Post on 04-Oct-2020
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.
3
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.
12
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