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SYNCHRONIZATION IN IMPULSE RADIO ULTRAWIDEBAND COMMUNICATION SYSTEMS By SANDEEP AEDUDODLA A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2006 Copyright 2006 by Sandeep Aedudodla ACKNOWLEDGMENTS I would like to express my gratitude to Prof Tan Wong, for his constant encourage ment, support and patience throughout the course of my graduate study. His meticulous guidance and frank advice have been instrumental in bringing this work to its present form. I am thankful to Saravanan Vijayakumaran for the numerous insightful discussions and his help in reviewing some of this work. I would also like to thank the members of my supervisory committee, Prof. John Shea, Prof. Liuqing Yang and Prof Ye Xia, for their guidance, suggestions and interest in my work. I would like to thank my parents and my sister and brotherinlaw for their uncon ditional love, support and patience, which have constantly motivated me to confront the challenges faced during my graduate studies. TABLE OF CONTENTS page ACKNOW LEDGM ENTS ...................... . ........ iii LIST OF FIGURE S . . . . . . . . vii ABSTRACT .... ............................... .. ix CHAPTER 1 INTRODUCTION .............................. 1 1.1 Objectives and Main Contributions ........ . ...... 2 1.2 D issertation Outline . . . . . . . 4 2 BACKGROUND AND RELATED RESEARCH ........ . .. 5 2.1 Acquisition Methods in Traditional Spread Spectrum Systems . 5 2.2 Signal Acquisition in UWB Systems ....... . ...... 9 2.2.1 System M odel ...... .. ..... ... .... ......... 9 2.2.2 Current Approaches Towards UWB Signal Acquisition . 12 2.2.2.1 Detectionbased approaches .... . ..12 2.2.2.2 Efficient search strategies . . . . 15 2.2.2.3 Search space reduction techniques . ..... 16 2.2.2.4 Estimationbased schemes . . . . 17 2.2.2.5 Miscellaneous approaches . . . . 20 3 ISSUES AND CHALLENGES IN THE DESIGN OF UWB ACQUISITION SY STEM S . . . . . . . . . 21 3.1 H it Set . . . . .. . .. . 2 1 3.2 Asymptotic Acquisition Performance of Thresholdbased Schemes . 23 3.3 The Search Space in UWB Signal Acquisition . . 25 3.4 Generalized Likelihood Ratio Test for UWB Signal Acquisition . 26 4 ACQUISITION WITH HYBRID DSTH UWB SIGNALING . . 31 4.1 System M odel . . . . . . . . 31 4.1.1 Hybrid DSTH Signal Format . . . . . 31 4.1.2 Received Signal . . . . . . 32 4.2 Hit Set Formulation . . . . . . . 33 4.3 Stage 1: TH Acquisition . . . . . . 36 4.3.1 The Decision Statistic . . . . . . 38 4.3.2 Mean and Variance of the Decision Statistic . . . 39 4.3.2.1 Averaging over the DS sequence . . 41 4.3.2.2 Averaging over the TH sequence . . 41 4.3.3 False Alarm and Detection Probabilities . . . 43 4.4 Stage 2: DS Acquisition . . . . . . 44 4.5 Setting Thresholds 71 and 72 . . . . . 46 4.6 Mean Detection Time . . . . . . . 49 4.7 Numerical Results . . . . . . . 51 4.8 System Design and Complexity Considerations . . . 55 5 ACQUISITION IN TRANSMITTED REFERENCE UWB SYSTEMS . 57 5.1 TRUWB Systems . . . . . . . 57 5.2 System M odel . . . . . . . . 58 5.3 Twolevel DS Signaling Structure . . . . . 59 5.4 Hit Set Definition . . . . . . . 62 5.5 Twostage Acquisition Scheme for TRUWB Signaling . . 67 5.5.1 Decision Statistic of the First Stage . . . . 69 5.5.2 Decision Statistic for the Second Stage . . . 71 5.5.3 Probabilities of False Alarm and Detection ... . ...... 71 5.6 Decision Threshold Selection . . . . . . 72 5.7 Mean Detection Time . . . . . . . 77 5.8 Numerical Results . . . . . . . 80 6 FINE TIMING ESTIMATION . . . . . . 83 6.1 System M odel . . . . . . . . 84 6.2 PilotAssisted Timing Estimation . . . . . 85 6.2.1 Maximum Likelihood Timing Estimation . . . 86 6.2.2 CramerRao Lower Bound . . . . . 88 6.3 Nonpilotassisted Timing Estimation . . . . 90 6.3.1 Maximum Likelihood Timing Estimation . . . 91 6.3.2 Cramer Rao Lower Bound . . . . . 93 6.4 Suboptimal Timing Estimation Methods . . . . 97 6.4.1 Timing with Dirty Templates (TDT) . . . . 97 6.4.2 Transmitted Reference (TR) Signaling ..... . . 97 6.5 Numerical Results . . . . . . . 98 6.5.1 Channel Parameter Extraction . . . . . 98 6.5.2 Computation and Simulation Results . . . . 99 7 CONCLUSIONS . . . . . . . . 110 7.1 C conclusions . . . . . . 7.2 Open Problem s . . . . . REFERENCES . ..................... . . . 1 10 .. . 111 .. . 112 BIOGRAPHICAL SKETCH ............................. 119 LIST OF FIGURES Figure page 21 Block diagram of a parallel acquisition system for directsequence spread spec trum system s . . . . . . . . 7 22 Block diagram of a serial acquisition system for directsequence spread spec trum system s . . . . . . . . 7 23 Block diagram of the acquisition scheme proposed by Blazquez et al. .. 13 24 Block diagram of the acquisition scheme proposed by Soderi et al. ...... 14 25 Autocorrelation function (ACF) of correlator outputs z[n] or its Fourier series (FS) coefficients estimated via sample averaging and used to estimate timing o ffset . . . . . . . . . 1 8 31 Effect of received SNR on size of hit set H for NR 5 and NR 10. .. 22 32 The ROCs when the threshold is set for a singleton hit set containing only the true phase and for a hit set defined in (31) with An 10. .. . 23 33 Generalized likelihood ratio test for evaluation of phase .. .. . .30 41 The hybrid DSTH signal format . . . . . . 32 42 Conceptual block diagram of the hybrid DSTH twostage acquisition scheme. . . . . . . . . . . 3 2 43 Effect of deviation from the true phase on BER for NR = 5 . .... 37 44 Squaring loop for TH pattern acquisition .................. .. 38 45 Acquisition system for DS stage .................. ...... .. 45 46 Flowgraph to determine mean detection time for Hybrid DSTH Acquisition S y stem . . . . . . . . . 5 1 47 Effect of received SNR on size of hit set H for NR = 5 and NR= 10. .. 53 48 Mean detection time for hybrid DSTH and double dwell systems for NR = 5. 54 49 Mean detection time for hybrid DSTH and double dwell systems for NR = 10. 54 51 Illustration of the delay and multiply operation on the received signal. .. 61 52 Illustration of the margin of error tolerable in the despreading of the inner code in the first stage . . . . . . . 63 53 Block diagram of the autocorrelation receiver. ........... ........ 64 54 Illustration of search strategy used by the twostage TR acquisition system. 69 55 Flowgraph illustrating the proposed twostage acquisition scheme. ...... 78 56 Effect of received SNR on hit set size H... ............ . 82 57 Mean detection time for twostage and singlestage TRUWB acquisition sys tem s . . . . . . . . . . 82 61 Comparison between noiseless received signal in CM1 model and reconstructed signal after parameter extraction . . . . . . 100 62 CRLB and simulation results of ML, TDT and TR estimators in a typical CM1 channel w ith = 16 . . . . . . . 102 63 CRLB and simulation results of ML, TDT and TR estimators in a typical CM2 channel with N = 26. Note the greater number of symbol observations com pared to CM 1 . . . . . . . . 103 64 CRLB and simulation results of ML, TDT and TR estimators in a typical CM3 channel with N = 42 . . . . . . . 104 65 CRLB and simulation results of ML, TDT and TR estimators in a typical CM4 channel with N = 99 . . . . . . . 105 66 CRLB and simulation results of pilotassisted ML estimator in a typical CM1 channel with N = 16, assuming Ntmax = 100....... . . 106 67 CRLB and simulation results of pilotassisted ML estimator in a typical CM2 channel with N = 26, assuming Ntmax = 100....... . . 107 68 CRLB and simulation results of nonpilotassisted ML and TDT estimators in a typical CM1 channel with N = 16, assuming Ntmax = 100. . . 108 69 CRLB and simulation results of nonpilotassisted ML and TDT estimators in a typical CM2 channel with N = 26, assuming Ntmax = 100. . . 109 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy SYNCHRONIZATION IN IMPULSE RADIO ULTRAWIDEBAND COMMUNICATION SYSTEMS By Sandeep Aedudodla December 2006 Chair: Tan F. Wong Major Department: Electrical and Computer Engineering Ultrawideband (UWB) communication has recently generated a significant amount of interest and is a primary physical layer candidate for wireless personal area network (WPAN) standards aimed at both high data rate and low data rate applications. Ultrawideband spread spectrum systems employing impulse radio signaling are being considered for these applications. The UWB channel is a dense multipath channel with a large delay spread and highly resolvable multipath components, owing to the narrow pulses employed. In any communication system, the receiver requires knowledge of the timing information of the received signal to accomplish demodulation. The process of obtaining this timing information is known as synchronization. Spread spectrum systems typically achieve synchronization in two stages: acquisition and tracking. The acquisition stage attempts to achieve coarse synchronization to within a chip duration by evaluating the phases in the search space and the tracking stage employs a code tracking loop to maintain fine synchronization. Ultrawideband systems employ long spreading sequences to eliminate spectral lines resulting from pulse repetition. Moreover, the short pulses employed result in a fine resolution of the search space. Due to these reasons, UWB acquisition systems face a large search space which results in a large amount of time for the system to successfully acquire the signal. This results in long preambles in a packet based WPAN which adversely affects the network throughput. The dense resolvable multipath results in the existence of multiple phases which can be considered to be a good estimate of the true phase. In this dissertation, we define a hit set to be the set of phases which guarantee a nominal demodulation performance subsequent to acquisition. We focus on signal design and the development of efficient acquisition schemes which significantly reduce the mean detection time through search space reduction. One such design is the proposed hybrid direct sequencetime hopping (DSTH) signaling format and a twostage acquisition scheme which combats the large search space problem in impulse radio UWB systems. We study the acquisition problem in transmitted reference (TR) UWB systems with DS signaling and design a twostage acquisition scheme which greatly reduces the mean detection time. We also consider the fine timing estimation problem in impulse radio systems in which the receiver does not have knowledge of the received pulse shapes or the channel. We derive the CramerRao lower bound and maximum likelihood estimators for pilotassisted and nonpilotassisted cases, and compare their performance to suboptimal methods with lower complexity. CHAPTER 1 INTRODUCTION A class of spread spectrum techniques known as ultrawideband (UWB) commu nication [14] has recently received a significant amount of attention from academic researchers as well as from the industry. Ultrawideband signaling is being considered for high data rate wireless multimedia applications for the home entertainment and personal computer industry as well as for low data rate sensor networks involving lowpower devices. It is also considered a potential candidate for alternate physical layer protocols for the highrate IEEE 802.15.3 and the lowrate IEEE 802.15.4 wireless personal area network (WPAN) standards [5, 6]. In any communication system, the receiver needs to know the timing information of the received signal to accomplish demodulation. The subsystem of the receiver which performs the task of estimating this timing information is known as the synchronization stage. Synchronization is an especially difficult task in spread spectrum systems which employ spreading codes to distribute the transmitted signal energy over a wide band width. The receiver needs to be precisely synchronized to the spreading code to be able to despread the received signal and proceed with demodulation. In spread spectrum systems, synchronization is typically performed in two stages [7, 8]. The first stage achieves coarse synchronization to within a reasonable amount of accuracy in a short time and is known as the acquisition stage. The second stage is known as the tracking stage and is respon sible for achieving fine synchronization and maintaining synchronization through clock drifts occurring in the transmitter and the receiver. Tracking is typically accomplished using a delay locked loop [7]. Timing acquisition is a particularly acute problem faced by UWB systems as explained in the sequel. This dissertation addresses the significance of the acquisition problem in UWB systems and the ways to efficiently tackle it. Accurate timing estimation following coarse acquisition is required to perform demodulation and is useful in applications requiring precise localization. Impulse radio systems often do not have knowledge of the received pulse shape and the channel. We present pilotassisted and nonpilotassisted maximum likelihood (ML) timing estimators in the absence of such information. We also derive the CramerRao lower bound (CRLB) on the performance of any unbiased estimator. We compare the simulation performance of the ML estimators to the CRLB and to the performance of suboptimal timing estimators that do not require knowledge of the pulse shape and the channel. 1.1 Objectives and Main Contributions Short pulses and low duty cycle signaling [1] employed in UWB systems place stringent timing requirements at the receiver for demodulation [9, 10]. The wide band width results in a fine resolution of the timing uncertainty region thereby imposing a large search space for the acquisition system. Typical UWB systems also employ long spreading sequences spanning multiple symbol intervals in order to remove spectral lines resulting from the pulse repetition present in the transmitted signal. In the absence of any side information regarding the timing of the received signal, the receiver needs to search through a large number of phases1 at the acquisition stage. This results in a large acquisition time if the acquisition system evaluates phases in a serial manner and results in a prohibitively complex acquisition system if the phases are evaluated in a parallel manner. Moreover the relatively low transmission power of UWB systems requires the receiver to process the received signal for long periods of time in order to obtain a reliable estimate of the timing information. In a packetbased network, each packet 1 Traditionally, in directsequence spread spectrum systems the chiplevel timing of the PN sequence is referred to as the phase of the spreading signal. In this document, we use phase and timing interchangeably. has a dedicated portion known as the acquisition preamble within which the receiver is expected to achieve synchronization. However for the high datarate applications envisaged for UWB signaling, long acquisition preambles would significantly reduce the throughput of the network. The transmitted pulse can be distorted through the antennas and the channel and hence the receiver may not have exact knowledge of the received pulse signal waveform [11]. The short pulses used in UWB systems also result in highly resolvable multipath with a large delay spread, at the receiver [12]. The UWB receiver could therefore synchronize to more than one possible arriving multipath component (MPC) and still perform satisfactorily. This means that there could exist multiple phases in the search space which could be considered acceptable and could be exploited to speed up the acquisition process. These challenges arising from the signal and channel characteristics unique to UWB systems indicate the significance of the acquisition problem in UWB communication and the need to address it efficiently. Addressing some of these issues is the focus of this dissertation. We propose a hybrid direct sequencetime hopping (DSTH) signaling format and a twostage acquisition scheme for UWB systems which enables in a significant reduction in the size of the search space. We evaluate the performance of the twostage scheme in terms of the mean detection time. We define the hit set as the set of phases, which, following acquisition, result in a satisfactory receiver BER performance. The hit set thus characterizes the effect of the dense resolvable multipath on the acquisition system performance. The acquisition problem in direct sequence transmitted reference (TR) UWB systems is analyzed, where we observe that there is a significant relaxation in the timing requirement which can be exploited by the use of a twostage acquisition scheme presented. We propose to study the tracking problem for UWB systems as future work. 1.2 Dissertation Outline This dissertation is organized as follows. In Chapter 2, we briefly summarize the acquisition approaches adopted by traditional spread spectrum systems and also classify and discuss the current research on UWB signal acquisition. Through a discussion of existing spread spectrum techniques we seek to distinguish the issues and challenges unique to UWB acquisition system design, which are presented in Chapter 3. An understanding of these issues, particularly the existence of multiple acquisition phases and the asymptotic acquisition performance, enables better UWB acquisition system design. In Chapter 4, we present the hybrid DSTH signaling format and a twostage acquisition scheme to combat the large search space problem in UWB acquisition systems. In Chapter 5 we discuss the acquisition problem in TRUWB systems with DS signaling and present a twostage acquisition scheme which significantly reduces the mean detection time. We discuss pilotassisted and nonpilotassisted fine timing estimation methods for impulse radio, in the absence of received pulse shape and channel information in Chapter 6. The dissertation is concluded in Chapter 7. CHAPTER 2 BACKGROUND AND RELATED RESEARCH In this chapter, acquisition methods in conventional spread spectrum systems are summarized and acquisition techniques for UWB communication systems are discussed. Through a discussion of spread spectrum acquisition techniques, we wish to highlight that UWB signaling is a class of spread spectrum communication and thus acquisition techniques for UWB systems could employ similar concepts with appropriate modifications. We also discuss the existing work on acquisition in UWB systems. 2.1 Acquisition Methods in Traditional Spread Spectrum Systems Ultrawideband communication falls in the category of spread spectrum communica tion systems. In this section, we briefly review the main features of acquisition methods used in traditional spread spectrum systems to put the current approaches to UWB signal acquisition in perspective. There has been extensive research on spreading code acquisition and tracking for spread spectrum systems with directsequence, frequency hopping and hybrid modulation formats [7, 8, 13]. We will bring out the main issues by considering the timing acquisition of directsequence spread spectrum systems. In a directsequence spread spectrum system, the receiver attempts to despread the received signal using a locally generated replica of the spreading waveform. Despreading is achieved when the received spreading waveform and the locally generated replica are correctly aligned. If the two spreading waveforms are out of synchronization by even a chip duration, the receiver may not collect sufficient energy for demodulation of the signal. As mentioned before, the synchronization process is typically divided into two stages: acquisition and tracking. In the acquisition stage, the receiver attempts to bring the two spreading waveforms into coarse alignment to within a chip duration. In the tracking stage, the receiver typically employs a code tracking loop which achieves fine synchronization. If the received and locally generated spreading waveforms go out of synchronization by more than a chip duration, the acquisition stage of the synchronization process is reinvoked. The reason for this two stage structure is that it is difficult to build a tracking loop which can eliminate a synchronization error of more than a fraction of a chip. A typical acquisition stage attempts to bring the synchronization error down to be within the pullin range of the tracking loop by searching the timing uncertainty region in increments of a fraction of a chip. A simplified block diagram of an acquisition stage which is optimal in the sense that it achieves coarse synchronization with a given probability in the minimum possible time is the parallel acquisition system [7] shown in Fig. 21. This stage checks all the candidate phases in the uncertainty region simultaneously and the phase corresponding to the maximum correlation value is declared to be the phase of the received spreading waveform. In an additive white Gaussian noise (AWGN) channel, this acquisition strategy produces the maximumlikelihood estimate (from among the candidate phases) of the phase of the received spreading waveform. However, the hardware complexity of such a scheme may be prohibitive since it requires as many correlators as the number of candidate phases being checked, which may be large depending on the size of the timing uncertainty region. A widely used technique for coarse synchronization, which trades off hardware complexity for an increase in the acquisition time, is the serial search acquisition system shown in Fig. 22. This system has a single correlator which is used to evaluate the candidate phases serially until the true phase of the received spreading waveform is found. Hybrid methods such as the MAX/TC criterion [14], have also been developed which employ a combination of the parallel and serial search acquisition schemes and reduce the acquisition time at the cost of increased hardware complexity. All the acquisition schemes employ a verification stage [15] which is used to confirm the coarse estimate of the true phase before the control is passed to the tracking loop. X filter deter corresponding Verification o code Received Ito largest stage tracking loop signal IN energy Bandpass Energy X filter detector s(tt,) Figure 21: Block diagram of a parallel acquisition system for directsequence spread spectrum systems which evaluates the candidate phases 1 12, ... I. In the ith arm, the decision statistic corresponding to the candidate phase t, is generated by correlating the received signal with a delayed version of the locally generated spreading waveform s(t). Received signal Figure 22: Block diagram of a serial acquisition system for directsequence spread "p~im ilS systems which evaluates the candidate 1 .l.  tl, 12, ... ,n serially until the threshold is exceeded. The decision statistic corresponding to the candidate phase 1i is generated by correlating the received signal with a delayed version of the locally gener ated , .; ,vi. waveform s(t). If the threshold is not exceeded, the search updates the value of the candidate phase and the process continues. STo code tracking loop In traditional spread spectrum acquisition schemes, the signaltonoise ratio (SNR) of the decision statistic improves with an increase in the dwell time, which is the integration time of the correlator. Thus the probability of correctly identifying the true phase of the received spreading waveform can be increased by increasing the time taken to evaluate each candidate phase. This tradeoff has been identified and exploited by several researchers for the development of more efficient acquisition schemes and has led to their classification into fixed dwell time and variable dwell time schemes [7, 8]. The fixed dwell time based schemes are further classified into single and multiple dwell schemes [16]. The decision rule in a single dwell scheme is based on a single fixed time observation of the received signal whereas a multiple dwell scheme comprises multiple stages with each stage attempting to verify the decision made by a previous stage by observing the received signal over a comparatively longer duration. Variable dwell time methods are based on the principles of sequential detection [17] and are aimed at reducing the mean dwell time. The integration time is allowed to be continuous and incorrect candidate phases are dismissed quickly which results in a smaller mean dwell time. Several performance metrics have been used to measure the performance of acquisition systems for spread spectrum systems. The usual measure of performance is the mean acquisition time which is the average amount of time taken by the receiver to correctly acquire the received signal [7, 8, 18]. The variance of the acquisition time is also a useful performance indicator, but is usually difficult to compute. The mean acquisition time is typically computed using the signal flow graph technique [19]. For parallel acquisition systems, a more appropriate performance measure is the probability of acquisition or alternatively the probability of false lock [20]. In the presence of multipath, there could exist more than one phase which could be considered to be the true phase of the received signal. However, few acquisition schemes for spread spectrum systems [21, 22] have taken this into consideration. 2.2 Signal Acquisition in UWB Systems As discussed in Section 1.1, the distinguishing feature of UWB systems is the wide bandwidth and the relatively low transmission power constraint imposed by regulatory bodies. The wide bandwidth enables fine timing resolution resulting in a large number of resolvable paths in the UWB channel response. There may be more than one path where a receiver lock could be considered successful acquisition. The stringent power constraint necessitates the use of long spreading sequences which together with fine timing resolution results in a large search space for the acquisition system. So the main difference between the acquisition problems for UWB systems and traditional spread spectrum systems is the presence of multiple acquisition states and the relatively large search space in the former. The large search space obviates the use of a fully parallel acquisition system due to its high hardware complexity. Hence much of the existing work on UWB signal acquisition has focused on serial and hybrid acquisition systems. Several researchers have tackled the large search space problem by proposing schemes which involve more efficient search techniques. However, the existence of multiple acquisition states has received relatively less attention and has not been sufficiently exploited. Furthermore, a significant portion of the existing work assumes an AWGN channel model for the UWB channel and neglects the effect of multipath in the development and evaluation of the proposed acquisition schemes. In the next subsection, we describe general models for the propagation channel and the acquisition signal for UWB systems. This model will be used in the later subsections to describe the main features of some of the proposed schemes for UWB signal acquisition. 2.2.1 System Model The transmitted UWB signal consists of a train of short pulses monocycless) which may be dithered by a timehopping (TH) sequence to facilitate multiple access and to reduce spectral lines. The polarities of the transmitted pulses may also be randomized us ing a DS spreading code to mitigate multiple access interference (MAI). The generalized UWB signal transmitted during the acquisition process for a single user can be expressed as a series of UWB monocycles b(t) of width Tp each occurring once in every frame of duration Tf as 00 x(t) = bi1/Nbja[l/Nds] t(t Tf C[INh]T), (21) 1 00 where Nb is the number of consecutive monocycles modulated by each data symbol bi, Tf is the pulse repetition time, T, is the chip duration which is the unit of additional time shift provided by the TH sequence and [.], [L denote the integer division remainder operation and the floor operation respectively. The pseudorandom TH sequence {ct(l);h 1 has length Nth where each cl takes integer values between 0 and Nh 1 where Nh is less than the number of chips per frame Nf = Tf/Tc. The DS sequence {ai)}N has length Nds with each al taking the value +1 or 1. Some UWB systems may employ only TH or DS spreading and may not send any data during the acquisition stage. In those cases, the transmitted signal is obtained by setting c = 0, al = +1 and bi = +1 accordingly. The UWB indoor propagation channel can be modeled by a stochastic tapped delay line [12, 23] which can be expressed in the general form in terms of its impulse response Ntap h(t) S pkhfk(t tk), (22) k0 where Ntap is the number of taps in the channel response, hk is the path gain at excess delay tk corresponding to the kth path and pk denotes the polarity of the path gain. Due to the frequency sensitivity of the UWB channel, the pulse shapes received at different excess delays are pathdependent [24]. The functions fk(t) model the combined effect of the transmit and receive antennas and the propagation channel corresponding to the kth path on the transmitted pulse. To enable tractable analysis, we assume that the pulses are distorted identically in all the MPCs, i.e., fk(t) = f(t) and that the excess delays tk = kTe. Under these assumptions, the impulse response can be expressed as Ntap1 h(t) S pkhkf(t kTe), (23) k0 where f(t) represents the distortion to the transmitted pulse due to the antennas and the channel, Ntap is the number of MPCs and hk is the path gain at the excess delay (k + 1)T, normalized by the received power at one meter. The path gains are modeled as independent, but not identically distributed random variables with Nakagamim distributions. The pk's are modeled as independent random variables equally likely to take the values 1 [12]. The average energy gain for the kth path [23] is given by Etot for k = 0 k l+rF(c) (24) Et re(k1)Tc/ fork 1...Ntap  l+rF(c) " where Etot is the total average energy in all the paths normalized to the total average energy El received at a distance of Im, r is the ratio of the average energy of the second MPC to the average energy of the direct path, c is the decay constant of the power delay profile and F(e) = 1exp[ (Nti1)T The Nakagamim random variable hk has the probability density function given by 2mmk h2mc1 mITk h2 Pk (h) exp (25) k F(mfk) Q k According to Cassioli et al. [23], Etot, r and c are all modeled by lognormal distributions. The Nakagami fading figures mk for hk are distributed according to truncated Gaussian distributions whose mean and variance vary linearly with excess delay. These longterm statistics are treated as constants over the duration of the acquisition process. The received signal from a single user can then be expressed as oO r(t) = b /Nbja[l/Nd]Wr(t ITf C[/Nth]Tc T) + n(t), (26) 1 oc where Ntap  Wr(,t)= ,, ,'.(t kTr) (27) k0 is the received waveform corresponding to a single pulse. Here r (t) f(t) ((t) is the received UWB pulse. The duration of the received pulse T, is assumed to be less than the chip duration T,. The propagation delay is denoted by 7 and n(t) is a zero mean noise process. Given the received signal, the acquisition system attempts to retrieve the timing offset 7. 2.2.2 Current Approaches Towards UWB Signal Acquisition Acquisition schemes for UWB systems in the literature can be broadly classified into those which follow detectionbased approaches and those which rely on estimation theoretic strategies. The acquisition methods which employ a detection based approach typically evaluate a candidate phase by first obtaining a measure of correlation between the received signal and a locally generated template signal offset by the candidate phase. This measure of correlation is then compared to a threshold in order to make a decision. These candidate phases could be evaluated in a serial, parallel or hybrid manner. Among the detectionbased schemes for UWB acquisition some schemes focus exclusively on the development of efficient search strategies to quickly evaluate the candidate phases in the search space and certain other schemes propose twostage acquisition methods that achieve a reduction in the search space itself. In the estimationbased methods, an estimate of the timing is typically obtained by maximizing a statistic over a set of candidate phases. This statistic is usually obtained from correlation of the received signal with a template signal. These schemes thus do not involve a threshold comparison. Most of the estimationbased schemes attempt to exploit the cyclostationarity inherent in UWB signaling owing to pulse repetition. 2.2.2.1 Detectionbased approaches Some of the acquisition schemes proposed for UWB signal acquisition involve the straightforward application of traditional spread spectrum acquisition techniques. In Blazquez et al. [25], the traditional coarse acquisition scheme where the search space is searched in increments of a chip fraction is analyzed for the acquisition of TH UWB signals in AWGN noise. Fig. 23 shows a block diagram of the scheme where a particular phase ti in the search space is checked by correlating the received signal with a locally generated template signal with delay ti. If the integrator output exceeds the threshold, the phase ti is declared to be a coarse estimate of the true phase of the received signal. If the threshold is not exceeded, the search control updates the phase to be checked as ti+ = ti + cTp where c < 1 and Tp is the pulse width. This process continues until the threshold is exceeded. Received Is threshold Yes Declare ti to be Signal Integrator exceeded? coarse estimate signal y exceededr of true phase s(tt No Template signal Search generator control Figure 23: Block diagram of the acquisition scheme proposed by Blazquez et al. A parallel acquisition scheme is presented in Yuanjin et al. [26] for UWB signals spread by a Barker code of length 4, which is unreasonable considering that long spreading sequences are needed in UWB systems to eliminate spectral lines. The output of a matched filter matched to the received pulse is sampled at the chip rate and the samples are then passed through four psuedonoise (PN) matched filters corresponding to the four possible delays of the Barker sequence. The delay corresponding to the output with largest energy is chosen as the coarse estimate of the true phase. In Soderi et al. [27], the output of a matched filter, whose impulse response is a timereversed replica of the spreading code, is integrated over successive time intervals of size mTo where 1 < m < Ntap and To is the chip duration in an attempt to combine the energy in the multipath. The integrator output is then sampled at multiples of mTo and compared to a threshold as illustrated in Fig. 24. The performance of this scheme is evaluated in static multipath channels with 2 and 4 paths and is shown to improve mean acquisition time performance. mTc Received PN matched ith Threshold To code signal PN matched Integrator with Threshold To code filter dwell time mT c comparison tracking loop Figure 24: Block diagram of the acquisition scheme proposed by Soderi et al. In Ma et al. [28], the nonconsecutive search proposed in Shin et al. [21] and a simpler version of the MAX/TC scheme [14] called the global MAX/TC are applied to the acquisition of UWB signals in the presence of multipath fading and MAI. In the nonconsecutive search, only one phase in every D consecutive search space phases is tested by correlating the received signal with a template signal with that particular phase. The decimation factor D is chosen to be not larger than the delay spread Ntap. In the global MAX/TC, a parallel bank of correlators is used to evaluate all the nonconsecutive phases and the phase corresponding to the correlator output with maximum energy is chosen as the coarse estimate of the true phase. In Zhang et al. [29], a hybrid acquisition scheme called the reduced complexity se quential probability ratio test (RCSPRT) is presented for UWB signals in AWGN, which is a modification of the multihypothesis sequential probability ratio test (MSPRT) for the hybrid acquisition of spread spectrum signals [30]. In the MSPRT, if the sequential test in one of the parallel correlators identifies the phase being tested as a potential true phase the control is passed to the verification stage which verifies its decision. In the RCSPRT, the sequential test in each of the parallel correlators is used only to reject the hypotheses being tested as soon as they become unlikely and replaces them with new hypotheses. The RCSPRT stops when all the phases except one have been rejected. This scheme has merit at low SNRs where the time required to reject incorrect phases may be much smaller than the time required to identify the true phase. In Vijayakumaran et al. [31, 32], the effect of equal gain combining (EGC) on the acquisition of UWB signals with TH spreading is investigated in a multipath environ ment. The acquisition problem is formulated as a binary composite hypothesis testing problem where the set of phases where a receiver lock results in a nominal uncoded bit er ror probability constitute the alternate hypothesis. Two schemes based on EGC called the squareandintegrate (SAI) and the integrateandsquare (IAS) are analyzed and compared in [32]. The IAS scheme is similar to the one shown in Fig. 23 with the exception that the template signal is given by Nth1 (t)= v(t ITf ciT), (28) l=0 where v(t) = O t r(t kTc), G is the length of the EGC window and rb(t) is the receiver's estimate of the received pulse shape. Thus in IAS, EGC is done first and then the correlator output is squared to generate the decision statistic. In SAI, the received signal is first squared to eliminate the pulse inversion and then EGC is performed to utilize the energy in the multipath. In this case, the template signal is once again given by (28) with v(t) = J o 22(t kT). It is shown that even though EGC improves the acquisition performance in SAI at low SNRs, the performance of IAS with no EGC is superior to the SAI at all SNRs. 2.2.2.2 Efficient search strategies A search strategy specifies the order in which the candidate phases in the timing uncertainty region are evaluated by the acquisition system. When there are more than one acquisition phases in the uncertainty region the serial search which linearly searches the uncertainty region is no longer the optimal search strategy. More efficient nonconsecutive search strategies called the "lookandjumpbyKbins" search and bit reversal search are analyzed in the noiseless scenario with mean stopping time as the performance metric in Homier et al. [33]. A generalized flow graph method is presented in Homier et al. [34, 35] to compute the mean acquisition time for different serial and hybrid search strategies. For the case when the acquisition phases are K consecutive phases in the uncertainty region, it has been claimed that the lookandjumpbyKbins search is the optimal serial search permutation when K is known and the bit reversal is the optimal search permutation when K is unknown. Under the assumption that the probability of detection in all the K consecutive acquisition phases is the same and with mean detection time as the performance metric, the optimum permutation search strategy has been found in [36] using techniques in majorization theory. Suppose that the timing uncertainty region is divided in to bins indexed by 0,1, ,... 1. The ith position in the optimal permutation is given by R (i 1)K (mod N) + +1, (29) where i E {1, 2,... N} and d is the greatest common divisor (GCD) of Ns and K. 2.2.2.3 Search space reduction techniques Some acquisition schemes attempt to solve the large search space problem by employing a twostage acquisition strategy [3741]. The basic principle behind all these schemes is that the first stage performs a coarse search and identifies the true phase of the received signal to be in a smaller subset of the search space. The second stage then proceeds to search in this smaller subset and identifies the true phase. In Bahramgiri et al. [37], such a twostage scheme is proposed for the acquisition of timehopped UWB signals in AWGN noise and MAI. The search space is divided in to Q mutually exclusive groups of M consecutive phases each. In the first stage, each one of the Q groups is checked by correlating the received signal with a sum of M delayed versions of the locally generated replica of the received signal. Once a group is identified as containing the true phase, the phases in the group are searched by correlating with just one replica of the received signal. A scheme based on the same principle has been developed independently in Gezici et al. [38]. Both of these schemes have been developed under the assumption of an AWGN channel and their performance is likely to suffer in the presence of multipath. In Reggiani et al. [39], an acquisition scheme for UWB signals with TH spreading called nscaled search is presented, where the search space is divided into groups of M = Nf/2 where n > 1. The TH sequence used to generate the replica of the received signal is also modified by neglecting the n least significant bits of each additional shift cl. Although the actual scheme involves chiprate sampling of a matched filter output, it is equivalent to correlating the received signal with M delayed versions of the modified replica of the received signal. In this sense, it is similar in spirit to the schemes described above. 2.2.2.4 Estimationbased schemes Certain approaches towards acquisition in UWB systems have employed estimation theoretic methods to obtain timing information of the received signal. The nondata aided timing estimation approaches [42, 43] exploit cyclostationarity, inherent in UWB signaling due to pulse repetition, to estimate timing information of the received signal. These schemes require framerate sampling in the acquisition stage and pulserate sampling during the tracking stage. The signal model assumes only TH spreading and no polarity randomization of the pulses, i.e., al 1. It is also assumed that the received pulses from all paths bk = (t), for k = 0, 1,..., Ntap 1, and the period of the TH sequence is equal to a symbol duration, i.e., Nb = Nth. The timing offset is assumed to be confined to a symbol duration and is expressed as 7 = N6Tf + c, where N, E [0, Nth 1] and c E [0, Tf) represents the pulselevel offset. The acquisition system estimates the framelevel timing offset by estimating N6. To do this, a sliding correlator correlates the received signal with the template b(t) and framerate samples z(n) = J( Tf !(t nTf)r(t) are obtained. Under certain conditions, it is observed that the autocorrelation Rz(n; v) = E{z(n)z(n + v)} of z(n) is periodic in n with period Nth and hence z(n) is a cyclostationary process. Estimates R (n; v) of R (n; v) are obtained by sample averaging and the framelevel timing estimate is obtained by picking the peak of the periodically timevarying correlation of the sampled correlator output [43] and is given by N = round{[argmax (n, v) + f]N]th (210) where round{} denotes the rounding operation. A slightly more robust approach [42, 43] estimates the Fourier coefficients 7Ri(n, v) of the periodic sequence R (n; v) via sample averaging which are then used to estimate the framelevel timing as N= round [(v (n; v) Nth) (211) L2 7I Nth where 0(n; v) = Z, (n, v). The estimation of the pulselevel timing offset C, is done using a similar method but however requires pulserate sampling of the correlator output. These schemes are conceptually illustrated in Fig. 25. Sliding z(n) Estimate r(t) Correlator ACF/FS Coefts. Estimate via sample averaging N and E V (t) Figure 25: Autocorrelation function (ACF) of correlator outputs z[n] or its Fourier series (FS) coefficients estimated via sample averaging and used to estimate timing offset. In Tian et al. [44], a maximum likelihood (ML) timing estimation scheme is presented for data aided and nondata aided methods and a tradeoff between acquisition accuracy and complexity is discussed. A dataaided timing estimation scheme employing EGC is analyzed in [44], assuming the timing offset to be less than a symbol duration, which estimates the framelevel timing offset from the observation of M symbol durations of the received signal as M1 N, argmax Y z,(N,, bl) (212) l=0 where zi(N, bl) = E1 f oJ r(t\. (t INbTf NTf gT)dt denotes the output of the correlator with the EGC window of length G. Another similar dataaided timing estimation scheme is developed in [45] where the timing estimation problem is translated to an ML amplitude estimation problem and a generalized likelihood ratio test to detect the presence or absence of a UWB signal is developed which makes use of the ML timing estimates in the likelihood ratio test. Least squares estimates of the timing and the channel impulse response, using Nyquist rate samples of the received signal, are obtained in Carbonelli et al. [46], under the restrictive assumption that the r < Tf and is thus far from being practical. A nondata aided timing estimation method called timing with dirty templates (TDT) is presented in Yang et al. [47] which in the absence of intersymbol interference (ISI), makes use of crosscorrelations between adjacent symbols to estimate timing information of the received signal. In this scheme, a symbollength segment of the received waveform is used as a template and correlated with the subsequent symbollength segment, and the symbolrate correlator output samples are summed over K pairs of symbols to estimate the timing information 7, which is assumed to be within a symbol duration, as K /2kNbTf \ 2 7 arg max S r(t)r(t NbTf)d (213) E[O,NbTf) ~=1 (2k1)NbTf A training sequence design method for a similar dataaided scheme is presented in Yang et al. [48]. In Wu et al. [49], a method is presented for optimizing allocation of pulses in training and information symbols used for acquisition, channel estimation and symbol detection. Transformdomain methods, which obtain estimates of channel parameters employ ing subNyquist sampling rates, are presented in MAravic et al. [5052] where the joint channel and timing estimation problem is translated into a harmonic retrieval problem. These methods obtain samples Fr [n] of the Fourier transform, Fr () of the received sig nal and use them to estimate the excess delays tk employing standard spectral estimation techniques. However, these schemes can estimate tkS only after the timing offset 7 is known and hence cannot be used for timing acquisition. In Zhang et al. [53], the CramerRao lower bounds (CRLBs) for the time delay estimation problem are derived for UWB signals in AWGN and multipath channels. It is shown that a larger number of multipath results in higher CRLBs and a potentially inferior performance for unbiased estimators. 2.2.2.5 Miscellaneous approaches An acquisition strategy for impulse radio which makes use of relative timing between pulses in specially chosen TH sequences is presented in Zhang et al. [54] in the absence of multipath. This scheme may not be applicable in the presence of multipath which is usually the case with UWB systems. An acquisition scheme implemented on UWBbased positioning devices which use a coded beacon sequence in conjunction with a bank of correlators is presented in Fleming et al. [55] and assumes absence of multipath. A distributed synchronization algorithm for a network of UWB nodes, motivated by results from synchronization of pulsecoupled oscillators in biological systems such as synchronized flashing among a swarm of fireflies and synchronous spiking of neurons, is presented in Hong et al. [56]. CHAPTER 3 ISSUES AND CHALLENGES IN THE DESIGN OF UWB ACQUISITION SYSTEMS We present, in this chapter, the significance of the acquisition problem in UWB communication systems and discuss the issues that distinguish the acquisition problem in UWB systems from traditional spread spectrum systems. We discuss some of the issues and challenges in UWB signal acquisition which may not have received sufficient attention in the existing literature. 3.1 Hit Set In a multipath channel, the energy corresponding to the true signal phase is spread over several MPCs. The primary difference between the acquisition problems in a multi path channel and a channel without multipath is that there is more than one hypothesized phase which can be considered a good estimate of the true signal phase. In a multipath environment, the receiver may lock onto a nonlineofsight (nonLOS) path and still be able to perform adequately as long as it is able to collect enough energy. From the viewpoint of postacquisition receiver performance, a receiver lock to any one of such paths can be considered successful acquisition. Thus we require a precise definition of what can be considered a good estimate of the true signal phase. A typical paradigm for transceiver design is the achievement of a certain nominal uncoded bit error rate (BER) An. Then all those hypothesized phases such that a receiver locked to them achieves an uncoded BER of An can be considered a good estimate of the true signal phase. We define the hit set to be the set of such hypothesized phases. For a given true phase 7, let PE(AT) denote the BER performance of the receiver when it locks to the hypothesized phase 7 where AT = 7 7. Let Tm be the minimum SNR at which the receiver achieves a BER of An when it locks to the LOS path, that is, PE(O) < An when the SNR is Tn and PE(0) > An for all SNRs less than Tn. Then for an SNR T > To and true phase r, the hit set is given by {H = { : PE(AT) < A}. (31) The hit set when a partial Rake (PRake) receiver [57] is employed for demodulation has been derived in [32, 58] and is treated in detail in Chapter 4. Fig. 31 shows a plot of the number of phases in the hit set as a function of the SNR when An = 102 and the PRake receiver has N = 5 and N = 10 fingers. It is observed that the cardinality of the hit set could be significantly large depending upon the operating SNR. 50 C 45 NR= 40  35  30 T 25 U) Eb/No (dB) Figure 31: Effect of received SNR on size of hit set H for NR 5 and NR = 10. A design for an acquisition system which does not take the hit set into account can result in a significant performance degradation. For instance, in serial acquisition schemes, such as the one shown in Fig. 23, the decision threshold is usually set such that the average probability of false alarm is constrained by a small positive constant 6 < 1, i.e., d = argminmax Eh[PFA(7, AT)] < (32) 7 T~r ri Fig. 32 shows two receiver operating characteristics (ROCs) for an acquisition scheme where the received signal is correlated with a template signal and the correlator output is squared and compared to a threshold. The detailed derivation of the performance analysis can be found in [32]. For one of the ROCs, the threshold was set assuming that the hit set consists of only the true phase 7 and for the other the hit set definition in (31) was used assuming a PRake receiver with NR = 5 fingers with the nominal BER requirement An = 103 and the average energy received per pulse to noise ratio equal to 5 dB. When the hit set contains only the true phase 7, the threshold needs to be set much higher in order to prevent the decision statistics for the other phases in the multipath profile, which have significant energy, from exceeding it. This causes the degradation in the probability of detection when = r. 09 08 S06 05 04 2 03 01 0 ROC for singleton hit set ROC for defined hit set 0 01 02 03 04 05 06 07 08 09 1 Figure 32: The ROCs when the threshold is set for a singleton hit set containing only the true phase and for a hit set defined in (31) with An 103. 3.2 Asymptotic Acquisition Performance of Thresholdbased Schemes A typical thresholdbased timing acquisition system consists of a verification stage in which a threshold crossing at a candidate phase is checked to see if it was a false alarm or a true detection event. The usual procedure for implementing the verification stage is to have a large dwell time for the correlator [7]. The large dwell time increases the effective SNR of the decision statistic and in the absence of channel fading, this results in accurate verification, i.e., the probabilities of a false alarm and a miss can be made arbitrarily small. However, for thresholdbased acquisition schemes in multipath fading channels it was shown [59] that no matter how large the SNR is or how we choose the threshold it may not be possible to make the probabilities of detection and false alarm arbitrarily small. In particular, the asymptotic performance of two typical thresholdbased acquisition schemes for TH UWB signals was calculated in [60]. It was shown that if the threshold is such that the average probability of false alarm is less than a given tolerance, then there is a nontrivial lower bound on the asymptotic average probability of miss. This lower bound translates to an upper bound on the asymptotic average probability of detection. These results suggest that it may not be possible to build a good verification stage for UWB signal acquisition systems by just increasing the dwell time. They also suggest that the principles underlying the design of efficient UWB signal acquisition schemes may be very different from the traditional spread spectrum acquisition schemes. In traditional spread spectrum acquisition systems, the decision threshold is chosen such that the probability of false alarm in each of the nonhit set phases is small. The verification stage helps the acquisition system recover from false alarm events when they occur. Considering that the construction of a verification stage in some UWB signal acquisition systems may be difficult, a more appropriate choice of decision threshold is one which restricts the probability that the acquisition process encounters a false alarm to be small. So if PF(7) is the average probability that the acquisition process ends in a false alarm, then the decision threshold 7d is chosen such that Pp (7) is constrained by a small positive constant 6 < 1, 7d = argmin PF(7) < 6. (33) The performance of spreadspectrum acquisition systems has typically been characterized by the calculation of mean acquisition time [7, 19]. In mean acquisition time calculations, a false alarm penalty time is assumed which is the dwell time of the verification stage, i.e., the time required by the acquisition system to recover from a false alarm event. Thus mean acquisition time calculations implicitly assume the existence of a verification stage. For UWB signal acquisition systems, if the threshold is set according to (33) the mean detection time is a reasonable metric for system performance. The mean detection time is defined as the average amount of time taken by the acquisition system to end in a detection, conditioned on the nonoccurrence of a false alarm event. The calculation of the mean detection time thus does not require any assumption on the verification stage. Finally, several detectionbased schemes for UWB signal acquisition have proposed using some form of EGC to improve the acquisition performance by combining the energy in the multipath [3739]. The asymptotic performance of thresholdbased UWB signal acquisition schemes using EGC has been calculated in [60]. It has been shown that EGC may lead to a significant performance degradation. 3.3 The Search Space in UWB Signal Acquisition The large search space in UWB signal acquisition poses significant challenges in the design and implementation of practical systems. Most estimationbased schemes are based on the ML principle and hence involve the simultaneous calculation of the likelihood function corresponding to each one of the phases in the search space followed by a maximum operation. When the search space is large, a fully parallel implementation of this scheme is not feasible and one may have to resort to a serial or hybrid implementation where the system calculates the likelihood functions for small groups of phases in the search space sequentially. The likelihood functions calculated at each intermediate step need to be stored until all the phases are evaluated. The likelihood functions calculated at each step correspond to different noise realizations and so a simple maximum operation may not be a good method to find the true phase especially at low SNRs. A more robust approach might be repeated calculation of the likelihood function at each phase followed by averaging to reduce the variations due to noise. This effectively amounts to trading off hardware complexity for an increase in the acquisition time to achieve similar acquisition performance. However, the performance of such reduced complexity estimationbased acquisition schemes in terms of estimation accuracy and acquisition time is still an open research direction. Although detectionbased schemes which evaluate the phases in the search space one at a time have a simpler hardware implementation, they may suffer from a large mean detection time which makes them unsuitable for high data rate applications. For instance, the mean detection time of the serial acquisition scheme in [32] was found to be of the order of one second. Furthermore, it was shown that the time spent by the acquisition system in evaluating and rejecting the nonhit set phases was the dominant part of the mean detection time causing it to decrease only marginally with increase in SNR. Thus acquisition techniques capable of reducing the search space are crucial in the design of efficient acquisition schemes. For example, the twostage hybrid DSTH scheme [58] described in Chapter 4 achieves a mean detection time of the order of a millisecond. Another approach to solve the search space problem is by designing the higher layers in the network architecture carefully. A multiple access protocol which employs continuous physical layer links in the network in order to avoid repeated acquisition is presented in Kolenchery et al. [61]. The timing uncertainty region may be reduced significantly if a beaconenabled network is employed, where the medium access is coordinated by a central node which periodically transmits beacons to which other nodes synchronize and follow a slotted medium access approach. 3.4 Generalized Likelihood Ratio Test for UWB Signal Acquisition There has not been much effort in the direction of finding the optimal detectors for the acquisition problem in UWB systems. Most detectionbased schemes for UWB signal acquisition have been ad hoc schemes based on the principles of traditional spread spectrum acquisition systems. In the context of the hit set and the dense multipath in UWB systems, a reasonably systematic approach to detector design is the generalized likelihood ratio test (GLRT). It is instructive to examine the structure of the GLRT de tector used by a serial acquisition system which tries to find the true phase by evaluating the phases in the search space one at a time. Although the GLRT is not an optimal test, it has been known to work quite well in general [62]. The GLRT has been shown to be asymptotically uniformly most powerful among the class of invariant tests [63]. The received signal is observed over a duration of M periods of the DS sequence, which is assumed without loss of generality to be longer than the TH sequence, and this observation is denoted by r. The acquisition system is to determine whether a hypothesized phase r can be considered the true phase of the received signal. It is assumed that the hypothesized phase is a multiple of the chip duration T,. To enable tractable analysis, it assumed that the true phase is also a multiple of T. The number of phases the search space is thus NdsNf. From the definition of the hit set earlier, it is clear that there exist many ways in which r can be considered to be the true phase. Without loss of generality, suppose that the hit set is {r AbTc, r (Ab 1)T,..., r + AfTc} where Ab and Af are integers between 0 and NdsNf/2. Also suppose that an allones data training sequence is sent in the acquisition preamble. This results in a composite hypothesis testing problem whose hypotheses can be formulated as follows: Ho : r is not an acceptable phase, i.e., 7r SO() HI : r is an acceptable phase, i.e, r E S(r), whereS(r) {r AfTc, r (Af 1)T,..., + AbTc}. The GLRT is given by max{h,vS(f)}p(r h, v) H1 A (r) = (34) ) max{h,v~s(f)}p(r h, v) Ho ( where the vector h, of channel gains {hk}, is assumed to be deterministic but unknown and 7 is the decision threshold. It can be shown easily using techniques similar to those used in [64, 65] that when n(t) is an AWGN process with power spectral density No/2, the choices of v and h which maximize p(r I h, v) in the numerator in (34) are given by T = arg max CT(v)C(v) VES(t) 1 hi =C(Tr) (35) MNds Rs, and similarly for the denominator in (34) To argmaxCT(v)C(v) vS(t) 1 ho C(To), (36) MNd&s R, where R,, foTw (t) and C(u) = [Co(), Ci(),..., C~a (u) with /MNdTf Ck ) r(t)Sk(t ) (37) where MNdsl Sk(t V) a[1l/Nds]r(t kTc ITf cTc v)dt. (38) l=0 Also, it can be easily shown that the test in (34) can be written as MNdsR T hT H' No0 A(r) [h ) h o Nds [hih hhoh] >I <7. (39) 2 Ho 2 Using (35) and (36), the GLRT in (39) reduces to Ntap Ntap1 H A(r) = max C (v) max > C(v) d N'7 (310) VES(T) v S() Ho 2 The threshold 7' can be set such that the probability of false alarm PFA Pr{A(r) > 7' I Ho} < 6, where 6 is a specified false alarm tolerance. It can be observed from (310) that the test statistic given by the GLRT amounts to correlating the received signal with Ntap different templates, each corresponding to a different MPC, summing the squared outputs of each of these correlators, maximizing this sum for two disjoint sets of phases, and comparing the difference to a threshold as illustrated in Fig. 33. This test statistic thus attempts to collect the energy from all the MPCs through a form of equal gain combining. 29 However it is immediately clear that such an implementation is prohibitively complex to realize. Thus other suboptimal strategies need to be explored which would collect energy from the MPCs in an alternative way. Simple energy detection approaches thus need to be considered and other techniques to reduce the search space and thus the mean detection time need to be designed. We propose and analyze two such techniques in the following two chapters. S+ ATc) 2 s (tt + AT) Np1 : MAX s tt A o) S(trtT) s t t 1 )  Ntap MAX s t 4+ (A+)+ ) s (t + +T ) Figure 33: Generalized likelihood ratio test for evaluation of phase The upper and lower MAX operations evaluate the maximum of "I Z C k (v) over v S(T) and v S(I), respectively. CHAPTER 4 ACQUISITION WITH HYBRID DSTH UWB SIGNALING In this chapter we present a hybrid DSTH signaling format [40, 58] for UWB systems which enables in a significant reduction in the search space faced by the acquisition system. The hybrid DSTH signaling format allows the acquisition to be performed in two stages which results in small values of the mean detection time. 4.1 System Model 4.1.1 Hybrid DSTH Signal Format The proposed hybrid DSTH signaling format for UWB uses two levels of spreading. The data symbols are first spread using a TH sequence of period Nth. The resulting time hopped signal is further spread using a DS sequence of period Nds. It is noted that Nds is chosen to be relatively long compared to Nth and is a multiple of Nth, i.e. Nds = DNth where D is an integer. The transmitted signal is a train of monocycles b(t) of energy E/P and can be expressed as 00 l=oO where b(t) = (t)/ /p is a unit energy monocycle, [] and [J denote the integer divi sion remainder operation and the floor operation, respectively. The sequence {ai}1' is a random periodic DS spreading sequence with period Nds, where al is equally likely to be 1 or +1 and a,, and at, are independent if /1 / 12. The sequence {c} th1 is a random periodic TH sequence with period Nth where each sequence element ci is equally likely to take any value in the set {0, 1,..., Nh 1} and c, and Q, are independent if 11 / 12. The data bits are denoted by the sequence {bi}. The number ofmonocycles modulated by one bit of data is Nb. The duration of a timehopping frame is Tth = NhTo where To is the chip duration which is greater than the width of the transmitted mono cycle. The hybrid DSTH format for UWB is illustrated in Fig. 41. Binary phase shift keying (BPSK) data modulation is assumed, i.e., bi E {1, 1}. The twolevel spreading allows us to divide the acquisition process into two stages: one for the TH sequence and another for the DS sequence as shown in Fig. 42. As a result, the search space can be significantly reduced as will be shown in the following sections. In addition, the DS spreading on top of the TH spreading smoothes any spectral lines caused due to the shorter periodic TH sequence. Na h TNd Th Figure 41: The hybrid DSTH signal format. Received Squaring TH spreading Ht DS spreading it signal operation code acquisition code acquisition Figure 42: Conceptual block diagram of the hybrid DSTH twostage acquisition scheme. 4.1.2 Received Signal We assume that the channel is modeled as detailed in Section 2.2.1. The received pulse waveform is given by br (t) = f(t) (t). It is assumed that the duration of the received pulse does not exceed the chip duration T,. With the channel response in (23) the received waveform corresponding to a single pulse is Ntap1 (t)= ,k,,' (t kT). (42) k0 The received signal as a sum of the signal and noise components is given by r(t) = rs(t) + n(t). The signal component is given by 00 Ntap1 rs(t)= E > bL[ja[i]Lr(t lTthc[1TC T') = >E pkhksk(t), (43) 1=oo k0 where So00 Sk(t) = bL ja [](t kTe ITth c[ Tc T), (44) 7b [Nd [Nth} denotes the received signal from the kth multipath component and r is the signal delay through the channel. The noise component, n(t), is assumed to be a zeromean AWGN process with twosided power spectral density m. It is assumed that the received pulse waveform r (t) and the TH and DS spreading sequences are known at the receiver. LE A NbEjEtotR b(0) The received bitenergytonoise ratio is then given by a NbE Et l(0) where R1n (u) = j~0 ,, (t ,, (t + v)dt denotes the autocorrelation of a power of the received pulse waveform. 4.2 Hit Set Formulation When the transmitted signal is passed through a multipath channel, more than one hypothesized phase at the receiver may be considered a good estimate of the true phase. A hypothesized phase can be considered as belonging to the hit set if a receiver locked to that phase performs successful demodulation in the sense of achieving a relatively low probability of bit error following acquisition. This way of defining the hit set by considering postacquisition performance of the receiver of the hit set was previously done in [31]. It is assumed that the channel estimation block following acquisition estimates the channel coefficients perfectly and that a partial Rake demodulator [57], which estimates the first arriving NR (NR < Ntap) paths is used. Perfect channel estimation is assumed to make the analysis amenable and to focus on the definition of the hit set. Suppose that the receiver locks onto the hypothesized phase 7, which is an integer multiple of T. To make the analysis tractable, the actual delay 7 is also assumed to be an integer multiple of T,. Let AT = aTth +/3T, where Nds 1 < a < Nd, 1 and 0 < ,3 < Nh 1. The decision statistic of the partial Rake demodulator for a particular data bit (without loss of generality, consider the demodulation of bit bo) following perfect channel estimation is given by yb(AT; h) Rb(AT; h) + nb, (45) where Rb(AT; h) is the signal component and nb is the noise component. The signal component of the correlator output is given by min{Ntap1,aNh+3+NR1} Rb(Ar;h) = boNbVE h R(O) k=max{0,aNh+3} Ntap1 minNtapl,aNh+/+NR1} S, Nb1 U(1 + ,i,0) + Y ZY I ,o) k1=0 k2 max{0,aNh+/3} i=Sm 1=0 kick2 i 0 b a ar ~ ar+i]PklPk2hk" hkiX2 k( + ( +i)Nh+ C[ ,]k2 Nh+ C[ ) (46) where X2(a, b) = 1 if a = b and zero otherwise, U(a, b) = 1 if a > b and zero otherwise, Sm = [N~j 1 + 1 denotes the number of timehop frames the multipath spread occupies and S, = [Nl 1 + 1 denotes the number of frames occupied by the NR taps. The vector h is a vector containing the channel parameters {pkhk}. The first term on the righthand side of (46) denotes the contribution to the signal part of the correlator by the desired bit bo and the second term denotes the contribution arising from the intersymbol and interframe interference. Conditioned on h, the noise component of the correlator output, nb, is a Gaussian random variable of zero mean and variance given by min{Ntap ,aNh ++NR1} I 2 S hkaJ (O) k=max{0,aNh+3} min{Ntap1,aNh+3+NR1} Nb1 1 7 L Ids I dsI + Y a[ il la[ 12 1Pk Pk2hk hk2 kj,k2 max{O,cNh+P} 11,12=0 k l k2 11712 SX2(ki + llNh + c[i ], k2 + 2Nh + C[]). (47) Nth Nth Due to the random nature of the DS spreading sequence {al} for sufficiently large Nds, the mean values of Rb(AT; h) and or averaged over all possible random sequences {a} reasonably approximate their true values. When this averaging is done, the second term in both (46) and (47) would be zero. Then the decision statistic yb conditioned on the channel coefficients h, is a Gaussian random variable with mean ((AT, h)) / A NbR (0)0 (AT, h) if bo 1 E /NbR9 (0)e(AT, h) if bo 1 and variance ab (0(AT, h)) NbR9 (0)0(AT, h), where e(AT, h) is the sum representing the channel energy gains collected by the partial Rake demodulator, min{Ntap l,aNh ++NR1} O(AT, h) hk. k=max{0,aNh +3} The probability of bit error (for BPSK modulation) can then be expressed as Pe(O(AT, h), Eb/A ) Q (p((AT h)) Q 2Ebo(AT, h) (48) (yb (e(AT, h)) NoEt where Q(x) 1/ /2 J:0 et/2dt. Using the result from [66, pp. 268270], the average probability of error can now be expressed as a single integral Ee P (AT, h), 1 j e O E dii, (49) E P N( Nh),o o NoEtot sin2 where min{Ntap 1,aNh +3+NR1} mlk e(s)= n (1  (410) k=max{0,aNh+3} 'k is the moment generating function of e(AT; h). For a fixed number, NR, of Rake taps, the minimum received bitenergytonoise ratio, (k required to achieve a certain nominal uncoded BER performance when the receiver locks to the strongest path is determined. In the channel model considered, the power delay profile exhibits a decaying path loss and the strongest path corresponds to the first arriving path. Hence Eo [P (0((0, h), )i =) An, where An is the nominal desired uncoded BER. Then, for Eb > (b ) the hit set is defined, for a given value of true phase 7, as the set of phases H {= :Ee [Pe' (AT,h), ) < An (411) This definition of the hit set implies that at = () the hit set consists of one element, i.e., the true phase of the received signal. However, for higher 1, there could exist multiple phases in 'H which would still guarantee at least the nominal BER performance. Hence, the cardinality of 'H increases in general with increasing . Fig. 43 illustrates the effect of deviation from the true phase on the BER performance of the system for NR = 5. From this figure, if A, were to be chosen to be 102, the corresponding (o) is approximately 11 dB. 4.3 Stage 1: TH Acquisition The first step of the acquisition process is to acquire timing information of the time hopping pattern. Without loss of generality, an allones training sequence (i.e. bi = 1) is assumed to be transmitted in the acquisition preamble. The acquisition of the TH pattern can be accomplished by the squaring loop illustrated in Fig. 44. The DS spreading in the signal is removed by the squaring operation. A serial search strategy is employed by the clock control when searching for the true phase. In order to evaluate the hypothesized phase 7, the squared received signal is correlated with a locally generated reference signal 10, 102 4 6 8 10 12 14 16 18 20 22 (Eb/No) dB Figure 43: Effect of deviation from the true phase on BER for NR = 5. s(t T). The squared received signal can be written as r2(t) r(t) + n(t) + 2r,(t)n(t). (412) The squared signal component can be obtained from (43) as Ntap1 o 2(E) ,1 ) h 7 2 krc 1rth U[,]rc I) th N h I k0 ooth Nt,,ap1 Ntp1 +E1 ) )1 PkPmhkhmSk(t)Sm(t). (413) kO0 mO0 kmrn The reference signal for the first stage is given by the train of squared pulses MNth1 s(t ) 2 '(tlTth c C T ), (414) l=0th where the dwell time of the correlator for the first stage is M periods of the TH sequence. Constructing the reference signal for correlation in (414) requires knowledge of the received pulse shape and the spreading codes {cl} and {al} at the receiver, and this assumption has been made throughout the document. r(t) z z' hit s(t T) miss Reference TH Clock Signal Generator Control Figure 44: Squaring loop for TH pattern acquisition 4.3.1 The Decision Statistic The decision statistic z(Ar; h) for the first stage, conditioned on the channel coefficients h and the sequences {a } and {cl}, can be expressed as a sum of three random variables resulting from the output of the correlator of Fig. 44 as T+MNthTth z(A; h) r2(t)s(t )dt T +MNth Tth fT+MNthTth zI (AT;h) Z2 (Ar;h) fT+MNthTth + n2(t)s(t T)dt. (415) Z3(AT;h) We can compute z (Ar; h) using (413) as Ntap1 Ntap1 Ntap1 zl(ATr;h) MNthE lR2(0) > rk (AT)hk+EIR 2(0) pkPm khmAk,m(AT), k=0 k=0 m=0 kmrn (416) where rk(Ar) Nth1 S X2 ( + k + + and Ak,m(AT)  MNth1 1 1 Sa a +iadi,+j x3(C[+i+] + k + k+i, hc[,+j] +m+ jN, c[I +,3), l=0 i=Sm j=Sm Ldh with X3(a, b, c) = if a b c and zero otherwise. The term zl(Ar; h) denotes the contribution to the decision statistic due to the signal part alone. The number of times the kth multipath component is collected by the correlator is thus given by MNthrk(AT) and Ak, (AT) represents the coefficient of the cross term corresponding to the kth and mth multipath components. The second term on the right hand side of (416) denotes the contribution from interframe interference caused due to the multipath of one pulse spilling over into the adjacent TH frame. Similarly, we obtain the second term of the decision statistic, the contribution from the signalnoise cross term resulting from the squaring operation, as Ntap 1 Tc z2(AT; h) 2El, pkhkBk(AT) j '(t))n t)dt k= where MNth1 1 Bk(AT) ar[+i+.]X2(crl+i+] + k + iNh, c[L I +/3). (417) [ Nd,\ Nth [Nth\ l= 0 i= m Finally, the third term denoting the pure noise component is given by MNth1 z3(Ar; h)= n2 (t) ''(tlTth 1 )dt. (418) 1=0 4.3.2 Mean and Variance of the Decision Statistic For a given AT, conditioned on a particular channel realization h and the sequences {al} and {cl} we approximate the decision statistic z(AT; h) by a Gaussian distribution with mean p (Ar; h) and variance oa,(Ar; h) which are to be evaluated. Suppose that the bandwidth of the UWB communication system is B. The above Gaussian approximation for the output of the integrator is quite accurate when the timebandwidth product of the integrator, MNthTthB is large [7, pp. 240250], which is usually true for a UWB system. The mean of the decision statistic conditioned on {ai} and {cl} can be shown to be pI(Ar;h {al}, {c}) =E[z(Ar; h) {a}, {cl}] NtMp1 NtMp1 NtMp1 MNthEIR,2(O) ri(AT)h + EIR,2(O) S PkOhkhmAk,(A) i=O k=0 m=0 kem SMNthR, (0) N S 2(419) 2 The following (conditional) moments will be useful in computing the conditional variance of the decision statistic: E[z(AT; h) {ai}, {ci}] (E[zi(AT; h)  {al}, {c}])2, /Ntap 1 \ 2 E[z2(AT; h) {az}, {ci}] 2NoR93 (0) p PkhkBk(AT)) , \ k m2 2 2 T2 2 (A1; h) 1 {a}, {cfl] 4 + 0^02 MNthlM 2(O). (420) It can also be shown that E[zz2] = 0 = E[z2Z3] and E[ziZ3] = E[zi]E[z3]. Using the above results, we obtain the conditional variance of the decision statistic as Nt/pl )2 2(AT; h {a}, {c}) =2NoEIR 3 (0) pkhkBk(AT) + MNthR2(0). \ k=o0 (421) It is noted that the averaging done in the preceding analysis was only with respect to the noise process n(t). To simplify the analysis, we make use of the assumptions that the TH and DS sequences are random sequences and that Nth and Nds are sufficiently large. Under these assumptions, the values of rk(AT), p (AT; h) and oa,(AT; h), averaged over {al} and {cI}, reasonably approximate their actual values. 4.3.2.1 Averaging over the DS sequence The mean and variance of the decision statistic, averaged over the DS sequence {al}, are to be evaluated. It can be shown that for k / m, E[Ak,m(AT)] = 0 where the averaging is done over the DS sequence. Hence the mean of the decision statistic conditioned on {cl} for a given h, is given by Ntp1 p,(AT; h I {cf}) =MNthEIR2(0) rk(AT)h + MNthR9(O). (422) k0 Similarly it can be shown that the variance of the decision statistic conditioned on {cl} for a given h, is given by 2(AT;h {cI}) 2 MNaRN2(O) Ntap1 Ntp 1 + M2(Nth + Sm + 1)NoEIR0 3(0) Y Y pkpmnhkhCkmn(AT) k=0 m=0 (423) where Nth Nth 1 1 CkmA7) Nth +Sm+ 1 X2(c[]+ k+ilN, c[i] +3) s Sm iI=sm i2=sm (424) X2(C[Rl +fl+i2Nh2 C[ris+1 3). LNth] [Nth J 4.3.2.2 Averaging over the TH sequence The mean and variance in (422) and (423) averaged over the TH sequence is to be found. From (422) and (423), it is clear that all we need is to evaluate E[rk(AT)] and E[Ckm(AT)], where the averaging is done over the sequence {cl}. The average values E[rk(A)] have been calculated in [31] and are given by 1 min{#,k+iNh}+N/hI1 if a G ISI X2(3, k aNh) + Ei min{ /3m=max,k+iNh} 1N if { S , E[rk(AT) i i = 1i Ymin{3,k+iNh}+Nhi1 2 otherwise. li=S,,m m=max{4,k+iNh} N (425) The values E[Ckm,(A)] can be calculated similarly, from the observation that Ckm(AT) in (424) is a sum of Bernoulli random variables X2(C[lI+ + k + ilNh, C [il + Nth Nth 3)X2 (c[ ] + m + i2Nh, c _ + 3), and are given by Nth Y1 min{/3, k+ilNh, m+i2Nh}+Nhl 1 2 ,_hm .Zr=max{3, k+ilNh, m+i2Nh} Nh I1 YminP{, k+iNh, m+iNh}+Nh +iN1 +h i ,, Zrmax{3, kiNh, Tm m+iN,} if .. [Ckm (AT) 1 2(3, k aNh) Z!2 S^ 2Zlmax{?3, m+4;iN} 7 E[CkM) X2(0 k 1 a Ymin{f3 m +iNh }+Nh 1 I +X2(P, m alIh) Yi_ 2ina3, m+iNhl+NhI 1 i= Sr Eq=maxf{3, m+iNhl } +X2(P, k aNh)X2(/3, m aNh) otherwise. Then the average variance of the decision statistic for a given channel realization h can be expressed as o2(AT; h) S2 Ntap1 Ntap1 o M NR2(0) + 2M2(Nth + S + NOEIR3() PkP khE[Ckm(AT) k=O m=0 (426) The typical values of E[Ckm(AT)] are observed to be small causing the second term of the average variance on the right hand side of (426) to be negligible compared to the first term at the values of 1 considered and the second term can be safely ignored, greatly simplifying further analysis. 4.3.3 False Alarm and Detection Probabilities As shown in Fig. 44, a hit for the first stage is the event when the correlator output exceeds the decision threshold 71 for some E c H. A miss for the first stage occurs when the decision statistic does not exceed 71 for all E c H. Conditioning on a particular channel realization h, with the approximate Gaussian distribution for the decision statistic of the first stage and given the decision threshold y7, the false alarm and detection probabilities for a particular value of AT can be computed as Pfal(7, AT) = Pr{z(AT; h) > 71  i , Pdl(7, AT) = Pr{z(Ar; h) > 71  e H}. (427) Then the false alarm probability averaged over all channel realizations is given by Eh[Pfal.l(71, AT)] [Q (7 /1(AT;h)) V H. (428) The average false alarm probability is computed using a similar technique employed in [31], which expresses the false alarm probability averaged over all channel realizations in the form of a single integral 1 1 foo IM 1 t2 t \ ji Eh[Pf1(71, AT) Im e J dt, (429) h /T 2 7T.I F where Ntp1 0 m OR,(W ) IH I jw.MNthElR2(O)E[rk(ATr)] Qk (430) k is the characteristic function of Ri(AT; h) = MNthER1 2(0) Z~o 1 E[rk ( h , which is a weighted sum of the squares of channel taps and thus a weighted sum of independent Gamma random variables. A similar expression can be obtained for the average probability of detection, Eh [Pdl (1, Ar)]. The decision threshold 71 for the first stage is chosen as discussed in Section VI. 4.4 Stage 2: DS Acquisition When the first stage encounters a hit for a particular hypothesized phase Ti, the acquisition process enters the second stage which deals with the acquisition of the DS sequence. The search space for the second stage of the acquisition system consists only of the D phases: Ti + (d 1)NthT, d 1,, 2,..., D, because one period of the DS sequence encompasses D periods of the TH sequence and one of the above D phases would be the true phase of the received signal unless a false alarm has occurred in the TH stage. A false alarm occurring in the first stage would mean a false alarm event for the hybrid DSTH system, since the search space for the DS stage would then comprise only nonhit set phases. The acquisition system for the second stage is shown in Fig. 45 in which the received signal is correlated with a reference signal and the output of the correlator is squared and compared to a threshold 72. The reference signal for the second stage is given by M'Nds1 s'(t f E a[l/Nds]r(t lTth C[/Nh]To ), (431) l=0 where the dwell time of the correlator is M' periods of the DS sequence. The output of the integrator can be expressed as r+M'NdsTth Ntap1 y(AT; h) ] r(t)st(t f)dt M'Nd VEIR9(O) r'(AT)hk + ny, Sk=0 (432) where r (AT) E Z1 N L a ]a[ ]X2 (c[ k i+ N, i + ). Again, when the DS and TH sequences are sufficiently long, r'(AT) can be replaced by its mean value, E[r'(AT)], averaged over the sequences {al} and {cl}, where it can be shown that E[Ir(AT)] = 2(3, k aNh) if a {Sm,...,1} and zero otherwise. Conditioning on a given channel realization h, y(AT; h) is a Gaus sian random variable with mean p,(Ar; h) R2(AT; h), where R2(AT; h) = M'Nds ER (0) Zj"N1 E[rI(A)]pkhk. It is straightforward to show that the average Figure 45: Acquisition system for DS stage noise variance is given by a2 M'NdsR) (0) . Conditioned on the channel coefficients h and given the threshold 72, the false alarm and detection probabilities for the second stage are given by Pf2 72, AT) Pd2 (72, AT) Pr{y2(AT; h) > 72 H}, Pr{y2(AT; h) > 72 T }. The false alarm probability averaged over the channel realizations for the second stage is given by, Eh [Pfa2 where ti (2, AT)] Eh (AT;h)Y (\//2 ++(AT;h)  71 Im e tv  dt + Im {e 2 dr, J \1 2 t (__ t _ te t2 12t) 35) :1 e O2 in ) dt (435) 7J t ty ) ie third equality in (435) is obtained from the fact that Ntap1 t ( I (M'NdsVE7R(O)E[r'(AT)]u) + ( (M'NdVER()E[K(AT)w)) k=0 (436) is the realvalued characteristic function of R2(Ar; h). In (436), ., (w) is the charac teristic function of the Nakagamim random variable hk [66]. The average detection probability, Eh [Pd2 (72, AT)], can be obtained similarly. The decision threshold 72, for the second stage is chosen as described in the following section. (433) (434) 4.5 Setting Thresholds 71 and 72 The thresholds 71 and 72 are chosen such that the average probability of the two stage acquisition system ending in a false alarm, PFA < 6, where 6 is chosen to be some small number. The reason for imposing this constraint on the system is made clear in the following section which describes the use of mean detection time, instead of mean acquisition time [7], as the system performance metric for UWB acquisition systems. The calculation of PFA is summarized in the following. An upper bound on this probability PjA, is also derived since PjA is easier to calculate than PFA. Suppose that the hit set H consists of H consecutive phases and can be assumed without loss of generality to be the set {0, Te, 2T, ..., (H 1)Tc}. Given the thresholds 71 and 72, let the average probabilities of detection for the jth phase in the hit set for the first (TH) and second (DS) stages be denoted by PD1(j) and PD2 (j) respectively where j E {1, 2,... H}. The average false alarm probability for the (n H)th phase in the set of nonhit set phases for the TH stage, {HTe, (H + 1)T, ..., (N 1)To}, is denoted by PFA1 (n) and the average false alarm probability for the (m H)th phase in the set of nonhit set phases for the DS stage, {HTe, (H + 1)T,..., (ND 1)T}, is denoted by PFA2(m) where n E {H + 1, H + 2,... N}, m E {H +1, H + 2,... ND} and N = NthNh. The average probabilities of false alarm for the first and second stages can be bounded by PFA (n) < 61 and PFA2(m) < 62 where 61 = max, PFA1(n) and 62 = maxm PFA2(m). The maximum probabilities of false alarm 61 and 62 are typically the false alarm probabilities corresponding to phases which lie just outside the hit set. We note that 61 and 62 are determined by the choices of 71 and 72. An overall false alarm event for the hybrid DSTH acquisition system can occur in two ways: 1. A false alarm event in the TH stage would mean a false alarm event for the entire system, since following a false alarm in the TH stage, the DS stage would evaluate only nonhit set phases. 2. There could occur a detection in the TH stage followed by a false alarm in the DS stage. First, the average probability that a false alarm event would occur in the DS stage, following a detection in the TH stage at the jth hit set phase, is computed. This average probability is denoted by PF2(j). Following the detection in the TH stage at the jth hit set phase, the phase evaluation in the DS stage is equally likely to begin at any one of the phases {(j 1)Tc, (j 1 + N)T,,..., (j 1 + (D 1)N)T}, and PF2(j) is given by: D1 1 1 DI PF2(j) PM2(J)Pf2(, 1) + Pf2(j, d) (437) d=l D1 < PM2(j)P2(j ) + E 2( d) 2(j). (438) d=2 where PM2(j) 1 PD2(j) is the probability of miss in the DS stage at the jth hit set phase and Pf2 (j, d) is the probability of occurrence of a false alarm in the DS stage conditioned on a detection in the TH stage at the jth hit set phase and the first evaluation in the DS stage beginning at the phase (j 1 + dN)To (d = 1,..., D 1), and can be shown to be 1D Pf2, d) 1 PFA2(j + kN)) 1 PM2(j) k=d i PM2(J) [ i ( PFA2(j + kN)) < 1 (1 Dd PD2 j1 P2( d). (439) Now, the average probability PF (n), of the acquisition system ending in a false alarm conditioned only on the starting phase (n 1)To of the serial search of the TH stage, is to be determined. The false alarm probability conditioned on each starting phase will be the sum of the probabilities of mutually exclusive events leading to a false alarm. There arise two cases: Case 1: Search starts outside hit set, i.e., n E {H + 1, H + 2,..., N} It can be shown that S) 1 [( 1 F12(1) PM1 PFI(M) 1 (1 PFA I(k)) ( k=n 1 PM1 [kH+1( PFAI(k)) < 1 (1 6)Nn+l 1 2() M1 P (), (440) 1 M1( 61)NH where PF12(j) is the average probability of the DSTH system ending in a false alarm in the DS stage, following a detection in the TH stage, conditioned on the phase evaluation in the TH stage starting in the jth hit set phase and is given and bounded as H i1 PFI2(j)= l D(k)) PDPF2( i=j k=j H i1 S (1 PDk)) PD D1 F((i) A (j), (441) i=j k=j and PM1 I= I(1 PDl(j)) is the average probability of miss for the TH stage. Case 2: Search starts inside hit set, i.e., n E {1, 2,..., H} In this case, it can be shown that PiF(n) PF12(n)+ F PD INIH 1 PF1()) PMN [ (1 PDI() (1 PFAl(k )) 1  2  M1 k=n k=H+1 1 PM1 (I fkH+(1 PFAI(k4)) < P N \ / H _1 Pl2 APM1 pb1(). l pMI(t 61)NH] F (442) From the bounds Pb1(n), an upper bound can be obtained on the average probability of the hybrid DSTH acquisition system ending in a false alarm, PFA, as N N PFA PF (n) < p(n) A b n=l n=l (443) Given this upper bound, the thresholds 71 and 72 are chosen such that PjA < 6, where 6 denotes the tolerance on the average probability of the occurrence of a false alarm for the hybrid DSTH system. 4.6 Mean Detection Time Typically the performance of spreadspectrum acquisition systems has been characterized by the calculation of mean acquisition time [7, 19]. In such calculations, a false alarm penalty time is assumed, which is the time required by the acquisition system to recover from a false alarm event. In typical acquisition systems, this penalty time is the long dwell time of a subsequent verification stage which attempts to confirm detection and false alarm events with high probability. However, it has been shown in [59] that the construction of such a verification stage for thresholdbased UWB acquisition systems in multipath fading channels is difficult. This is because of the existence of an upper bound on the probabilities of detection even as the SNR is asymptotically increased. Hence it is not immediately apparent how one should assign a penalty time for a false alarm event in UWB acquisition systems. For this reason we use the mean detection time as the metric for system performance instead of the mean acquisition time. The mean detection time is defined as the average amount of time taken by the acquisition system to end in a detection, conditioned on the nonoccurrence of a false alarm event. The calculation of the mean detection time thus does not require any assumption on the false alarm penalty time. However, to ensure that the acquisition system rarely encounters a false alarm, the probability of the acquisition system ending in a false alarm event PFA is constrained by 6 as explained in the previous section. The search space for the first stage consists of N phases: S1 {(n 1)Tc : 1 < n < N}. The search space for the second stage is the set of D phases: S2 = 1i + (d 1)NthTc : d = 1, 2,..., D}, where Ti is the hypothesized phase which causes a hit to occur in the first stage. Thus the effective size of the search space of the proposed two stage UWB acquisition process is N + D, which is significantly smaller compared to the search space size of ND when the spreading is done using only either a DS or TH sequence of period Nds. The proposed hybrid DSTH UWB signal format and the proposed twostage acquisition system which exploits this format result in a significant reduction in search space. Using the flowgraph technique [7, 19], the mean detection time for the overall system can be easily calculated and the final expression has been provided. Fig. 46 shows the flowgraph for determining the mean detection time. In the flowgraph, the jth hit set phase is shaded and labeled j. The dwell times of the first and second stages are denoted by Ti = MNthTth and T2 = M'NadsTh. Then the mean detection time from the flowgraph is given by Tdet [= G(z)] (444) dz with G(z) = L 1 Gi(z), where the functions Gi(z) represent the sum of the branch labels of all paths leading from the ith cell in the first stage to the acquisition (ACQ) state, and are given by NI Glom(z) jH [H G1Mm() [GDj(z)] G2(z) ifi {1,. 1 G IM (Z)GIO (Z) j "n "'l 2" Gi(z) I ( z)o() i [ i GlMm(Z) GlDj(z)G2j(z) + 1M()GIO() C 1 [H 1 G1Mm(Z)] G1Dj(z)G2(z),if {1,... (445) The functions corresponding to the first stage used in the above are given by GiM(z) =l G1Mmn(z) and Gio(z) _= HNH+1 Giom(z), where the branch labels for the first stage are given by G1Mmr () G1Dm (Z) Giom(z) GloDrnW (446) (447) (448) (1 PDI(m))z1, me {,..., H} PDI(m)T1, me {1,..., H} ZT1 mC{ H + 1t,... N}. The functions corresponding to the second stage are as follows. The function G2j(z) represents the sum of the branch labels of all paths leading to the ACQ state conditioned ,H}. H}. , on a detection event in the first stage occurring at the phase j E {1,..., H} and is given by 1 G2Dj(z) D d GD 1 G2 ()Go() T2(Dd+l) (449) D 1 G22Mj(z)G20j[) t+ where the branch labels and functions for the second stage are given by G2Dj(Z) D2(j)z, j {1,...,H} (450) G2Mj() (1 PD2(j))T2, j {1,...,H} (451) G20j(z) T2(D1), j {H+ 1,...,ND}. (452) C. e ION( /1" z) G z 1+(d) N GD \i H+(d1)N Sj+(d1)N Stg IZ z / U J i/,T i G /D 1+D1)N G ) H+(D )N G2Dj)Z) ( ACQ Figure 46: Flowgraph to determine mean detection time for Hybrid DSTH A i ii .iti, *i' System. 4.7 Numerical Results The performance of the proposed UWB acquisition system has been evaluated through the calculation of the mean detection time of the overall system. The following values for the system parameters were chosen for the calculations. The TH sequence period Nth = 128, the period of the DS sequence Nds = 1024, i.e., D = 8, the number of channel taps Ntap 100, Nh = 16, Nb = 8, M = 1, M' = 1/2 and To = 2 ns. The required nominal uncoded BER performance has been set as AX = 102 which corresponds to () = 11 dB when the number of taps in the Rake demodulator NR = 5 and ( ) 8.5 dB when NR 10. The tolerance on the average probability No mmi of the acquisition system ending in a false alarm has been set has 6 = 0.05. The power ratio has been set as r = 4 dB, the decay constant e = 16.1 dB and Etot = 20.4 dB which corresponds do the distance of 10m between the transmitter and receiver [23]. The Nakagami fading figures mk = 3.5 , 0 < k < Ntap 1 are their mean values given in [23]. In order to verify the approximations employed in the calculation, a Monte Carlo simulation was run on the acquisition system and the average detection time was noted. The simulations make use of the analytically computed thresholds 71 and 72. The performance of the proposed twostage, hybrid DSTH acquisition system was compared to a conventional double dwell acquisition system [7], with mean detection time being the performance metric. In order to make a fair comparison between the two systems, the same hybrid DSTH signaling format was used in the double dwell system. Thus, both the systems under comparison have similar hit sets. Each of the two stages in the double dwell system is similar to the second (DS) stage of our proposed hybrid twostage system with a serial search employed. However, unlike the proposed twostage hybrid DSTH system, the ordinary double dwell system does not exploit the structure inherent in the hybrid DSTH signaling format to aid acquisition. Moreover, the performance of the doubledwell system is optimized by a twodimensional search over the variable dwell times of the two stages. Both the systems under comparison constrain the average probability of the acquisition system ending in a false alarm, PFA < 6 = 0.05. Fig. 47 shows the effect of the received m on the size of the hit set. As the received b is increased, the system achieves the nominal desired BER at more phases other than the true phase and this results in increasing sizes of the hit set. The mean detection time results for the hybrid DSTH and the double dwell systems are shown in Fig. 48 and Fig. 49 for NR = 5 and NR = 10 respectively where the hybrid DSTH system is seen to outperform the optimum double dwell system by a gain which is of the order of D. The mean detection time of the hybrid DSTH system obtained analytically as well as from simulation are presented. The close accordance between the simulation and the results from the calculation indicate that the approximations made in the analysis are justified. Even a suboptimum choice of dwell times for the two stages in the hybrid DSTH acquisition system results in a much smaller mean detection time when compared to the optimum double dwell acquisition system as seen from Fig. 48 and Fig. 49. This advantage is primarily due to the reduction in the size of the search space achieved by using the hybrid DSTH signaling format and the proposed hybrid twostage acquisition scheme. 50  N R5 45  N =10 40 35  30 S25 12 13 Eb/No (dB) Figure 47: Effect of received SNR on size of hit set H for NR = 5 and NR =10. Optimum Double Dwell Hybrid Simulation Hybrid Analytic 14 15 16 17 18 19 20 21 22 23 E /N (dB) Figure 48: Mean detection time for hybrid DSTH and double dwell systems for Nj 5. 100 SHybrid Simulation Hybrid Analytic S Optimum Double Dwell j3 101 10  ~ _ 0i 10 14 15 16 17 18 19 20 21 22 23 E .' (dB) Figure 49: Mean detection time for hybrid DSTH and double dwell systems for Ni\ 10. 4.8 System Design and Complexity Considerations The number of monocycles modulated by one bit of data, Nb would determine the transmission rate of the system. However the DS and TH spreading codes can span multiple symbols, which is typically the case with UWB systems which use long spreading codes to keep the transmit power spectral density under the FCC limit. The ratio D = Nda/Nth would determine the reduction of the search space achieved by using hybrid DSTH spreading compared to a system that only uses a single long spreading code of length Nas. The value of Nb can be independent of the ratio D. The hit set has been defined assuming that the receiver comprises a partial Rake demodulator which estimates NR consecutive channel taps. It is crucial to note that although channel estimation cannot be done during the acquisition phase, the hit set can still be defined assuming perfect channel estimation would be done subsequent to acquisition. The effect of estimation errors on the performance of the acquisition system is subject to further research. The threshold calculation in both the stages of the acquisition system has been constructed such that the channel need not be known exactly at the acquisition stage; only knowledge of the distribution of the channel taps, the received pulse shape and the received signaltonoise ratio 1 would suffice. The design parameter M is the number of periods of the TH sequence NthTth comprising the dwell time of the correlator of the first stage and M' is the number of periods of the DS sequence NdaTth comprising the dwell time corresponding to the second stage. In any acquisition system there is always a tradeoff in increasing the dwell time of the correlator. Although increasing the dwell time would improve the SNR of the decision statistic thereby resulting in lower probabilities of miss and false alarm, it would also contribute towards an increase in the detection time. In the TH stage, the designer would typically need to choose values of M > 1 to obtain a reliable decision statistic, due to the short period of the TH sequence. However M' can be a fraction less than 1 due to the relatively longer period of the DS sequence. The computational complexity of the proposed acquisition system is minimal. The computation of the thresholds for the TH and DS stages can be done using singlevariable integration as in (429) and (435) respectively. On a final note, the hardware required to implement both the stages of the acquisition system as shown in Fig. 44 and Fig. 45 would need both analog and digital components. The squaring operation in Fig. 44 would need to be implemented in analog whereas the squaring operation in Fig. 45 could be done digitally after sampling the integrator output. The sampling rate required for the first stage would be of the order of 1 Hz and that of the second stage would be of MNthNhTc the order of 1 Hz which are significantly lower than the symbol sampling rate of Hz due to the TH and DS spreading sequences spanning multiple symbols. 1 Hz due to the TH and DS spreading sequences spanning multiple symbols. Nb Nh Tc CHAPTER 5 ACQUISITION IN TRANSMITTED REFERENCE UWB SYSTEMS In this chapter, we discuss the acquisition problem in TRUWB systems employing DS spreading. When DS spreading is employed, we show that there exists a significant relaxation in the timing requirement of the autocorrelation receiver. We exploit this relaxation by designing a twostage acquisition scheme for TRUWB systems. 5.1 TRUWB Systems Recently, there has been a significant amount of interest in TR schemes for UWB communication systems [64, 6775]. In TRUWB systems, a reference pulse is transmit ted for every datamodulated pulse which is then used by the receiver for demodulation. Demodulation is typically performed using an autocorrelation receiver [69] which cor relates the received signal with a delayed version of itself. The delay is chosen such that channel response corresponding to the reference pulse is correlated with the channel response corresponding to the datamodulated pulse resulting in a capture of all the energy in the multipath. Although the TR scheme suffers a performance degradation at low SNRs due to usage of the noisy received signal as the correlator template [70, 75], the fact that it enables the use of a lowcomplexity receiver capable of exploiting the dense multipath in the UWB channel makes it an attractive alternative to systems which use conventional Rake receivers. Much of the existing literature on TRUWB systems has focused on the performance evaluation of optimal and suboptimal receivers [64, 69, 71, 72, 75]. The problem of timing acquisition in TRUWB systems has received little attention. It has been claimed by some authors [67, 71] that the use of TR schemes mitigates the timing acquisition problem in UWB systems. However, there has not been any analysis to support these claims. In this chapter, we investigate the problem of acquisition for TRUWB signals with DS signaling. We show a twolevel DS signaling structure is essential to achieve good acquisition performance. We also propose a twostage acquisition scheme which exploits the TR signal structure to reduce the acquisition search space. 5.2 System Model We assume that the propagation channel is modeled as detailed in Section 2.2.1. In TRUWB systems, the transmitted signal consists of a sequence of pulse pairs, each pair consisting of a reference pulse followed by a datamodulated pulse. The modulation format can be either pulse amplitude modulation [64] or pulse position modulation [69]. The signal may also be spread using timehopping or directsequence signaling to eliminate spectral lines and help combat MAI. In this chapter, we consider TRUWB systems with DS signaling and antipodal modulation. The transmitted signal is given by 00 x(t) = d[/t i, 1 [(t ITf) + b/Nbja[l/Nds]( ITf Td)] (51) 1 00 where b(t) is the UWB monocycle waveform, P is the transmitted power, Tf NfTo is the frame duration containing one pulse pair and [.], [L denote the integer division remainder operation and the floor operation, respectively. The two DS sequences {di} and {al} take values in {1, 1} and have lengths Mds and Nds, respectively. The number of frames modulated by a bit bi E {1, 1} is given by Nb. The delay Td = NdTc between the reference and modulated pulses is chosen to be larger than the multipath spread Tm = NtapTc to avoid interference between the multipath responses of the reference and datamodulated pulses. Here we assume that the duration of the received pulse ,', (t) = fk(t) (t) corresponding to the kth path is less than T,. The frame duration Tf is chosen to be equal to 2Td to avoid interframe interference. The received signal is given by r(t) = h(t) x(t) + n(t) = rs(t) + n(t) l=00 (52) where Ntap1 Wr(t) = ,.,,,,.', (t kTe). (53) k0 Here El is the total received energy at a distance of one meter from the transmitter, ,', (t) is the received UWB pulse corresponding to the kth path which is normalized to have unit energy, 7 is the propagation delay, and n(t) is an AWGN process with zero mean and power spectral density _. For analytical simplicity, the MAI is modeled as a white Gaussian random process whose effect is included in n(t). The received bit energy to noise ratio is thus given by Eb A 2NbElEtot No No 5.3 Twolevel DS Signaling Structure In this section, we explain why a twolevel structure on the DS signaling is essential for achieving good acquisition performance. We also briefly describe a twostage acquisition system which exploits the TR signal format to reduce the search space. The main advantage of using a TR signal format for UWB systems is that the energy in the multipath can be collected using a simple autocorrelation receiver, which correlates the received signal with a delayed version of itself. The DS signaling consists of an outer code {di} which modulates both the reference pulse and the datamodulated pulse and an inner code {al} which modulates only the datamodulated pulse. To see the advantage of this twolevel DS signaling, we need to consider the effect of the delay and multiply operation on the received signal. Fig. 51 illustrates the polarities due to DS signaling and modulation of the received signal, the delayed version of the received signal and the product of the the received signal and its delayed version1 in the absence of noise. Note that the polarity of every alternate received pulse in the product signal depends only on the modulating bit and the inner code {al} and is independent of the outer code {di}. Thus if the phase of the inner code is known, the bit can be demodulated without knowledge of the phase of the outer code {d,} by using the returntozero (RZ) gating waveform shown in the figure. The product signal is multiplied with the RZ gating waveform and integrated over a bit duration to get the decision statistic for a particular bit. The purpose of the RZ gating waveform is twofold: to despread the inner code {al} and to restrict the period of integration to those times where the product signal does not depend on the outer code. Thus the RZ gating waveform accomplishes a function similar to the multiple integration windows employed in [69]. This method of achieving demodulation without knowledge of the outer code {di} is advantageous because of the following reason. The distinguishing feature of UWB systems is the wide bandwidth and the stringent spectral mask constraint imposed by regulatory bodies. The wide bandwidth results in a fine resolution of the timing uncertainty region while the spectral mask constraint necessitates the use of long spreading sequences spanning multiple symbol intervals in order to remove spectral lines resulting from the pulse repetition present in the transmitted signal. These two features together result in a large space for the acquisition system which results in a large acquisition time if the acquisition system evaluates phases in a serial manner and in a prohibitively complex acquisition system if the phases are evaluated in a parallel manner. However, by employing the twolevel DS signaling the burden of eliminating the spectral lines can be placed on the outer code {d,} by choosing its length Mds to be large. This does not increase the size of the search space which is proportional to the 1 We shall henceforth refer to the product of the received signal and its delayed version as the "product signal". Polaritie i v A \t I \A A J s Transmitted K : do b d d1 a I I I I Received Polarities d d2 b d d a Delayed Received SDelayed Received Polaritier aobo d doaobo ab dda b a 1 a2b0 d3d2a2bo a3b Product Si Polaritie; ao a1 a2 a3 SI I I I I I I I Gating Waveform Figure 51: Illustration of the delay and Itmuliptl operation on the received signal s s d, d, a, t1 d da, l d3 d3 a, I s gnal length of the inner code Nds. The inner code {al} results in a peaky ambiguity function as in traditional spread spectrum systems and hence is essential for good acquisition performance. In addition, the inner code helps combat MAI. Thus it can be chosen to be of relatively smaller length compared to the outer code. From the above discussion, it is clear that the purpose of the acquisition system is to align (at least approximately) the RZ gating waveform with the useful part of the product signal. Although the size of the search space is now proportional to the length of the relatively short inner code, it can still be large owing to the fine timing resolution of the UWB system. So we propose a twostage acquisition system which solves the problem ofRZ gating waveform alignment in two steps. In the first stage, the acquisition system attempts to find the phase of the inner code modulating the product signal by correlating the product signal with a locally generated replica of the nonreturntozero (NRZ) inner code DS waveform with chip duration Tf. As illustrated in Fig. 52, the phase of the locally generated NRZ inner code DS waveform can suffer a large margin of error and still be successful in despreading the inner code corresponding to the useful part of the product signal, when the DS codes have ideal autocorrelation properties. This allows the first stage to evaluate phases in increments proportional to the margin of error. Once the approximate phase of the inner code is found in the first stage, the second stage of the acquisition system proceeds to align the RZ gating waveform with the useful part of the product signal by searching serially through the phases around the estimated phase. A more precise description of the twostage acquisition system can be found in Section 5.5. 5.4 Hit Set Definition In a multipath channel, there may be multiple phases where a receiver lock can be considered successful acquisition. Thus we require a precise definition of what can be considered a good estimate of the true signal phase. A typical paradigm for transceiver design is the achievement of a certain nominal uncoded bit error rate (BER) An. Then all those hypothesized phases such that a receiver locked to them achieves an uncoded BER aobo dldoaobo albe I I I d2dialbo a2bo d3d2a2bo a3b I I I I Polarities I I Product Signal l I/ / ,Polarities a, a2 a3 I I I I Inner code DS waveform Figure 52: Illustration of the margin of error tolerable in the despreading of the inner code in the first stage. Margin of error \, of An can be considered a good estimate of the true signal phase. We define the hit set to be the set of such hypothesized phases. This way of defining the hit set was first proposed in [32, 58]. We assume that the autocorrelation receiver of Fig. 53 is used for demodulation. For a given true phase 7, let Pe(AT) denote the BER performance of the autocorrelation receiver when it locks to the hypothesized phase T, where AT = 7. Let (b mi be the minimum received bit energy to noise ratio at which the autocorrelation receiver achieves a BER of An when it locks to the LOS path, i.e., Pe(O) < An when the SNR is E )b and Pe(0) > An for all SNRs less than b) Then for an SNR > is ) min N min No (b )i and true phase 7, the hit set is given by H { : Pe(AT) < An}. (54) Received Integrate over \ Decision Signal bit duration Delay Td x) Gating waveform Figure 53: Block diagram of the autocorrelation receiver. To completely characterize the hit set, we need to calculate the error performance of an autocorrelation receiver which is locked to a particular hypothesized phase 7. Since the goal of the acquisition stage is coarse synchronization, the hypothesized phase can be assumed to be an integer multiple of T. To make the analysis tractable, we assume that the true phase is also an integer multiple of T. Thus AT = (Nf + 3)Tc where a, f3 are integers and 0 < f < Nf 1. This corresponds to the assumption that the tracking loop locks to the true phase 7 if the hypothesized phase 7 is within T, of it, i.e., AT < T,. If AT > T, it is assumed that the tracking loop locks to the nearest multipath component. In order to demodulate the bit bk, the receiver correlates the received signal with a product of the delayed received signal and the RZ gating waveform Ck(t 7) lktNb a[L/Nds]Pm(t Tf Td T), where 7 is the receiver's estimate of the true phase and pm(t) = 1 for t E [0, Tm) and zero otherwise. As explained in Section 5.3, the RZ gating waveform does not depend on the outer DS code {di}. This is because {di} modulates both the reference and datamodulated pulse (see (52)) and is thus removed by the delay and multiply operation. To determine the hit set, it is necessary to obtain the probability of bit error as a function of the timing offset AT. Without loss of generality, consider the demodulation of bit bo. The decision statistic at the correlator output, conditioned on a particular channel realization h, is given by 1 +NbTf Y(A; h) r(t)r(t Td)co(t T)dt (55) Yb Jf which can be expressed as a sum of signal and noise components as Y(AT;h) r(t)(t Td)co(t T)dt + +NbTf (t)n(t Td)CO(t )dt Yi(Ar;h) Y2(AT;h) 1 f+NbTf +NbTf Y3(Ar;h) Y4(ALr;h) (56) with Yi(Ar; h) denoting the signal component and {Y (Ar; h)}i2 comprising the noise components. Suppose that the bandwidth of the UWB system is B. When the time bandwidth product of the integrator NbTfB is large, the decision statistic Y(AT; h) can be approximated by a Gaussian distribution [69, 73] with mean pb(AT; h) and variance ob(AT; h), which are to be evaluated. The mean of the decision statistic, conditioned on the channel realization h, can be shown to be approximately /fb (AT; h) El Nb l b ,N tap1 2 Nb 1= 0 b(l+)/NbJa[l/Nds]a[(l+ )/Nds] zk=7t3 E NNb tap b 2 l=0o 1 b[(l+ /N /Nbbja[l \ [ I "/Nd ]d[( / i I k=)/3 E b1 b bjNf+Ntap1 2 E E 1o b(l(++l)/Nbja[l/Nd,]a[(+a+l )/Nds] Zk=N+Ntap 0 P3Nd+Ntap1 if 0 < 3 < Nd, 1 Nd h2 Nd nk, if Nd <3Nf t, otherwise. To simplify the analysis, we assume that the codes {ai} and {di} have been ran domly generated with each element equally likely to be 1. Then when Nb is sufficiently large, the mean of the decision statistic, conditioned on the channel realization h, can be approximately expressed as boEl YNta h2 if a = 0 and 0 < 3 < Nd 1 Pb(AT; h) boE, ENf+Nti1 h2 if a 1 and Nf Ntap < 3 < Nf 1 0 otherwise. The variance of the decision statistic, conditioned on the channel realization h can be expressed as o1 (AT; h) EN [Y22(AT; h) + Y2 (A; h) + 2(AT; h)] +2EN [Y2(AT; h)Y3(AT; h) + Y2(AT; h)Y4(AT; h) + Y3(AT; h)Y4(ATr()7) since EN[Y, (AT; h)] EN[Y3 (AT; h)] EN [Y4(AT; h)] 0. Note that EN[] in the above equation denotes expectation with respect to the noise distribution. It can be easily shown that NT ++NbTf 2t tN E Ntap1 E[Y22(AT; h)] E[Y32(AT; h)] 2= r (tct )dt = 2 h 2Nb 2 Nb k0 (58) and E[Y42 (AT; h)] NTm. Also it is easy to show that E[Y2(AT; h)Y4(AT; h)] = E[Y3(AT; h)Y4(AT; h)] E[Y2(AT; h)Y3(AT; h)] = 0. The noise variance of the decision statistic conditioned on a particular channel real ization is thus given by CT (A; h) NT + E : h The conditional probability of bit error is then given by Pe(Ar; h) Q b b(T ;h) )where Q(x) (1/ ) f et2 /2dt. The average probability of bit error Pe(ArT) P[ b (AT;h) Ifpo i LH [ b (z.T;h) )J  Here, we briefly describe the general method used to evaluate this average probabil ity of error. The average probabilities of false alarm and detection in the following section are also calculated similarly. Let the function I(A, B) = Q (A + be a function of p( v'q( )I the two positive random variables B z= y Nt" h and A(AT) = AT) h2, where 0 < p(AT), q(A) < Ntap 1. The average value of the function I(.) is thus given by E[I(.)] fo0 fo0 I(a, b)fAB(a, b)dadb, where fAB(a, b) is the joint probability density function (p.d.f) of the random variables A and B. The random variables A and B are each a sum of independent gamma random variables. The joint p.d.f of A and B can be evaluated easily as fAB(a, b) = fBA(b a)fA(a) by noting that B can be expressed as B P(A) h2 + A + Nt%"p1 h . 1 k0 kk kq(A)+l 5.5 Twostage Acquisition Scheme for TRUWB Signaling In this section, we give a detailed description of the twostage acquisition scheme mentioned in Section 5.3. We also derive the decision statistics for the first and second stages. Since the demodulation stage does not depend on the phase of the outer code {di}, the timing uncertainty region is equal to the duration of the inner code and is given by NsT, where Ns = NdsNf. As in Section 5.4, both the true phase r and the hypothesized phase r are assumed to be integer multiples of To to make the analysis tractable. Then r, S E Sp {nT : n E Z and 0 < n < N, 1} and Ar = r = aTf + 3To where a and 3 are integers such that Nds + 1 < a < Nds 1 and 0 < 3 < Nf 1. The hit set 'H typically consists of hypothesized phases in the neighborhood of the true phase, i.e.,  = {17 AbTc, T (Ab 1)Tc,..., r + AfTc}, where Ab and Af are nonnegative integers. The hit set thus consists of H = Ab + Af + 1 phases. The phases in the search space can be labeled without loss of generality as Sp {0, 1,... N 1}, where the label i denotes the phase 7 + iT. When labeled in this manner, the hit set is given by H= {Ns Ab,..., Ns 1,0, 1,...,Af}. The first stage of the acquisition system evaluates the phases in the search space Sp in increments of J = 2Ntap Nd + 1. The reason for choosing this value for J will become apparent in the next subsection, where it will be shown that this is exactly the size of the margin of error tolerable in despreading the inner code. Thus by evaluating the search space in increments of size J the first stage will evaluate at least one phase where the inner code is despread. If the first stage begins its search in phase j of the search space Sp, the phases evaluated by the first stage are given by S(j) = j, (j + J), (j + 2J) s,..., (j + DJ)N}, where D = [(Ns 1)/JJ and the notation (i), = i mod Ns. The phase j where the search begins can be any element of the search space Sp. A particular phase i in the search space is evaluated by correlating the product signal with a replica of the inner code DS waveform which is delayed by iTc and the correlator output is compared to a threshold. Thus the effective search space (ESS) Si(j) of the first stage consists of D + 1 phases in the search space which are evaluated sequentially until the threshold is exceeded. Once the threshold is exceeded at particular phase i E Sp in the first stage, the control is passed to the second stage which then evaluates the phases in the set S2(i) Nd, Nd + 1,... + ,..., + d 1}. ThesetS2(i) consists of 2Nd phases centered at the phase i and is the ESS for the second stage. In the second stage, a particular phase i is evaluated by correlating the product signal with a gating waveform with delay iT, and comparing the correlator output to a threshold. The rationale behind the fine search in the second stage is that the threshold crossing in the first stage may occur at a phase where the inner code is despread but the gating waveform is not sufficiently aligned. The fine search tries to align the gating waveform such that a significant portion of the useful part of the product signal is collected. The overall search strategy is illustrated in Fig. 54. First stage search strategy  2J Search space H s(j) 2 Second stage ESSs S2 (+J)  Figure 54: Illustration of search strategy used by the twostage TR acquisition system. 5.5.1 Decision Statistic of the First Stage The first stage of the acquisition system attempts to despread the inner code {al} modulating the product signal by correlating it with a replica of the inner code DS waveform with chip duration Tf given by 00 sa(t) = a[l/Nds]pT(t ITf) (59) where pT(t) = 1 for t E [0, Tf) and zero otherwise. The decision statistic of the first stage for a particular hypothesized phase r, conditioned on a particular channel realization h, is given by Z(Ar; h) + dTf r(t)r(t Td)Sa(t r)dt (510) M V Nds ? where the dwell time of the correlator is equal to M1 periods of the inner code. Again, the decision statistic Z(AT; h) can be approximated by a Gaussian distribution with mean pI(Ar; h) and variance Cr2(AT; h), which are to be evaluated. Once again, by the random sequence assumption on the DS signaling sequences {al} and {di} when Ad, is sufficiently large, the mean of the decision statistic is zero for a I {1, 0} and can be shown to be approximately El Y t h2 if a 0 and 0 < f3 < Nd E Np d h2 ifa 0 and Nd < < Nd + ap 1 z(AT; h) < E Nd'1 h if a = 1 and Nd < 3 < Nd + Nap 1 El ZNtp1 hI ifa = 1 and Nd + Ntap < 3 < f 1 0 otherwise. From the above expression, it is clear that the signal part of the decision statistic is the maximum when a = 0, 0 < 3 < Nd and a = 1, Nd + Ntap < 3 < Nf 1. Thus the signal part of the decision statistic is maximum when Tf + Tm + Td < AT < Td, i.e. for all 7 E [7 (Td Tm), r + Td]. This set of hypothesized phase around the true phase also corresponds to the region in the search space where the inner code modulating the useful part of the product signal is despread. So the estimate of the true phase of the inner code can be anywhere in this contiguous set of phases and still enable successful despreading of the inner code. This property of first stage decision statistic allows us to perform a coarse search of the search space in increments of size J = 2Nd Ntap + 1, which is the number of phases in this maximum energy region. This coarse search is then guaranteed to hit the maximum energy region atleast once no matter where in the search space it begins. By a calculation similar to Section 5.4, the noise variance of the decision statistic for the first stage, conditioned on h, can be shown to be given by NTf 3NoE1 Ntap1 S4(AT; h) 1N 2 13 h 2 (511) kO0 5.5.2 Decision Statistic for the Second Stage In the second stage, the acquisition system tries to align the gating waveform with the product signal so that a significant portion of the energy in the useful part of the product signal is collected. The decision statistic for the second stage is given by Y(AT; h) 1 J N r(t)r(t Td)c(t )dt (512) 1 Nds Jr where c(t) is the gating waveform given by M2Ndsl c(t) = a[i/Nd]Pm(t ITf Td) (513) 1=0 and the dwell time of the correlator of the second stage is 3 _. periods of the DS sequence {al}. Following an analysis similar to the one in Section 5.4, the decision statistic for the second stage Y(AT; h) conditioned on h has a Gaussian distribution with mean E Ntp1 h ifa 0 and <3 < Nd  py(Ar; h) El f+Nt10 h2 ifa 1 and Nf Ntap < < Nf 1 0 otherwise. and variance N2T N01 Ntap 1 2c(AT; h) h (514) 411.'Nds 1. Nd k=0 5.5.3 Probabilities of False Alarm and Detection The hit set for the acquisition system and hence the second stage is H. The hit set for the first stage is defined as the set of all those hypothesized phases which, upon a threshold crossing event in the first stage, result in at least one phase from ' being evaluated in the second stage. The hit set for the first stage is thus a larger set of 2Nd + H 1 elements given by 'H' = {N, Nd Ab+ i,..., N 1,0,1,..., Nd+Af}. Conditioned on the channel coefficients h and given the threshold 71, the probabili ties of false alarm and detection for the first stage are given by Pfl(71, Ah) = Pr{Z(Ar; h) > <71 I ,H'}, Pdl(71, Arh) = Pr{Z(Ar; h) > <7 1 e H'}. (515) For the first stage, the probabilities of false alarm and detection averaged over the channel realizations are given by, Pfl(71, AT) EH[Pfl(71, ATrh)] EH Q (71 z(AT; h) H/ Pddl((11, AT) EH Pd1 (71, AT ] h)] EH 7 (A; h) IT . (516) Similarly, for the second stage the conditional probabilities of false alarm and detection for a threshold 72 are given by Pf2(72, Arlh) = Pr{Y(AT; h) > 72 i T }, Pd2(72, Arlh) = Pr{Y(AT; h) > 72  e H}. (517) Then the average probabilities of false alarm and detection are given by Pf2(72, AT) = EH [Pf2 (72, Arh)] EH 2 (Ar; h) Pd2 (72, AT) = EH[Pd2(72, Arh)] = EH Q 72 py(Ar; h) (518) 5.6 Decision Threshold Selection The thresholds of the two stages, 71 and 72, are chosen such that the average probability of the twostage acquisition system ending in a false alarm, PFA(71, 72), is small. The reason for setting the thresholds in this manner is made clear in the following section which describes the use of mean detection time, instead of mean acquisition time [7], as the system performance metric for UWB acquisition systems. The use of the average probability of the acquisition system ending in a false alarm as the criterion for decision threshold selection was first done in [32, 58]. The calculation of PFA(71,72) is summarized in the following. In order to allow ease of notation in the sequel, we relabel, without loss of generality, the phases in the search space as follows. The phases in the hit set of the first stage are now denoted by 7' = {0, 1,..., 2Nd + H 2}. The other consecutive phases in the search space are denoted by {2Nd + H 1,... Ns 1}. The phases belonging to the hit set of the second stage can then be denoted by H = {Nd 1, Nd,... Nd + H 2}. The average probability of false alarm for a nonhit set phase i 7' for the first stage is denoted by Pfl (i) and that for a nonhit set phase j H for the second stage is denoted by Pf2 (j). The average probability of detection for the hit set phase m E {0,..., 2Nd + H 2} for the first stage is denoted by Pdl(m) and the average probability of detection for the hit set phase n E {Nd 1,..., Nd + H 2} for the second stage is denoted by Pd2 (n). An overall false alarm event for the twostage TR acquisition system can occur in two ways: 1. A false alarm event in the first stage would mean a false alarm event for the entire system, since following a false alarm in the first stage, the second stage would evaluate only nonhit set phases. 2. There could occur a detection event in the first stage followed by a false alarm in the second stage. These events need to be enumerated while computing PFA(71, 72). First, the average probability that a false alarm event would occur in the second stage, following a detection event in the first stage at the jth hit set phase (0 < j < 2Nd + H 2), is computed. This average probability is denoted by PF2(j) and can be shown to be given by 1 PM2(j)Phf PF2(j) 1 Pnf2(j)1 P(i)P(i) (519) where Pnf2(j) denotes the probability of nonoccurrence of a false alarm in the second stage prior to the evaluation of a hit set phase, following a detection event in the first stage at its jth hit set phase, and is given by S2Ndj1 Pf2 ((j Nd + k)N)], if0 < j < 2Nd 2 Pnf2(j) 1, if2Nd 1 < < 2Nd+H2 and Phf2(j) denotes the probability of nonoccurrence of a false alarm in the second stage during one complete evaluation of all nonhit set phases in its search space and is given by Nd1 [1 f2 (( Nd + k)N)] if0 < j < H 1 2 (H2Ndj 1 [ Pf2 Nd+ k) )] (n [1 P( ( 1 + k)) if H S2j3Nd [1 ((Nd + k)j)], if2Nd 1 Following a detection event at the jth hit set phase in the first stage, the probability of miss for the second stage PM2(j) is the probability of missing all the hit set phases falling in the search space of the second stage and is given by n 0 o[1 Pd2(Nd+k1)], if0 k j2Nd+I [1 Pd2(Nd + k 1)] if2Nd < j < 2Nd + H 2. Using the values PF2(j), the probability of the twostage acquisition system encountering a false alarm can now be calculated. The first phase k evaluated by the first stage is assumed to lie randomly in the set of phases {0,, .., N, 1} which corresponds to the timing uncertainty region in the absence of any timing sideinformation. The first stage then evaluates the D + 1 phases in the set Si(k). Since the number of phases in the hit set of the first stage is 2Nd + H 1 and since the first stage evaluates only one phase among every J phases, the number of phases evaluated that lie in the hit set of the first stage is L + 1 where L = 2Nd+Hl. Now, the average probability PFI(k), of the acquisition system ending in a false alarm conditioned only on the starting phase k of the first stage, is to be determined. The false alarm probability conditioned on each starting phase will be the sum of the probabilities of mutually exclusive events leading to a false alarm. There arise two cases: Case 1: Search starts outside hit set, i.e., k E {2Nd + H 1,..., Ns 1}. It can be shown that 1 P 1PF2(1, k) Mi(k) PFI(k) = 1 fl(k), (520) 1 PMl( ) Phfl(k) where PfI (k) denotes the probability of nonoccurrence of a false alarm in the first stage prior to the evaluation of a hit set phase, conditioned on the first phase evaluation of the first stage being at the kth phase, and is given by Pnfl(k) (1 Pfl ((k+iJ)N)) (521) i=0 and PhfI (k) denotes the probability of nonoccurrence of a false alarm in the first stage during one complete evaluation of all nonhit set phases in its search space, conditioned on the first phase evaluation of the first stage being at the kth phase, and is given by Phfl(k) (I P fl((k + iJ))) ] (1 P((k iJ)N)) (522) i=n(k)+L+l i=0 The number of nonhit set phases evaluated by the first stage prior to the evaluation of the first phase in the hit set is n(k) + 1 where n(k) = [LN and PF12(j, k) is the average probability of the acquisition system ending in a false alarm in the second stage, following a detection in the first stage, conditioned on the phase evaluation in the first stage starting in the jth hit set phase (k + (n(k) + j)J)Ns and is given by PF12(j, k) = Pd ((k (n(k) + 1)J)N)) S ((k + ((k) + N P((k + ((k) + ). (523) Pdl ((k + (n(k ) + i)J)N ) PF2((k + (n(k ) + i)J)N ). (523) The probability of missing all the hit set phases falling in the search space for the first stage, conditioned on the first phase evaluation beginning at phase k, is given by L PMi(k) = (1 Pdl ((k + (n(k) + 1)J)N)). (524) l 1 Case 2: Search starts inside hit set, i.e., k {0, 1,..., 2Nd + H 2}. In this case, the number of hit set phases evaluated before the first stage evaluates a nonhit set phase is L'(k) + 1 where L'(k) = L2Nd+k ] It can then be shown that the probability of the system ending in a false alarm is given by L'(k) PF(k) =Pa(k) + [1 dl (( +J))] b(k)) 1(k) o 0 t PMio(k)Pb(k) (525) where the probability that the system ends in a false alarm prior to a nonhit set phase evaluation is given by L'(k) i1 Pa(k) (1 dl ((k 1J)Ns)) Pdl ((k + iJ)Ns) PF2 ((k + iJ)N) (526) i=0 L.=0 and the probability of nonoccurrence of a false alarm during one complete evaluation of all the nonhit set phases in the first stage is given by L'(k)+DL Pb (k) n (1 Pf (( + J)N)). (527) i=L'(k)+1 The probability of the system ending in a false alarm during one complete evaluation of all the hit set phases for the first stage is given by L'(k)+D+1 il e(k) ( i d ((k + 1J)N)) Pd ((k + J)N) PF2 ((k + J)N) i=L'(k)+DL+l l=0 (528) and the probability of missing all the hit set phases in the search space given by D PMI(k) [1 Pdl ((k + (n(k) + ) J)N)] (529) l=L'(k)+DL+1 The average probability of the twostage acquisition system ending in a false alarm, PFA (1, 72) can now be expressed as SNis1 PFA(71,l72) Y PFl(k). (530) k0 The thresholds 71 and 72 are chosen such that PFA(71, 72) < 6, where 6 denotes the tolerance on the average probability of the occurrence of a false alarm for the twostage acquisition system. 5.7 Mean Detection Time The mean detection time is the average amount of time taken by the acquisition system to end in a detection, conditioned on the nonoccurrence of a false alarm event. The calculation of the mean detection time thus does not require any assumption on the false alarm penalty time. However, to ensure that the acquisition system rarely encounters a false alarm, the probability of the acquisition system ending in a false alarm event PFA(71, 72) is constrained by 6 as explained in the previous section. The first phase evaluated by the first stage is assumed to be picked equally likely from any of the elements in the set Sp. Conditioned on the starting phase k E Sp, the set of phases evaluated by the first stage is thus given by Si (k), which consists of D + 1 phases. Fig. 55 shows the flowgraph for the twostage acquisition scheme. As discussed in Section 5.5.3, the hit set for the first stage comprises those phases where a threshold crossing in the first stage results in at least one phase from 'H being evaluated in the second stage. The dwell times of the first and second stages are denoted by T1 = M1NdsTf and T2 3= .VNdsTf. Then the mean detection time is given by Tdet= IG(z) (531) dz z=1 Stage 1 Ns1 0 1 Ndl Nd+H2 2Nd+I2 J Hit Set / \ \ N +H2 d Nd+i_2 Stage 2\ Stae2 Stage2 Stage 2 ; Nd1 ACQ Figure 55 Flowgraph ill itr..i;, the proposed twc J.t..r, e iq Ir.;it;,,,, scheme. with G(z) = ,'o Gk(z), where the functions Gk(z) represent the sum of the branch labels of all paths leading from the kth cell in the first stage to the acquisition (ACQ) state, and are given as follows Case 1: k e {2Nd + H 1, 2Nd + H,..., Ns 1}. G(n(k)+l)TI G1Dk (Z) Gk(z) (532) 1 z(D+1L)TIG1Mk(Z)' The functions corresponding to the first stage used in the above are given by L i1 G1Dkc() ()Pdk(k)+i)) ( d(k((k)+)+)), (533) i= 1 l 1 and G1MkM(Z) ZTPM1(k), where for convenience of notation, we denote k = (k + iJ)Ns. Conditioned upon a hit in the first stage in the phase j E {0, 1,..., 2Nd + H 2}, the sum of the branch labels leading to the ACQ state are given by G" znl(J)T2G2Dj ( (34 G2() ()T2G2 (534) 1 zR3(j)T G2Mj(X where ni(j) denotes the number of nonhit set phases evaluated by the second stage prior to the evaluation of a hit set phase, conditioned on a hit occurring in the first stage at phase j, and is given by i() 2Ndj if0 0 if2Nd 1 The total number of nonhit set phases evaluated by the second stage, conditioned on a hit occurring in the first stage at phase j, is given by ni () if0 < < H n3(j) n(j) + j H + 2 if H < j < 2Nd 1 3Nd j H + 2 if 2Nd The functions corresponding to the second stage used in the above are given by G2Dj(z) o z(k+)TPdNd + 2 k 1)H Io 1 [1Pd2(Nd +k 1)], if0 :2+l ^PdNd + k I i j2Nd+l [1 Pd2(NVd + i 1)], if2N2d < < 2d + H 2. and Sz(J+)T2PM2(j) if0 z2Ndj+H M2(j) if 2Nd < j < 2Nd + H 2. Case 2: k e {0,1,...,2Nd + H 2}. G1M k(z)G1D'k(Z) (5 Gk(z) G1Dpk(z) Z (D+1L)TIG1M'k()' ( where the transfer function corresponding to the paths leading to the ACQ state until the first nonhit set phase is encountered is given by L'(k) i1 G1Dpk(Z) zG1, i(z)Pdl(kj) H (1 Pdl(j1)) (536) i=0 1=0 and the transfer function corresponding to the paths leading to the ACQ state during one sweep of all the hit set phases encountered for the first stage is given by D i1 G1D'k(Z) = zG (z)Pdl(kJ) H T1P(1 Pdl(kj)). (537) i=L'(k)+DL+1 l=L'(k)+DL+1 The transfer function corresponding to missing the hit set phases, until the first nonhit set phase is encountered, is given by L'(k) G1Mpk(Z) ZT1( Pdl(kj)), (538) 1=0 and the function corresponding to missing all the phases in the hit set during a sweep is given by L'(k)+D+l GIM'k(Z) H ZT(l Pdl(kf)). (539) l=L'(k)+DL+2 5.8 Numerical Results The mean detection time Tdet is used as a performance metric to evaluate the performance of the proposed twostage acquisition scheme. The performance of the proposed system is compared to that of a singlestage acquisition system which performs a serial search over the whole search space Sp utilizing a detector which has a structure similar to the second stage of the proposed system. It is noted that the hit set for this singlestage acquisition system is the same as the proposed twostage system. The following values of the system parameters were chosen during the calculations. The period of the inner DS sequence Nds = 32, the period of the outer DS sequence Mds = 256, Nd = 110, Ntap = 100, M1 = 1, = 1 and To = 2 ns. The tolerance on the average probability of the acquisition system ending in a false alarm has been set as 6 = 0.05. The power ratio has been set as r = 4 dB, the decay constant c = 16.1 dB and Etot = 20.4 dB which corresponds do the distance of 10m between the transmitter and receiver [23]. The Nakagami fading figures mk = 3.5 0[, 0 < k < Ntap 1 are their mean values given in [23]. The required nominal uncoded BER performance has been set as An 103. The number of phases in the hit set is shown as a function of m in Fig. 56. As b increases, the system achieves the nominal desired BER at more phases other than the true phase and this results in increasing sizes of the hit set. The mean detection time performance of the proposed twostage system and the singlestage acquisition system is shown in Fig. 57. In order to verify the approximations made in the analysis, we also determine the mean detection time for the two systems through Monte Carlo simulation, in which we make use of the analytically computed thresholds 71 and 72. We observe that the approximations made in the analysis are reasonable. The improvement in performance achieved by the twostage scheme is due to a significant reduction in the size of the effective search space. The singlestage system faces a large search space of size Ns and its mean detection time is dominated by the time spent by the search in evaluating and rejecting the large number of nonhit set phases. Thus even though the probabilities of detection for the singlestage scheme improve with SNR, the improvement in the mean detection time is imperceptible. On the other hand, the size of the effective search space of the two stage scheme is just 2Nd + D + 1, an improvement of the order of Nds. Thus the time spent by the acquisition system in evaluating and rejecting the nonhit set phases is much smaller and the improvement in the mean detection time with increase in SNR is apparent. 0' 10 12 14 16 18 20 22 EbN0 (dB) Figure 56: Effect of received SNR on hit set size H. 100 24 26 28 30 Singlestage Analytic Twostage Analytic Singlestage Simulation .... Twostage Simulation 10'1 E 0 h S 102 103 10 12 14 16 18 20 22 24 26 28 30 Eb 0 (dB) Figure 57: Mean detection time for twostage and singlestage TRUWB acquisition systems. CHAPTER 6 FINE TIMING ESTIMATION Accurate timing estimation is crucial to the performance of impulse radio ultra wideband (UWB) systems. The narrow pulses and low duty cycle signaling in UWB systems place stringent timing requirements at the receiver for demodulation. In addition to affecting the receiver's bit error rate performance [76], accurate timing information is also essential in UWB systems incorporating precise ranging capabilities. An important challenge faced by UWB systems is that the transmitted pulse can be distorted through the antennas and the channel. Due to the frequency selectivity of the UWB channel, the pulse shapes received at different excess delays are pathdependent [24]. Moreover, the short pulses used in UWB systems result in highly resolvable multipath with a large delay spread [12]. In this chapter, we address the timing estimation problem in UWB systems when the receiver does not have knowledge of the received pulse shapes and the channel. We derive maximum likelihood (ML) timing estimators and the CramerRao lower bound (CRLB) for both pilotassisted and nonpilotassisted scenarios. We focus on fine timing estimation, where the timing uncertainty region is within a pulse width, and assume that coarse synchronization has already been achieved. Efficient timing acquisition schemes for UWB systems have been developed [77], which achieve coarse synchronization to within a pulse duration. We compare the CRLB to the performance of the ML timing estimator and suboptimal timing estimation methods such as the dirtytemplate method (TDT) [78] and transmitted reference (TR) [67, 69, 71] signaling, both of which do not require knowledge of the pulse shapes or the channel at the receiver. We evaluate the CRLB and simulate the performance of the ML timing estimator and the suboptimal schemes under the IEEE 802.15.3a UWB channel models described in [12]. 6.1 System Model The transmitted UWB signal consists of a train of short pulses monocycless) which may be dithered by a timehopping (TH) sequence to facilitate multiple access and to reduce spectral lines. The polarities of the transmitted pulses may also be randomized using a directsequence (DS) spreading code to mitigate multiple access interference (MAI). Such a signal can be expressed as a series of UWB monocycles 0(t) of width Tp, each occurring once in every frame of duration Tf as Co x(t) = bL /Nbja[l/Nd]O(t ( d Tf C[/Nth]TC), (61) where Nb is the number of consecutive monocycles modulated by each data symbol bl, Tf is the pulse repetition time, To is the chip duration which is the unit of additional time shift provided by the TH sequence and [.], [L denote the integer division remainder oper ation and the floor operation, respectively. The pseudorandom TH sequence {ci}l 01 has length Nth where each Qc takes integer values between 0 and Nh 1 with Nh less than the number of chips per frame Nf Tf/T,. The DS sequence {a}lIN1 has length Nds with each al taking the value +1 or 1. Some UWB systems may employ only TH (al = +1) or only DS (c = 0) spreading and may not send any data (bl = +1) during the acquisition stage. The UWB indoor propagation channel can be modeled by a stochastic tapped delay line [12, 23] which can be expressed in the general form in terms of its impulse response Nt1 h(t) = hkfk(t kT), (62) k0 where Nt is the number of taps in the channel response, hk is the path gain at excess delay kT, corresponding to the kth path. The functions fk(t) model the combined effect of the transmit and receive antennas and the propagation channel corresponding to the kth path on the transmitted pulse. We assume that Nf > Nt, so that there is no interframe interference. The received signal from a single user can then be expressed as r(t) = rs(t) + n(t) with 00 rT(t) = blb/Nbja[l/Nds]Wr(t Tf C[/Nh]Tc 7), (63) 1 00 where Nt1 wr(t)= ,, ', (t kTc) (64) k0 is the received waveform corresponding to a single pulse. Here ,', (t) = fk(t) (t) is the received UWB pulse from the kth path normalized to unit energy. The duration of the received pulse Tw is assumed to be equal to the chip duration Tc. The propagation delay is denoted by 7 and n(t) is a zeromean additive white Gaussian noise process with variance .2. Given the received signal, the synchronization system attempts to retrieve the timing offset 7. We consider a singleuser system during the following development. The analysis can be easily extended to the case of multiple users, if the multipleaccess interference can be modeled as an additional Gaussian noise component. 6.2 PilotAssisted Timing Estimation In this section, we derive a maximum likelihood timing estimation algorithm for a UWB radio receiver in the absence of information regarding the channel and received pulse shape, but having knowledge of the training symbols {bl}. We ignore timehopping for notational simplicity and the analysis can easily be extended to incorporate time hopping. We approximate the received pulse in the kth path by a truncated Fourier series expansion using L' = 2L + 1 coefficients as L'1 St) p, ,.(t) (65) i=0 where the orthonormal basis functions are given by Oi(t) are given by wTpw(t) ifi 0 S(t) cos(2rit/Tw)pw(t) if 1 < < L (66) S sin(2r(i L)t/Tw)pw(t) if L + 1 < i < 2L. where Eci = fT cos2(2rit/Tw)dt, Esj = fT sin2(27rit/Tw)dt for i = 1,..., L and pw(t) = 1 for t E [0, Tw] and zero otherwise. The unit energy condition on ,', (t) therefore translates to ppkP = 1 where the vector of L' pulse coefficients Pk [= k,0,Pk,cl, ,Pk,cL,Pk,sl, ,Pk,sL]. The received waveform corresponding to a single transmitted pulse can then be modeled as Nt1 Nt1 L'1 wr(t) ,,(t kTw) hk .. (t kT,), (67) k=O kO0 i=O where the pulse received from the kth path is normalized to unit energy. The received signal, given by (63) with c = 0 for all 1, is observed over a duration of N, symbols and this observation is denoted by r. 6.2.1 Maximum Likelihood Timing Estimation We denote the channel coefficient vector by h =[ho, hi,. ht _]T. Recall that the pulse coefficient vector of the kth path is pk [Pk,o,Pk,1, .. Pk,L'1]T for k E {0, 1,..., Nt 1}. For ease of notation we denote the matrix of pulse coefficients by p = [po, P, ... PNt1]. We note that during the development of the ML timing estimators and the CRLB for the pilotassisted and nonpilotassisted cases, we assume that the number of channel taps is known at the receiver. When Nt is not known at the receiver, the analysis can be easily extended by assuming a maximum number Ntmax such that Ntmax is always larger than Nt. The conditional loglikelihood function of r, given the parameters 7, p and h is 1 1 AL(r; r, p, h) = const + hTB(p,7) NNh'h (68) O 2 where the vector B(p, 7) = [Bo(po, T), B(po, r),..., B, N1(pN, 1, T)]T and Bk(pk 7) pk Ck(T). In the above, Ck(T) [Ck,O(r), Ck,(T),..., CLk,'IT with Ns  Ck,i(r) = b,(k,i,n(r) (69) where (n+l)Nb1 (k,i,n(T) rt) a[/\, jt f kTw )dt. (610) "To l=nNb The ML estimates for the timing and the pulse and channel coefficients are then given by [T0, po0.,I Nt1, h] arg max [2hTB(p, r) NNbhTh] {T,Po,Pl,...,PNt ,h : T>0, IIP 1 ll1} argmax U(, p,h) V(r,p), (611) {T,Po,Pl,...,P Nt ,h : r>0, HIIP7 1} where U(r, p, h) NN h N B(p, T) h N B(p, )] and V(T, p) N BT(p, r)B(p, r) are quadratic forms and U(r, p, h) is minimized for h l1Nb B(p, 7). Hence, the ML estimates for the pulse coefficients and the timing are given by [T, o, p1, PNt1] arg max {TP,PO,P1...,P 1: r>0, HIIP llk 1} Nt1 arg max p Ck()]2 (612) {T,po,P ...,PNt : T>0, Hpklll 1} k 0 We solve the optimization problem in (612) by first maximizing the argument over po, Pi,., PNt1 and then over r. For any value of > 0, Nt1 arg max C T] 2 {pO,Pl,...,PNtllPk 1} k 0 Nt1 argmax p Ck(T)C(T) Pk with p k 1 Vk. (613) {PO,P1,...,PNt1: T>0, p kll 01} k=o Ak (T) In (613), we note that for any value of for each k E {0, 1,..., Nt 1}, the matrix Ak T() is real, symmetric and positive definite. Therefore the kth term in the summation is V(r, p) maximized by setting pk = Ulk(T), where Ulk(T) is the eigenvector corresponding to the maximum eigenvalue Alk (T) of Ak (). The optimum value 7 is thus given by Nt1 7= argmax Ai~lk(). (614) e[o0,Tw] k 0 The ML estimates of the pulse coefficients are then given by pk = Ulk(T) for 0 < k < Nt1. 6.2.2 CramerRao Lower Bound We now compute the CRLB for estimating the timing information. The parameters to be estimated are the timing 7, the pulse coefficients po, pl,... p1 and the channel gain coefficients h. Owing to the unitenergy condition on the pulses, we have pk,L' /1 t_ 2 p ,j, and hence we instead consider the reduceddimension vector of pulse coefficients qk = Pk,o, ... Pk,L'2]T for 0 < k < Nt 1. The vector of parameters is denoted by E = [7, q qT,..., qT_ hT]T. The parameters are assumed to be deterministic but unknown. The Fisher information matrix for the estimation of E is given by Jo,o Jo,i .. Jo,Nt Jo,Nt+l Jl,0 J1,l J1,Nt J1,Nt+l J = (615) JNt,O JNt,1 .. JNt,Nt JNt,Nt+l JNt+1,0 JNt+l,1 JNt+l,Nt JNt+l,Nt+l The elements of the Fisher information matrix in (615) are as follows: 1. Jo, = E a 2 ] : Since n(t) is a zeromean noise process, Jo,o is given by Joo = r,(t; E)r,(t; O)dt. (616) a, 0 To We can obtain a simplified expression for Jo,o by observing the second derivative, with respect to time, of the basis functions of (66) to be 0 ifi 0 i( = T i (t) if l 4w2(iL)2 i L(t) ifL+l Using (616) and (617), we can show that 4 2 Nt1 L 2L Jo,o = Ns22 b [ L) p (618) k=O j=1 j=L+1 2. JOkl E l(r ) for 0 < k < Nt 1 Making use of the relation 2 J o___l LL [q'L 'c l J  Pk,L'l 1 2/ 0 j, we can show that Jo,kc+l 1 l C T J+1,0 is a 1 x L' 1 vector whose ith element is given by NsN.hb P, 2LPL if i = ,7 /_1 Tw Ns,,. 1, 2 +ip+ p 2LpL) if 1 < i < L [Jo,k+l1i T2 Tw Pk Tw (619) 1 NNbh h 2k7(iL)pkiL + Pki 27LPk,L T2 TPk,L 1Tw if L +1 where we have made use of the fact that Opk,L' 1/pk,i Pk,i/Pk,L'  3. Jo,Nt+l = E 2AL ( J It can be shown that Jo,Nt+l JT+, ,0 0. 4. Ji+1+i E A forO < kl,k2 < t t 1 It can be shown that 4. Jkl+l,k2+l L q q J   Jk1+1,k2+1 = 0 for ki / k2. For k = k2 k we can show that 1 F 1 Jk+1,k+ 1 s 1 I + qkqk (620) S Pk,L'1 5. Jk+1,N+[ E A(r;l] for0 < k < N 1 : We can show that Jk+1,Nt+l Nt+l,k+l 0. 6. JNt+l,Nt+l E a2l(r : It is straightforward to show that the Nt x Nt matrix, JNt+i,Nt+l= 2I. From the above, the Fisher information matrix in (615) can then be expressed in the simplified partitioned form Jo,o E 0 J ET G 0 (621) S0OH where E = [Jo,, Jo,2,... Jo,Nt], H = JNt+,Nt+l and Jl,1 0 0 ... 0 0 J2,2 0 ... 0 G (622) 0 0 ... 0 JNt,N, By using a standard result on the inversion of partitioned matrices, the CRLB on the variance of any unbiased timing estimator can be expressed as CRLB1() [J1], Jo,o EGET. (623) The inverse of the matrix G can be computed by noting from (620) that for 0 < k < Nt, 2 ^1,f+2 [I + q(PkL'i q q Y) qk] NNb 2 [I qkq] (624) Hence (623) can be expressed as 1 CRLB(T) Nt (625) k JO,kJk,k O,k 6.3 Nonpilotassisted Timing Estimation In this section we consider the timing estimation problem, when the training symbols are unknown and derive the ML timing estimator and compute the CRLB. 