DESIGN AND PERFORMANCE OF ULTRA-WIDEBAND ACQUISITION SYSTEMS

By

SARAVANAN VIJAYAKUMARAN

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2007

1

Copyright 2007

by

Saravanan Vijayakumaran

2

To my teachers.

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ACKNOWLEDGMENTS

I would like to thank my advisor, Dr. Tan F. Wong, for his guidance and encouragement

throughout my graduate studies at UF. I have learned a great deal from him not only

through his instruction but also by imitation. I feel very fortunate for having had the

opportunity to work with him for the past few years.

I would also like to thank Dr. Michael Fang and Dr. John Shea for their guidance and

the many interesting discussions.

Finally, I would like to thank the Dr. Paul Robinson and Dr. Alexander Turull of the

UF Mathematics Department for encouraging my interest in their courses and for their

unlimited patience in answering my questions.

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TABLE OF CONTENTS

page

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

CHAPTER

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.1 Brief Review of Spread Spectrum Acquisition Systems . . . . . . . . . . . . 131.2 Previous Work on UWB Acquisition . . . . . . . . . . . . . . . . . . . . . 15

1.2.1 Efficient Search Strategies . . . . . . . . . . . . . . . . . . . . . . . 171.2.2 Search Space Reduction Techniques . . . . . . . . . . . . . . . . . . 18

1.3 Previous Work on UWB Time-of-Arrival Estimation . . . . . . . . . . . . . 201.4 Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2 PROBLEM DEFINITION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2.1 Channel Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2.2 Transmitted and Received Signals . . . . . . . . . . . . . . . . . . . 26

2.3 Hit Set Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.4 The UWB Acquisition Problem . . . . . . . . . . . . . . . . . . . . . . . . 30

3 ACQUISITION OF TIME-HOPPING UWB SIGNALS . . . . . . . . . . . . . . 32

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.2 Analysis of SAI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2.1 Derivation of the Decision Statistic . . . . . . . . . . . . . . . . . . 333.2.2 Average Probabilities of Detection and False Alarm . . . . . . . . . 36

3.3 Analysis of IAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.3.1 Derivation of the Decision Statistic . . . . . . . . . . . . . . . . . . 373.3.2 Average Probabilities of Detection and False Alarm . . . . . . . . . 38

3.4 Mean Detection Time Analysis of Serial Search . . . . . . . . . . . . . . . 393.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4 ASYMPTOTIC PERFORMANCE OF THRESHOLD-BASED ACQUISITIONSYSTEMS IN MULTIPATH FADING CHANNELS . . . . . . . . . . . . . . . . 54

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

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4.3 Asymptotic Performance of Threshold-based Acquisition Systems . . . . . 574.4 Asymptotic Performance of Threshold-based UWB Signal Acquisition . . . 61

4.4.1 Asymptotic Performance of the SAI Approach . . . . . . . . . . . . 624.4.2 Asymptotic Performance of the IAS Approach . . . . . . . . . . . . 654.4.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5 A SEARCH STRATEGY FOR UWB SIGNAL ACQUISITION . . . . . . . . . 72

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.3 Mean Detection Time Calculation . . . . . . . . . . . . . . . . . . . . . . . 745.4 The Jump-by-H Permutation Search Strategy . . . . . . . . . . . . . . . . 755.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6 UWB TIME-OF-ARRIVAL ESTIMATION STRATEGIES . . . . . . . . . . . . 81

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.2 UWB TOA Estimation: Known Channel Statistics . . . . . . . . . . . . . 826.3 UWB TOA Estimation: Unknown Channel Statistics . . . . . . . . . . . . 846.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

6.4.1 Dense UWB Channels . . . . . . . . . . . . . . . . . . . . . . . . . 876.4.2 Sparse UWB Channels . . . . . . . . . . . . . . . . . . . . . . . . . 89

6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

APPENDIX

A AVERAGE NUMBER OF MPCS COLLECTED . . . . . . . . . . . . . . . . . 101

B AVERAGE PROBABILITY THAT THE ACQUISITION PROCESS WILLEND IN A FALSE ALARM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

C PROOF THAT Aε(γ; ∆τ) AND Bε(γ; ∆τ) DEFINED IN (4–25) SATISFY THECONDITIONS OF THEOREM 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 105

D PROOF THAT Aε(γ; ∆τ) AND Bε(γ; ∆τ) DEFINED IN (4–35) SATISFY THECONDITIONS OF THEOREM 1 . . . . . . . . . . . . . . . . . . . . . . . . . 107

E PROOF THAT Q (DEFINED IN (5–12)) IS THE VECTOR IN THE SET ACORRESPONDING TO THE PERMUTATION R . . . . . . . . . . . . . . . . 108

F THE PDF THE SUM OF A FLIPPED NAKAGAMI RANDOM VARIABLEAND A GAUSSIAN RANDOM VARIABLE . . . . . . . . . . . . . . . . . . . . 114

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

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LIST OF TABLES

Table page

5-1 Serial search for Ns = 8 and H = 3. . . . . . . . . . . . . . . . . . . . . . . . . . 80

5-2 Permutation search (1, 4, 7, 2, 5, 8, 3, 6) for Ns = 8 and H = 3. . . . . . . . . . . 80

5-3 Mean detection time (MDT) values for the serial search and hueristic searchstrategies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

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LIST OF FIGURES

Figure page

1-1 Block diagram of a parallel acquisition system for direct-sequence spread spectrumsystems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1-2 Block diagram of a serial acquisition system for direct-sequence spread spectrumsystems which evaluates the candidate phases t1, t2, . . . , tn serially. . . . . . . . 23

1-3 Block diagram of the acquisition scheme proposed by Blazquez et al. . . . . . . 23

1-4 Block diagram of the acquisition scheme proposed by Soderi et al. . . . . . . . . 23

1-5 Template signals used in the two-stage acquisition scheme proposed by Bahramgiriet al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1-6 Block diagram of the two-stage acquisition scheme proposed by Aedudodla et al. 24

1-7 Transmitted signal along with its component signals used by Furukawa et al. . 24

2-1 The hit set size as a function of the average energy received per pulse to noiseratio for Np = 5 and 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3-1 Block diagram of the SAI acquisition system. . . . . . . . . . . . . . . . . . . . 46

3-2 Block diagram of the IAS acquisition system. . . . . . . . . . . . . . . . . . . . 46

3-3 Effect of EGC window length on the probability of a miss for SAI when Np = 5 46

3-4 Effect of EGC window length on the probability of a miss for SAI when Np = 10 47

3-5 Effect of EGC window length on the probability of a miss for IAS when Np = 5 48

3-6 Effect of EGC window length on the probability of a miss for IAS when Np = 10 49

3-7 Effect of EGC window length on the mean detection time for SAI when Np = 5 50

3-8 Effect of EGC window length on the mean detection time for SAI when Np = 10 51

3-9 Effect of EGC window length on the mean detection time for IAS when Np = 5 52

3-10 Effect of EGC window length on the mean detection time for IAS when Np = 10 53

4-1 Best AROC of the SAI approach to UWB signal acquisition. . . . . . . . . . . . 70

4-2 Best AROC of the IAS approach to UWB signal acquisition. . . . . . . . . . . . 70

4-3 IAS AROC corresponding to hit set phases other than the LOS path when G = 1. 71

6-1 Illustration of the packet exchange scheme used to estimate the TOA. . . . . . . 92

6-2 Mobile positioning based on TOA measurements. . . . . . . . . . . . . . . . . . 92

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6-3 The location of the observations used for TOA estimation. . . . . . . . . . . . . 93

6-4 Probability of incorrect estimation in dense channels for the rule which minimizesthe error probability when the channel statistics are known. . . . . . . . . . . . 93

6-5 Mean estimation error in dense channels for the rule which minimizes the errorprobability when the channel statistics are known. . . . . . . . . . . . . . . . . 94

6-6 Probability of incorrect estimation in dense channels for the rule which minimizesthe average estimation error when the channel statistics are known. . . . . . . . 95

6-7 Mean estimation error in dense channels for the rule which minimizes the averageestimation error when the channel statistics are known. . . . . . . . . . . . . . 96

6-8 Probability of incorrect estimation in dense channels for the heuristic rule whenthe channel statistics are unknown. . . . . . . . . . . . . . . . . . . . . . . . . 97

6-9 Mean estimation error in dense channels for the heuristic rule when the channelstatistics are unknown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6-10 Probability of incorrect estimation in sparse channels for the heuristic rule whenthe channel statistics are unknown. . . . . . . . . . . . . . . . . . . . . . . . . . 99

6-11 Mean estimation error in sparse channels for the heuristic rule when the channelstatistics are unknown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

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Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy

DESIGN AND PERFORMANCE OF ULTRA-WIDEBAND ACQUISITION SYSTEMS

By

Saravanan Vijayakumaran

May 2007

Chair: Tan F. WongMajor: Electrical and Computer Engineering

The acquisition of ultra-wideband (UWB) signals is a potential bottleneck for system

throughput in a packet-based network employing UWB signaling format in the physical

layer. The problem is mainly due to the fine time resolution and the low received signal

power which forces the acquisition system to process the signal over long periods of time

before getting a reliable estimate of the timing of the signal. In this dissertation, we

focus on the development of more efficient acquisition schemes by taking into account the

signal and channel characteristics. The presence of dense multipath in the UWB channel

suggests the presence of multiple acquisition states which could be exploited to speed

up the acquisition process. In this dissertation, we give a precise characterization of the

set of phases in the uncertainty region where a receiver lock can be considered successful

acquisition. We call this set of phases the hit set. We design and compare the performance

of two schemes for the acquisition of time-hopping UWB signals which attempt to exploit

the energy in the multipath to improve the acquisition performance. We prove a general

result characterizing the asymptotic performance of threshold-based acquisition schemes

in multipath fading channels. We use this result to characterize the performance limits of

the aforementioned UWB acquisition schemes. We then consider the problem of finding

efficient search strategies when there are multiple elements in the hit set. We use the

insights gained in the design of UWB acquisition schemes in the development of efficient

schemes for the closely related problem of time-of-arrival estimation.

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CHAPTER 1INTRODUCTION

The Federal Communications Commision (FCC) defines ultra-wideband (UWB)

technology as any wireless transmission scheme that operates with a fractional bandwidth

of at least 20%, or occupies more than 500 MHz of absolute bandwidth. Ultra-wideband

signaling [1–4] is under evaluation as a possible modulation scheme for wireless personal

area network (PAN) protocols. The features of UWB radio which make it an attractive

choice are its multiple access capabilities [1, 5], lack of significant multipath fading [6–8],

ability to support high data rates [9] and low transmitter power resulting in longer battery

life for portable devices.

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 bandwidth.

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 [10, 11]. 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 responsible 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 [10].

Timing acquisition is a particularly acute problem faced by UWB systems due to

the following reasons. Short pulses and low duty cycle signaling [1] employed in UWB

systems place stringent timing requirements at the receiver for demodulation [12, 13].

The wide bandwidth results in a fine resolution of the timing uncertainty region, thereby

imposing a large search space for the acquisition system. Typical UWB systems also

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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 packet-based network, each packet has a

dedicated portion known as the acquisition preamble within which the receiver is expected

to achieve synchronization. However for the high data-rate applications envisaged for

UWB signaling, long acquisition preambles would significantly reduce the throughput of

the network . Hence there is a need to develop more efficient acquisition schemes by taking

into account the UWB signal and channel characteristics.

Closely related to the problem of timing acquisition in UWB systems is the problem

of localization using UWB signals. The absence of a carrier in UWB signals obviates the

use of energy-based localization methods. Localization based on round-trip time-of-flight

measurements is an ideal candidate for UWB localization systems due to its simplicity and

the high time resolution of the UWB signals. However, the dense multipath in the UWB

channel is a hindrance to the accuracy of such systems.

This chapter is organized as follows. In the next section, we briefly review the main

features of acquisition methods used in traditional spread spectrum systems to put the

problem of UWB signal acquisition in perspective. In Section 1.2, we briefly describe

1 Traditionally, in direct-sequence spread spectrum systems the chip-level timing of thePN sequence is referred to as the phase of the spreading signal. In this document, we usephase and timing interchangeably.

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the existing literature on UWB acquisition. In Section 1.3, we review the existing work

on time-of-arrival estimation of UWB signals. An outline of this dissertation is given in

Section 1.4.

1.1 Brief Review of Spread Spectrum Acquisition Systems

Ultra-wideband communication falls in the category of spread spectrum communication

systems. There has been extensive research on spreading code acquisition and tracking

for spread spectrum systems with direct-sequence, frequency-hopping and hybrid

modulation formats [10, 11, 14]. We will bring out the main issues by considering the

timing acquisition of direct-sequence spread spectrum systems.

In a direct-sequence 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 down the synchronization error to

within the pull-in 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

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probability in the minimum possible time is the parallel acquisition system [10] shown

in Fig. 1-1. This acquisition system checks all the candidate phases in the uncertainty

region simultaneously. In the ith arm, the decision statistic corresponding to the candidate

phase ti is generated by correlating the received signal with a delayed version of the

locally generated spreading waveform s(t) 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

maximum-likelihood 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. 1-2. 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. The

decision statistic corresponding to the candidate phase ti is generated by correlating the

received signal with a delayed version of the locally generated spreading waveform s(t).

If the threshold is not exceeded, the search updates the value of the candidate phase and

the process continues. Hybrid methods such as the MAX/TC criterion [15] 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 which is used to confirm the coarse

estimate of the true phase before the control is passed to the tracking loop.

In traditional spread spectrum acquisition schemes, the signal-to-noise 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

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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 [10, 11]. 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 [10, 11, 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.

1.2 Previous Work on UWB Acquisition

In this section, we describe the existing approaches to UWB acquisition which take a

detection-theoretic approach to the problem. 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

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large search space prevents 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 either an AWGN or a flat fading channel model for the UWB channel

and neglects the effect of multipath in the development and evaluation of the proposed

acquisition schemes.

In Blazquez et al. [23], the traditional coarse acquisition scheme where the search

space is searched in increments of a chip fraction is analyzed for the acquisition of

time-hopped UWB signals in AWGN noise. Fig. 1-3 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+1 = ti + εTp where ε < 1 and Tp is the pulse width. This process continues until the

threshold is exceeded.

In Soderi et al. [24], the output of a matched filter, whose impulse response is a

time-reversed replica of the spreading code, is integrated over successive time intervals of

size mTc, where m is an integer greater than one but not exceeding the number of taps in

the channel response and Tc is the chip duration, in an attempt to combine the energy in

the multipath. The integrator output is then sampled at multiples of mTc and compared

to a threshold as illustrated in Fig. 1-4. 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.

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In Ma et al. [25], the non-consecutive search proposed by Shin et al. [21] and a

simpler version of the MAX/TC scheme [15] called the global MAX/TC are applied to

the acquisition of UWB signals in the presence of multipath fading and multiple access

interference (MAI). In the non-consecutive 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 of the channel. In the global MAX/TC, a parallel bank of correlators is used

to evaluate all the non-consecutive 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. [26], a hybrid acquisition scheme called the reduced complexity

sequential probability ratio test (RC-SPRT) 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 [27]. 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

RC-SPRT, 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 RC-SPRT 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.

1.2.1 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 is more than one

acquisition phase 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 “look-and-jump-by-K-bins” search and bit reversal search

are analyzed in the noiseless scenario with mean stopping time as the performance

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metric in Homier et al. [28]. Suppose that the timing uncertainty region is divided in

to bins indexed by 0, 1, . . . , Ns − 1. In look-and-jump-by-K-bins search, starting in

bin 0, the search continues on to bin K, then to 2K and so on. So for Ns = 9 and

K = 3, the look-and-jump-by-K-bins search searches the bins in the following order

0, 3, 6, 1, 4, 7, 2, 5, 8. In bit reversal search, the order in which the bins are searched

is obtained by reversing the bits in the binary representation of the linear search

variable. For instance, when Ns = 9 the linear search has the binary representation

000, 001, 010, 011, . . . , 111 and the bit reversal search is obtained by ‘bit reversal’ by

000, 100, 010, 110, . . . , 111. It then corresponds to the search order 0, 4, 2, 6, 1, 5, 3, 7.A generalized flow graph method is then used to compute the mean acquisition time for

different serial and hybrid search strategies [29, 30]. For the case when the acquisition

phases are K consecutive phases in the uncertainty region, it has been claimed that the

look-and-jump-by-K-bins search is the optimal serial search permutation when K is known

and the bit reversal is the optimal search permutation when K is unknown.

1.2.2 Search Space Reduction Techniques

Some acquisition schemes attempt to solve the large search space problem by

employing a two-stage acquisition strategy [31–35]. 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. [31], such a two-stage scheme is proposed for the acquisition of time-hopped UWB

signals in AWGN noise and multiple-access intereference (MAI). The search space is

divided into 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. This is illustrated in Fig. 1-5 in the absence of

18

noise and MAI. A scheme based on the same principle has been developed independently

by Gezici et al. [34]. 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. [35], an acquisition scheme for UWB signals with time-hopping

(TH) spreading called n-scaled search is presented, where the search space is divided into

groups of M = Nf/2n where Nf is the frame size and 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 chip-rate

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.

A two-stage scheme which achieves search space reduction by employing a hybrid

DS-TH spreading signal format is described by Aedudodla et al. [32, 36]. In the first

stage, the DS spreading is removed by squaring the received signal and the timing of the

TH spreading code, which has a relatively small length, is acquired. Once this is done,

the acquisition of the DS spreading code is performed by searching the search space in

increments equal to the length of the TH code. Fig. 1-6 shows a conceptual block diagram

of this system.

Another two-stage acquisition scheme for UWB signals with DS spreading which

employs a special signal format is presented by Furukawa et al. [33]. The signal transmitted

during the acquisition process is a sum of two signals, a periodic pulse train and a pulse

train with DS spreading, as shown in Fig. 1-7. In the first stage, the timing of the periodic

pulse train is acquired by correlating the received signal with a replica of the periodic

pulse train. This is an easy task considering that the uncertainty region is just twice

the pulse repetition time Tf . Once this is done, the chip boundaries of the DS spreading

sequence are known and the second stage needs to only search in increments of 2Tf to

acquire the timing of the DS spreading sequence.

19

1.3 Previous Work on UWB Time-of-Arrival Estimation

There has been a recent explosion in the existing literature on UWB time-of-arrival

(TOA) estimation. However, a significant number of these papers describe experimental

results obtained from hardware testbeds employing UWB signals to perform TOA

estimation. The actual algorithms used to estimate the TOA of the UWB signals have also

been developed independently in papers which employ mathematical models and computer

simulations to make their case. We will now briefly present the key contributions in the

latter portion of the existing literature.

One of the earliest contributions to TOA estimation was made by Lee and Scholtz

[37], who used a generalized maximum likelihood approach to estimate the multitude of

nuisance parameters in addition to the TOA to get a better estimate.

In Gezici et al. [38], a survey of the UWB localization methods based on signal

strength (SS) measurements, angle-of-arrival (AOA) measurements and TOA measurements

is given. The problems arising out of the dense multipath in UWB channels are discussed

and the Cramer Rao lower bounds (CRLBs) for the TOA estimation problem are derived.

TOA estimation schemes based on correlation of the received signal with a noisy template

(which itself is a part of the received signal) are presented.

In Cardinali et al. [39], the CRLBs for the two high data rate signal formats proposed

by the IEEE 802.15.3a Task Group, i.e., the direct sequence UWB (DS-UWB) and the

multiband orthogonal frequency-division multiplexing (MB-OFDM), are calculated. By

optimizing over the set of synchronization sequences, it is shown that the MB-OFDM

format can provide potentially better performance. Also, the CRLB for the low data rate

signal format proposed by the IEEE 802.15.4a Task Group is analyzed as a function of the

pulse shape.

In Qi et al. [40], the CRLBs in the presence and absence of prior knowledge of the

non-line-of-sight (NLOS) delay statistics are calculated. The maximum likelihood and

20

maximum a posteriori detectors are presented and modified to account for the fact that

strong multipath components can help achieve better accuracy for TOA estimation.

A two-step TOA estimation scheme is presented by Gezici et al. [41], where the first

step uses an energy detector to coarsely estimate the position of the multipath profile

and the second step uses a hypothesis testing approach to locate the LOS path by casting

as a change detection problem. The unknown channel parameters are estimated using

maximum likelihood and method of moments estimators and these estimates are used in

the calculation of the likelihood ratios.

In Falsi et al. [42], several suboptimal algorithms based on detecting the peaks in the

matched filter output are analyzed. The first algorithm calculates the position of the N

matched filter outputs of largest magnitude and picks the earliest arriving position as the

TOA estimate. In the second algorithm, the largest matched filter output is estimated

and its contribution is subtracted from the received signal. The remaining signal is passed

through the matched filter and the largest output is calculated and its contribution

subtracted. This process is repeated N times and the earliest arriving position of the

N largest matched filter outputs is taken as the TOA estimate. The third algorithm is

similar to the second in the iterative process of estimation and subtraction, with the

exception that the ith step involves the estimation of the i largest matched filter outputs.

Energy detection-based approaches to TOA estimation are considered in Guvenc et

al. [43–45]. In the first paper by Guvenc et al. [43], the received signal is passed through

an energy detector and the samples of the energy detector output are compared to a

threshold. The threshold is selected to be between the maximum and minimum values

of the outputs and the first threshold crossing gives the location of the LOS path. In

the second paper by Guvenc et al. [44], for the same system model the threshold is

chosen using the kurtosis of the energy detector output samples. In the third paper [45],

the decision statistics and performance of stored-reference, transmitted-reference and

energy-detection based schemes are analyzed under the assumption of an AWGN channel.

21

For realistic multipath channels, a maximum likelihood approach is taken. In another

paper by the same authors [46], the received signal is either passed through an energy

detector or processed by correlating it with a stored reference signal or a transmitted

reference signal. In each case, the outputs are then used to perform TOA estimation via a

hypothesis testing approach.

1.4 Dissertation Outline

This dissertation is organized as follows. In Chapter 2, we describe the UWB system

model which will be used in the design and evaluation of the acquisition schemes proposed

in this document. We evaluate and compare two schemes for the acquisition of TH UWB

signals in Chapter 3. We prove a general result characterizing the asymptotic performance

of threshold-based acquisition schemes in multipath fading channels in Chapter 4. This

result is used to evaluate the asymptotic performance of the two schemes proposed in

Chapter 3. The problem of finding efficient search strategies in the set of all search

strategies which are permutations of the search space is addressed in Chapter 5. We

develop and evaluate schemes for time-of-arrival estimation of UWB signals in Chapter 6.

22

To code

Bandpass

filter

Energy

detector

s(t−t )

Bandpass

filter

Energy

detector

s(t−t )

Bandpass

filter

Energy

detector

s(t−t )

Received

signal

1

2

n

to largest

energy

corresponding iChoose t

Verification

stage tracking loop

Figure 1-1: Block diagram of a parallel acquisition system for direct-sequence spreadspectrum systems.

tracking loop

Received

signal

Bandpass

filter detector

Energy Is threshold

exceeded?

Yes

Search

control

Spreading waveform

generator

No

ti

is(t−t )

Verification

stage

Success

Failure

To code

Figure 1-2: Block diagram of a serial acquisition system for direct-sequence spreadspectrum systems which evaluates the candidate phases t1, t2, . . . , tn serially.

of true phase

i

tiReceived

signal

Search

control

is(t−t )

Is threshold

exceeded?

Yes

No

Integrator

Template signal

generator

Declare to be

coarse estimate

t

Figure 1-3: Block diagram of the acquisition scheme proposed by Blazquez et al.

signal

c

Tc

m

Integrator withdwell time m

Thresholdcomparison tracking loop

To codePN matched

filter

Received

T

Figure 1-4: Block diagram of the acquisition scheme proposed by Soderi et al.

23

Second stage template signal

Received signal

First stage template signal

Figure 1-5: Template signals used in the two-stage acquisition scheme proposed byBahramgiri et al.

HitSquaringoperation

TH spreadingcode acquisition

Receivedsignal

DS spreadingcode acquisition

Hit

Figure 1-6: Block diagram of the two-stage acquisition scheme proposed by Aedudodla etal.

Transmitted signal

T f

Periodic pulse train

Pulse train with DS spreading

Figure 1-7: Transmitted signal along with its component signals used by Furukawa et al.

24

CHAPTER 2PROBLEM DEFINITION

2.1 Introduction

The timing information of the received signal is essential for the performance of

a receiver in a wireless communication system. In a multipath channel, the energy

corresponding to the true signal phase is spread over several MPCs. The main difference

between the acquisition problems in a multipath channel and a channel without multipath

is that there are more than one hypothesized phases which can be considered a good

estimate of the true signal phase. In a dense multipath environment, which is the typical

scenario under which UWB systems operate, the receiver may lock onto a non-line-of-sight

(NLOS) path and still be able to perform adequately as long as it is able to collect enough

energy. From the viewpoint of post-acquisition 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.

In this chapter, we propose a definition of the set of hypothesized phases which

correspond to a good estimate of the true signal phase by considering the demodulation

performance subsequent to acquisition. We call this set of hypothesized phases the hit set.

The hit set concept enables us to give a precise definition of the acquisition problem for

UWB systems. We note that such a definition is applicable for any multipath channel.

In the next section, we describe the UWB system model. In Section 2.3, we calculate

the hit set for this system, followed by the definition of the UWB acquisition problem in

Section 2.4.

2.2 System Model

2.2.1 Channel Model

We assume that the propagation channel is modeled by the UWB indoor channel

model described in Cassioli et al. [47]. This model gives a statistical distribution for the

path gains based on a UWB propagation experiment but does not address the issue of

characterization of the received waveform shape. Due to the frequency sensitivity of the

25

UWB channel, the pulse shapes received at different excess delays are path-dependent

[48]. To enable tractable analysis, we assume that the pulse shapes associated with all the

propagation paths are identical. The channel is then a stochastic tapped delay line model

expressed as the impulse response

h(t) =

Ntap−1∑

k=0

pkhkf(t− kTc), (2–1)

where Ntap is the number of taps in the channel response, Tc = 2 ns is the tap spacing,

hk is the path gain at excess delay kTc, pk is equally likely to be ±1 to account for signal

inversion due to reflections [49] and f(t) models the combined effect of the transmitting

antenna and the propagation channel on the transmitted pulse. The path gains are

independent but not identically distributed with Nakagami-m distributions. The average

energy gains Ωk = E[h2k] of the path gains normalized to the total energy received at one

meter distance are given by

Ωk =

Etot

1+rF (ε), for k = 0

Etot

1+rF (ε)re−((k−1)Tc/ε), for k = 1, 2, . . . , Ntap − 1,

(2–2)

where Etot is the total average energy in all the paths normalized to the total energy

received at one meter distance, r is the ratio of the average energy of the second MPC and

the average energy of the direct path, ε is the decay constant of the power delay profile

and F (ε) = 1−exp[−(Ntap−1)kTc/ε]

1−exp(−kTc/ε). According to Cassioli et al. [47], Etot, r and ε are all

modeled by lognormal distributions. The Nakagami fading figures mk are distributed

according to truncated Gaussian distributions whose mean and variance vary linearly with

excess delay. These long-term statistics are treated as constants over the duration of the

acquisition process.

2.2.2 Transmitted and Received Signals

The transmitted signal is given by

x(t) =√

P

∞∑

l=−∞alψ(t− lTf − clTc), (2–3)

26

where ψ(t) is the UWB monocycle waveform, P is the transmitted power, Tf = NfTc is

the pulse repetition time, al is the pseudorandom direct-sequence (DS) code with period

Nds taking values ±1, cl is the pseudorandom time-hopping (TH) sequence with period

Nth taking integer values between 0 and Nh − 1, and Tc is the step size of the additional

time shift provided by the TH sequence. The pulse repetition time Tf is chosen to be not

less than (Nh + Ntap)Tc to avoid overlap between the multipath responses corresponding

to distinct transmitted pulses. Note that the transmitted signal is periodic with period

NperTf where Nper is the lowest common multiple of Nth and Nds.

If u(t) = h(t) ∗ x(t), the received signal is given by

r(t) = u(t) + n(t)

=√

E1

∞∑

l=−∞alw(t− lTf − clTc − τ) + n(t), (2–4)

where

w(t) =

Ntap−1∑

k=0

pkhkψr(t− kTc). (2–5)

Here E1 is the total received energy at a distance of one meter from the transmitter,

ψr(t) = f(t) ∗ ψ(t) is the received UWB pulse of duration Tw < Tc normalized to have unit

energy, τ is the propagation delay, and n(t) is an additive white gaussian noise (AWGN)

process with zero mean and power spectral density N0

2.

2.3 Hit Set Definition

As mentioned earlier, we will use demodulation performance subsequent to acquisition

to define the hit set. We need to describe the receiver structure in order to quantify

demodulation performance. The presence of a high degree of path diversity in the UWB

channel motivates the use of a Rake receiver to improve demodulation performance. The

three main Rake receiver structures considered for UWB signal demodulation are the all

Rake (ARake), the selective Rake (SRake) and the partial Rake (PRake) receivers [8, 50].

The large number of resolvable multipaths in the UWB channel obviates the use of the

ARake receiver due to the complexity involved in its implementation. We assume that

27

the receiver uses a partial Rake (PRake) receiver to perform demodulation. Our choice

is guided by the fact that the PRake receiver has lower complexity and still achieves

comparable bit error performance relative to the SRake receiver [50].

A typical paradigm for transceiver design is the achievement of a certain nominal

uncoded bit error rate (BER) λn. Then all those hypothesized phases such that a receiver

locked to them achieves an uncoded BER of λn can be considered a good estimate of

the true signal phase. We define the hit set to be the set of such hypothesized phases.

To simplify the analysis, we assume that the true phase τ is an integer multiple of Tc.

By the periodicity of the transmitted signal, we have 0 ≤ τ ≤ (NperNf − 1)Tc. The

hypothesized phase τ is also an integer multiple of Tc with the same range as τ . Then

∆τ = τ − τ = αTf + βTc where α and β are integers such that −Nper + 1 ≤ α ≤ Nper − 1

and 0 ≤ β ≤ Nf − 1. For a given true phase τ , let PE(∆τ) denote the BER performance

of the PRake receiver when it locks to the hypothesized phase τ . Let Υm be the minimum

SNR at which the PRake receiver achieves a BER of λn when it locks to the LOS path,

that is, PE(0) ≤ λn when the SNR is Υn and PE(0) > λn for all SNRs less than Υn. Then

for an SNR Υ ≥ Υn and true phase τ , the hit set is given by

Sh = τ : PE(∆τ) ≤ λn. (2–6)

To completely characterize the hit set, we need to calculate the error performance

of a partial Rake (PRake) receiver which is locked to a particular hypothesized phase τ .

We assume that the modulation format is BPSK with Nb consecutive UWB monocycles

modulated by one bit. The signal received during the demodulation stage is given by

rb(t) =√

E1

∞∑

l=−∞bb l

Nbcalw(t− lTf − clTc − τ) + n(t), (2–7)

where bi ∈ −1, 1 for each i, bxc is the largest integer not greater than x,

w(t) =

Ntap−1∑

k=0

pkhkψr(t− kTc), (2–8)

28

and n(t) is a zero-mean AWGN process with power spectral density N0

2. The PRake

receiver is assumed to have Np fingers where Np ≤ Ntap. When the receiver estimates τ

to be the true phase in the acquisition stage, the PRake receiver estimates and combines

the paths arriving at delays τ + kTc (k = 0, 1, . . . , Np − 1) to obtain the decision statistic.

Since τ = τ + (αNf + β)Tc, the PRake receiver is estimating the values of pαNf+β+ihαNf+β+i

for i = 0, 1, . . . , Np − 1, where we define pk = hk = 0 for k /∈ 0, 1, . . . , Ntap − 1. To

make the analysis tractable, we assume that PRake is able to estimate these path gains

and inversions perfectly. The decision statistic for the mth bit, Zm, can be obtained by

correlating the received signal with the following template signal,

sb(t) =

(m+1)Nb−1∑

l=mNb

alv(t− lTf − clTc − τ), (2–9)

where

v(t) =

Np−1∑i=0

pαNf+β+ihαNf+β+iψr(t− iTc). (2–10)

Then we have

Zm =1

Nb

∫ τ+[(m+1)Nb−1]Tf

τ+mNbTf

rb(t)sb(t)dt = bm

√E1

Np−1∑i=0

p2αNf+β+ih

2αNf+β+i + nb

= bm

√E1

Np−1∑i=0

h2αNf+β+i + nb (2–11)

where nb is a zero-mean Gaussian random variable with variance σ2b = N0

2Nb

∑Np−1i=0 h2

αNf+β+i.

Then from Simon et al. [51, pp.268–269], the average probability of error is given by

PE(∆τ) = Eh

Q

√2E1Nb

∑Np−1i=0 h2

αNf+β+i

N0

=1

π

∫ π2

0

αNf+β+Np−1∏

i=αNf+β

Mi

(− 2E1Nb

N0 sin2 θ

)dθ, (2–12)

29

where Mi(·), the moment generating function of h2i , is given by

Mi(s) =

(1− Ωis

mi

)−mi

for i ∈ 0, 1, . . . , Ntap − 11 otherwise.

(2–13)

Fig. 2-1 shows the hit set size as a function of the average energy received per pulse to

noise ratio E1Etot

N0for the nominal uncoded BER requirement λn = 10−3, the number of

monocycles modulated by one bit Nb = 8, the length of the channel response Ntap = 100,

Nf = 116 and Np = 5, 10. We assume that Etot = −20.4 dB which is its mean value

when the transmitter-receiver separation is 10 m [47]. We choose the power ratio r = −4

dB, decay constant ε = 16.1 dB and fading figures mk = 3.5 − kTc

73, 0 ≤ k ≤ Ntap − 1,

which are their mean values given in Cassioli et al. [47]. This plot confirms our claim in

the beginning of this chapter about the existence of multiple phases where a receiver lock

can guarantee adequate demodulation performance.

2.4 The UWB Acquisition Problem

For a particular value of τ , the hit set Sh is obtained from (2–6) using (2–12). The

acquisition process can then be formulated as a composite binary hypothesis testing

problem [52] with the following hypotheses:

H0 : τ /∈ Sh

H1 : τ ∈ Sh. (2–14)

Our goal is to design efficient acquisition schemes which take into account the UWB signal

and channel characteristics, and characterize their performance.

30

0 2 4 6 8 10 12 14 16 18 200

10

20

30

40

50

60

70

80

90

100

Average energy received per pulse to noise ratio (dB)

Num

ber

of h

it se

t pha

ses

Np = 5

Np = 10

Figure 2-1: The hit set size as a function of the average energy received per pulse to noiseratio for Np = 5 and 10

31

CHAPTER 3ACQUISITION OF TIME-HOPPING UWB SIGNALS

3.1 Introduction

The UWB channel is a dense multipath channel without significant fading [7, 53]. In

a dense multipath environment, there will be a considerable amount of energy available

in the multipath components (MPCs). It seems reasonable to expect that an acquisition

scheme which utilizes the energy in the MPCs would perform better than one which

does not. In this chapter, we consider the acquisition of UWB signals having only TH

spreading. The system model is same as that described in Chapter 2, except that DS code

is absent in the transmitted signal1 .

Considering that we have no information regarding the channel state, there are

essentially two ways in which we can attempt to utilize this energy in order to develop a

more efficient acquisition scheme. In the first approach, the received signal is first squared

to eliminate the channel inversion and then equal gain combining (EGC) is performed to

exploit the rich path diversity present in UWB channels. In the second approach, EGC is

performed first and the integrator output is then squared to generate the decision statistic.

In the sequel, we will refer to the former as square-and-integrate (SAI) and to the latter as

integrate-and-square (IAS).

It is not exactly clear which approach is more efficient. Also, the choice of the length

of the EGC window is not apparent. For instance, in SAI, a small window will not collect

enough energy and thus will result in a low probability of detecting the correct signal

phase. A large window may collect a considerable amount of energy even when the true

phase does not match the hypothesized phase, resulting in a high probability of false

1 The expressions for the transmitted and received signals can be obtained by settingal = 1.

32

alarm. In this chapter, we derive and compare the performance of both SAI and IAS as a

function of the EGC window length.

This chapter is organized as follows. We derive expressions for average probabilities

of detection and false alarm for SAI and IAS in Sections 3.2 and 3.3, respectively. In

Section 3.4, we give a design criterion for choosing the decision threshold and derive the

mean detection time for a serial search strategy as a function of the average probabilities

of detection and false alarm. In Section 3.5, the mean detection time and the probability

of a miss are used as performance metrics to compare the two approaches. Section 3.6 has

some concluding remarks.

3.2 Analysis of SAI

3.2.1 Derivation of the Decision Statistic

The acquisition system correlates the squared received waveform with a locally

generated replica and compares the correlator output to a threshold to determine whether

the hypothesized phase of the replica is correct (as shown in Fig. 3-1). If the threshold is

exceeded, the hypothesized phase becomes the estimate of the true phase. We assume that

the normalized received monocycle waveform ψr(t) and the TH sequence cl are known to

the receiver. The received signal is the same as in (2–4) with the exception of the DS code

and is given by

r(t) =√

E1

∞∑

l=−∞w(t− lTf − clTc − τ) + n(t). (3–1)

We propose to use an equal gain combiner of window size G. The receiver template signal

wr(t) is given by

wr(t) =G−1∑

k=0

ψ2r (t− kTc). (3–2)

The reference TH signal can be obtained by combining the receiver template signal wr(t)

and the known time hopping sequence as

s(t) =

MNth−1∑

l=0

wr(t− lTf − clTc − τ), (3–3)

33

where M specifies the number of TH waveform periods2 in the dwell time and τ is the

hypothesized phase. The correlator output is given by

y =1

MNth

∫ τ+MNthTf

τ

r2(t)s(t)dt

=1

MNth

∫ τ+MNthTf

τ

u2(t)s(t)dt +2

MNth

∫ τ+MNthTf

τ

u(t)s(t)n(t)dt

+1

MNth

∫ τ+MNthTf

τ

n2(t)s(t)dt. (3–4)

The first term in (3–4) can be simplified to

s(∆τ ;h) = E1Rψ2r(0)

Ntap−1∑

k=0

rk(∆τ)h2k, (3–5)

where h is an Ntap × 1 vector containing the channel gains, Rψnr(ν) =

∫∞−∞ ψn

r (t)ψnr (t + ν)dt

and rk(∆τ), the average number of times the energy in the kth MPC is collected by one

period of the reference TH signal, is given by

rk(∆τ) =1

Nth

Nth−1∑

l=0

1∑i=0

G−1∑j=0

χ(cl + j + β, cl+i+α + k + iNf), (3–6)

where χ(a, b) = 1 if a = b, and 0 otherwise. The value of rk(∆τ) depends on the particular

pseudorandom TH sequence chosen. To simplify the analysis we assume that the TH

sequence is random and that Nth is large. Under these assumptions, the mean value of

rk(∆τ) is a reasonable approximation to the actual value. The mean value of rk(∆τ) is

calculated in Appendix A by averaging over the set of random TH sequences.

2 In the absence of the DS spreading code, the period of the transmitted signal is Nper =Nth.

34

Conditioned on the random vector h, the second term in (3–4) is a zero-mean

Gaussian random variable with variance

σ2y(∆τ ;h) =

2N0

M2N2th

∫ τ+MNthTf

τ

u2(t)s2(t)dt (3–7)

=2E1Rψ3

r(0)N0

MNth

Ntap−1∑

k=0

rk(∆τ)h2k, (3–8)

where the second equality is obtained by exploiting the similarity between the integral in

(3–7) and the first term of (3–4). We have also used the fact that

s2(t) =

MNth−1∑

l=0

G−1∑

k=0

ψ4r (t− kTc − lTf − clTc − τ), (3–9)

which differs from s(t) only in the exponent of the received pulse waveform ψr(t).

We approximate the third term in (3–4) by a Gaussian random variable with mean µy

and variance ν2y which are given by

µy =1

MNth

E

[∫ τ+MNthTf

τ

n2(t)s(t)dt

]=

N0

2MNth

∫ τ+MNthTf

τ

s(t)dt

=GRψr(0)N0

2=

GN0

2(3–10)

and

ν2y =

1

M2N2th

E

[(∫ τ+MNthTf

τ

n2(t)s(t)dt

)2]− µ2

y

=1

M2N2th

E

[∫ τ+MNthTf

τ

∫ τ+MNthTf

τ

n2(t)n2(u)s(t)s(u)dtdu

]− µ2

y

=N2

0

2M2N2th

∫ τ+MNthTf

τ

s2(t)dt +N2

0

4M2N2th

(∫ τ+MNthTf

τ

s(t)dt

)2

− µ2y

=N2

0 Rψ4r(0)

2MNth

, (3–11)

respectively. Note that the expectation in the derivation of µy and ν2y is only with respect

to the noise process n(t). This approximation is accurate provided that the product of the

integration time MNthTf and the bandwidth of the system B is large [10, pp. 240–250],

which is the case for the scenarios we consider.

35

Then the correlator output can be written as

y = s(∆τ ;h) + ny, (3–12)

where, conditioned on h, ny is a Gaussian random variable with mean µy and variance

σ2y(∆τ ;h) + ν2

y .

3.2.2 Average Probabilities of Detection and False Alarm

For a particular channel realization h and fixed ∆τ , the decision statistic y in (3–12)

has a Gaussian distribution with probability density function

pY(y) =1√

2π(σ2y(∆τ ;h) + ν2

y)exp

[−(y − s(∆τ ;h)− µy)2

2(σ2y(∆τ ;h) + ν2

y)

]. (3–13)

The probabilities of false alarm and detection conditioned on the particular channel

realization and given the decision threshold γ are given as

PFA(γ, ∆τ |h) = Pr[y > γ|τ /∈ Sh] = Q

γ − s(∆τ ;h)− µy√

σ2y(∆τ ;h) + ν2

y

, τ /∈ Sh. (3–14)

PD(γ, ∆τ |h) = Pr[y > γ|τ ∈ Sh] = Q

γ − s(∆τ ;h)− µy√

σ2y(∆τ ;h) + ν2

y

, τ ∈ Sh. (3–15)

From (3–5) and (3–8), one sees that the conditional probabilities of false alarm and

detection depend on h only through s1(∆τ ;h) =∑Ntap−1

k=0 rk(∆τ)h2k, which is a scaled

version of s(∆τ ;h). Using (3–5) and (3–8) we define

I(s1(∆τ ;h)) = Q

γ − E1Rψ2

r(0)s1(∆τ ;h)− µy√

2E1Rψ3r(0)N0

MNths1(∆τ ;h) + ν2

y

. (3–16)

Since the path gains hk (k = 0, 1, . . . , Ntap − 1) are independent, the characteristic function

of s1(∆τ ;h) is given by

Φs(ω; ∆τ) =

Ntap−1∏

k=0

Mk(jrk(∆τ)ω), (3–17)

36

where Mk(·) is defined in (2–13). The probability density function (pdf) of s1(∆τ ;h) is

given by fs(x; ∆τ) = 12π

∫∞−∞ Φs(ω; ∆τ)e−jωxdω. Then for τ /∈ Sh, the probability of false

alarm averaged over the channel realizations is given by

EH[PFA(γ, ∆τ |h)] = EH[I(s1(∆τ ;h))] =

∫ ∞

0

I(t)fs(t; ∆τ)dt. (3–18)

Similarly, for τ ∈ Sh, the average probability of detection is given by

EH[PD(γ, ∆τ |h)] =

∫ ∞

0

I(t)fs(t; ∆τ)dt. (3–19)

The structure of Mk(·) prevents from evaluating fs(·) in closed form. So we resort to

numerical integration to calculate fs(·) and the average probabilities of false alarm and

detection.

3.3 Analysis of IAS

In this section, we analyze an acquisition system which takes the IAS approach. The

derivation of the decision statistic in this case is very similar to the decision statistic

derivation in the previous section. All the relevant assumptions made in the previous

section, to enable tractable analysis, still hold unless stated otherwise. To avoid repetition,

we only define those quantities which have not already been defined in the previous

section.

3.3.1 Derivation of the Decision Statistic

In this approach, the acquisition system correlates the received waveform with a

locally generated template signal and squares the integrator output to generate the

decision statistic (as shown in Fig. 3-2). The receiver template signal vr(t) is given by

vr(t) =G−1∑

k=0

ψr(t− kTc). (3–20)

The reference TH signal is given by

q(t) =

MNth−1∑

l=0

vr(t− lTf − clTc − τ). (3–21)

37

The decision statistic is given by

z =

[1

MNth

∫ τ+MNthTf

τ

r(t)q(t)dt

]2

=

√E1

Ntap−1∑

k=0

rk(∆τ)pkhk

︸ ︷︷ ︸V (∆τ ;h)

+nz

2

(3–22)

where nz is a zero-mean Gaussian random variable with variance σ2z = GN0

2MNthand rk(∆τ) is

given in (3–6).

3.3.2 Average Probabilities of Detection and False Alarm

For a particular channel realization h and fixed ∆τ , the decision statistic z in (3–22)

has a non-central chi-square distribution with probability density function

pZ(z) =1√

2πzσz

e−(z+V 2(∆τ ;h))/2σ2z cosh

(√zV (∆τ ;h)

σ2z

). (3–23)

The probabilities of false alarm and detection conditioned on the particular channel

realization and given the decision threshold γ ≥ 0 are given by

PFA(γ, ∆τ |h) = Pr[z > γ|τ /∈ Sh]

= Q

(√γ − V (∆τ ;h)

σz

)+ Q

(√γ + V (∆τ ;h)

σz

), τ /∈ Sh, (3–24)

PD(γ, ∆τ |h) = Pr[z > γ|τ ∈ Sh]

= Q

(√γ − V (∆τ ;h)

σz

)+ Q

(√γ + V (∆τ ;h)

σz

), τ ∈ Sh. (3–25)

Before we derive the average probabilities of detection and false alarm, it is instructive to

look at the characteristic function ΦV(ω; ∆τ) of V (∆τ ;h). Since the polarities pk and path

gains hk are independent, we have

ΦV(ω; ∆τ) =

Ntap−1∏

k=0

[φk(√

E1rk(∆τ)ω) + φk(−√

E1rk(∆τ)ω)

2

], (3–26)

where φk(·) is the characteristic function of the Nakagami-m distributed hk [51]. Since the

hk’s are real-valued, the φk(·)’s are conjugate symmetric functions and hence ΦV(·) is a

real-valued function.

38

The Gil-Pelaez lemma [54] gives an alternative form of the Q function as

Q(x) =1

2− 1

π

∫ ∞

0

1

te−t2/2 sin(tx)dt. (3–27)

Substituting this form of the Q function in (3–24), the probability of false alarm averaged

over the channel realizations, for τ /∈ Sh, is given by

EH[PFA(γ, ∆τ |h)] (3–28)

= 1− 1

π

∫ ∞

0

1

te−t2/2EH

[sin

t(√

γ − V (∆τ ;h))

σz

+ sint(√

γ + V (∆τ ;h))

σz

]dt

= 1− 2

π

∫ ∞

0

1

te−t2/2 sin

(√γt

σz

)EH

[sin

V (∆τ ;h)t

σz

]dt

= 1− 2

π

∫ ∞

0

1

te−t2/2 sin

(√γt

σz

)Im

EH

[ej

V (∆τ ;h)tσz

]dt

= 1− 2

π

∫ ∞

0

1

te−t2/2 sin

(√γt

σz

)ΦV

(t

σz

; ∆τ

)dt, (3–29)

where the last equality follows from our observation that ΦV(·) is real-valued. Similarly,

for τ ∈ Sh, the average probability of detection is given by

EH[PD(γ, ∆τ |h)] = 1− 2

π

∫ ∞

0

1

te−t2/2 sin

(√γt

σz

)ΦV

(t

σz

; ∆τ

)dt. (3–30)

3.4 Mean Detection Time Analysis of Serial Search

We define a hit or detection event as the event when the decision threshold is

exceeded for some τ ∈ Sh. We define a miss as the event when the decision threshold

is not exceeded for all τ ∈ Sh. Although the average probability of a miss is a potential

indicator of acquisition system performance, the mean acquisition time is usually the

metric used to evaluate the performance of acquisition systems [10]. The mean acquisition

time of an acquisition system depends on the particular search strategy used in evaluating

the phases in the search space. We consider a serial search strategy for the evaluation of

the acquisition schemes developed in this chapter. The design of better search strategies is

considered in Chapter 5.

39

The calculation of the mean acquisition time enumerates all false alarms which occur

before a detection event and associates a false alarm penalty time Tfa to each one of

them. The false alarm penalty time is equal to the dwell time of a verification stage in

the acquisition system which aids in the confirmation of detection events and rejection

of false alarm events with high probability. In other words, a good verification stage

simultaneously achieves low probabilities of miss and false alarm. The choice of the mean

acquisition time as the performance metric implicitly assumes that one can construct

such a verification stage. However, we will show in Chapter 4 that for threshold-based

UWB acquisition systems the average probabilities of false alarm and miss cannot be

made arbitrarily small even in the asymptotic scenario of the SNR approaching infinity.

Thus it is not apparent how one would build a good verification stage for such systems.

We propose to deal with this problem by choosing the decision threshold γ such that

the average probability that the acquisition process will end in a false alarm, PF(γ), is

small. The justification for this design is that a false alarm is a more serious problem in

the absence of a verification stage. Then if PF(γ) is small enough, we can use the mean

detection time as the performance metric. The mean detection time is defined as the

average time it takes for the acquisition process to end in a detection event in the absence

of false alarms.

In Appendix B, we calculate PF(γ) as a function of the average probabilities of

detection in the hit set and the average probabilities of false alarm at the phases not in

the hit set. We choose the decision threshold γd to be the minimum threshold such that

PF(γ) is not greater than a given positive constant δ ≤ 1,

γd = infγ|PF(γ) ≤ δ (3–31)

If the correlator outputs for different phase evaluations are assumed to be independent,

then the average probability of a hit for a particular τ is Eh[PD(γd, ∆τ |h)] and the average

40

probability of a miss is given by

PM =∏

τ∈Sh

(1− Eh[PD(γd, ∆τ |h)]). (3–32)

Owing to our definition, the hit set Sh consists of a contiguous set of H hypothesized

phases within the search space. The search space is the set Sp = nTc : n ∈ Z and 0 ≤ n ≤Ns − 1 where Ns = NthNf . Let the first phase of the hit set be at position A in the search

space Sp. Then the hit set consists of the phases (A − 1)Tc, ATc, . . . , (A + H − 2)Tc.The initial value of the hypothesized phase which corresponds to the starting point of the

search is chosen at random from the set Sp. Thus there is no loss of generality in assuming

that A = 1.

We need to consider all possible sequences of events leading to a hit or detection

event. The mean detection time can then be calculated as the average time taken for

each of the detection events. A detection event is defined by a particular position n of the

initial value of the hypothesized phase in Sp, the position i of the hypothesized phase in Sh

where we have a hit and a particular number of misses j of Sh. Let Tdet(n) be the mean

detection time conditioned on the event that the serial search starts at the nth position in

Sp i.e. the initial value of the hypothesized phase is (n − 1)Tc. Then the mean detection

time is

T det =1

Ns

Ns∑n=1

Tdet(n). (3–33)

First, suppose that the initial value of the hypothesized phase lies to the right of the

hit set, i.e., n ∈ H + 1, H + 2, . . . , Ns. The total detection time for a particular detection

event defined by (n, j, i) is then

T (n, j, i) = (Ns − n + 1)T + jNsT + iT

= (Ns − n + 1 + jNs + i)T (3–34)

where T is the dwell time for the evaluation of one hypothesized phase. Let Pd(i) denote

the average probability of detection of the ith phase of the hit set. The average probability

41

of the serial search missing the hit set is PM =∏H

i=1[1 − Pd(i)]. Then the probability of j

misses of Sh followed by a hit at the phase in Sh which is at the ith position of the hit set

is P jMPh(i) where Ph(i) = Pd(i)

∏i−1r=1[1 − Pd(r)]. The mean detection time conditioned on

the starting point of the serial search is given by

Tdet(n) =H∑

i=1

∞∑j=0

T (n, j, i)P jMPh(i)

=H∑

i=1

[Ns − n + i + 1

1− PM

+NsPM

(1− PM)2

]TPh(i)

= (Ns − n + 1)T +NsPMT

1− PM

+

∑Hi=1 iTPh(i)

1− PM

, (3–35)

where we have used the identities∑H

i=1 Ph(i) = 1− PM in obtaining the third equality.

Now suppose that the initial value of the hypothesized phase falls in the hit set,

i.e., n ∈ 1, 2, . . . , H. Let m be the total number of phases evaluated for a particular

detection event. We can partition the set of detection events into two sets, one containing

those events for which m ≤ H − n + 1 and the other containing those events for which

m > H − n + 1. The mean detection time for events in the first set is just mT and for

events in the second set it is Tdet(H + 1) + (H − n + 1)T where Tdet(H + 1) is obtained

from (3–35). Averaging over the total number of phases evaluated we get

Tdet(n) =H∑

i=n

(i− n + 1)TPd(i)i−1∏j=n

(1− Pd(j))

+(Tdet(H + 1) + (H − n + 1)T )H∏

j=n

(1− Pd(j)). (3–36)

From (3–35) and (3–36), we obtain the conditional mean detection times Tdet(n) for

all values of n ∈ 1, 2, . . . , Ns. The mean detection time is obtained by substituting these

values in (3–33).

3.5 Numerical Results

In this section, we compare the performance of SAI and IAS in terms of the average

probability of a miss PM and the mean detection time T det. We also investigate the effect

42

of increasing the EGC window length on these performance metrics for both schemes. We

choose the following values for the system parameters: the TH sequence period Nth = 256,

Nh = 16, the length of the channel response Ntap = 100, Nf = 116 and the number of

monocycles modulated by one bit Nb = 8. The nominal uncoded BER requirement is set

to be λn = 10−3. The decision threshold γd is chosen to be the minimum threshold such

that the bound on PF(γd) is δ = 0.05. We assume that Etot = −20.4 dB which is its mean

value when the transmitter-receiver separation is 10 m [47]. We choose the power ratio

r = −4 dB, decay constant ε = 16.1 dB and fading figures mk = 3.5− kTc

73, 0 ≤ k ≤ Ntap−1,

which are their mean values given in Cassioli et al. [47].

Figs. 3-3 and 3-4 show the effect of increasing G on the average probability of a miss

PM for SAI when the number of PRake fingers are Np = 5 and 10, respectively. For each

value of Np, we plot PM for the average energy received per pulse to noise ratio E1Etot

N0=

7, 10, 15 and 20 dB. When E1Etot

N0is low, PM decreases at first as G increases and then

begins to increase. When E1Etot

N0is low, increasing G helps combat the effect of the noise

by collecting more energy when the hypothesized phase belongs to the hit set. This is

the reason for the initial decrease in PM. At the same time, the energy collected in the

non-hit set phases begins to increase resulting in a much higher threshold being chosen to

ensure that PF(γd) does not exceed δ. Consequently, the probabilities of detection decrease

causing PM to increase. When E1Etot

N0is high, the detrimental effect of the noise is not

very significant and hence increasing G does not improve the probabilities of detection

significantly. But the probabilities of detection suffer from the stringent choice of threshold

required to keep PF(γd) small. This is the reason for the increase in PM with G for high

values of E1Etot

N0. The values of PM show a slight decrease for large G in the case when

Np = 10 compared to the case when Np = 5. This is because the hit set is larger when

Np = 10 but the probabilities of detection in the additional phases becomes significant

only for large G.

43

Figs. 3-5 and 3-6 show the effect of increasing G on the average probability of a miss

PM for IAS when the number of PRake fingers are Np = 5 and 10, respectively. For all

values of E1Etot

N0we considered, PM increases with G from a value close to zero to a value

close to one. Thus performing EGC in the IAS approach actually results in a degradation

in performance. As G increases, the EGC window collects multiple paths which may have

opposing polarities resulting in cancellations and hence a decrease in the probabilities of

detection in the hit set phases. This is the reason for the increase in PM with G. This

cancellation effect is also present in the non-hit set phases resulting in a less stringent

choice for the threshold needed to keep PF(γd) small. This effect is absent in the case of

SAI because the squaring operation eliminates the path polarities. The small values of

PM (for small G) suggest that IAS does a better job of averaging out the effect of the

noise than SAI. Squaring the received signal before integrating seems to be preventing this

averaging effect in SAI, resulting in higher values of PM when E1Etot

N0is low. When E1Etot

N0

is high, the less stringent threshold results in smaller values of PM for IAS in comparison

to SAI. Once again, the presence of a larger hit set for Np = 10 results in smaller values of

PM for IAS in comparison to the case of Np = 5.

To compare the performance of the two schemes in terms of the mean detection time

T det, we assume that the dwell time is equal to one period of the TH sequence i.e. M = 1

and T = NthNfTc. Figs. 3-7 and 3-8 show the mean detection time in seconds (at different

values of E1Etot

N0) for SAI as a function of G for Np = 5 and 10, respectively. Figs. 3-9 and

3-10 show the corresponding plots for IAS. For both schemes, the effect of increasing G

on the mean detection time mirrors its effect on PM for the same reasons mentioned in

the previous paragraph. Once again, performing EGC is beneficial in the SAI approach

and causes performance degradation in the IAS approach. For SAI, we observe that the

minimum mean detection time is achieved for some value of G larger than one. This value

of G changes with E1Etot

N0but is the same even when the number of PRake fingers Np is

increased from 5 to 10. Increasing Np keeping the E1Etot

N0fixed increases the size of the

44

hit set. But the probabilities of detection in the additional phases becomes significant

only for large values of G where the probabilities of detection have already been lowered

by the stringent choice of threshold. On the other hand, the IAS approach achieves

mean detection times which are significantly lower than the corresponding values for SAI

when E1Etot

N0is low and hence is a more efficient scheme. Even though the probabilities

of detection get better as E1Etot

N0increases, the mean detection time does not change

significantly since it is dominated by the time the acquisition system spends in the non-hit

set phases. The minimum mean detection time is seen to be of the order of a second which

is too high from a practical system viewpoint. This is due to the large search space and

the fact that the serial search has to evaluate a considerable number of phases on the

average before it encounters the hit set. This issue can be alleviated by a parallel search

strategy.

3.6 Conclusions

We have analyzed two approaches, namely SAI and IAS, for the acquisition of UWB

signals which perform EGC to utilize the energy in the multipaths. By considering system

performance subsequent to acquisition, the set of phases which can be considered a hit

was obtained. In the SAI approach, performing EGC improves acquisition performance

at low SNRs while it causes performance degradation in the IAS approach. With mean

detection time as the metric for system performance, we observe that the IAS approach

outperforms the SAI significantly. Thus EGC may not be a good method to utilize the

energy available in the multipaths to improve acquisition performance. Finally, the far

from practical values of the mean detection time obtained motivate the need for a parallel

search strategy and the development of acquisition schemes capable of reducing the search

space.

45

r(t)

∆τIs > ?

τSignal Generator

Reference

CorrelatorOperation

Squaring

τs(t− )

ControlClock

No

YesR( ;h)∆τ γ

R( ;h)

Figure 3-1: Block diagram of the SAI acquisition system.

γ

τ

τs(t− )

R( ;h)∆τIs > ?

Signal Generator

r(t) Correlator

Reference

Operation

Squaring

ControlClock

No

YesR( ;h)∆τ

Figure 3-2: Block diagram of the IAS acquisition system.

2 4 6 8 10 12 140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

EGC Window Length

Pro

babi

lity

of a

mis

s

E1E

totN

0−1 = 7 dB

E1E

totN

0−1 = 10 dB

E1E

totN

0−1 = 15 dB

E1E

totN

0−1 = 20 dB

Figure 3-3: Effect of EGC window length on the probability of a miss for SAI when Np =5

46

2 4 6 8 10 12 140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

EGC Window Length

Pro

babi

lity

of a

mis

s

E1E

totN

0−1 = 7 dB

E1E

totN

0−1 = 10 dB

E1E

totN

0−1 = 15 dB

E1E

totN

0−1 = 20 dB

Figure 3-4: Effect of EGC window length on the probability of a miss for SAI when Np =10

47

2 4 6 8 10 12 140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

EGC Window Length

Pro

babi

lity

of a

mis

s

E1E

totN

0−1 = 7 dB

E1E

totN

0−1 = 10 dB

E1E

totN

0−1 = 15 dB

E1E

totN

0−1 = 20 dB

Figure 3-5: Effect of EGC window length on the probability of a miss for IAS when Np =5

48

2 4 6 8 10 12 140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

EGC Window Length

Pro

babi

lity

of a

mis

s

E1E

totN

0−1 = 7 dB

E1E

totN

0−1 = 10 dB

E1E

totN

0−1 = 15 dB

E1E

totN

0−1 = 20 dB

Figure 3-6: Effect of EGC window length on the probability of a miss for IAS when Np =10

49

2 4 6 8 10 12 140

1

2

3

4

5

6

7

8

9

10

EGC Window Length

Mea

n D

etec

tion

Tim

e (s

econ

ds)

E1E

totN

0−1 = 7 dB

E1E

totN

0−1 = 10 dB

E1E

totN

0−1 = 15 dB

E1E

totN

0−1 = 20 dB

Figure 3-7: Effect of EGC window length on the mean detection time for SAI when Np =5

50

2 4 6 8 10 12 140

1

2

3

4

5

6

7

8

9

10

EGC Window Length

Mea

n D

etec

tion

Tim

e (s

econ

ds)

E1E

totN

0−1 = 7 dB

E1E

totN

0−1 = 10 dB

E1E

totN

0−1 = 15 dB

E1E

totN

0−1 = 20 dB

Figure 3-8: Effect of EGC window length on the mean detection time for SAI when Np =10

51

2 4 6 8 10 12 140

1

2

3

4

5

6

7

8

9

10

EGC Window Length

Mea

n D

etec

tion

Tim

e (s

econ

ds)

E1E

totN

0−1 = 7 dB

E1E

totN

0−1 = 10 dB

E1E

totN

0−1 = 15 dB

E1E

totN

0−1 = 20 dB

Figure 3-9: Effect of EGC window length on the mean detection time for IAS when Np =5

52

2 4 6 8 10 12 140

1

2

3

4

5

6

7

8

9

10

EGC Window Length

Mea

n D

etec

tion

Tim

e (s

econ

ds)

E1E

totN

0−1 = 7 dB

E1E

totN

0−1 = 10 dB

E1E

totN

0−1 = 15 dB

E1E

totN

0−1 = 20 dB

Figure 3-10: Effect of EGC window length on the mean detection time for IAS when Np =10

53

CHAPTER 4ASYMPTOTIC PERFORMANCE OF THRESHOLD-BASED ACQUISITION SYSTEMS

IN MULTIPATH FADING CHANNELS

4.1 Introduction

In this chapter, we investigate the asymptotic error performance of threshold-based

timing acquisition systems having fixed dwell time in multipath fading channels. We

restrict our attention to acquisition systems with fixed dwell time because it represents

the case of packetized mobile communication systems. This is a scenario where good

acquisition performance is crucial, since the timing needs to be repeatedly estimated for

every packet as it may change due to node mobility. And since throughput considerations

limit the length of the preamble which can be prepended to a particular packet, there

might be a limit to the accuracy with which the timing can be estimated. Thus it is of

interest to get an estimate of the best possible acquisition performance which can be

achieved by using a finite-length preamble.

In the absence of channel fading, it is a well-known result that the probabilities of

occurrence of false alarms and misses, which are due to the noise alone, can be made

arbitrarily small by operating at a higher SNR, which is typically done by increasing

the dwell time of the correlator [10]. As the SNR increases, even a sub-optimally chosen

threshold, located between the means of the distributions of the decision statistic when

the hypothesized symbol timing is correct and incorrect, forces the probabilities of false

alarm and miss to become arbitrarily small. It is, however, reasonable to expect that

the presence of channel fading can cause errors to occur, irrespective of how high the

average SNR is. This is due to the fact that a high average SNR only guarantees that the

detrimental effect of the noise is negligible and the channel fading can still induce errors in

the acquisition process.

In this chapter, we isolate the detrimental effect of the multipath channel fading on

the acquisition performance of a finite dwell time threshold-based acquisition system, by

considering the asymptotic performance as the average SNR increases without bound. We

54

show that no matter how large the average SNR is or how we choose the threshold, there

exist fading scenarios with a non-zero and sometimes restrictive average probability of

occurrence of false alarms and misses.

We describe the system model in Section 4.2 which is general enough to encompass

most threshold-based timing acquisition systems. In Section 4.3, we state and prove

the main result of the chapter which basically says that if there is a threshold which

restricts the average probability of false alarm to be smaller than a fixed tolerance, then

no matter how large the average SNR is, there is a possibly non-trivial lower bound on the

asymptotic average probability of miss. In Section 4.4, we apply the result to evaluate and

compare the asymptotic acquisition performance of the two acquisition schemes developed

in Chapter 3. Section 4.5 has some discussion and conclusions.

4.2 System Model

Let s(t) be the transmitted signal and h(t) be the channel response which is assumed

to be random but fixed during the acquisition process. Then the received signal is given

by r(t) = x(t − τ) + n(t) where x(t) = s(t) ∗ h(t) ( ∗ denotes convolution), τ is the true

symbol timing and n(t) is a zero-mean wide-sense stationary (WSS) additive noise process.

Let τ be the hypothesized symbol timing. Then the decision statistic generated by the

acquisition system is given by R(∆τ ;h) = g(r(t), τ) where g is some bivariate functional,

∆τ = τ − τ and assuming that the channel fading effects can be characterized by a

finite-dimensional vector h. Let FN(·; ∆τ |h) be the conditional cumulative distribution

function (CDF) of R(∆τ ;h) conditioned on h.

In a multipath channel, the receiver need not lock to the line-of-sight (LOS) path to

perform successful demodulation. Depending on the performance criteria chosen, there

will be a set of hypothesized symbol timings τ called the hit set (which we will denote by

Sh) where a receiver lock can be considered successful acquisition. Since the goal of the

acquisition process is to achieve coarse synchronization, the true symbol timing τ can be

assumed to belong to a finite set Sp of timings which is an adequately quantized version of

55

the timing ambiguity region. The hypothesized symbol timings are chosen from this finite

set and hence the hit set is also finite. Note that it is the distance ∆τ of a hypothesized

symbol timing τ from the true symbol timing τ which determines if τ belongs to Sh or

not. In this sense, the actual value of the true symbol timing is irrelevant. For a particular

value of τ , the acquisition process can be formulated as a composite binary hypothesis

testing problem with the following hypotheses:

H0 : τ /∈ Sh

H1 : τ ∈ Sh. (4–1)

The probabilities of false alarm and detection conditioned on the particular channel

realization h and given the decision threshold γ are given, respectively, by

PFA(γ; ∆τ |h) = Pr[R(∆τ ;h) > γ|h, τ /∈ Sh] = 1− FN(γ; ∆τ |h) = F ′N(γ; ∆τ |h), τ /∈ Sh,

PD(γ; ∆τ |h) = Pr[R(∆τ ;h) > γ|h, τ ∈ Sh] = 1− FN(γ; ∆τ |h)

= F ′N(γ; ∆τ |h), τ ∈ Sh, (4–2)

where F ′N(·; ∆τ |h) is the complementary conditional CDF of R(∆τ ;h) conditioned on h.

Then the probabilities of false alarm and detection averaged over the channel realizations

are given by

PFA(γ; ∆τ) = EH[PFA(γ; ∆τ |h)] = EH[F ′N(γ; ∆τ |h)], τ /∈ Sh, (4–3)

PD(γ; ∆τ) = EH[PD(γ; ∆τ |h)] = EH[F ′N(γ; ∆τ |h)], τ ∈ Sh. (4–4)

The average probability of miss is then given by

PM(γ; ∆τ) = 1− PD(γ; ∆τ), τ ∈ Sh. (4–5)

Henceforth, whenever we write PFA(γ; ∆τ) or PFA(γ; ∆τ |h) it is implicit that τ /∈ Sh.

Similarly, PD(γ; ∆τ), PD(γ; ∆τ |h), PM(γ; ∆τ) and PM(γ; ∆τ |h) all imply that τ ∈ Sh.

56

4.3 Asymptotic Performance of Threshold-based Acquisition Systems

Let σ2 be the power (variance) of the noise process n(t). Let H be the set of all

possible channel parameter vectors h. Note that PFA(γ; ∆τ) and PM(γ; ∆τ) defined

in the previous section are functions of σ. To avoid cumbersome notation, we write

limσ→0+ PFA(γ; ∆τ) to mean limσ→0+ PFA(γ, σ; ∆τ). Furthermore, for a positive sequence

σn with limit 0, we write lim supσn→0+ PM(γ; ∆τ) to mean lim supn→∞ PM(γ, σn; ∆τ).

The following theorem is the main result of this chapter.

Theorem 1. Consider a threshold-based acquisition system with decision statistic

R(∆τ ;h) with the property that for every threshold γ and ε > 0, there is an η(γ, ε) > 0

such that when σ2 < η(γ, ε) there exist subsets Aε(γ; ∆τ),Bε(γ; ∆τ) of H, for every τ ∈ Sp,

such that

(i) Pr(Aε(γ; ∆τ) ∪ Bε(γ; ∆τ)) > 1− ε.

(ii) For all h ∈ Aε(γ; ∆τ), FN(γ; ∆τ |h) > 1− ε.

(iii) For all h ∈ Bε(γ; ∆τ), FN(γ; ∆τ |h) ≤ ε.

For some δ > 0, if there exists an η1(δ) > 0 such that PFA(γ; ∆τ) < δ for all σ2 < η1(δ)

and for all τ /∈ Sh, then limσ→0+ PM(γ; ∆τ) ≥ limε→0+ Pr(Aε(γm(δ); ∆τ)) where γm(δ) =

infγ : limε→0+ Pr(Bε(γ; ∆τ)) ≤ δ, for all τ /∈ Sh. Furthermore, given ξ > 0 there exists a

κ(δ, ξ) > 0 such that PM(γ; ∆τ) ≥ limε→0+ Pr(Aε(γm(δ); ∆τ))− ξ for all σ2 < κ(δ, ξ).

Proof. By the hypothesis, for every γ and ε > 0 there is an η(γ, ε) > 0 such that when

σ2 < η(γ, ε) there exists a subset Aε(γ; ∆τ) of H such that for all h ∈ Aε(γ; ∆τ),

FN(γ; ∆τ |h) > 1− ε. Then for all σ2 < η(γ, ε), we have

PM(γ; ∆τ) ≥ EH[FN(γ; ∆τ |h)IAε(γ;∆τ)(h)] > (1− ε) Pr(Aε(γ; ∆τ))

≥ Pr(Aε(γ; ∆τ))− ε, (4–6)

57

where IAε(γ;∆τ)(·) is the indicator function of the set Aε(γ; ∆τ). Furthermore, when

σ2 < η(γ, ε) and Dε(γ; ∆τ) = Aε(γ; ∆τ) ∪ Bε(γ; ∆τ) we have

PM(γ; ∆τ) ≤ EH[FN(γ; ∆τ |h)(IDε(γ;∆τ) + IDcε(γ;∆τ))(h))]

≤ EH[FN(γ; ∆τ |h)(IAε(γ;∆τ) + IBε(γ;∆τ) + IDcε(γ;∆τ))(h))]

≤ Pr(Aε(γ; ∆τ)) + ε Pr(Bε(γ; ∆τ)) + ε

≤ Pr(Aε(γ; ∆τ)) + 2ε. (4–7)

Consider any convergent positive sequence σn with limit zero. For any ε > 0, from (4–6)

and (4–7), we have

Pr(Aε(γ; ∆τ))− ε ≤ lim infσn→0+

PM(γ; ∆τ) ≤ lim supσn→0+

PM(γ; ∆τ)

≤ Pr(Aε(γ; ∆τ)) + 2ε. (4–8)

Now consider a convergent positive sequence εn with limit zero. Since (4–8) holds for

every ε > 0, we have

lim supεn→0+

Pr(Aεn(γ; ∆τ)) = lim supεn→0+

[Pr(Aεn(γ; ∆τ))− εn] ≤ lim infσn→0+

PM(γ; ∆τ)

≤ lim supσn→0+

PM(γ; ∆τ) ≤ lim infεn→0+

[Pr(Aεn(γ; ∆τ)) + 2εn]

= lim infεn→0+

Pr(Aεn(γ; ∆τ)). (4–9)

But for any sequence εn,

lim infεn→0+

Pr(Aεn(γ; ∆τ)) ≤ lim supεn→0+

Pr(Aεn(γ; ∆τ). (4–10)

So we have

lim infεn→0+

Pr(Aεn(γ; ∆τ)) = lim supεn→0+

Pr(Aεn(γ; ∆τ)). (4–11)

Thus limεn→0+ Pr(Aεn(γ; ∆τ)) exists for every positive sequence εn converging to zero.

Furthermore, all the inequalities in (4–9) are actually equalities and limσn→0+ PM(γ; ∆τ)

exists for all sequences σn. By fixing the sequence εn and considering all possible

58

positive sequences σn converging to zero, we see that limσn→0+ PM(γ; ∆τ) = limεn→0+ Pr(Aεn(γ; ∆τ))

for all sequences σn. Thus by the definition of the limit of a function [55] we have

limσ→0+

PM(γ; ∆τ) = limεn→0+

Pr(Aεn(γ; ∆τ)). (4–12)

Since the left hand side in (4–12) is fixed for all sequences εn, by the definition of the

limit of a function we have

limσ→0+

PM(γ; ∆τ) = limε→0+

Pr(Aε(γ; ∆τ)). (4–13)

Similarly, for σ2 < η(γ, ε) we have

Pr(Bε(γ; ∆τ))− ε < PFA(γ; ∆τ) < Pr(Bε(γ; ∆τ)) + 2ε (4–14)

and hence we can show that

limσ→0+

PFA(γ; ∆τ) = limε→0+

Pr(Bε(γ; ∆τ)). (4–15)

Let γ∗ be a threshold such that PFA(γ∗; ∆τ) < δ for all σ2 < η1(δ) and τ /∈ Sh. Since the

complementary conditional CDF F ′N(γ; ∆τ |h) is a non-increasing function of the threshold

γ, the average probability of false alarm PFA(γ; ∆τ) is a non-increasing function of γ. For

all σ2 < η1(δ) we have

γ∗ ≥ infγ : PFA(γ; ∆τ) < δ, for all τ /∈ Sh. (4–16)

Since (4–16) holds for all σ2 < η1(δ), we have

γ∗ ≥ infγ : limσ→0+

PFA(γ; ∆τ) ≤ δ, for all τ /∈ Sh

= infγ : limε→0+

Pr(Bε(γ; ∆τ)) ≤ δ, for all τ /∈ Sh,

(4–17)

where the equality follows from (4–15). Note that the expression on the right hand side of

the equality in (4–17) is equal to γm(δ) defined in the statement of the theorem.

59

Since γ∗ ≥ γm(δ), for τ ∈ Sh we have

PM(γ∗; ∆τ) = EH[FN(γ∗; ∆τ |h)] ≥ EH[FN(γm(δ); ∆τ |h)]

= PM(γm(δ); ∆τ), (4–18)

where the inequality follows from the fact that FN(γ; ∆τ |h) being a conditional CDF is an

increasing function of γ. From (4–13) and (4–18), we have

limσ→0+

PM(γ∗; ∆τ) ≥ limε→0+

Pr(Aε(γm(δ); ∆τ)), (4–19)

for all τ ∈ Sh, which proves the first statement of the theorem.

From (4–13), given ξ > 0 and a threshold γ, there exists a η2(γ, ξ) > 0 such that

PM(γ; ∆τ) ≥ limε→0+

Pr(Aε(γ; ∆τ))− ξ (4–20)

for all σ2 < η2(γ, ξ). Then from (4–18) and (4–20), we have

PM(γ∗; ∆τ) ≥ PM(γm(δ); ∆τ) ≥ limε→0+

Pr(Aε(γm(δ); ∆τ))− ξ (4–21)

for all σ2 < η2(γm(δ), ξ) = κ(δ, ξ). This proves the second statement of the theorem.

We present some discussion regarding the conditions and statement of the above

theorem. From (4–2) and (4–2), it is clear that the set Aε(γ; ∆τ) corresponds to a

subset of H where PFA(γ; ∆τ |h) or PD(γ; ∆τ |h) (depending on whether τ /∈ Sh or

τ ∈ Sh) do not exceed ε. Similarly, the set Bε(γ; ∆τ) corresponds to a subset of H where

PFA(γ; ∆τ |h) or PD(γ; ∆τ |h) exceed 1 − ε. So the conditions of Theorem 1 require the

decision statistic to be such that when the noise variance is small enough (or equivalently

at high enough SNRs), the conditional probabilities of false alarm and detection are

(with probability close to one) either close to zero or close to one. Furthermore using

conditions (i)-(iii) of the theorem and (4–3)-(4–4), it is easy to see that at high SNRs

PFA(γ; ∆τ) ≈ Pr(Bε(γ; ∆τ)) and PM(γ; ∆τ) ≈ Pr(Aε(γ; ∆τ)). Any threshold γ which

restricts PFA(γ; ∆τ) to be less than some δ will be larger than the smallest threshold

60

γm(δ) which restricts Pr(Bε(γ; ∆τ)) to not exceed δ. So the theorem states that this

lower bound on the threshold translates to a lower bound on PM(γ; ∆τ) which may be

non-trivial even in the asymptotic scenario. If the threshold γ is chosen carefully, then

we have PM(γ; ∆τ) ≥ limσ→0+ PM(γ; ∆τ), but this is not true in general for all γ. So the

lower bound on the asymptotic average probability of miss may not always be a lower

bound on the average probability of miss at finite SNRs. Nevertheless, the last statement

of the theorem states that the lower bound in the asymptotic case is a good approximation

for the lower bound on the average probability of miss at large (finite) SNRs. Thus the

tradeoff between the PFA(γ; ∆τ) and PM(γ; ∆τ) at large SNRs can be characterized by the

tradeoff between δ and limε→0+ Pr(Aε(γm(δ); ∆τ)). The main advantage of the theorem

is that this tradeoff can be calculated using sets defined according to the conditional

probabilities of detection and false alarm, which are usually easier to obtain.

4.4 Asymptotic Performance of Threshold-based UWB Signal Acquisition

In this section, we evaluate and compare the asymptotic acquisition performance

of the SAI and IAS approaches to the acquisition of UWB signals with time-hopping

spreading.

We assume that the PRake receiver has Np fingers where Np ≤ Ntap. Then for true

phase τ , we choose the hit set as Sh = τ− (Np−1)Tc, τ− (Np−2)Tc, . . . , τ +(Ntap−1)Tc.The phases in the hit set correspond to those phases from which the PRake receiver can

collect at least one resolvable path of the channel response corresponding to the true

phase. This is not a reasonable definition for the hit set at finite SNRs since some of the

resolvable paths may be too weak to enable good demodulation performance. Hence a

receiver lock to such a path may not be considered successful acquisition. However, Sh

defined as above contains any path where good demodulation performance can be achieved

at a finite SNR. Thus it represents the largest possible hit set and consequently SCh is the

smallest possible non-hit set. This corresponds to the least restrictive choice of γm(δ) in

Theorem 1. For this choice of Sh, the lower bound on the asymptotic average probability

61

of miss is the smallest and hence it results in the best possible asymptotic acquisition

performance over all choices of Sh.

4.4.1 Asymptotic Performance of the SAI Approach

In this subsection, we derive the asymptotic performance of an acquisition system

which takes the SAI approach.

From (3–12), the decision statistic of the SAI approach is given by

R(∆τ ;h) = s(∆τ ;h) + ny, (4–22)

where, conditioned on h, ny is a Gaussian random variable with mean µy and variance

σ2y(∆τ ;h) + ν2

y . The expressions for the mean and variance can be found in Section 3.2.1.

The probabilities of false alarm and detection conditioned on the particular channel

realization and given the decision threshold γ are given as

PFA(γ, ∆τ |h) = F ′N(γ; ∆τ |h), τ /∈ Sh = Q

γ − s(∆τ ;h)− µy√

σ2y(∆τ ;h) + ν2

y

, τ /∈ Sh,

PD(γ, ∆τ |h) = F ′N(γ; ∆τ |h), τ ∈ Sh = Q

γ − s(∆τ ;h)− µy√

σ2y(∆τ ;h) + ν2

y

, τ ∈ Sh.

In order to be able to apply Theorem 1 to this case, we need to first verify that the

required conditions hold. Since the path gains are distributed according to Nakagami-m

distributions, h has an absolutely continuous distribution [56] and hence s(∆τ ;h) has an

absolutely continuous distribution. Since Sp is finite, for any threshold γ ≥ 0 and every

ε > 0, there exists a κ(γ, ε) > 0 such that for all τ ∈ Sp we have

Pr(h : γ − κ(γ, ε) ≤ s(∆τ ;h) ≤ γ + κ(γ, ε)) <ε

2. (4–23)

Note that s(∆τ ;h) is a non-negative random variable for all τ ∈ Sp. Then by choosing a

positive integer n such that n−1 < ε/2 and a positive real number Ks ≥ maxmean(s(∆τ ;h)) :

62

τ ∈ Sp, for all τ ∈ Sp we get

Pr(h : s(∆τ ;h) ≥ nKs) ≤ mean(s(∆τ ;h))

nKs

≤ 1

n<

ε

2. (4–24)

In Appendix C, we show that Aε(γ; ∆τ) and Bε(γ; ∆τ) defined below satisfy the

conditions of Theorem 1.

Aε(γ; ∆τ) = h : s(∆τ ;h) < γ − κ(γ, ε).

Bε(γ; ∆τ) = h : γ + κ(γ, ε) < s(∆τ ;h) < nKs. (4–25)

Then for all ε > 0,

Pr(Aε(γ; ∆τ)) = Pr(h : s(∆τ ;h) ≤ γ)− Pr(h : γ − κ(γ, ε) ≤ s(∆τ ;h) ≤ γ)

≥ Pr(h : s(∆τ ;h) ≤ γ)

−Pr(h : γ − κ(γ, ε) ≤ s(∆τ ;h) ≤ γ + κ(γ, ε))

> Pr(h : s(∆τ ;h) ≤ γ)− ε, (4–26)

where the last inequality follows from (4–23). Since Pr(Aε(γ; ∆τ)) ≤ Pr(h : s(∆τ ;h) ≤γ) for all ε > 0, we have

limε→0+

Pr(Aε(γ; ∆τ)) = Pr(h : s(∆τ ;h) ≤ γ). (4–27)

Similarly, we can show that

limε→0+

Pr(Bε(γ; ∆τ)) = Pr(h : s(∆τ ;h) ≥ γ). (4–28)

Then by Theorem 1, for any δ > 0 if there exists a threshold γ and an η1(δ) > 0 such that

PFA(γ; ∆τ) < δ for all σ2 < η1(δ) and for all τ /∈ Sh, then

limσ→0+

PM(γ; ∆τ) ≥ Pr(h : s(∆τ ;h) ≤ γm(δ)) (4–29)

where γm(δ) = infγ : Pr(h : s(∆τ ;h) ≥ γ) ≤ δ, for all τ /∈ Sh.

63

Note that the lower bound on the asymptotic average probability of miss in (4–29)

results in the following upper bound on the asymptotic average probability of detection,

limσ→0+

PD(γ; ∆τ) ≤ Pr(h : s(∆τ ;h) ≥ γm(δ)), (4–30)

where τ ∈ Sh. By evaluating this upper bound as a function of δ, we obtain an asymptotic

receiver operating characteristic (AROC) which characterizes the best achievable trade-off

between the average probabilities of false alarm and detection. From the definition of

γm(δ) and the expression for the upper bound in (4–30), we observe that the AROC for

a particular τ ∈ Sh depends on the separation between the corresponding distribution of

s(∆τ ;h) and the distributions of s(∆τ ;h) for all τ /∈ Sh. For instance, if the distribution

of s(∆τ ;h) for some τ ∈ Sh is close to the distribution of s(∆τ ;h) for any τ /∈ Sh, then

the upper bound on the asymptotic average probability of detection for that τ ∈ Sh will be

close to δ.

The CDF of s(∆τ ;h) is needed to calculate the AROC. From (3–5), s(∆τ ;h) is a

linear combination of independent random variables and hence its characteristic function is

given by

φs(ω; ∆τ) =

Ntap−1∏

k=0

φk(E1Rψ2r(0)rk(∆τ)ω), (4–31)

where φk(·)’s are the characteristic functions of the Gamma distributed h2k’s [51]. By the

Gil-Pelaez lemma [54], the CDF of s(∆τ ;h) is given by

Fs(x) =1

2+

1

π

∫ ∞

0

Im

ejtxφs(−t; ∆τ)

t

dt

=1

2+

2

π

∫ π2

0

Im

ejx tan θφs(− tan θ; ∆τ)

sin 2θ

dθ,

(4–32)

where the second equality is obtained by the change of variable t = tan θ. The second

integral has finite limits of integration and hence is more suitable for numerical evaluation.

64

4.4.2 Asymptotic Performance of the IAS Approach

In this subsection, we derive the asymptotic performance of an acquisition system

which takes the IAS approach. From (3–22), the decision statistic is given by

R(∆τ ;h) = [V (∆τ ;h) + nz]2 (4–33)

where nz is a zero-mean Gaussian random variable with variance σ2z = Gσ2

MNthand rk(∆τ) is

given in (3–6). The probabilities of false alarm and detection conditioned on the particular

channel realization and given the decision threshold γ ≥ 0 are given by

PFA(γ, ∆τ |h) = F ′N(γ; ∆τ |h), τ /∈ Sh

= Q

(√γ − V (∆τ ;h)

σz

)+ Q

(√γ + V (∆τ ;h)

σz

), τ /∈ Sh,

PD(γ, ∆τ |h) = F ′N(γ; ∆τ |h), τ ∈ Sh

= Q

(√γ − V (∆τ ;h)

σz

)+ Q

(√γ + V (∆τ ;h)

σz

), τ ∈ Sh.

As before, V (∆τ ;h) has an absolutely continuous distribution and hence for any

threshold γ ≥ 0 and every ε > 0, there exists a κ(γ, ε) > 0 such that for all τ ∈ Sp we have

Pr(h : |V (∆τ ;h)−√γ| ≤ κ(γ, ε) or |V (∆τ ;h) +√

γ| ≤ κ(γ, ε)) < ε. (4–34)

In Appendix D, we show that Aε(γ; ∆τ) and Bε(γ; ∆τ) defined below satisfy the

conditions of Theorem 1.

Aε(γ; ∆τ) = h : −√γ + κ(γ, ε) < V (∆τ ;h) <√

γ − κ(γ, ε).

Bε(γ; ∆τ) = h : V (∆τ ;h) >√

γ + κ(γ, ε) or V (∆τ ;h) < −√γ − κ(γ, ε). (4–35)

We can also show that

limε→0+

Pr(Aε(γ; ∆τ)) = Pr(h : −√γ ≤ V (∆τ ;h) ≤ √γ),

limε→0+

Pr(Bε(γ; ∆τ)) = Pr(h : V (∆τ ;h) ≥ √γ or V (∆τ ;h) ≤ −√γ).

65

Then by Theorem 1, for any δ > 0 if there exists a threshold γ and an η1(δ) > 0 such that

PFA(γ; ∆τ) < δ for all σ2 < η1(δ) and for all τ /∈ Sh, then

limσ→0+

PM(γ; ∆τ) ≥ Pr(h : −√

γm(δ) ≤ V (∆τ ;h) ≤√

γm(δ)) (4–36)

where γm(δ) = infγ : Pr(h : V (∆τ ;h) ≥ √γ or V (∆τ ;h) ≤ −√γ) ≤ δ, for all τ /∈ Sh.

Finally, we have the following upper bound on the asymptotic average probability of

detection,

limσ→0+

PD(γ; ∆τ) ≤ Pr(h : V (∆τ ;h) ≥√

γm(δ) or V (∆τ ;h) ≤ −√

γm(δ)),

where τ ∈ Sh. Once again, the AROC calculation requires the CDF of V (∆τ ;h). Since the

polarities pk and path gains hk are independent, the characteristic function of V (∆τ ;h), in

this case, is given by

φV(ω; ∆τ) =

Ntap−1∏

k=0

[φk(√

E1rk(∆τ)ω) + φk(−√

E1rk(∆τ)ω)

2

], (4–37)

where φk(·) is the characteristic function of the Nakagami-m distributed hk [51].

Substitution of the above equation in (4–32) yields the CDF of V (∆τ ;h).

4.4.3 Numerical Results

To calculate the AROC for the SAI and IAS acquisition schemes, we choose the

following values for the system parameters: the TH sequence period Nth = 1024, Nh = 16,

M = 1, the length of the channel response Ntap = 100, the number of PRake fingers

Np = 5 and Nf = 116. We assume that Etot = −20.4 dB which is its mean value when

the transmitter-receiver separation is 10 m [47]. We choose the power ratio r = −4 dB,

decay constant ε = 16.1 dB and fading figures mk = 3.5 − kTc

73, 0 ≤ k ≤ Ntap − 1, which

are their mean values given in Cassioli et al. [47]. The best AROC, which is again a plot

of limσ→0+ PD(γm(δ); 0) versus δ, does not depend on the received power and hence we set

E1 = 1.

66

For the UWB channel model we have chosen, the asymptotic average probability of

detection is largest when ∆τ = 0. Thus the best AROC is a plot of limσ→0+ PD(γm(δ); 0)

versus δ. Fig. 4-1 shows the best AROC of the SAI approach for EGC window sizes

G = 1, 2, 5, 10 and 15. The AROC becomes worse as the EGC window length increases

and is best for G = 1, which is equivalent to the case when there is no EGC. This is

consistent with the finite SNR results where we found that performing EGC for acquisition

is not advantageous at high SNRs. As G increases the signal energy collected by the EGC

window s(∆τ ;h) increases both when τ = τ and τ /∈ Sh. For τ = τ , the additional

energy collected is from the NLOS paths which are weaker in comparison to the LOS path

and thus the increase in signal energy is relatively small. The increase is more significant

when τ /∈ Sh since the additional energy is comparable to the energy collected when

G = 1. Thus the separation between the distributions of s(∆τ ;h) when τ = τ and τ /∈ Sh

decreases, causing the AROC to get worse.

Fig. 4-2 shows the best AROC of the IAS approach for EGC window sizes G =

1, 2, 5, 10 and 15. The upper bound on the asymptotic average probability of detection

is almost trivial for G = 1 and becomes significantly restrictive as G increases. As G

increases, for τ = τ the EGC window collects multiple paths which may have opposing

polarities resulting in cancellations and hence a decrease in the probability of detection.

For G = 1, this cancellation is absent when τ = τ but still occurs when τ /∈ Sh since the

random time-hopping sequence facilitates collection of multiple paths. Thus the signal

energy collected when τ /∈ Sh is much smaller than the signal energy collected when τ = τ ,

resulting in a significant separation between the corresponding distributions of V (∆τ ;h).

Hence the AROC is not restrictive for G = 1. Since the best AROC is just an upper

bound on the AROCs of all the hit set phases, we plot for G = 1 the AROCs of the phases

τ ∈ Sh corresponding to ∆τ = 5Tc, 10Tc, 15Tc, 20Tc and 30Tc in Fig. 4-3. We see that even

for IAS with G = 1 the bound on the asymptotic average probability of detection becomes

increasingly restrictive as the distance of the hit set phase from the LOS path increases.

67

This is because the energy in the paths decays with increase in distance from the LOS

path.

4.5 Conclusions

A typical 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 [10]. The large dwell time increases the

effective SNR of the decision statistic and in the absence of channel fading, this results

in accurate verification. In this chapter, we evaluated the asymptotic performance of

threshold-based timing acquisition systems in the presence of multipath fading and found

that, no matter how large the SNR is or how we choose the threshold, there are fading

scenarios in which false alarms and misses occur with non-zero and sometimes significant

average probability. Thus it may not be possible to build a good verification stage for

threshold-based acquisition systems operating in such channels by just increasing the dwell

time.

We found that if we choose a threshold such that the average probability of false

alarm is less than a given tolerance, then there is a possibly non-trivial lower bound on the

asymptotic average probability of miss. This lower bound translates to an upper bound

on the asymptotic average probability of detection. We evaluated this upper bound for

two threshold-based approaches, namely SAI and IAS, for the acquisition of UWB signals

with time-hopping spreading. For SAI, we found that the upper bound on the asymptotic

average probability of detection was significantly restrictive for all values of EGC window

size. But for IAS, the upper bound was almost trivial atleast for some hit set phases when

there was no EGC being done. Nevertheless, there were still some hit set phases where

the upper bound was restrictive. These results seem to suggest that EGC may not be a

good strategy to improve acquisition performance. More importantly, they suggest that

68

acquisition might be a potential bottleneck on throughput in any UWB-based packet

network employing threshold-based acquisition systems.

69

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

δ

Asy

mpt

otic

PD

(γm

(δ);

0)

G = 1G = 2G = 5G = 10G = 15

Figure 4-1: Best AROC of the SAI approach to UWB signal acquisition.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

δ

Asy

mpt

otic

PD

(γm

(δ);

0)

G = 1G = 2G = 5G = 10G = 15

Figure 4-2: Best AROC of the IAS approach to UWB signal acquisition.

70

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

δ

Asy

mpt

otic

PD

(γm

(δ);

∆τ)

∆τ = 5Tc

∆τ = 10Tc

∆τ = 15Tc

∆τ = 20Tc

∆τ = 30Tc

Figure 4-3: IAS AROC corresponding to hit set phases other than the LOS path whenG = 1.

71

CHAPTER 5A SEARCH STRATEGY FOR UWB SIGNAL ACQUISITION

5.1 Introduction

When there are multiple elements in the hit set, the serial search may no longer be

the optimal sequential search strategy. In this chapter, we consider the problem of finding

efficient search strategies in the set of all search strategies which are permutations of

the search space. Finding the optimal permutation search strategy which minimizes the

mean detection time when the search space is large and the probabilities of detection of

the hit set elements are arbitrary turns out to be prohibitively complex. However, if we

assume the probabilities of detection of all the hit set phases to be equal then there exists

a permutation search strategy which minimizes the mean detection time. Since the actual

probabilities of detection are not equal, this search strategy although not optimal serves as

a useful heuristic solution to an otherwise intractable problem. Furthermore, we see that

this search strategy has a simple Jump-by-H structure and improves the mean detection

time by a significant amount compared to the serial search.

The features of the UWB system model relevant to the problem considered are briefly

described in Section 5.2. The mean detection time of an arbitrary permutation search

strategy is calculated in Section 5.3 and the best permutation search strategy under the

assumption of equal probabilities of detection is found in Section 5.4. We present some

numerical results in Section 5.5 quantifying the improvement in mean detection time

performance followed by some concluding remarks in Section 5.6.

5.2 System Model

In this section, we briefly describe those aspects of an UWB acquisition system

which are relevant to the problem of finding efficient search strategies. A more detailed

description can be found in earlier chapters. It was found that the IAS approach without

EGC (i.e. with EGC window size equal to one) was the better strategy suggesting that

EGC may not be a good method to utilize the energy in the multipath to improve

acquisition performance.

72

In this chapter, we consider the IAS acquisition system without EGC which has

the structure shown in Fig. 3-2. The transmitter transmits a periodic signal with period

NsTc during the acquisition process, where Tc is the UWB pulse duration and Ns is a

positive integer. We assume that the pull-in range of the tracking loop is Tc and hence

the acquisition search only needs to search the timing ambiguity region in increments

of Tc. The timing ambiguity region is equal to the period of the transmitted signal

and hence the search space, which is the set of all hypothesized phases, is given by

0, Tc, 2Tc, . . . , (Ns − 1)Tc. The received signal is correlated with a locally generated

reference signal and the correlator output is squared to generate the decision statistic

R(∆τ ;h) where ∆τ = τ − τ , the difference between the hypothesized phase τ and the

true phase τ of the received signal, and h is a random vector containing the channel taps.

The decision statistic R(∆τ ;h) is compared to a threshold γ and the hypothesized phase

τ used to generate the reference signal is accepted as an estimate of the true phase of the

received signal if the threshold is exceeded. If the threshold is not exceeded, the process

is repeated with a new value for the hypothesized phase. A search strategy is then the

sequence of hypothesized phases which are checked until the threshold is exceeded. We will

find it convenient to represent the search space by Sp = 1, 2, 3, . . . , Ns, where the integer

n indexes the hypothesized phase (n− 1)Tc.

As mentioned earlier, there may be multiple phases in a dense multipath environment

which can be considered a good estimate of the true phase. Once again, we assume that

a partial Rake (PRake) receiver [50] is employed for demodulation and hit set in this

case has been derived in Chapter 3. The hit set Sh is typically a block of H consecutive

phases in the search space Sp where two elements i, j are considered to be consecutive if

|i − j| (mod Ns) = 1 or Ns − 1. For a particular value for the true phase of the received

signal, the position of the first element of the hit set block is p, which is assumed to be

equally likely to be any element of Sp. Given p, the positions of all the hit set elements are

completely specified. When p > Ns −H + 1, the last p − Ns + H − 1 hit set phases wrap

73

around and are represented by the first p−Ns + H − 1 phases of the search space. This is

due to the periodicity of the transmitted signal.

5.3 Mean Detection Time Calculation

The problem of finding the optimal permutation search strategy when the probabilities

of detection are arbitrary is complex. But if we assume that the probabilities of detection

in the hit set elements are equal, we are able to find a suboptimal search strategy which

reduces the mean detection time significantly. This permutation search strategy serves as a

useful heuristic solution to the otherwise intractable problem.

So we proceed to find the permutation search strategy which minimizes the mean

detection time under the assumption of equal detection probabilites in all hit set elements.

We first calculate the mean detection time when the search strategy is an arbitrary

permutation R of the search space. Let Pd be the average probability of detection in any

hit set element. For a particular initial position p of the hit set in the search space, let

the positions of appearance of elements of hit set elements in the sequential search be

tp,i : i = 1, 2, . . . , H. So the first appearance of a hit set element is at tp,1, the second

appearance is at tp,2 and so on. Table 5.6 illustrates this for the serial search starting in

position 1 of Sp when Ns = 8 and H = 3, where the positions in boldface indicate the

presence of a hit set element. The last three columns of the table contain the positions

of the first, second and third appearances of a hit set element for a particular value of p.

Table 5.6 shows the positions of appearance of the hit set elements for the permutation

search strategy (1, 4, 7, 2, 5, 8, 3, 6) when Ns = 8 and H = 3. Note that the columns

indicating the presence of hit set elements in Table 5.6 are obtained by permuting the

corresponding columns of Table 5.6. Also note that a hit set element appears in every

position of the permutation exactly H times where each appearance corresponds to a

distinct value of p in Sp. It is easy to see that this is true for any permutation search

strategy and for all values of Ns and H.

74

A detection event is defined by the position tp,i where we have a hit and a particular

number of misses j of Sh. Let T be the dwell time of the correlator. The time taken for a

miss event is Ns. The time for a particular detection event defined by (p, i, j) is then

T (p, i, j) = tp,iT + jNsT. (5–1)

The probability that there is a hit in position tp,i is given by Ph(i) = Pd(1 − Pd)i−1. The

probability of j misses of Sh is equal to P jM where PM = (1 − Pd)

H . The mean detection

time conditioned on the fact that the first element of the hit set is in position p of the

search space is given by

Tdet(p) =H∑

i=1

∞∑j=0

T (p, i, j)P jMPh(i)

=H∑

i=1

∞∑j=0

[tp,iT + jNsT ]P jMPh(i)

=T

∑Hi=1 tp,iPh(i)

1− PM

+NsTPM

∑Hi=1 Ph(i)

(1− PM)2

=T

∑Hi=1 tp,iPh(i)

1− PM

+NsTPM

1− PM

. (5–2)

The mean detection time is then given by

Tdet =1

Ns

Ns∑p=1

Tdet(p) =T

∑Hi=1(

∑Ns

p=1 tp,i)Ph(i)

Ns(1− PM)+

NsTPM

1− PM

. (5–3)

Note that the second term in the right hand side of (5–3) does not depend on the

permutation R. Then any optimization with respect to R can only hope to minimize

the first term.

5.4 The Jump-by-H Permutation Search Strategy

We want to minimize g(s) =∑H

i=1 siPh(i), where s = (sH , sH−1, . . . , s1) and

si =∑Ns

p=1 tp,i, over all permutations of S. Note that sH > sH−1 > . . . > s1. By the fact

that∑H

i=1 si = Ns(Ns+1)H2

and that Ph(i) is a decreasing function of i for all Pd, we have

the following result.

75

Lemma 1. g(s) is Schur-concave [57] on A = s = (sH , . . . , s1) : si =∑Ns

p=1 tp,i,

i = 1, 2, . . . , H, for some permutation R of S.

Proof. Let D = (x1, . . . , xn) : x1 ≥ · · · ≥ xn. Note that A is a subset of D. For all s ∈ Dand k = H,H − 1, . . . , 2,

g(sH , . . . , sk+1, sk + ε, sk−1 − ε, sk−2, . . . , s1) =H∑

i=1

siPh(i)− ε[Ph(k − 1)− Ph(k)],

which is decreasing in ε since Ph(k − 1) ≥ Ph(k). The Schur-concavity of g(s) on Dfollows from Lemma 3.A.2 in Marshall and Olkin [57]. Since A is a subset of D, g(s) is

Schur-concave on A.

Thus g(s) is minimized if s is the maximal vector of A. If x ≺ y, i.e. if x is majorized

by y for some x,y ∈ A, then

k∑i=1

xi ≥k∑

i=1

yi, k = 1, . . . , H − 1, (5–4)

andH∑

i=1

xi =H∑

i=1

yi. (5–5)

where xi and yi are the (H − i + 1)th components in x and y respectively.

Lemma 2. Let rk be not greater than the minimum value of∑k

i=1 si over all permutations

of S for k = 1, . . . , H−1 and rH = Ns(Ns+1)H2

. If ri+2− ri+1 ≥ ri+1− ri for i = 0, . . . , H−2,

then the vector

q = (rH − rH−1, rH−1 − rH−2, . . . , r2 − r1, r1) , (5–6)

majorizes all the vectors in A.

Proof. By hypothesis, we have q1 ≤ q2 ≤ . . . ≤ qH and∑H

i=1 qi = Ns(Ns+1)H2

, where qi is the

(H − i + 1)th component of q. Let s ∈ A and let si be its (H − i + 1)th component. Since

s1 ≤ s2 ≤ . . . ≤ sH , the sum of the k smallest components of s is

k∑i=1

si ≥ rk = (rk − rk−1) + . . . + (r2 − r1) + r1 =k∑

i=1

qi (5–7)

76

for k = 1, 2, . . . , H − 1. Furthermore,∑H

i=1 si = Ns(Ns+1)H2

. Thus q majorizes s and since

the choice of s was arbitrary, q majorizes all the vectors in A.

We now proceed by finding one particular set of rk’s which satisfy the conditions of

Lemma 2 and then exhibit a permutation R of S whose corresponding vector x ∈ A is

equal to the vector q defined by these rk’s. This vector x then majorizes all the vectors in

A. Hence the permutation search strategy R minimizes the mean detection time.

Theorem 2. The minimum value of∑k

i=1 si over all permutations of S is Nk(Nk+1)H2

+

(Nsk − NkH)(Nk + 1) for k = 1, . . . , H, where Nk = bNskHc. These minima are all

simultaneously achieved by a permutation R of S given by

Ri = (i− 1)H (mod Ns) +

⌊i− 1

(Ns

d)

⌋+ 1, (5–8)

where Ri is the element in its ith position and d is the greatest common divisor (GCD)

of Ns and H. Thus the search strategy R is optimal in the set of permutation search

strategies.

Proof. First, we note that it is not entirely obvious but easy to show that (5–8) does

indeed define a permutation. Suppose Ri and Rj are equal for some integers i, j such that

i = l(Ns

d) + ni and j = m(Ns

d) + nj where 1 ≤ ni, nj ≤ Ns

dand 0 ≤ l, m ≤ d − 1. If l = m,

then Ri − Rj = 0 if and only if (i − j)H (mod Ns) = 0. Since |ni − nj| ≤ (Ns

d) − 1, this

implies i = j. Now suppose (without loss of generality) that l > m. Then

Ri −Rj = [(l −m)(Ns

d) + ni − nj]H (mod Ns) + l −m

= (ni − nj)H (mod Ns) + l −m, (5–9)

which is not equal to zero since the first term is a multiple of H and the second term l−m

is not greater than d − 1 < H. Thus the Ri are distinct for i = 1, . . . , Ns and it then

follows by the pigeonhole principle that R is a permutation of S.

77

There are Ns number of tp,i’s in∑k

i=1 si, where each tp,i is in the set 1, 2, . . . , Nswith the restriction that each distinct value of tp,i appears at most H times. We can

obtain a lower bound of∑k

i=1 si by assigning the smallest values in 1, . . . , Ns to the tp,i’s

such that each value is assigned H times. Then the elements in the set 1, . . . , Nk are

each assigned H times and Nk + 1 is assigned Ns −NkH times where Nk = bNs

Hc. Thus we

have

k∑i=1

si ≥ H + · · ·+ NkH + (Nk + 1)(Ns −NkH)

=Nk(Nk + 1)H

2+ (Nsk −NkH)(Nk + 1), (5–10)

for k = 1, . . . , H.

Let rk be equal to the lower bound obtained in (5–10), i.e.,

rk =Nk(Nk + 1)H

2+ (Nsk −NkH)(Nk + 1) (5–11)

for k = 1, . . . , H. Then rH = Ns(Ns+1)H2

and

rk+1 − rk = (Nk + 1) · (H −Nsk + NkH) + (Nk + 2) ·H + · · ·+ Nk+1 ·H

+(Ns(k + 1)−Nk+1H) · (Nk+1 + 1),

for k = 0, 1, . . . , H − 1. For each k ∈ 0, 1, . . . , H − 1, rk+1 − rk is a sum of Ns terms

belonging to the set Nk + 1, . . . , Nk+1 + 1 with each distinct value appearing at most H

times. Since Nk+1 ≥ Nk + M , rk+2 − rk+1 ≥ rk+1 − rk for i = 0, 1, . . . , H − 2. Thus the rk’s

satisfy the conditions in Lemma 2 and consequently

q = (rH − rH−1, rH−1 − rH−2, . . . , r2 − r1, r1) , (5–12)

majorizes all the vectors in A. In Appendix E, we show that the vector x ∈ Acorresponding to the permutation R defined in (5–8) equals q.

78

It is easy to see that R has a Jump-by-H structure. R consists of d consecutive

blocks each containing Ns

delements, where the ith block consists of the elements (i,H +

i, . . . , (Ns

d− 1)H + i) for 1 ≤ i ≤ d.

5.5 Numerical Results

In order to compare the mean detection time performance of the heuristic permutation

search strategy with the serial search strategy, we chose the following values for the system

parameters: Size of the search space Ns = 29696, Tc = 2 ns, dwell time T = Nsc, and

SNR = 7,10 dB. The hit set was obtained under the assumption that the PRake receiver

has 5 fingers and the nominal uncoded BER requirement is λn = 10−3. The threshold was

set according to (3–31) with δ = 0.05. Table 5.6 shows the mean detection times for the

serial search and heuristic search strategies. The mean detection time of the serial search

strategy does not change much with increase in SNR even though the size of the hit set

increases significantly. This is because the mean detection time is dominated by the time

spent by the acquisition system in evaluating and rejecting the non-hit set phases before it

reaches the hit set. The heuristic permutation strategy provides an improvement of more

than 70% in the mean detection time compared to the serial search.

5.6 Conclusions

We began with the observation that the serial search may no longer be the optimal

search strategy when the hit set consists of multiple phases which is the case for the dense

UWB channel. We provided a heuristic suboptimal solution to the generally intractable

problem of finding the permutation search strategy which minimizes the mean detection

time by assuming that the detection probabilities of all hit set elements are equal. We also

found that the heuristic search strategy has a simple Jump-by-H structure and hence it

can be generated easily obviating the need to store the whole permutation.

79

p tp,1 tp,2 tp,3

1 1 2 3 4 5 6 7 8 1 2 32 1 2 3 4 5 6 7 8 2 3 43 1 2 3 4 5 6 7 8 3 4 54 1 2 3 4 5 6 7 8 4 5 65 1 2 3 4 5 6 7 8 5 6 76 1 2 3 4 5 6 7 8 6 7 87 1 2 3 4 5 6 7 8 1 7 88 1 2 3 4 5 6 7 8 1 2 8

Table 5-1: Serial search for Ns = 8 and H = 3.

p tp,1 tp,2 tp,3

1 1 4 7 2 5 8 3 6 1 4 72 1 4 7 2 5 8 3 6 2 4 73 1 4 7 2 5 8 3 6 2 5 74 1 4 7 2 5 8 3 6 2 5 85 1 4 7 2 5 8 3 6 3 5 86 1 4 7 2 5 8 3 6 3 6 87 1 4 7 2 5 8 3 6 1 3 68 1 4 7 2 5 8 3 6 1 4 6

Table 5-2: Permutation search (1, 4, 7, 2, 5, 8, 3, 6) for Ns = 8 and H = 3.

SNR Hit set size Serial Search MDT Heuristic Search MDT7 dB 25 0.8808 s 0.2498 s10 dB 40 0.8803 s 0.1993 s

Table 5-3: Mean detection time (MDT) values for the serial search and hueristic searchstrategies.

80

CHAPTER 6UWB TIME-OF-ARRIVAL ESTIMATION STRATEGIES

6.1 Introduction

The high time resolution of UWB signals suggests the possibility of building precise

location estimation systems based on time-of-arrival (TOA) measurements. Impulse

radio is one of the two optional physical layer specifications identified by the IEEE

802.15.4a standardization group for low data rate communications combined with high

precision ranging/location capability (1 meter accuracy and better). The low-power and

low-cost implementation of UWB ranging systems will enable a wide range of applications,

including logistics (package tracking), security applications (localizing authorized persons

in high-security areas), medical applications (monitoring of patients), search and rescue

(locating survivors in avalanche/earthquake rubble), and military applications.

The simplest method of calculating the distance between two asynchronous

transceivers consists of using a packet exchange to get a measure of the signal round-trip

time-of-flight (TOF) and using this time to calculate the distance. A terminal (the

requester) which wants to estimate the round-trip TOF sends packets to the other

terminal (the responder) which responds after a predetermined delay. The delay enables

the requester terminal to switch from the transmitting mode to the receiving mode. Once

the responder terminal’s packets are received by the requester terminal, it can estimate the

round-trip TOF and hence the TOA. This scheme is illustrated in Fig. 6-1. If the TOAs

between a mobile terminal and three distinct anchors (nodes whose positions are known

a priori) are available at a fusion center, the mobile position can be easily computed

in the two-dimensional plane by calculating the intersection of the circles with radii

corresponding to the individual distance estimates of the mobile terminal from the anchors

(as shown in Fig. 6-2).

81

The goal of the TOA estimation algorithm is to find the TOA of the earliest path,

which we will henceforth refer to as the LOS path1 . In a packet-based TOA estimation

protocol, the acquisition of the packet is the first operation which is performed. As we

have seen in earlier chapters, the receiver may not lock to the LOS path. In this chapter,

we propose strategies to locate the LOS path after successful acquisition under different

assumptions about the knowledge of channel statistics.

The chapter is organized as follows. In Section 6.2, we develop a TOA estimation

algorithm under the assumption that the channel statistics are completely known. For the

case of unknown channel statistics, we develop a heuristic TOA estimation algorithm in

Section 6.3. In Section 6.4, we evaluate the performance of these estimation algorithms

using probability of incorrect estimation and mean estimation error as performance

metrics. Section 6.5 has some concluding remarks.

6.2 UWB TOA Estimation: Known Channel Statistics

Like most receiver operations, the TOA estimation algorithm will be executed after

the acquisition operation. We proceed with the design of the TOA estimation algorithms

under the assumption that the acquisition system has successfully locked to a multipath

component in the hit set. This simplifies the design and enables isolation of the TOA

estimation algorithm performance from the performance of the acquisition step preceding

it.

In the case of known channel statistics, the hit set is known and the successful

acquisition assumption leads us to a M -ary hypothesis testing problem where M is

the number of multipath components in the hit set. We will be collecting a number of

observations around the path the acquisition system has locked to. The distribution of

these observations depends on which hit set element was captured by the acquisition

1 The earliest path may be one which passes through several obstacles and hence is nota LOS path in the conventional sense.

82

system. A correct resolution of the true hypothesis generating the observations results

in the identification of the hit set element the acquisition system has locked to. We can

then obtain the location of the beginning of the multipath profile since the position of the

earliest path relative to each hit set element is known.

We obtain a vector of Wl + Wr + 1 observations with a chip spacing of Tc seconds

around the path the acquisition system has locked to. As shown in Fig. 6-3, Wl of these

observations are taken to the left of the acquisition lock position and Wr of them are taken

to the right of the acquisition lock position. Including the observation taken at the lock

position, we have a total of Wl + Wr + 1 observations. Each observation is obtained by

correlating the received signal at the observation position with a reference signal over a

duration of Npl frames. As in the derivation of the IAS decision statistic, the reference TH

signal is given by

q(t) =

Npl−1∑

l=0

ψr(t− lTf − clTc − τ), (6–1)

where τ is chosen such that the pulse ψr(t) is aligned with the observation location. The

observation is given by

y =1

Npl

∫ τ+NplTf

τ

r(t)q(t)dt =√

E1

Ntap−1∑

k=0

qk(∆τ)pkhk + ny (6–2)

where ny is a zero-mean Gaussian random variable with variance σ2y = N0

2Npland qk(∆τ)

is equal to 1 if the observation location corresponds to the kth path of the multipath

profile and it is 0 otherwise. Since the channel taps are placed Tc apart, qk(∆τ) can be 1

for at most one value of k. Thus each observation is either Gaussian distributed or has a

distribution of a random variable which is the sum of a flipped Nakagami random variable

and a Gaussian random variable. The pdf of the observation for the latter case is derived

in Appendix F.

Let Nobs = Wl + Wr + 1. Let the observation vector be Y = [y1 y2 . . . yNobs]T . Let

M be number of multipath components in the hit set the acquisition system can lock to.

83

Let the likelihood of observing Y when the acquisition system has locked to the ith hit set

element be pi(Y), i = 1, 2, . . . , M . Thus we have a M -ary hypothesis testing problem with

hypotheses

Hi : Y ∼ pi. (6–3)

for 1 ≤ i ≤ M . Then under the assumption that the acquisition system is equally likely

to lock to any one of the M hit set elements, the decision rule d(Y) which minimizes the

probability of incorrect decision is

d(Y) = j if pj(Y) ≥ pi(Y) ∀ 1 ≤ i ≤ M. (6–4)

If Li is the location of the ith hit set element, the error induced by deciding on Hi when

Hj is the true hypothesis is given by Cij = |Li − Lj|. Under the assumption of equally

likely hypotheses, the average error induced by deciding on Hi when Y is observed is given

by

Ci(Y) =M∑

j=1

Cijpj(Y ) (6–5)

for 1 ≤ i ≤ M . Then the decision rule which minimizes the average error is given by

d(Y) = j if Cj(Y) ≤ Ci(Y) ∀ 1 ≤ i ≤ M. (6–6)

6.3 UWB TOA Estimation: Unknown Channel Statistics

In this case, we once again assume that the acquisition system has locked to a

multipath component in the hit set. Using the method described in the previous section,

we collect Nobs observations around this position. When the channel statistics are not

known, the distribution of the observation vector Y can be modeled as a Gaussian vector

with mean µ and covariance matrix σ2yI, where I is the Nobs × Nobs identity matrix.

The ith component of the mean vector µ, µi, is non-zero if the ith observation location

contains a multipath component and is zero otherwise. Since the hit set is not known, we

cannot use the M -ary hypothesis testing formulation of the previous section. However, if

84

the number of observations to the left of the locked path, Wl, is large enough, we can hope

that the observation window starts from a position which is to the left of the multipath

profile. In this case, the observations to the left of the multipath profile will have zero

mean.

We could try to cast this problem as a (Nobs − 1)-ary hypothesis testing problem with

the following hypotheses.2

Hi : µj = 0 for 1 ≤ j ≤ i, µi+1 6= 0 (6–7)

for 1 ≤ i ≤ Nobs − 1. However, the values of the non-zero mean components are still

unknown. One way to solve this problem is an extension of the generalized likelihood ratio

test (GLRT) to multiple hypothesis testing. In the GLRT, one substitutes the values of

the unknown parameters with their maximum likelihood estimates. Unfortunately, this

approach is not viable for the situation here. To see this, consider the following hypotheses

for the situation Nobs = 3.

H1 : µ = [ 0 µ2 µ3 ]T , µ2 6= 0

H2 : µ = [ 0 0 µ3 ]T , µ3 6= 0.

The density of the observation vector Y = [ y1 y2 y3 ]T under the hypotheses is

p1(Y; µ2, µ3) =1

(2πσ2y)

32

exp

[−y2

1 + (y2 − µ2)2 + (y3 − µ3)

2

2σ2y

](6–8)

p2(Y; µ3) =1

(2πσ2y)

32

exp

[−y2

1 + y22 + (y3 − µ3)

2

2σ2y

](6–9)

For the case when both y2 and y3 are non-zero, the maximum likelihood estimates of the

unknown parameters are µ2 = y2 and µ3 = y3. Then p1(Y; µ2, µ3) > p2(Y; µ3), for all such

2 Note that the formulation implicitly assumes that at least one observation falls to theleft of the multipath profile.

85

Y. Since a non-zero observation vector occurs with probability one under both hypotheses,

for equally likely hypotheses the test which minimizes the probability of error will always

choose H1. Thus the GLRT approach to dealing with the unknown means is not feasible

for this situation.

We propose to deal with the problem of the unknown parameters by performing local

decisions on each of the observation vector components and using these local decisions

to locate the LOS path. Even though this is a heuristic solution, it performs reasonably

well as evidenced by the numerical results in the next section. For each component of the

observation vector, we consider the following binary hypothesis testing problem

H0 : µi = 0

H1 : µi 6= 0

where yi ∼ N (µi, σ2y). Under the usual conventions, a false alarm is the event of choosing

H1 when H0 is true and a detection event is the event of choosing H1 when H1 is true.

We constrain the probability of false alarm to a small value δ < 1 and seek the uniformly

most powerful (UMP) test, i.e., a test which maximizes the probability of detection for all

non-zero values of the unknown parameter µi. Unfortunately, a UMP test does not exist

for this situation. This is because the most powerful test for positive values of µi does not

coincide with the most powerful test for negative values of µi. However, if we restrict our

attention to unbiased tests, i.e. tests for which the probability of detection is at least δ for

all values of the unknown parameter µi, there exists a UMP unbiased test which is given

by the following

Choose H1 if |yi| > γ (6–10)

where γ = σyQ−1(δ/2).

Applying the above binary test on each component of the observation vector Y results

in a vector of binary decisions where the positions corresponding to the hypothesis H1

give an approximate indication of the location of the multipath components. One way

86

to estimate the location of the LOS path is to choose it to be the position of the earliest

H1 decision in the binary decision vector. In the next section, we evaluate this and other

heuristic methods to locate the beginning of the multipath profile using this binary vector.

6.4 Numerical Results

In this section, we evaluate the performance of the TOA estimation schemes

developed in the previous sections using the probability of incorrect estimation and

the mean estimation error as the performance metrics. We investigate the effect of the

channel model and the location and size of the observation window on the estimation

performance. For each channel model and observation window specification, the number of

pulses used in generating the correlation statistic, Npl, is increased which in turn results in

a linear increase in the signal-to-noise ratio.

6.4.1 Dense UWB Channels

For dense UWB channels, the multipath profile can be modeled as a tapped delay

line with regular tap spacings. The channel model described in Section 2.2 is an example

of a dense UWB channel and will be used for evaluating the performance of the TOA

estimation schemes. We choose the following values for the system parameters: the length

of the channel response Ntap = 100, Nf = 116, Nh = 16. The hit set is obtained under

the assumption that the PRake receiver has 5 fingers and the nominal uncoded BER

requirement is λn = 10−3. The channel statistics are set to the values used in Section 3.5.

For the case of known channel statistics, Figs. 6-4 and 6-5 show the probability of

incorrect estimation of the LOS path location and the mean estimation error as a function

of the number of pulses in the correlation, Npl, for the decision rule described in (6–4),

respectively. These results are for the case when the average energy received per pulse to

noise ratio E1Etot

N0= 5 dB. The number of multipath components in the hit set for this

case is 13. The performance metrics are plotted for different values of Wl and Wr, the

number of observations taken to the left and to the right of the acquisition lock position,

respectively. The performance does not vary with changes in Wr as long as the value of

87

Wl is larger than the hit set size. But once the value of Wl falls below the hit set size the

performance degradation is significant, as seen in the cases when Wl = 10 and Wl = 5.

This is because the left edge of the multipath profile is the location of a sudden change in

channel statistics in dense multipath channels. For the values of Wl which are larger than

the hit set size, this left edge always falls within the observation window. Thus a decision

rule based on the channel statistics is able to perform better for these values of Wl. A

similar trend is observed in Figs. 6-6 and 6-7 which show the performance metrics for the

decision rule described in (6–6).

For the case of unknown channel statistics, we perform the binary hypothesis test

of (6–10) on each observation vector component with δ = 0.01. The location of the LOS

path is chosen to be the leftmost position in the observation vector where the binary

test chooses H1 three times consecutively.. This is a valid heuristic in a dense UWB

channel where the LOS path is immediately followed by other multipaths. We evaluate

this decision rule using the same channel model as the previous decision rules to enable

a fair comparison. Figs. 6-8 and 6-9 show the performance metrics for this rule which

requires a larger number of pulses in the correlation to achieve performance comparable

to the previous decision rules. Once again, the performance is severely degraded if the

value of Wl is smaller than the hit set size. This is because the test will always fail if the

beginning of the multipath profile does not fall in the observation window and the chance

of this event occurring increases when Wl is smaller than the hit set size. The problem,

however, is that the size of the hit set is unknown when the channel statistics are not

known. So larger than necessary observation window sizes might be required to guarantee

good performance of this heuristic estimator.

88

6.4.2 Sparse UWB Channels

A sparse UWB channel consists of clusters of arriving paths [49]. The impulse

response of a sparse UWB channel can be expressed as

hsp(t) =L∑

l=1

Kl∑

k=0

pk,lhk,lδ(t− Tl − τk,l), (6–11)

where L is the number of clusters, Kl is the number of multipath components in the lth

cluster, pk,l, hk,l are the sign and amplitude of the kth component of the lth cluster, Tl

is the arrival time of the lth cluster and τk,l is the delay of the kth component of the lth

cluster relative to the lth cluster arrival time.

The number of clusters L is Poisson distributed with probability mass function

pL(l) =(L)l exp(−L)

l!, (6–12)

where L is the mean of L. The distribution of the cluster arrival times is given by the

Poisson process

p(Tl|Tl−1) = Λ exp[−Λ(Tl − Tl−1)], l > 0, (6–13)

where Λ is the cluster arrival rate. The distributions of the ray arrival times are given by

p(τk,l|τ(k−1),l) = λ exp[−λ(τk − τ(k−1),l)], k > 0, (6–14)

where λ is the ray arrival rate within each cluster. As in the dense UWB channel case,

we model the sign of the a ray component, pk,l, to be equally likely to be 1 or -1 and its

amplitude hk,l to be Nakagami distributed.

For this channel model, the decision rule which assumes knowledge of the channel

statistics becomes prohibitively complex. To see this, let T denote the set of all possible

cluster delay and ray delay realizations. The fading figures and energies of the rays are

assumed to be delay dependent with this dependence known. Given the realization τ ∈ Tand observation vector Y, the likelihood of the ith hypothesis is given by pi(Y; τ). Then

the actual likelihood of the ith hypothesis is given by∑

τ∈T pi(Y; τ). However, the number

89

of realizations in T increases exponentially with the observation window size. Also, the

sparseness of the channel warrants a large observation window as the acquisition lock

might occur quite far from the LOS path.

So for the case of sparse UWB channels, we restrict our attention to decision rules

which do not assume knowledge of channel statistics. Once again, we perform the binary

hypothesis test of (6–10) on each observation vector component with δ = 0.01. We locate

the leftmost position in the observation vector where see a pattern of three consecutive

H0 decisions followed by a H1 decision and at least one more H1 decision in the next two

binary decisions. The location of the LOS path is decided to be the first H1 decision in

this pattern. Figs. 6-10 and 6-11 show the probability of incorrect estimation of the LOS

path location and the mean estimation error as a function of the number of pulses in the

correlation for this decision rule. We choose the mean number of clusters L = 3, the

cluster arrival rate Λ = 0.047 ns−1 and the ray arrival rate λ = 1.54 ns−1. The decay

constant of the energy of a path is Γ = 12.53 ns, i.e., a path at delay τ is weaker than

the LOS path by a factor of exp(τ/Γ). We neglect paths which are 30 dB weaker than the

LOS path. The probability of incorrect estimation is higher than that for the case of dense

channels and increases significantly for values of Wl less than 50. The mean estimation

error has the same trend but there is a slight increase for large values of Npl. An increase

in the number of pulses used in the correlation reduces the variance of the noise and hence

smaller thresholds are sufficient to constrain the probability of false alarm under H0 by

δ. But a smaller threshold results in weaker paths being detected. When the value of Wl

is small, the observation window might start in a position between two clusters and the

weaker paths of the first cluster prevent the consecutive H0 decisions from occurring until

the beginning of the second cluster. This results in the increase in estimation error since

the second cluster is farther from the LOS path.

90

6.5 Conclusions

We have developed TOA estimation schemes under the assumptions of known and

unknown channel statistics. These schemes have been evaluated in dense and sparse

UWB channels. For the case of dense UWB channels, the schemes developed under

the assumption of known channel statistics are capable of achieving probabilities of

incorrect estimation less than 0.01 and mean estimation error less than 0.1 ns, while

the schemes developed under the assumption of unknown channel statistics achieve a

probability of incorrect estimation of less than 0.02 and mean estimation error less than

1 ns. However, sparse UWB channels turn out be challenging with the schemes using the

channel statistic information becoming prohibitively complex and the schemes which do

not use this information resulting in probabilities of incorrect estimation around 0.1 and

mean estimation error around 7 ns.

91

Response packet

TOA

TOA

Round−trip TOF

Response Delay

Transmitted packets

Received packetsCommunication payload

Acquisition header

Request packet

Figure 6-1: Illustration of the packet exchange scheme used to estimate the TOA.

i

Anchor 1Anchor 2

Anchor 3

D

D

D

1

2

3

D = Speed of light X TOAi

Figure 6-2: Mobile positioning based on TOA measurements.

92

Acquisition lock

lW bins W binsr

Multipath profile

Figure 6-3: The location of the observations used for TOA estimation.

50 100 150 200 250 3000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Number of pulses in correlation

Pro

babi

lity

of in

corr

ect e

stim

atio

n

Wl = 20, W

r = 15

Wl = 20, W

r = 10

Wl = 20, W

r = 5

Wl = 15, W

r = 15

Wl = 10, W

r = 15

Wl = 5, W

r = 15

Figure 6-4: Probability of incorrect estimation in dense channels for the rule whichminimizes the error probability when the channel statistics are known.

93

0 50 100 150 200 250 3000

1

2

3

4

5

6

7

8

9

10

Number of pulses in correlation

Mea

n es

timat

ion

erro

r (n

s)

Wl = 20, W

r = 15

Wl = 20, W

r = 10

Wl = 20, W

r = 5

Wl = 15, W

r = 15

Wl = 10, W

r = 15

Wl = 5, W

r = 15

Figure 6-5: Mean estimation error in dense channels for the rule which minimizes the errorprobability when the channel statistics are known.

94

50 100 150 200 250 3000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Number of pulses in correlation

Pro

babi

lity

of in

corr

ect e

stim

atio

n

Wl = 20, W

r = 15

Wl = 20, W

r = 10

Wl = 20, W

r = 5

Wl = 15, W

r = 15

Wl = 10, W

r = 15

Wl = 5, W

r = 15

Figure 6-6: Probability of incorrect estimation in dense channels for the rule whichminimizes the average estimation error when the channel statistics are known.

95

0 50 100 150 200 250 3000

1

2

3

4

5

6

7

Number of pulses in correlation

Mea

n es

timat

ion

erro

r (n

s)

Wl = 20, W

r = 15

Wl = 20, W

r = 10

Wl = 20, W

r = 5

Wl = 15, W

r = 15

Wl = 10, W

r = 15

Wl = 5, W

r = 15

Figure 6-7: Mean estimation error in dense channels for the rule which minimizes theaverage estimation error when the channel statistics are known.

96

50 100 150 200 250 3000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Number of pulses in correlation

Pro

babi

lity

of in

corr

ect e

stim

atio

n

Wl = 20, W

r = 15

Wl = 20, W

r = 10

Wl = 20, W

r = 5

Wl = 15, W

r = 15

Wl = 10, W

r = 15

Wl = 5, W

r = 15

Figure 6-8: Probability of incorrect estimation in dense channels for the heuristic rulewhen the channel statistics are unknown.

97

0 50 100 150 200 250 3000

10

20

30

40

50

60

Number of pulses in correlation

Mea

n es

timat

ion

erro

r (n

s)

Wl = 20, W

r = 15

Wl = 20, W

r = 10

Wl = 20, W

r = 5

Wl = 15, W

r = 15

Wl = 10, W

r = 15

Wl = 5, W

r = 15

Figure 6-9: Mean estimation error in dense channels for the heuristic rule when thechannel statistics are unknown.

98

50 100 150 200 250 3000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Number of pulses in correlation

Pro

babi

lity

of in

corr

ect e

stim

atio

n

Wl = 100, W

r = 20

Wl = 75, W

r = 20

Wl = 50, W

r = 20

Wl = 25, W

r = 20

Figure 6-10: Probability of incorrect estimation in sparse channels for the heuristic rulewhen the channel statistics are unknown.

99

0 50 100 150 200 250 3000

10

20

30

40

50

60

70

80

Number of pulses in correlation

Mea

n es

timat

ion

erro

r (n

s)

Wl = 100, W

r = 20

Wl = 75, W

r = 20

Wl = 50, W

r = 20

Wl = 25, W

r = 20

Figure 6-11: Mean estimation error in sparse channels for the heuristic rule when thechannel statistics are unknown.

100

APPENDIX AAVERAGE NUMBER OF MPCS COLLECTED

The calculation of the expected value of rk(∆τ) is made easy by the observation that

it is a sum of Bernoulli distributed random variables.

rk(αTth + βTc) =1

Nth

Nth−1∑

l=0

1∑i=0

G−1∑j=0

χ(cl + j + β, cl+i+α + k + iNf)

︸ ︷︷ ︸Bernoulli random variable

(A–1)

Hence the expected value is just the sum of the probabilities of the events of each

Bernoulli random variable taking the value 1.

Pr[G−1∑j=0

χ(cl + j + β, cl+i+α + k + iNf) = 1]

= Pr[G−1⋃j=0

(cl+i+α + k + iNf = cl + j + β)]

=G−1∑j=0

Pr[cl+i+α + k + iNf = cl + j + β]

=G−1∑j=0

j+β+Nh−1∑

m=j+β

Pr[cl+i+α + k + iNf = m | cl + j + β = m]Pr[cl + j + β = m]

=G−1∑j=0

j+β+Nh−1∑

m=j+β

1

Nh

Pr[cl+i+α + k + iNf = m | cl + j + β = m]

=G−1∑j=0

j+β+Nh−1∑

m=j+β

1

Nh

Pr[cl+i+α + k + iNf = m]

=G−1∑j=0

minj+β+Nh−1,k+iNf+Nh−1∑

m=maxj+β,k+iNf

1

N2h

=G−1∑j=0

minj+β,k+iNf+Nh−1∑

m=maxj+β,k+iNf

1

N2h

. (A–2)

The sixth equality in the above calculation is due to the random sequence assumption

which does not hold if i + α = 0 (mod Nth). If there is an i1 ∈ 0, 1 such that i1 + α = 0

101

(mod Nth), we have

Pr[G−1∑j=0

χ(cl + j + β, cl+i1+α + k + i1Nf) = 1]

= U(β + G− 1, k + i1Nf)U(k + i1Nf , β), (A–3)

where U(a, b) = 1 if a ≥ b and 0 otherwise. In general, we have

E[rk(αTf + βTc)] = U(β + G− 1, k + i1Nf)U(k + i1Nf , β)

+1∑

i=0i6=i1

G−1∑j=0

minj+β,k+iNf+Nh−1∑

m=maxj+β,k+iNf

1

N2h

, (A–4)

where the expectation is over the set of random TH sequences.

102

APPENDIX BAVERAGE PROBABILITY THAT THE ACQUISITION PROCESS WILL END IN A

FALSE ALARM

Most of the notation used in the following has been defined in Section 3.4 during the

calculation of the mean detection time. As in the calculation of the mean detection time,

we assume that the hit set consists of the phases 0, Tc, 2Tc, . . . , (H − 1)Tc in the search

space Sp = nTc : n ∈ Z and 0 ≤ n ≤ Ns− 1. Let PF(γ, n) be the average probability that

the acquisition process will end in a false alarm conditioned on the event that the serial

search starts at the nth position in Sp. Then we have

PF(γ) =1

Ns

Ns∑n=1

PF(γ, n). (B–1)

First, suppose that the initial value of the hypothesized phase lies to the right of the

hit set, i.e., n ∈ H + 1, H + 2, . . . , Ns. Let Pf(i) denote the average probability of false

alarm at the ith position of the search space when H + 1 ≤ i ≤ Ns. Then

PF(γ, n) = 1−Ns∏i=n

(1− Pf(i)) +

[Ns∏i=n

(1− Pf(i))

][1−

Ns∏i=H+1

(1− Pf(i))

]PM

+

[Ns∏i=n

(1− Pf(i))

][Ns∏

i=H+1

(1− Pf(i))

][1−

Ns∏i=H+1

(1− Pf(i))

]P 2

M

+

[Ns∏i=n

(1− Pf(i))

][Ns∏

i=H+1

(1− Pf(i))

]2 [1−

Ns∏i=H+1

(1− Pf(i))

]P 3

M

+ · · ·

= 1−Ns∏i=n

(1− Pf(i)) +

[∏Ns

i=n(1− Pf(i))] [

1−∏Ns

i=H+1(1− Pf(i))]PM

1− PM

∏Ns

i=H+1(1− Pf(i))

= 1−[

Ns∏i=n

(1− Pf(i))

][1− PM

1− PM

∏Ns

i=H+1(1− Pf(i))

]. (B–2)

Now suppose that the initial value of the hypothesized phase falls in the hit set, i.e.,

n ∈ 1, 2, . . . , H. Then

PF(γ, n) = PF(γ, H + 1)H∏

i=n

(1− Pd(i)). (B–3)

103

The average probability that the acquisition process will end in a false alarm is now

obtained by substituting (B–2) and (B–3) in (B–1).

104

APPENDIX CPROOF THAT Aε(γ; ∆τ) AND Bε(γ; ∆τ) DEFINED IN (4–25) SATISFY THE

CONDITIONS OF THEOREM 1

The first condition on the sets is verified in the following manner using (4–23) and

(4–24).

Pr(Aε(γ; ∆τ) ∪ Bε(γ; ∆τ)) = 1− Pr(Acε(γ; ∆τ) ∩ Bc

ε(γ; ∆τ))

= 1− Pr(h : |s(∆τ ;h)− γ| ≤ κ(γ, ε)

or s(∆τ ;h) ≥ nKs)

≥ 1− Pr(h : γ − κ(γ, ε) ≤ s(∆τ ;h) ≤ γ + κ(γ, ε))

−Pr(h : s(∆τ ;h) ≥ nKs)

> 1− ε

2− ε

2= 1− ε. (C–1)

From (3–5) and (3–6), we see that σ2y(∆τ ;h) = σ2K1s(∆τ ;h) where K1 =

4Rψ3r(0)

Rψ2r(0)MNth

.

Then for h ∈ Aε(γ; ∆τ), we have

FN(γ; ∆τ |h) = 1−Q

γ − s(∆τ ;h)− µy√

σ2K1s(∆τ ;h) + ν2y

> 1−Q

κ(γ, ε)− µy√

σ2K1(γ − κ(γ, ε)) + ν2y

.

We can assume, without loss of generality, that γ−κ(γ, ε) ≥ 0. Then the Q-function in the

above equation is a decreasing function of σ2 and converges to 0 as σ2 → 0. So there exists

an η1(γ, ε) > 0 such that FN(γ; ∆τ |h) > 1− ε for all h ∈ Aε(γ; ∆τ) whenever σ2 < η1(γ, ε).

For h ∈ Bε(γ; ∆τ), we have

FN(γ; ∆τ |h) = 1−Q

γ − s(∆τ ;h)− µy√

σ2K1s(∆τ ;h) + ν2y

= Q

µy + s(∆τ ;h)− γ√

σ2K1s(∆τ ;h) + ν2y

< Q

(κ(γ, ε) + µy√nσ2K1Ks + ν2

y

). (C–2)

105

As before, the Q-function is a decreasing function of σ2 and hence there exists an

η2(γ, ε) > 0 such that FN(γ; ∆τ |h) < ε for all h ∈ Bε(γ; ∆τ) whenever σ2 < η2(γ, ε).

By choosing η(γ, ε) = min(η1(γ, ε), η2(γ, ε)), we see that the second and third conditions of

Theorem 1 hold for σ2 < η(γ, ε).

106

APPENDIX DPROOF THAT Aε(γ; ∆τ) AND Bε(γ; ∆τ) DEFINED IN (4–35) SATISFY THE

CONDITIONS OF THEOREM 1

The first condition on the sets is verified in the following manner using (4–34).

Pr(Aε(γ; ∆τ) ∪ Bε(γ; ∆τ)) = 1− Pr(Acε(γ; ∆τ) ∩ Bc

ε(γ; ∆τ))

= 1− Pr(h : |V (∆τ ;h)−√γ| ≤ κ(γ, ε)

or |V (∆τ ;h) +√

γ| ≤ κ(γ, ε))

> 1− ε. (D–1)

Furthermore for σ2 < η(γ, ε) = κ2(γ,ε)

2 ln 2ε

and h ∈ Aε(γ; ∆τ),

FN(γ; ∆τ |h) = 1−Q

(√γ − V (∆τ ;h)

σz

)−Q

(√γ + V (∆τ ;h)

σz

)

≥ 1− 1

2exp

[−(√

γ − V (∆τ ;h))2

2σ2z

]− 1

2exp

[−(√

γ + V (∆τ ;h))2

2σ2z

]

> 1− 1

2exp

[−κ2(γ, ε)

2σ2z

]− 1

2exp

[−κ2(γ, ε)

2σ2z

]

= 1− exp

[−κ2(γ, ε)

2σ2z

]> 1− ε. (D–2)

For h ∈ Bε(γ; ∆τ) such that V (∆τ ;h) >√

γ + κ(γ, ε),

FN(γ; ∆τ |h) = 1−Q

(√γ − V (∆τ ;h)

σz

)−Q

(√γ + V (∆τ ;h)

σz

)

= Q

(V (∆τ ;h)−√γ

σz

)−Q

(√γ + V (∆τ ;h)

σz

)

< Q

(V (∆τ ;h)−√γ

σz

)≤ exp

[−(V (∆τ ;h)−√γ)2

2σ2z

]

< exp

[−κ2(γ, ε)

2σ2z

]<

ε

2(D–3)

Similarly for h ∈ Bε(γ; ∆τ) such that V (∆τ ;h) < −√γ − κ(γ, ε), we can show that

FN(γ; ∆τ |h) < ε2.

107

APPENDIX EPROOF THAT Q (DEFINED IN (5–12)) IS THE VECTOR IN THE SET A

CORRESPONDING TO THE PERMUTATION R

We first show that∑k

i=1 si is equal to rk defined in (5–11), for k = 1, 2, . . . , H, for all

possible values of d = GCD(Ns, H).

Case 1: d = H

Let Ns = MH for some positive integer M . Then the ith position in the permutation

R is given by

Ri = (i− 1)H (mod Ns) +

⌊i− 1

M

⌋+ 1, (E–1)

for i = 1, 2, . . . , Ns. It is easy to see that the permutation consists of H consecutive blocks

each having M elements where the kth block can be written as

(k, H + k, 2H + k, . . . , (M − 1)H + k), (E–2)

for k = 1, 2, . . . , H. Since any two positions in a block are at least H − 1 phases apart

in Sp, for any position p of the first element of the hit set there is exactly one position

in the block where a hit set element appears. Thus, for a particular value of p ∈ Sp, the

ith appearance of a hit set element is in the ith block, i.e., (i − 1)M + 1 ≤ tp,i ≤ iM

for i ∈ 1, 2, . . . , H. Furthermore, a hit set element appears in every position of a block

exactly H where each appearance corresponds to a distinct value of p in Sp. Then we have

si =Ns∑p=1

tp,i = ((i− 1)M + 1) ·H + ((i− 1)M + 2) ·H + · · ·+ iM ·H, (E–3)

and

k∑i=1

si = 1 ·H + 2 ·H + · · ·+ kM ·H =kM(kM + 1)H

2

=Nk(Nk + 1)H

2+ (Nsk −NkH)(Nk + 1), (E–4)

where the last equality follows from the fact that Nk = Mk and Ns = NkH = MHk.

108

Case 2: d = 1

In this case, the ith position in the permutation is given by

Ri = (i− 1)H (mod Ns) + 1, (E–5)

for i = 1, 2, . . . , Ns.

We first calculate s1 =∑Ns

p=1 tp,1. Thus we need to enumerate the positions of the

first appearance of a hit set element in the permutation for all p ∈ Sp. Consider the first

N1 + 1 positions of the permutation, namely R1, R2, . . . , RN1+1. Note that the locations of

these positions in the search space are such that any two consecutive positions are located

H − 1 phases apart and any block of H consecutive phases in the search space Sp contains

one of these positions. We claim that for all p ∈ Sp, the first appearance of a hit set

element occurs in one of these positions, i.e., tp,1 ≤ N1 + 1. Suppose this is false. Then

there exists a p ∈ Sp such that when the first element of the hit set block is in p, none of

the H consecutive hit set phases appear in R1, R2, . . . , RN1+1. This implies that there is

a block of H consecutive phases in Sp which does not contain any of R1, R2, . . . , RN1+1,

which is a contradiction. Any two positions in the first N1 positions of the permutation,

namely R1, R2, . . . , RN1 , are at least H − 1 phases apart in the search space. Thus for a

particular value of p, a hit set element can appear only in one of these positions. So every

appearance of a hit set element in these N1 positions is a first appearance and corresponds

to a distinct p ∈ Sp. Since a hit set element appears exactly H times in every Ri, there are

exactly H first appearances in each one of R1, R2, . . . , RN1 and there cannot be any more

appearances. This accounts for the first appearance of a hit set element corresponding

to N1H distinct p’s in Sp. By the fact that tp,1 ≤ N1 + 1 for all p ∈ Sp, the remaining

Ns −N1H appearances have to occur in RN1+1. Thus

s1 =Ns∑p=1

tp,1 = 1 ·H + 2 ·H + · · ·+ N1 ·H + (N1 + 1) · (Ns −N1H)

=N1(N1 + 1)H

2+ (Ns −N1H)(N1 + 1). (E–6)

109

The remaining H − (Ns − N1H) appearances of a hit set element in RN1+1 are second

appearances of a hit set element since these correspond to p’s in Sp for which the first

appearance of a hit set element is not in RN1+1 but in one of R1, R2, . . . , RN1 .

In order to calculate s2 =∑Ns

p=1 tp,2, we consider the positions RN1+2, RN1+3, . . . , RN2+1.

The locations of these positions in the search space are such that any two consecutive

positions are located H − 1 phases apart and any block of H consecutive phases in the

search space Sp contains one of these positions. Hence by the argument in the previous

paragraph, a hit set element appears in one of these positions for all p ∈ Sp. Since

the first appearance of a hit set element occurs in R1, R2, . . . , RN1+1 for all p ∈ Sp,

each appearance of a hit set element in RN1+2, RN1+3, . . . , RN2+1 is either the second or

third1 appearance of a hit set element. Thus we have tp,2 ≤ N2 + 1 for all p ∈ Sp.

Furthermore, any two positions in the positions RN1+1, RN1+2, . . . , RN2 are at least

H − 1 phases apart in the search space. Thus for a particular value of p, a hit set

element can appear only in one of these positions. So each of the H appearances of a

hit set element in RN1+2, RN1+3, . . . , RN2 is a second appearance of a hit set element and

corresponds to a distinct p ∈ Sp. None of these appearances is a third appearance of

a hit set element because the only way this can happen is that for a particular p in Sp

the second appearance occurs in RN1+1 and for that same p a hit set element appears in

RN1+2, RN1+3, . . . , RN2 . We already know that H− (Ns−N1H) second appearances of a hit

set element occur in RN1+1. So far we have enumerated H − (Ns−N1H) + (N2−N1− 1)H

(= N2H − Ns) second appearances of a hit set element in RN1+1, RN1+2, . . . , RN2

corresponding to distinct p’s in Sp and there cannot be more any more appearances in

these positions. Since tp,2 ≤ N2 + 1, the remaining Ns − (N2H − Ns) second appearances

1 Due to the fact that some second appearances occur in RN1+1.

110

have to occur in RN2+1. Thus

s2 =Ns∑p=1

tp,2 = (N1 + 1) · (H − (Ns −N1H)) + (N1 + 2) ·H + (N1 + 3) ·H +

· · ·+ N2 ·H + (N2 + 1) · (2Ns −N2H) (E–7)

and using (E–6) we have

2∑i=1

si =2∑

i=1

Ns∑p=1

tp,i = 1 ·H + 2 ·H + · · ·+ N1 ·H + (N2 + 1) · (2Ns −N2H)

=N2(N2 + 1)H

2+ (2Ns −N2H)(N2 + 1). (E–8)

The value sk =∑Ns

p=1 tp,k for k > 2 can be calculated using arguments very similar to those

used in calculating s2. Finally, we get

k∑i=1

si =Nk(Nk + 1)H

2+ (Nsk −NkH)(Nk + 1). (E–9)

for k = 1, 2, . . . , H.

Case 3: 1 < d < H

Let M = Ns

d. Then the ith position in the permutation R is given by

Ri = (i− 1)H (mod Ns) +

⌊i− 1

M

⌋+ 1, (E–10)

for i = 1, 2, . . . , Ns. As in Case 1, the permutation consists of d consecutive blocks each

having M elements where the kth block can be written as

(RM(k−1)+1, RM(k−1)+2, . . . , RMk) = (k, H + k, 2H + k, . . . , (M − 1)H + k), (E–11)

for k = 1, 2, . . . , d. The ith position in the first block (R1, R2, . . . , RM) is given by

Ri = (i− 1)H (mod Ns) + 1, (E–12)

for i = 1, 2, . . . , M . Note that the structure of the permutation in the first block is the

same as the structure of the permutation in Case 2, i.e., any two consecutive positions are

111

located H − 1 phases apart in the search space. Let K = Hd. Then NK = b Ns

Hdc = M .

Since in Case 2, the value of sk depended only on the relative locations of the positions

R1, R2, . . . , RNk+1, we have

k∑i=1

si =Nk(Nk + 1)H

2+ (Nsk −NkH)(Nk + 1), (E–13)

for k < K. This argument cannot be extended for the case when k = K because the

positions 1, 2, . . . , RNK+1 go beyond the first block and hence their relative positions

are not as in Case 2. Nevertheless, the positions RNK−1+1, RNK−1+2, . . . , RNKare such

that any two positions are at least H − 1 phases apart in the search space and any

block of H consecutive phases contains exactly one of these positions. So each of the H

appearances of a hit set element in RNK−1+2, RNK−1+3, . . . , RNKis a Kth appearance of

a hit element and corresponds to a distinct p in Sp. We know from Case 2 that a Kth

appearance occurs H − (Ns(K − 1) − NK−1H) times in RNK−1+1. This accounts for

H − (Ns(K − 1) − NK−1H) + (NK − NK−1 − 1)H (= Ns) Kth appearances of a hit set

element. Since these Kth appearances all correspond to distinct p’s in Sp, this accounts

for all possible Kth appearances and hence we have

K∑i=1

si =NK(NK + 1)H

2=

NK(NK + 1)H

2+ (Ns −NKH)(NK + 1), (E–14)

where the second equality follows from the fact that Ns = Ns

d= MH = NKH. Thus

the first K appearances of a hit set element occur in the first block. Since the relative

locations of the positions in any of the d blocks are the same as those in the first block,

the kth K appearances of a hit set element occur in the kth block. Thus

kK∑i=1

si =Mk(Mk + 1)H

2, (E–15)

for k = 1, 2, . . . , d. Furthermore, the ith appearance of a hit set element in the kth block

occurs in the same positions within the block as the ith appearance of a hit set element

in the first block and the ith appearance of a hit set element in the kth block is the

112

((k − 1)K + i)th appearance overall of a hit set element in the permutation. Thus we have

(k−1)K+i∑

l=(k−1)K+1

sl = iM(k − 1)HNs +i∑

l=1

sl, (E–16)

for k = 1, 2, . . . , d. Note that for j = (k − 1)K + i, we have

Nj =

⌊Ns((k − 1)K + i)

H

⌋=

⌊Ns(k−1)H

d+ Nsi

H

⌋=

⌊M(k − 1) +

Ns

H

⌋= M(k − 1) + Ni.

(E–17)

Thus for j = (k − 1)K + i, from (E–15) and (E–16) we have

j∑

l=1

sl =

(k−1)K∑

l=1

sl +

(k−1)K+i∑

l=(k−1)K+1

sl

=M(k − 1)(M(k − 1) + 1)H

2+ iM(k − 1)HNs +

i∑

l=1

sl

=M(k − 1)(M(k − 1) + 1)H

2+ iM(k − 1)HNs +

Ni(Ni + 1)H

2

+(Ns −NiH)(Ni + 1)

=Nj(Nj + 1)H

2+ (Ns −NjH)(Nj + 1) (E–18)

Thus for all possible values of d we have shown that

k∑i=1

si =Nk(Nk + 1)H

2+ (Ns −NkH)(Nk + 1) = rk, (E–19)

for k = 1, 2, . . . , H. Then the vector in the set A corresponding to the permutation R is

given by

(sH , sH−1, . . . , s1) = (rH − rH−1, rH−1 − rH−2, . . . , r2 − r1, r1) = q. (E–20)

113

APPENDIX FTHE PDF THE SUM OF A FLIPPED NAKAGAMI RANDOM VARIABLE AND A

GAUSSIAN RANDOM VARIABLE

We wish to evaluate the pdf of the following random variable

y =√

E1pkhk + ny (F–1)

where pk is equally likely to be +1 or -1, hk is a Nakagami random variable with

parameters m, Ω and ny is a zero-mean Gaussian random variable with variance σ2y.

It is also given that pk, hk and ny are independent.

For convenience, let A0 =√

E1. If hk has pdf pH and ny has pdf pN , the pdf of y is

given by

pY (y) =1

2

∫ ∞

0

1

A0

pH

(x

A0

)pN(y − x)dx +

1

2

∫ ∞

0

1

A0

pH

(x

A0

)pN(y + x)dx

=1

A0Γ(m)

(m

Ω

)m 1√2πσy

[∫ ∞

0

(x

A0

)2m−1

e−mx2

A20Ω e

− (y−x)2

2σ2y dx

+

∫ ∞

0

(x

A0

)2m−1

e−mx2

A20Ω e

− (y+x)2

2σ2y dx

]

=1

A2m0 Γ(m)

(m

Ω

)m e− y2

2σ2y

√2πσy

[∫ ∞

0

x2m−1e−

m

A20Ω

+ 1

2σ2y

x2+ xy

σ2y dx

+

∫ ∞

0

x2m−1e−

m

A20Ω

+ 1

2σ2y

x2− xy

σ2y dx

]

= A1

∫ ∞

0

x2m−1e−A2x2+A3xdx + A1

∫ ∞

0

x2m−1e−A2x2−A3xdx

= A1(2A2)−mΓ(2m)e

A23

8A2

[D−2m

(− A3√

2A2

)+ D−2m

(A3√2A2

)]

= A1(2A2)−mΓ(2m)2−m+1

√π

Γ(m + 0.5)1F1

(m, 0.5,

A23

4A2

)

The last two equalities follow from (3.462.1) and (9.240) in [58], respectively. Here

Dp(z) is the parabolic cylinder function and 1F1(α, β; z) is the confluent hypergeometric

function.

114

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BIOGRAPHICAL SKETCH

Saravanan Vijayakumaran received the B.Tech. degree in electronics and communication

engineering in 2001 from the Indian Institute of Technology at Guwahati and the M.S.

degree in electrical engineering in 2004 from the University of Florida, Gainesville. He

is currently pursuing the Ph.D. degree at the University of Florida. From Jan. 2006 to

July 2006, he was a research intern at the Laboratory for Computer Communications

and Applications, Ecole Polytechnique Federale de Lausanne, Lausanne, Switzerland. His

research interests include wireless communications and information theory.

120