performance of ANSD with a = 0.6 and 0 = 0.9, with respect to the performance with known h at receiver is negligible with \H\ = 4 and Ne = 50, while, the loss is only 0.6 dB with \7i\ — 2 and Ne = 50. As was observed in the previous section the corresponding values for ANSD with \Ti\ = 4 and \'H\ = 2 without frequency estimation are 0.75 dB and 1.1 dB, respectively. Therefore application of ANSD allows the use of only two hypotheses for a power efficient performance under varying h conditions. Furthermore, comparison of the performance curves of ANSD using Eq. (3.30) with a = 0.6 and Eq. (3.32) with a = 0.6 and 0 = 0.9 in Fig. 4.17 reveals that under varying h conditions with \7i\ = 4 and Ne = 50, the performance of the two is comparable inspite of the phase noise due to Eq. (3.32) in the latter case. From these results the best values for a and 0 which give good performance under frequency offset and varying modulation index conditions are a — 0.6 and 0 = 0.9, respectively. The LDI detector is quite sensitive to frequency offset as can be seen in Figs. 4.13 and 4.14. As the variance of phase jitter increases, the performance of LDI detector becomes even worse. The performance of the LDI detector is approximately 4 dB inferior to ANSD with a = 0.6 and 0 = 0.9 under varying modulation index conditions. The performance of the MLM-LDI detector at h = 0.28 and h = 0.35 assuming known h at receiver is inferior to ANSD with A^ = 50 and \H\ = 4 by less than 0.5 dB as can be observed from Fig. 4.17. 4.2 Performance for Uncoded. Transmission 78 7 8 9 101og 1 0 (E/N 0 )-Figure 4.15: Performance of proposed NSD receiver for different values of B with a = 0.6 and h = 1/3 in an AWGN channel with constant channel phase. 0.28 0.29 0.3 0.31 h at transmitter 0.32 0.33 0.34 0.35 Figure 4.16: Performance the proposed NSD receiver for unknown h in an AWGN channel with constant phase. Eq. (3.32) for phase estimation has been used with a = 0.6 and B = 0.9 if not stated otherwise. 4.2 Performance for Uncoded Transmission 79 0.29 0.3 0.31 h at transmitter 0.32 0.34 0.35 Figure 4.17: Performance of ANSD with \H\ = 4, Ne = 50 as a function of h in an AWGN channel with constant channel phase. 4.2.3 Performance in the Presence of Interference The performance of the proposed ANSD in the presence of interference introduced by other Bluetooth devices (cf. Section 2.1.2) is now considered. Fig. 4.18 shows the graphs for required 101og 1 0(£ s/A 7o) for BER = 10 - 3 vs. h at transmitter for interferer shifted by A / c i from the desired signal. he(t) is used as receiver filter if not stated otherwise. The power ratios of the desired signal and the interferer are given in Table 2.1. The ANSD parameters are a = 0.6, 3 = 0.9, \H\ = 4, and Ne = 50. Since the loss due to interference depends mainly on the filter characteristics, the performance of ANSD in the presence of interference is fairly independent of a and 3. The loss due to adjacent channel interference with the interferer 1, 2 and 3 MHz apart from the desired signal is within 1 dB for the entire range of h. The curves for A / C i i = 2 and 3 MHz are coinciding indicating a similar performance of ANSD for these two cases 4.2 Performance for Uncoded Transmission 80 of ACL To compare the effectiveness of the applied receiver filter he(t) in suppressing ACI with that of the matched filter h^\t), the performance of h^\t) is also shown in Fig. 4.18 with A/ C i i = 1 MHz. The observed performance of matched filter is 1 dB inferior to that of he(t), verifying the better effectiveness of he(t) in suppressing ACI. The performance loss experienced in the presence of co-channel interference (CCI) is 4 to 6 dB, which is quite high. The LDI detector also suffers from a high loss of approximately 4 dB in the presence of CCI. For ANSD a low value of a — 0.2 helps suppressing CCI as shown in Fig. 4.18 achieving a gain of up to 2 dB compared to a = 0.6. Higher values of a provide better phase estimation only if the channel phase is constant. However, the presence of CCI introduces a random time-varying phase, hence, deteriorating the performance at higher values of a. _i \ i i i 1 1 0.29 0.3 0.31 0.32 0.33 0.34 0.35 h at transmitter »-Figure 4.18: Performance of the proposed ANSD with \7i\ = 4, Ne = 50, and 6 = 0.9 in the presence of interference using the representation IF(A/C ] i in MHz,CIR in dB). he(t) is used if not stated otherwise. 4.3 Performance with Channel Coding 81 4.3 Performance with Channel Coding As discussed in Section 3.6, GFSK modulation memory influences the error-gap distribution and hence, affects the coding performance. The high probability of occurrence of DAEs motivated the application of joint NSD and decoding for the RC and SEC-DAEC decoding for the expurgated HC. The BER performance of NSD with coded transmission and different decoding schemes for the RC and the HC are shown in Figs. 4.19 and 4.20, respectively, with a = 0.6, (3 = 0.9, and h = 1/3. The channel is assumed to be AWGN with constant channel phase. Fig. 4.19 shows the performance results for repetition coding for (a) the LDI detector with hard decoding, (b) the LDI detector with soft decoding, (c) NSD with hard decoding, (d) NSD with joint decoding, (e) and NSD with modified joint decoding. For comparison, the uncoded transmission results are also included along with the MLSD bounds according to Eq. (2.25). For the MLSD bound for repetition coded transmission rfmin(^)/5c for h = 1/3 is 3.1 corresponding to the joint NSD and decoding curve in Fig. 3.17. As observed from Fig. 4.19, at BER = 10~3 hard decoding for NSD gives approximately 4 dB performance gain which is further improved by approximately 2 dB using the joint NSD and decoding. The additional gain of 2 dB with joint NSD and decoding is due to the fact that joint NSD and decoding helps to reduce the occurrence of DAE which otherwise is very high (cf. Section 3.6). The performance of the modified joint NSD and decoding which employs the trellis shown in Fig. 3.16 (b) with reduced calculations is only slightly inferior to the original joint NSD and decoding but requires a lower computational complexity, hence suggesting the implementation of modified joint NSD and decoding in the proposed receiver. The performance of NSD closely approaches the corresponding MLSD bound. For the LDI detector, hard decoding achieves the performance gain of approximately 7dB. 4.3 Performance with Channel Coding 82 Application of soft decoding does not yield any further improvement as the frequency of DAE is not as high as in the case of NSD (cf. Figs. 3.12, 3.14). The performance results for the expurgated Hamming coded transmission for (a) the LDI detector with SEC decoding, (b) the LDI detector with SEC-DAEC decoding, (c) NSD with SEC decoding and (d) NSD with SEC-DAEC decoding are shown in Fig. 4.20 along with the MLSD bound for SEC-DAEC decoder. The performance gain achieved by the SEC decoder for the NSD receiver is negligible. This is because of the high occurrence of DAE due to the GFSK modulation memory and sequence detection which cannot be corrected by the SEC decoder. Application of SEC-DAEC decoder, however, yields a performance gain of 1 dB. The LDI detector achieves a higher coding gain of approximately 2 dB with SEC decoding because of the higher occurrence of single errors compared to NSD as may be observed from Fig. 3.12. A further performance gain of 1 dB is also achieved for the LDI detector if the SEC-DAEC decoder is applied. To achieve BER = 10~3, the required 10logw(Es/N0) for Hamming coded transmission is 10.5 dB, while, for repetition coded transmission it is close to 5.5 dB. In a Bluetooth packet, the packet header is repetition coded, while, the payload is uncoded or Hamming coded except for the HV1 packet type. Therefore, for reliable transmission of the packet payload the minimum required 10\og10(Es/N0) is 10.5 dB implying that the probability of receiving a corrupted repetition coded packet header is negligibly low. The proposed receiver outperforms the LDI detector with repetition coded and Ham-ming coded transmission by 5 dB and 3 dB, respectively. The modified decoding strategies help the proposed receiver to perform better for coded transmission. 4.3 Performance with Channel Coding 83 Figure 4.19: Performance of the proposed NSD receiver with a = 0.6, (3 = 0.9, h = 1/3 and rate 1/3 RC for an AWGN channel with constant phase. lOlog 1 0 ( E / N 0 ) Figure 4.20: Performance of the proposed NSD receiver with a = 0.6, (3 = 0.9, h = 1/3 and (15,10) HC for an AWGN channel with constant phase. 4.4 Packet Transmission over Fading Channel 84 4.4 Packet Transmission over Fading Channel Finally, the performance of the proposed NSD receiver for fading channels is studied. Since in the channel under consideration individual packets experience static fading, the packet error rate (PER) is the appropriate performance parameter. In the Bluetooth specifications three sizes of payload data packets have been defined, i.e., a packet can occupy a single, three, or five time slots. To observe the behavior of the proposed receiver in fading channels, data medium (DM3) and data high (DH3) packets which occupy three time slots, have been considered. The payload of DM3 packets is protected with the Hamming code, while the payload of DH3 packets is uncoded. The packet length is 976 and 1480 symbols, respectively, excluding packet header and the access code. A minimum of 75 packet errors are observed by transmitting over 1000 packets in each run of the simulation. According to the results in Section 4.3, the repetition coded packet header is safely assumed error free after decoding at the required SNR for reliable transmission of uncoded and Hamming coded payload. Figs. 4.21, 4.22, and 4.23 depict the simulated PER vs. 101og10(Ea/JV0) for the relevant examples of Rayleigh fading and Ricean fading with Ricean factors of K — 3 and 10, respectively. The performance of NSD and the LDI detector are shown for coded and uncoded transmission. A comparison of the performance curves reveals that the gains of the proposed NSD receiver over the LDI detector for the AWGN channel are well reflected in an improved PER for fading channels. NSD receiver outperforms the LDI detector by 5 to 6 dB for uncoded transmission and by approximately 3 dB for Hamming coded transmission. As shown in Fig. 4.21, the performance gains of NSD with SEC decoder over uncoded transmission are small while with the DAEC-SEC decoder gains of up to 1.5 dB are achieved. The LDI detector also shows better gains with the DAEC-SEC decoder over the SEC decoder. The performance gain in Rayleigh faded channel over uncoded transmission is close to 4 dB with the DAEC-SEC decoder, which is 1 dB higher than that with the SEC 4.5 Summary 85 decoder (cf. Fig. 4.21). Similar results are observed in the Ricean faded channel with K = 3 and 10 in Figs. 4.22 and 4.23, respectively. Hence, the DAEC-SEC decoder is advantageous for the NSD receiver and the LDI detector in the faded channels as well. 10° I I I I ' I i i i i i - I 10 15 20 25 30 35 40 lOlog l 0 ( E / N 0 ) • Figure 4.21: Packet error rate in Rayleigh fading channel for DM3 and DH3 packet types [BTS03]. h = 1/3, NSD with phase reference (Eq. (3.32)) with a = 0.6 and Q = 0.9. 4.5 Summary The performance results shown in this chapter demonstrate the enhanced power efficiency of the NSD receiver which is directly translated into an increased throughput and/or an improved coverage for Bluetooth systems. The performance of coded and uncoded trans-mission in (a) the entire range of possible modulation indices h, (b) time-variant channel phase conditions, (c) presence of interference, and (d) fading channels has been investigated. The proposed decomposition based NSD receiver with phase estimation using the a-metric, provides a near optimum performance with a simple receiver structure. The reduction to a two-state trellis achieves the desired modulation-index-invariant trellis with negligible loss 4.5 Summary 86 15 20 101og,0(E/N0) — Figure 4.22: Packet error rate in Ricean fading channel with K = 3 for DM3 and DH3 packet types [BTS03]. h= 1/3, NSD with phase reference (Eq. (3.32)) with a = 0.6 and 8 = 0.9. 10° * n 10 12 14 16 lOlog 1 0(E/N 0) Figure 4.23: Packet error rate in Ricean fading channel with K = 10 for DM3 and DH3 packet types [BTS03]. h = 1/3, NSD with phase reference (Eq. (3.32)) with a = 0.6 and 8 = 0.9. 4.5 Summary 87 in performance compared to the modulation-index-variant full-state trellis. The novel re-ceiver filter he(t) provides better robustness against interference compared to matched filter hJ?\t). An increase of 1 dB in the required 101og1 0(£' s/A /o) is observed in the presence of ACI. Adaptive NSD with just two hypotheses and an estimation period of 50 symbols gives close to optimum performance at the expense of a small increase in the complexity of the receiver. The proposed frequency estimation technique enables excellent performance of the NSD at the high frequency offsets permitted in the Bluetooth systems. However, its application also incurs 0.5 dB loss in performance, which is negligible compared to the gain achieved in the presence of frequency offsets. The modified decoding methods achieve 1 to 2 dB gain in performance compared to the conventional decoding methods, with only negli-gible added complexity. The proposed receiver outperforms the conventional LDI detector by 3 to 5 dB for typical Bluetooth channels. Chapter 5 Conclusions and Future Work 5.1 Conclusions In the present research work, the design of a simple and power-efficient receiver for Blue-tooth devices has been investigated. The conventional LDI detector employed in Bluetooth devices provides a very cost-effective and robust receiver solution. However, this receiver is very power-inefficient and it cannot cope with the high frequency offsets of up to ±100 kHz permitted in Bluetooth systems. Other state-of-the-art techniques which are more power efficient compared to the LDI detector are the MLM-LDI and the coherent MAP receivers. However, these receivers do not meet the requirement of a simple receiver structure. Fur-thermore, the performances with transmitter h varying from 0.28 to 0.35 and under high frequency offset conditions required for Bluetooth systems have not been investigated for MLM-LDI and MAP receivers. Considering the practical applicability for Bluetooth devices, the proposed receiver has been designed to achieve (a) high power-efficiency, (b) cost-effective implementation, (c) good performance under varying h conditions, (d) good performance under high frequency offset conditions, (e) effective interference suppression, and (f) efficient decoding for coded transmission. To achieve near-optimum performance the Rimoldi/Huber&Liu decomposi-88 5.1 Conclusions 89 tion approach was applied to GFSK, which allows independent desigining of the receiver filter and the Viterbi processor. The proposed receiver consists of one filter, he(t), followed by a two-state Viterbi processor. The proposed NSD receiver design and the performance results can be summarized as follows: • The one-dimensional matched filter h^\t) proposed by Huber&Liu has been modified by cascading it with a Gaussian pre-filter to obtain the novel filter he(t). The modified receiver filter he(t) has better stop-band characteristics compared to h[l\t) that are helpful in suppressing ACI. The improved interference suppression ability of he(t) is verified by the performance results shown in Fig. 4.18, see also [MLS04], [LSM04]. • A Viterbi processor is employed for noncoherent sequence detection, where the phase is implicitly estimated using the a-metric. A value of a close to one gives near-optimum performance as shown in Fig. 4.4. However, a high value of a cannot cope with the high frequency offsets. A simple and very effective frequency estimation algorithm is devised to provide good performance under high frequency offset con-ditions. The performance of designed NSD receiver with frequency estimation using Eq. (3.33) is almost independent of the frequency offset as shown in Fig. 4.14, see also [MLS04], [LSM04]. • The number of states in the Viterbi processor is reduced to 2 from 2p by applying per-survivor processing, thereby transforming the modulation-index-variant trellis struc-ture into a modulation-index-invariant trellis. The performance loss incurred due to state reduction is negligible (cf. Fig. 4.6), however, the complexity of the receiver is reduced to a great extent. • To ensure good performance when the modulation index h at the transmitter is un-known at the receiver, an h estimation algorithm is devised which estimates h based 5.1 Conclusions 90 on the best path metrics for a duration of 50 symbols for 2 to 4 hypotheses of h. At the expense of a small added complexity, the adaptive NSD allows for fast adaptation of NSD to the actually used h, cf. Fig. 4.17, [MLS04], and [LSM04]. • For efficient decoding of coded transmission, modified decoding strategies are sug-gested which specifically address the high occurrence of DAE due to the memory introduced by the GFSK modulation. For repetition coded transmission it is pro-posed to perform modified joint NSD and decoding that provides close to 2 dB gain at a BER of 10 - 3 (cf. Fig. 4.19) over a conventional hard decoding. A modified decoding algorithm for (15,10) Hamming coded transmission provides approximately 1 dB gain in performance to achieve 10 - 3 BER with only negligible added complexity, cf. Fig. 4.20, [MLS04], and [LSM04]. The performance results under typical channel conditions for Bluetooth systems show that the proposed NSD receiver outperforms the LDI detector by 3 to 5 dB for coded and uncoded transmission. Furthermore, the designed receiver has a simpler structure compared to the MLM-LDI detector and it outperforms the MLM-LDI detector by 0.5 to 1 dB. The performance of the NSD receiver for high frequency offsets is better than that of any of the state-of-the-art techniques for Bluetooth receivers available in literature. Therefore, the present receiver design is an attractive solution for low-complexity yet power-efficient Bluetooth devices. The proposed modified decoding scheme for the (15,10) Hamming code also enhances the performance of the LDI detector by 1 dB. As the implementation of the designed decoder does not demand any substantial increase in complexity, the improved decoding method can benefit current Bluetooth devices that employ LDI detectors, at no extra cost, cf. [LMS04a], [LMS04b]. 5.2 Recommendations for Future Work 91 5.2 Recommendations for Future Work • For practical applicability of the proposed NSD receiver, it is suggested to develop a cost-effective hardware implementation. • The interference from co-existing WLAN networks may be investigated as both Blue-tooth and WLANs operate in the ISM band. • The coherent MAP receiver, which employs Laurent's decomposition approach to GFSK, also shows high performance gains over the LDI detector. The application of implicit phase estimation and the proposed estimation algorithms for h and the fre-quency offset to the MAP receiver may be considered in future research on Bluetooth systems. Glossary Abbreviations ACL Asynchronous Connectionless ANSD Adaptive Noncoherent Sequence Detection ARQ Automatic Repeat Request AWGN Additive White Gaussian Noise BER Bit Error Rate BSC Binary Symmetric Channel CIR Carrier to Interferer (power) Ratio CPM Continuous Phase Modulation CRC Cyclic Redundancy Check DAE Double Adjacent Error DH Data High rate (packet) DM Data Medium rate(packet) DV Data Voice (packet) ECB Equivalent Complex Baseband EGD Error Gap Distribution FCC Federal Communications Commission FEC Forward Error Correction FHS Frequency Hop Synchronization FHSS Frequency Hopping Spread Spectrum GFSK Gaussian Frequency Shift Keying HC Hamming Coding HV High Quality Voice (packet) FEC Forward Error Correction I&D Integrate and Dump ISI Inter-Symbol Interference ISM Industrial, Scientific and Medical LDI Limiter Discriminator Integrate & Dump 92 GLOSSARY 93 LOS Line of Sight MAP Maximum a Posteriori Probability Mbps Mega bits per second MLM Max-Log-Maximum likelihood MLSD Maximum Likelihood Sequence Detection NSD Noncoherent Sequence Detection PER Packet Error Rate PSP Per Survivor Processing RC Repetition Coding SCO Synchronous Connection Oriented SEC Single Error Correcting SEC-DAEC Single Error Correcting-Double Adjacent Error Correctin SD Sequence Estimation SIG Special Interest Group SNR Signal to Noise (power) Ratio VA Viterbi Algorithm WER Word Error Rate WLAN Wireless Local Area Network WPAN Wireless Personal Area Network GLOSSARY 94 Variables a Forgetting factor for phase estimation using Eq. (3.30) 3 Forgetting factor for frequency estimation using Eq. (3.34) (j)(t,a) Information carrying CPM phase 00 Constant channel phase Time varying channel phase Phase of the CPM signal modulated by information bits a W,b) Modified information carrying phase A<pc(t,r) Phase jitter aA = aA(T) Standard deviation of phase jitter in degrees 6[n - L] Phase state at time instant nT *[n - L] Modified normalised phase state at time instant nT p(d\i}) Vector of D elements representing correlation of signal elements p(t, d[i]) with the matched filters ho(t) multiplied by matrix C~l (cf. Eq. (3.16)) p(t,d\i]) Signal element corresponding to address vector d[i] A[t] Branch metric at instant iT m Sequence metric at instant iT IN Difference sequence of two sequences of data symbols of length iV a Input binary data stream of bits, a[i] £ {±1} a Estimated binary bit stream, a[i] G {±1} b Unipolar information sequence, b[i] G {0,1} BT Time bandwidth product D Dimension of the filter bank at the receiver front end d[i] Address vector of L + 1 coordinates corresponding to a signal element at instant iT,d[i] = {^[n - L + 1], b[i - L],b[i]} d[i] Address vector corresponding to a hypothetical information sequence d[t] Address vector corresponding to the estimated information sequence m^in Minimum Euclidean distance between two sequences dmm{h) Upper bound for minimum Euclidean distance between two sequences as a function of h E, Average signal energy per transmitted information symbol GLOSSARY 95 fc Passband carrier frequency fr Reference carrier frequency A / Frequency offset between transmitter and receiver A/ C i j Frequency offset of the interferer from carrier g Static fading gain g(t) CPM frequency baseband pulse h = k/p Modulation index for CPM where k and p are relatively prime integers h Hypothetical Modulation index h Estimated Modulation index TC Set of hypothetical modulation indices ho(t) Impulse reponse of the bank of matched filter at the receiver with dimen-sion D, hD{t) = {h%\...,h[g]} he(t) Impulse reponse of the novel filter given by Eq. (3.21) i(t) Collective interferecnce K Ricean factor L Phase memory in terms of information symbols M Modulation level iVo Single sided power spectral density of noise NE Estimation period in symbols for ANSD n{t) Continuous time complex baseband white Gaussain noise process q[t) Baseband phase response (phase pulse) for CPM qref[i] Estimated phase reference at instant iT q"ei[i] Phase reference at instant iT estimated using a-metric and Eq. (3.30) or Eq. (3.32) 9ref[*] Phase reference at instant iT estimated using A -^metric Rc Code rate r(t) Noisy random-phase complex baseband receiver signal s(t, a) Complex baseband transmitted signal corresponding to information se-quence a s(t, d) Complex baseband transmitted signal corresponding to sequence of ad-dress vectors d Sn State vector for full-state trellis at time instant nT GLOSSARY 96 S\ State vector for one-state trellis at instant nT S„ State vector for two-state trellis at instant nT •SRF(^) Passband transmitted signal t Continuous time T Modulation interval Td Root mean square delay spread introduced by channel Tp Packet duration x[i] Vector of D elements representing filtered samples of r(t) at time instant iT,x[i\ = {xi[i],...,xD[i]} GLOSSARY 97 Definitions mx n a[i] XT X* x H = XT* ?R{x} g(t) = i / v / 2 7 7 / t 0 ° e " T 2 / 2 d r <t)*y(t) = J™oox(T)y(t-T)dT x[i] * y[i] = Efc=_oo x[k]y[i - k] Natural logarithm of x Ceiling function Complement of a[i] Zeroth order modified Bessel function of the first kind Transpose of matrix x Complex congulate of x Hermitian transpose of matrix x Real part of the complex number x The standard Gaussian Q function Continuous-time convolution of the functions x(t) with y(t) Discrete-time convolution of the functions x[i] with y[i] Bibliography [AAS86] J. 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