«Ltd.

“an. .

. kn... . (.il.
.. ii. \ . .1...)
.JPC}! 1W“ 5...:
. vamfifi 5.“.
. . . a
HIM]... a}... !
. . : .. .
341, 3:19 flung .

«4

.3212:

Wt . 1

. v.8

42.34341... .
:31...... L...
5 1.3.13 3..
51.». Inf-:7 u}

rascl; if .. . 21...
. 3.33 .1... 5. gfiuh.» kammmfiwgm. .umwtgfiiaww

 

 

 

I

#0)}?

This is to certify that the
thesis entitled

Spatial Diversity in Wireless Sensor Networks

presented by

Sivanvitha Devarakonda

has been accepted towards fulfillment
of the requirements for the

Master of degree in Electrical Engineering
Science

 

 

    

4"” - ’ f/ ’L
/ (V'Najor PKersgor’s Signature

04/09/2007

 

Date

MSU is an affirmative-action, equal-opportunity employer

-.--.-.-.—.—.—~—.-.--.—.--;--o—-

 

 

LIBRARY '
Michigan State

University

 

 

PLACE IN RETURN BOX to remove this checkout from your record.
To AVOID FINES return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

DAIEDUE

DAIEDUE

DAIEDUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6/07 p:/ClRC/DateDue.indd-p.1

SPATIAL DIVERSITY IN WIRELESS SENSOR NETWORKS

By

Sivanvitha Devarakonda

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE
Department of Electrical and Computer Engineering

2007

ABSTRACT
SPATIAL DIVERSITY IN WIRELESS SENSOR NETWORKS
By
Sivanvitha Devarakonda

This thesis focuses on the applications of spatial diversity for (i) distributed data
fusion and (ii) improving estimation accuracy in sensor topologies that can be modeled as
correlated Gaussian random fields. Distributed sensor networks, are essentially multi-
terminal systems. This motivates the application and extension of recent advances in
Multiple Input Multiple Output (MIMO) systems to Wireless Sensor Networks (W SNs).
The capacity increase that diversity promises translates to energy efficiency and latency
reduction in distributed sensor networks. We apply Space Time Block Codes (STBC) for
joint transmission and data fusion, a method that enables high rate information retrieval.
The fusion schemes are based on orthogonal space-time block codes, i.c. the Alamouti
(Quatemion) and the Octom'on codes. The reduction in latency of the proposed space-
time fusion is demonstrated through comparison with optimal fusion Schemes under
identical constraints.

In the latter part we consider distributed estimation for correlated Gaussian random
fields in wireless fading channels. The optimum estimator is derived based on a Multiple
Input Multiple Ouput (MIMO) abstraction that integrates data correlation with spatial
diversity. The estimator performance is characterized in terms of multiuser diversity,
channel signal-to-noise ratio and receive diversity of the system. By integrating spatial
correlation with receive diversity, we show that the field estimation accuracy can be

significantly improved while ensuring energy and bandwidth efficiency.

Copyright by
SIVANVITHA
DEVARAKONDA
2007

“Guru Brahma Guru. Vishnu Guru Devo Maheshwara

Guru Sakshath Parambrahma Tasmai Sm' Gurave Namaha”

- Adi Sankara

This work is dedicated to Sri Viswayogi Viswamji and all my Gurus

iv

ACKNOWLEDGMENTS

“Whatever you can do or dream you can, begin it. BoldneSs has genius, power and magic
in it. Begin it now ” - Goethe

Words are not enough to express my gratitude to my parents for kindling the “dream
it and achieve it” spirit in me, for their constant encouragement and indefatigable support
of all my endeavors. I thank my advisor Dr. Radha Hayder for always keeping in
perspective the big picture, his excellent research insight, for his constant support and his
continued confidence in me. I thank Dr. Naofal Al-Dhahir, for introducing me to the
great expectations of spatial diversity and for his guidance and support. I owe special
thanks to Kiran Misra for his guidance, interest and his eye for technical detail. Thanks to
Shirish Karande for his supply of interesting ideas and references. I owe many thanks to
Syed Ali Khayam, Naveen Nair, Yong Ju, and Sushantha Das for their timely help. My
thanks will not be complete without mentioning my husband Chandrasekhar for being my

rock of Gibraltar and my family for always being there.

TABLE OF CONTENTS

 

 

 

 

 

LIST OF FIGURES VIII
LIST OF TABLES - XI
1. INTRODUCTION 1
2. RELATED WORK 6
2.1. THE PARALLEL FUSION NETWORK .............................................................. 7

2.2. DESIGN OF FUSION RULES ........................................................................... 8

2.3. CHANNEL UNAWARE DATA FUSION IN PARALLEL NETWORKS .................... 9

2.4. CHANNEL AWARE FUSION IN PARALLEL NETWORKS ................................. 1 1

2.5. CHANNEL AWARE DECISION FUSION IN WIRELESS FADING ....................... 12

2.5.1. Optimal likelihood-ratio (LR) based fusion rule .............................. I 4

2.5.2. Sub-optimum fusion .......................................................................... I 4

2.5.2.1. Chair-Varshney fusion rule ............................................................ 15

2.5.2.2. Maximum ratio combining fusion statistic .................................... 15

2.5.2.3. Equal gain combining fusion rule .................................................. 16

3. BACKGROUND 17
3.1. SPACE-TIME PROCESSING IN WIRELESS NETWORKS .................................. 17

3.2. THE MIMO CHANNEL MODEL .................................................................. 20

3.3. REVIEW OF INFORMATION THEORETIC RESULTS ........................................ 22

3.3.1. Capacity ofA/HMO channels ............................................................. 23

3.3.1.1. Low SNR approximation of the ergodic capacity ......................... 24

3.3.1.2. High SNR approximation of the ergodic capacity ......................... 25

3.3.1.3. Outage capacity .............................................................................. 25

3.3.1.4. Effective diversity order ................................................................ 26

3.4. INTRODUCTION TO SPACE-TIME CODES ..................................................... 28

3.4.1. Space- Time Block Codes (S TBCs) .................................................... 28

3.4.1.1. Orthogonal space-time block codes (O-STBC) ............................. 29

3.4.1.2. The Alamouti space-time block code ............................................ 29

3.4.1.3. Real orthogonal space-time block code for four transmit antennas3l
3.4.1.4. Complex orthogonal space-time block code for four transmit

antennas ......................................................................................... 32
3 .4. 1.5. Quasi-orthogonal designs .............................................................. 32
3.4.1.6. Diversity embedding in space-time codes ..................................... 33

4. CHANNEL AWARE DATA FUSION AND DISTRIBUTED DETECTION 34
4.1. CHANNEL AWARE DATA FUSION I — UNCERTAINTY VIA BINARY

vi

SYMMETRIC CHANNELS ............................................................................. 36

4.1.1. System representation and problem formulation .............................. 36
4.1.1.1. Source uncertainty — binary source ................................................ 36
4.1.1.2. Transmission channel .................................................................... 38

4.1.2. Data fitsion based on space-time coding for two sensors ................. 38

4.1.3. Simulation results and performance analysis ................................... 42

4.2. CHANNEL AWARE DATA FUSION II — SOURCE UNCERTAINTY VIA GAUSSIAN
NOISE ......................................................................................................... 46

4.2.1. Problem formulation — Gaussian observation noise ........................ 46

4.2.2. Data Fusion using space-time block codes ....................................... 49

4.2.3. Space-time fusion for four sensors .................................................... 50

4.2.4. Detection performance and comparison with optimal fusion schemes

.......................................................................................................... 53

4.2.5. Review of comparative optimal fusion rules in fading channels ...... 54

4.2.6. Detection performance of space-time data fitsion ............................ 56
4.2.6.1. Results for two sensor space-time data fusion ............................... 5 7
4.2.6.2. Results for four sensor Space-time fusion ...................................... 62

4.2. 7. Performance results of two sensor fitsion under equal energy and

time constraints ................................................................................. 66

5. DISTRIBUTED ESTIMATION - IMPACT OF RECEIVE DIVERSITY AND

 

 

CORRELATION 71
5.1. BACKGROUND AND OVERVIEW .................................................................. 71

5.2. THE PHYSICAL MODEL AND DATA CORRELATION ..................................... 75

5.2.1. Modeling the physical phenomenon ...................... ; ........................... 75

5.2.2. The correlation model ....................................................................... 76

5.3. SYSTEM MODEL AND OPTIMAL ESTIMATION ............................................. 77

5.3.1. The optimum estimator ..................................................................... 77

5.3.2. The performance of the estimator ..................................................... 79

5.4. SIMULATION RESULTS AND ANALYSIS ....................................................... 81

5.4.1. MSE as a fimction of spatial correlation (node separation) ............. 82

5.4.2. MSE as a fimction of multiuser diversity (joint transmissions) ........ 84

5.4.3. MSE as a function of receive diversity .............................................. 85

6. CONCLUSIONS AND FUTURE WORK - - - 90
REFERENCES _ - - - 93

 

Vii

LIST OF FIGURES

FIGURE 1. CANONICAL DISTRIBUTED DETECTION SYSTEM: THE PARALLEL FUSION
NETWORK ..................................................................................................................... 7
FIGURE 2. PARALLEL FUSION NETWORK IN THE PRESENCE OF FADING AND RECEIVER
NOISE ............................................................................................ 13
FIGURE 3. IMPLEMENTATION OF TRANSMIT AND RECEIVE LEVEL DIVERSITY IN MISO
AND SIMO SYSTEMS .................................................................................................. 18
FIGURE 4. REPRESENTATION OF THE ALAMOUTI SPACE-TIME BLOCK CODING
SCHEME .......................................................................................... 30
FIGURE 5. SYSTEM REPRESENTATION OF THE PARALLEL FUSION NETWORK WITH THE
BINARY SYMMETRIC CHANNEL (BSC) MODELING THE SOURCE UNCERTAINTY ......... 36
FIGURE 6. PLOT OF MUTUAL INFORMATION VS. CROSSOVER PROBABILITIES OF
BERNOULLI DISTRIBUTED LOCAL SENSOR DECISIONS ................................................. 43
FIGURE 7. PROBABILITY OF ERROR (P6. ) AS A FUNCTION OF CHANNEL SNR AND
MUTUAL INFORMATION ............................................................................................. 44
FIGURE 8. PROBABILITY OF ERROR VS. CHANNEL SN R FOR TWO SENSORS WITH EQUAL
SOURCE UNCERTAINTY (CROSS OVER PROBABILITIES OF THE BSCS EQUAL) .............. 45
FIGURE 9. SYSTEM REPRESENTATION OF A PARALLEL FUSION NETWORK OBSERVING A
PARAMETER o IN THE PRESENCE OF GAUSSIAN NOISE AT THE SENSORS ...................... 46

FIGURE 10. RECEIVER OPERATING CHARACTERIan OF LOCAL SENSORS AS FUNCTION
OF NOISE VARIANCE ................................................................................................... 48

FIGURE I I. SPACE-TIME TRANSMISSION AND FUSION FOR TWO SENSORS WITHOUT CO-
OPERATION AND SINGLE RECEIVE ANTENNA AT THE FUSION CENTER. ........................ 49

FIGURE 12. RECEIVER OPERATING CHARACTERISTICS OF SPACE-TIME FUSION VS.
OPTIMUM FUSION BASED ON MRC RULE WITH 0?, = 0.1 FOR THE TWO SENSOR

SCENARIO. .................................................................................................................. 59

FIGURE 13. RECEIVER OPERATING CHARACTERISTICS OF SPACE-TIME FUSION VS.
OPTIMUM FUSION BASED ON MRC RULE WITH 0,27 = 0.2 FOR THE Two SENSOR
SCENARIO. .................................................................................................................. 59

FIGURE 14. RECEIVER OPERATING CHARACTERISTICS OF SPACE-TIME FUSION VS.

viii

OPTIMUM FUSION BASED ON MRC RULE WITH 0,27 = 0.3 FOR THE Two SENSOR
SCENARIO. .................................................................................................................. 60

FIGURE 15. RECEIVER OPERATING CHARACTERISTICS OF SPACE-TIME FUSION VS.
OPTIMUM FUSION BASED ON MRC RULE WITH 0% = 0.4 FOR THE Two SENSOR
SCENARIO. .................................................................................................................. 60

FIGURE 16. PROBABILITY OF DETECTION OF THE SYSTEM AS A FUNCTION OF CHANNEL
SNR FOR DIFFERENT LEVELS OF SENSOR NOISE FOR Two SENSOR DATA FUSION ........ 61

FIGURE 17. RECEIVER OPERATING CHARACTERISTICS OF SPACE-TIME FUSION VS.
OPTIMUM FUSION BASED ON MRC RULE WITH 0% = 0.1 FOR FOUR SENSOR DATA

FUSION .......................................................................................... 63

FIGURE 18. RECEIVER OPERATING CHARACTERISTICS OF SPACE-TIME FUSION VS.
OPTIMUM FUSION BASED ON MRC RULE WITH 0% = 0.2 FOR FOUR SENSOR DATA
FUSION .......................................................................................... 63

FIGURE 19. RECEIVER OPERATING CHARACTERISTICS OF SPACE-TIME FUSION VS.
OPTIMUM FUSION BASED ON MRC RULE WITH 0% = 0.3 FOR FOUR SENSOR DATA
FUSION .......................................................................................... 64

FIGURE 20. RECEIVER OPERATING CHARACTERISTICS OF SPACE-TIME FUSION VS.
OPTIMUM FUSION BASED ON MRC RULE WITH 0% = 0.5 FOR FOUR SENSOR DATA

FUSION .......................................................................................... 64

FIGURE 21. SYSTEM PROBABILITY OF ERROR VS. CHANNEL SNR FOR FOUR SENSOR DATA
FUSION .......................................................................................... 65

FIGURE 22. THE TRANSMISSION TIME AND ENERGY PER DECISION FOR SPACE-TIME
FUSION VS. ROUND ROBIN TRANSMISSIONS FOR THE TWO SENSOR FUSION FOR
IDENTICAL ENERGY AND DELAY CONSTRAINTS .......................................................... 67

FIGURE 23. RECEIVER OPERATING CHARACTERISTICS FOR DATA FUSION SCHEMES FOR
Two SENSORS WITH IDENTICAL ENERGY AND DELAY CONSTRAINTS WITH SENSOR

VARIANCE, 0,2, = 0.1 ................................................................................................ 68

FIGURE 24. RECEIVER OPERATING CHARACTERISTICS FOR DATA FUSION SCHEMES FOR
TWO SENSORS WITH IDENTICAL ENERGY AND DELAY CONSTRAINTS WITH SENSOR

VARIANCE, 0?, = 0.2 ................................................................................................ 68

FIGURE 25. RECEIVER OPERATING CHARACTERISTICS FOR DATA FUSION SCHEMES FOR

ix

TWO SENSORS WITH IDENTICAL ENERGY AND DELAY CONSTRAINTS WITH SENSOR
VARIANCE, 0,2, = 0.4 ................................................................................................ 69

FIGURE 26. RECEIVER OPERATING CHARACTERISTICS FOR DATA FUSION SCHEMES FOR
TWO SENSORS WITH IDENTICAL ENERGY AND DELAY CONSTRAINTS WITH SENSOR

VARIANCE, 0,27 = 0.5 ................................................................................................ 69

FIGURE 27. REPRESENTATION OF OPERATING MODES IN A SENSOR NETWORK .............. 72

FIGURE 28. MULTIPLE INPUT MULTIPLE OUTPUT SYSTEM REPRESENTATION OF A
GAUSSIAN RANDOM FIELD WITH MULTIUSER AND RECEIVE ANTENNA DIVERSITY ...... 78

FIGURE 29. MEAN SQUARE ERROR OF THE ESTIMATOR AS A FUNCTION OF SENSORS’
SPATIAL SEPARATION, 7 (INVERSELY PROPORTIONAL To DEGREE OF CORRELATION)83

FIGURE 30. LOGARITHM OF MSE AS A FUNCTION OF SPATIAL SEPARATION OF SENSORS
HIGHLIGHTING THE THRESHOLD RANGE FOR HIGH DATA CORRELATION ..................... 83

FIGURE 31. PERFORMANCE OF THE ESTIMATOR AS A FUNCTION OF MULTIUSER
DIVERSITY IN THE PRESENCE OF CORRELATION VS. INDEPENDENT TRANSMISSIONS 85

FIGURE 32. MEAN SQUARE ERROR As A FUNCTION OF RECEIVE DIVERSITY FOR VARYING
INTER-NODE SPACING LE. 7 e { 1.0.1} ...................................................................... 86

FIGURE 33. MEAN SQUARE ERROR AS A FUNCTION OF RECEIVE DIVERSITY FOR VARYING
INTER-NODE SPACING LE. 7 e {0.01, 0.001 } ............................................................. 86

FIGURE 34. LOGARTIHM OF THE MEAN SQUARE ERROR AS A FUNCTION OF RECEIVE
DIVERSITY, DEMONSTRATING THE DIMINISHING RETURN ON MSE AS ANTENNAS
INCREASE ....................................................................................... 87

Images in this thesis are presented in color

LIST OF TABLES

TABLE I. COMPARISON OF TRANSMISSION AND DECISION TIMES FOR SPACE-TIME
FUSION VS. ROUND ROBIN SCHEMES FOR Two SENSORS ............................................ 42
TABLE H. COMPARISON OF TRANSMISSION TIMES FOR SPACE—TIME FUSION SCHEME VS.

FUSION USING ROUND ROBIN DATA COLLECTION SCHEME FOR FOUR SENSORS .......... 52

xi

1. INTRODUCTION

Wireless Sensor Networks (WSNS) are being deployed for applications such as
habitat monitoring, environmental sensing, target detection, localization and tracking [9]
[39] [40], situations in which human intervention may often be undesired, invasive and
most likely expensive. Due to dense deployment, the sensor measurements exhibit a high
degree of Spatial and temporal correlation. Energy efficiency, which directly translates to
increasing the lifetime of the network, continues to be the holy grail of Wireless Sensor
Networks. Although the overall network design itself is highly fault tolerant, the
individual sensors are subject to very low utilization in order to maximize the network
lifetime. Therefore the overall performance of the system depends critically on:

I Reliable event detection given a sparse number Of active sensors per unit area.

I Providing high data fidelity or high rate. information retrieval during event

triggers, i.c. when the density Of active sensors is high.

I Maximizing network lifetime, which relates directly to energy efficiency
Since the sensors transmit their measurements in a round robin scheme or hop-by-hop to
the fusion center, this exacerbates the latency in data collection and therefore event
detection, particularly in high rate applications since current sensor network protocols
cater to low bandwidth and low rate requirements.

Data fusion and distributed detection schemes are crucial to reliable event detection
and this problem has been well studied for various sensor topologies ([10] and references
therein). Traditional approaches assume the availability of transmission error-free

decisions or Observations at the fusion center. In practice the accuracy of event detection

depends on reliability of the received data and implicitly is a function of the channel
conditions. Therefore, sensor network design has to contend with both sensor
measurement errors as well as channel transmission errOrs. Channel aware distributed
detection and fusion schemes, [24] address the above issues by adecating an approach
that combines data fusion with the signal processing at the transceiver, rather than a
separation based approach (known to yield sub-optimal performance in typical cases ).

The basic detection problem is that of discerning the presence of a parameter(s) of
' interest in noise. This leads to the conclusion that multiple sensors observing the same
parameter(s) would reach similar decisions under low sensor noise and is related closely
to the multi-terminal inference problem. Such a system has built-in diversity; it is dual to
traditional MIMO systems which artificially incorporate correlation into independent data
streams through space-time coding. Motivated by this close parallel, the use of space-
time codes in manner that combines delay and spatial diversity for data fusion of co-
located sensors’ decisions is proposed in this work. The application of-Space-time coding
in Wireless Sensor Networks (WSNS) is an active field, but most approaches have
focused on the context of relay based transmissions [29] [30] [31] where all the
participating nodes assumed availability of error-free instances of the packets or data.

In the first part of this work proposes the use of space-time block code structures in a
‘truly’ distributed manner for the dual purposes of joint transmission and fusion of local
sensor decisions. The method is based on harnessing the inherent correlation in the sensor
decisions due to geographic proximity. The proposed schemes impact on global detection
is determined in terms of the overall probability of event detection versus false alarm.

This approach was motivated in part by the need for channel aware signal processing and

detection at the fusion center and in part to determine the conditions under which,
distributed Space-time schemes (without explicit co-operation) perform as well as
equivalent co-operative schemes. Even cO-operation cOmeS at the cost of increased
energy consumption; therefore the conditions under which non-cooperative schemes
perform as well as cooperative schemes were analyzed. The effect of varying levels of
data correlation given sensor noise on the proposed schemes has been presented in this
work. This is essential since the data transmitted by sensors tends to be highly correlated.
The global system performance has been characterized as function of sensor noise (or
correlation in decisions), local sensor thresholds, number of active sensors and the link
conditions. The performance of the proposed schemes has been compared to equivalent
fusion schemes that employ independent transmissions [23] [24]. The findings indicate
that the global detection performance of the space-time based fusion schemes is similar to
optimal fusion schemes with independent sensor transmissions (under identical energy
constraints). But the distinction of space-time fusion based methods is the dramatic
reduction in latency they achieve, especially during the data aggregation phase. This is
particularly relevant to triggered event collection since the sensors often Stop sampling
due to limited buffer capacity (until transmission of previously buffered data). This
potentially results in the loss of data fidelity and may lead to undetected events. In
addition, the proposed approach promises efficient use of bandwidth by utilizing the
multi-sensor diversity and joint transmissions while avoiding scheduling and contention
issues; all of which result in overall energy efficiency.

In the latter part of this thesis we extend the multi-terminal problem for binary

hypothesis testing to a broader framework where the correlation between distributed

sensors is more general and phenomenon under consideration is modeled as a spatially
correlated Gaussian Random field. The modified problem falls into the category of
Distributed Estimation in Wireless Sensor Networks since. the Space is no longer discrete.
Yet again the aim of the work is to understand the impact and the role of spatial diversity
on estimation accuracy in the presence of correlation. An integrated framework that
incorporates Spatial diversity and data correlation and a system abstraction in terms of
MIMO parameters is formulated in this work. For the underlying physical phenomenon
the sensor correlation is assumed to be a function of the spatial separation or node
density. The optimum estimator for the overall system is derived and the Mean Squared
Error (MSE) is analyzed as a function of the system parameters. The combined effects of
data correlation, node density, and multiuser and receive diversity on the overall
estimation accuracy are investigated.

The benefits of incorporating receive diversity into Wireless Sensor Network (W SN)
applications that require high data fidelity and resolution are demonstrated in the
performance results. Such an approach is particularly applicable to active sensing
scenarios, where the distribution of active sensors across the network varies or can be
adaptively varied. By integrating spatial correlation in wireless sensor networks with
receive diversity, we Show that the field estimation accuracy can be Significantly
improved while ensuring energy and spectrum efficiency.

Chapters 2 and 3 review the developments and related background that lay the
foundation to this work. The overview of contributions is as follows. The distributed
detection and channel aware data fusion problem are discussed in chapter 4. As a Starting

point the proposed Space-time fusion schemes are applied to a simplistic discrete model

in which the source is modeled as a binary random source and its associated uncertainty
is modeled via Binary symmetric Channels (BSCS) between the source and the sensors in
4.1. Space-time fusion schemes are then extended to the general model of additive
Gaussian noise at the sensors (for a Binary hypothesis testing problem) in 4.2. The
overall system performance in terms of the global probability of detection and false
alarm, local sensor thresholds and the transmission channel conditions is analyzed. In
chapter 5 the distributed estimation problem and the motivation and effects of integrating
diversity with correlation for accuracy and energy efficiency are discussed. The system
formulation and model is presented in 5.2 and the optimum estimator and the error
performance are derived in 5.3. The overall performance as a function of MIMO system
parameters and their impact on field reconstruction accuracy are presented in 5.4. The

concluding remarks are mentioned in chapter 6 along with a few insights to future work.

2. RELATED WORK

Wireless Sensor Networks (WSNS) are an emerging technology that can be used for
communications, control and instrumentation in industrial, environmental, scientific and
personal/home applications [9]. In this paradigm, the physical world can be monitored
through tiny devices often referred to as sensor nodes that can sense in multiple
modalities such as acoustic, seismic, thermal etc. These nodes communicate Via wireless
links and in theory are capable of self-configuration. The key feature Of such networks is
that they are inherently distributed systems and can be deployed densely thereby offering
feature resolution and data fidelity that surpasses single point sensing mechanisms. A key
function of distributed sensor networks to identify/detect physical phenomena or targets
of interest in situation awareness tasks [10]. From a system perspective this reduces to the
problem of the distributed network distinguishing a hypothesis in effect given the
possibility of multiple hypotheses. This requires information exchange between multiple
sensors which sense the environment independently and motivates the need for
Collaborative Signal Processing (CSP) and data (or decision) fusion depending on the
statistical nature of the sensor data. Sensor nodes are battery powered and this severely
constrains the energy resources for information exchange. For dense deployments
bandwidth is also at a premium. These factors affect overall network lifetime and play a
crucial role in the design of information exchange, sensor signal processing and data
fusion schemes. This section covers related work in the areas of distributed detection and

data fusion in wireless sensor networks relevant to this work.

2.1. The Parallel Fusion Network

In wireless sensor networks, information collected bydistributed nodes or devices is
relayed to a fusion center (which may also be referred to as the sink or the base station).

From a statistical decision theory perspective such a topology fits that of a parallel fusion

Phenomenon

Source Uncertainty

 

 

 

 

 

 

 

 

 

 

 

X1 X2 XN

i—HIK "02$

3 I I .
: ...... l .............. i

I . I
Local I Local Local
Detector1 . Detector 2 °°°°° Detector N
2 oz <I>N El

    

Ideal Transmission Channel

nnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnn

 

[ Fusion Center I

 

Figure 1. Canonical distributed detection system: The parallel fusion network
network [10], which consists of a number of local detectors (LD) and a fusion center,
which employs a global fusion strategy. This topology assumes that the sensors or the
local detectors observe the same phenomena. The focus is on the binary hypothesis

testing problem where H0 and H1 are the two hypotheses with a priori probabilities 71'0

and 7T1 respectively. This is particularly applicable to target detection/identification of

Specific physical phenomenon. Some instances include forest fires, monitoring volcanic

activity, detection of vehicles in hostile territories etc.

Let I, denote the sensor Observation at each node and p(:1:z- in) denote the

conditional probability density function given hypothesis H k where k = 0,1. The

detection of a Signal, S in additive Gaussian noise, i.c. given N observations made by
the local sensors is considered. Therefore the observation at the ith sensor during the jth

instant corresponds to,

Hozzrz-J =772-j (1)

H1:a:z-j=S+nz-j (2)
where i = 1, 2,...N and 5' corresponds to the observed arbitrary signal level (that may be

estimated a priori in a non-deterministic case) and 77,- 's are independently distributed

according to W(O, 07271) where 0,272. is the noise variance at the ith sensor.

2.2. Design of Fusion Rules

Each sensor makes a local binary decision a,- E {HO,H1} based on the observation(s)

mi] where

0, if detector decides H0
1, if detector decides H1

492: = 77(13): [

3
0, if fusion rule 7g10bal(7) yields H0 ( )

¢global : l

1, otherwise

where the local decision rule 72' is based on the likelihood ratio test at the local detector,

___I/I
2 < Z
p(a:,~/H0) ¢i=H0=0

The sensor thresholds r,- are based on the sensing modalities, topology constraints,
sensor noise variance levels etc. and are chosen such that they jointly minimize the local
as well as the global probability of detection error.

For the binary hypothesis case, the fusion rule is essentially a logical function with N
binary inputs and one binary output. One of the methods to choose a fusion rule is to

select a commonly used logical function such as the AND, OR or the MAJORITY rule

but they may not guarantee optimality.

2.3. Channel Unaware Data Fusion in Parallel Networks

This section reviews some of the optimum fusion rules in a Bayesian framework for
the case of an ideal/error-free communication Channel between the sensors and the fusion
center. The sensor performance indices are parameterized in terms of probabilities of
false alarm and detection and are used as weights in formulating the fusion rule. Let

Pfa, Pdi & P72 denote the probabilities of false alarm, detection and miss at the ith

local detector respectively, i.c.

P; = P(¢,- =1|H1) (5)
p3, = P(¢,- = 0|H1) (6)
Pfia = P(¢,- =1|H0) (7)

The probabilities of false alarm, detection and miss at the global level, i.c. at the fusion

center are represented by,

PgObal = P(¢global = llHO) (8)
Pdglobal = p<¢global = llHI) (9)

The optimum fusion rule has been formulated as a binary hypothesis test where the local
sensor decisions serve as the observations and is equivalent to a likelihood ratio test. The

Optimum fusion rule [10] for N sensors is given by,

¢globalzl
P(¢1.¢2,---a¢N|H1) > ”Moro—000);

P(¢I,¢2.---,¢N IHO) < 711(001- Cir)
global=0

 

 

(11)

where Ck) denotes the cost of global decision being H k given H1 is in effect given prior
probabilities vrknrl such that k,l = {0,1}. Due the independence of local decisions and

taking the logarithm of the likelihood ratio, the above can be expressed as,

 

 

N z- ,- ¢>gzobaz=1 N ,-
2:108 ,- ,- .i < 10s CH 2. (12)
1:1 PmP a . i=1 m

9°global=0

Therefore the Optimum fusion rule is a weighted sum of the individual sensor decisions
and the global threshold depends on the probabilities of miss and false alarm of the local

detectors.

10

2.4. Channel Aware Fusion in Parallel Networks

The distributed detection (DD) problem closely corresponds to the multi-terminal
inference problem [24] where the hypothesis testing problem is cast in an information
theoretic framework; where multiple terminals are subject to rate constraints while
communicating their information to the fusion center. In a classical DD framework the
two critical design issues are:

I Design of the fusion rule at the fusion center that ensures optimal performance
given local sensor data. In other words it is the data combining strategy that
ensures the best performance in terms of identification of the true hypothesis.

I Design of optimal and energy efficient local sensor signal processing algorithms.

In Wireless Sensor Networks (WSNS) the communication links are interference rich

and error prone and given the resource constraints on the in-situ sensors (that typically
run on an on-board battery); there is an inherent need for the channel-aware design of
signal processing and detection [24]. There are mainly two levels of uncertainty that
arise in the design of signal processing algorithms for WSNs:

I Source uncertainty which arises from the noise in sensor observations while
monitoring the source of interest.

I Channel Uncertainty due to noisy transmission links between the sensors and
the fusion center. Some of the factors include signal attenuation through
effects such as fading, interference and receiver noise.

AS discussed in the earlier section, the design of the optimum fusion rules for the

canonical parallel fusion network where there is no channel uncertainty is relatively

11

straightforward and has been addressed by classical detection theory. In WSNS applying
this theory leads to a separation based approach where the design of the fusion rules and
the communication mechanisms is addressed separately. This is Where the Distributed
Detection (DD) problem departs from the classical Multi-Terminal (MT) inference
problem. The key distinction between the two is that the MT problem tends to be data-
centric whereas the DD problem is inference-centric. In the former case the objective is
only to reconstruct the information transmitted by the terminals as faithfully as possible
given rate constraints; whereas in the latter case the ultimate goal is to draw an inference
about a parameter, phenomenon or a target of interest. Therefore in WSNS where
distributed detection is crucial, a separation based strategy may not be optimal. At this
point we note that the data processing inequality [1] suggests that joint processing of the
output of channels in a multi-terminal system rather than processing the output of the
receivers avoids potential information loss. This supports the need for integrating

transceiver design with fusion algorithms.

2.5. Channel Aware Decision Fusion in Wireless Fading
We present some of the results that extend classical distributed detection to the case
of channel fading effects in wireless sensor networks. Some of the decision fusion
schemes discussed in [22] [23] will be used later for benchmarking the performance of
the schemes proposed in this work. It was noted in [22] [23] [24] that channel aware
fusion strategies are much more energy efficient than the separation based approaches in
wireless networks. Figure 2 represents the parallel fusion network that incorporates the

source and the transmission channel uncertainty, i.e. fading in wireless links and additive

12

2?
7/ \A

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

_n1
Sensor 1 Sensor 2 Sensor N
l l I
CD1 (DZ
01 : I12 5:.
n1 “2
+

3’1\V 3' Y3
CFusion center jyx

Figure 2.Parallel fusion network in the presence of fading and receiver noise

 

receiver noise.

As in the earlier case the sensor Observations are based on the binary hypothesis
H0 (target absent) or H1 (target present) in effect. The decision fusion rule is derived
after incorporating all the channel effects for optimality. In this case it is assumed that the
‘h sensor with a binary decision transmits,

1 if H1 in effect

451' = (13)

— 1 otherwise

Therefore the received Signal at the fusion center is (independent sensor transmissions),

y'i = highi + 712' (I4)

13

where h.) is a real valued fading envelope with h,- > 0, and n,- E W(0,o,2,)

2.5.1. Optimal likelihood-ratio (LR) based fusion rule

Given the complete channel state information and local sensor performance indices
such as the probabilities of detection and false alarm, the goal is to obtain a fusion
strategy based on y,- for i = 1,2, ..N , that can determine the hypothesis in effect with the
best performance. Based on the conditional independence assumption of local sensor
observations, the optimum fusion Strategy was derived in [22] and is based on the
Likelihood Ratio (LR) test, indicated by A; which is compared to a global threshold in

order to yield the global decision, ¢global- It is referred to as the “LR-fusion rule” [22]

which is based on the received vector y , indicating the measurements of the N sensors

corrupted by transmission channel noise.

 

_<y.-h.->2 _<y.+h.~>2
N i 202 i 202
A(y)=f_(3_’E’1_)=H Pd 8 n + (1‘31” " (15)
f(y|Ho) M 417.43.)? _(y.-+I2z.-I2
Pfia e 20" + (1— Pia) e 20"
where y = [y1 3,717] is the received vector at the fusion center. It is noted that the

conditional density function f(y|Hk) with k = 0,1 is Gaussian distributed, hence the

likelihood ratio is of the form of equation(15).

2.5.2. Sub-optimum fusion
In this section we discuss the high SNR and low SNR approximations [23] [22] of the

optimal fusion rule and other statistics that require minimal a priori information. It is

14

interesting'to note that they reduce to familiar forms and demonstrate the close parallels

to diversity combining [17] and distributed detection.

2.5.2.1. Chair-Varshney fusion rule

This rule [22] is a two Stage approximation to the optimum fusion where an inference
is made about the decision qu' based on y,- (first step), and the fusion rule is based on the
inferences (or the estimates) of the decision (13, (second step). The Maximum Likelihood
estimate of the decision ¢i is given by,

@=“WW) (w)
Therefore based on the canonical DD system, the fusion rule [22] reduces to,

P1 I—fl
_ d d

A1 —— ’ E log —Pz. + 2 log 1 — Pi (17)
sngI(I/t)=1 fa si9n('yt)=-1 fa

 

2.5.2.2. Maximum ratio combining fusion statistic

This is a low SNR approximation [22] to the LR fusion rule and the limiting statistic
resembles that of the Maximum Ratio Combining (MRC) statistic used for diversity

combining [17] [23],

. N . . ,
A2 = DP; — P},)h.-y.- (18)
i=1
1” .
A2=—Zh.-y.- (19)
Ni=1

Here we assume the availability of the channel information and the local sensor

15

performance indices at the fusion center. However, in the case of identical sensor
performance indices at the fusion rule (18) reduces to the Simple form of (19) which is

the Maximum Ratio Combining (MRC) rule.

2.5.2.3. Equal gain combining fusion rule

This fusion rule requires the least amount of information and is in the form of an

equal gain combiner [22] [14],
1 N
A3 = — 2 y.- (20)
N i=1

Although this approach is suboptimal to the MRC based fusion (19); this approach has

been Shown to perform at par with the fusion rules (17) and (19), i.c. A1 and A2 for a

wide range of SNR in terms of its detection performance. It is noted that this is applicable

only for the particular case of real non-negative fading coefficients.

l6

3. BACKGROUND

Fundamental theoretical advances in multi-antenna communications are the key to
realizing the significant gains that diversity promises through increased transmission
rates, reliability and spectral efficiency in Wireless Networks. This section covers the

necessary terminology and background that relate to this work.

3.1. Space-Time Processing in Wireless Networks

Space-time (ST) communications today utilize the “spatial” dimension in addition to
the temporal and frequency dimensions. Diversity in context of communication refers to
multiple independent instantiations of information. This diversity could be utilized in
time, space, angle, link, polarization, angle and frequency. It was recognized early on as a
powerful tool for mitigating fading in wireless networks. The increasingly crowded
wireless spectrum and the need for higher data rates for broadband applications led to a
fresh look at diversity. Introduction of correlated information Streams in diverse spatial
modes led to an area of coding and signal processing across multiple antenna elements. In
particular, the information-theoretic results [3] [7] [8] [11] [12] pertaining to the impact
of spatial diversity i.c. transmit and receive diversity are a key motivation to this work.
Another form of diversity that is used extensively is that of multiuser diversity [14],
which is utilized via independent links between different users that are geographically
separate.

The taxonomy of antenna configurations [17] in ST wireless systems is as follows,

I Single Input Single Output (8180): A system with a single transmit (TX) antenna

and a single receive (Rx) antenna.

17

I Single Input Multiple Output (SIMO): A system with multiple receive (MR)

antennas and a Single transmit antenna.

I Multiple Input Single Output (MISO): A system with multiple transmit (MT)

antennas and Single receive antenna

I Multiple Input Multiple Output (MIMO): A system with multiple transmit (MT)

and receive (MR) antennas. The MIMO-MU refers to a multiuser system equipped

with multiple antennas (M) at the base-station communicating with multiple users

with one or more antennas.

Some of the key leverages of ST communication include array gain, diversity gain

Space-time Signal Processing

|____.—..

 

 

;’ " .\ MISO SIMO / ‘I’
1 . 1\ I 1 . z
o m
. \ / 5.
m—f’flhz 9\ 7 T'é’T—hzw’F-7 5
\\ v/i/ \l// \/ \\ \/ a
__ -2.._I y__ __.XJ \\ L—ZH g- m—Y»
: W h". : g
. \\ . 21
° ' 2.
v‘fi" \V :3
i L__NR___L
ir—NFJ

(a) Given transmitter side information

2
y = (ih1i2 + ih2|2 + + IhNTI )9:

 

 

 

(b) Given CSl at receiver

2
y =(lh1|2 +lh2|2 + + |hNRI )3

Figure 3. Implementation of transmit and receive level diversity in M180 and SIMO

systems

and spatial multiplexing gain. Array gain is defined as the improvement in the Signal-to-

Noise Ratio (SNR) obtained by coherently combining the signal transmitted or received

using multiple antennas. Receive diversity maximizes the signal level by combining

independent faded versions of the signal given the channel state information through

18

maximum ratio combining. The number of independent faded versions of a signal is
characterized as the diversity order, for instance in SIMO channels the diversity order is
equivalent to number of receive antennas (MR). Transmit diversity can be implemented
by transmitting a Signal across multiple antennas such that the Signal power is maximized
at the receiver. This is achievable with or without the availability of channel state
information at the transmitter. When CSI is available, the signal power across antennas
can be allocated in order to maximize the received power level. In the absence of CSI the
transmission scheme must be carefully designed to maximize the diversity order. In

MISO channels with independent fades the diversity order achievable is MT. MIMO

channels combine transmit and receive diversity and maximal diversity order MRMT is

realizable provided independent fading across antennas [17]. The advantage of diversity
gain is that it has the effect of stabilizing the link with fewer fluctuations and thereby
combats fading.

Spatial Multiplexing (SM) on the other hand opens up independent data pipes which
enable transmission of more data streams, thereby enhancing spectral efficiency. We will
shortly note the tension between multiplexing and diversity. Spatial multiplexing gain

however, requires multiple antennas on the transmitter as well as the receiver side unlike

transmit or receive diversity. SM provides a linear increase in capacity [17] [18] min

(M T, MR) without additional bandwidth requirements, the related information theoretic

results will be discussed briefly.

l9

3.2. The MIMO Channel Model

The attenuation in wireless channels can mainly be classified as path loss, which is
due to the distance between the communicating nodes, shadowing loss, which is because
of absorption of radio energy due to the surrounding structures and fading loss, which is
due to multiple reflections which interfere, causing signal level fluctuations. In a
multipath environment with rich scattering, the envelope of the received signal is
modeled by a Gaussian distribution. If there is a line of sight (LOS) component available
then the amplitude is assumed to be Ricean distributed Since the mean is non-zero.
However if there is no line of sight (NLOS) component available the received envelope is
well modeled as zero mean Gaussian distributed, i.c. we have Rayleigh fading. In the
case of a complex baseband Signal the in-phase and quadrature components can be
modeled as zero mean Gaussian distributed. This effect of rapid fluctuations in time,
frequency and space is commonly termed as microscopic fading. For MIMO channels
fading leads to time selectivity, frequency selectivity and space selectivity, which are
characterized in terms of coherence time, coherence bandwidth and coherence distance
respectively [55]. The coherence time is the time duration over which the channel
impulse response can be assumed to be constant. Coherence bandwidth is the frequency
Spread over which the channel frequency response can be assumed flat. Coherence
distance is the maximum Spatial response over which the channel response is assumed to
be constant.

Consider MIMO channels with MT transmit and MR receive antennas; the classical
MIMO channel model is that of a frequency flat fading channel with independent

identically distributed (i.i.d.) channel coefficients, i.c. spatially white. The elements of the

20

matrixH of dimension M R x MT, represent the fading channel coefficients which are

modeled as independent zero mean circularly symmetric complex Gaussian (ZMCSCG)

random variables. The received signal model in discrete time is,

y[k] = JEHsUI] + n[k] (21)

where y[k]€ CMRXI is the received signal vector with dimension M R x1,

 

s[lc] E (CMT X1 is the transmitted Signal vector of dimension MT x1 and
n[k] E CIT/(0,11%) is the additive zero mean circularly symmetric complex Gaussian

noise vector of dimension M R x 1with variance N 0 in each spatial dimension. Some the

properties of H include,
5 {[Hjm} = 0
2

e [11],, j

E [Hji [H]:l =0ifi¢korj¢l
v.7 .

 

} = 1 (22)

Throughout this work the input is assumed to be zero mean, e{s} = 0 with1

5{ss*}= R33 and Tr{R38} = MT in order to satisfy the transmission power

constraint Es .

 

' Throughout this work 5 represents the expectation operator.

21

3.3. Review of Information Theoretic Results

In this section we review some theoretical foundations [18] that give fundamental
limits to achievable gains in multiple antenna systems. The capacity of the additive white
Gaussian noise channel with variance N0 and input power constraint E8 is given by [l],

C=llog
2 0

 

1 + {ii} (23)

For scalar flat fading channels (i.c. no channel memory) given perfect channel state
information at the receiver (without channel feedback), the capacity is given by ([18]

references therein),

N ( )

Czle
2 0

log

 

 

 

 

For complex channels (With in-phase and quadrature transmissions) utilizing the
spatial diversity, these results are appropriately modified and the factor 1/2 disappears.

Considering the SIMO channel with MT = 1 and receive diversity orderMR, the

capacity was shown to be the following ([18] and references therein),

 

 

 

 

2
h
C=€log1+fl-—“-£S- (25)
No
where h 2: ha h] lid is the channel vector. The capacity of multiple antenna

fading channels with M R, MT >1 was first established in [3] and other supporting

results include [6] [8] [12]. Assuming that the current output has no memory of previous

inputs the time index can be removed and the signal model is represented by (26).

22

 

[E3
= H+ 2
y MT 8 n (6)

The mutual information [1] between the vectors 3 andy , i.c. I (s; y) is given by

I(any) = My) - h(Ills) (27)
Where h(y) is the differential entropy of y and h(yls) is the conditional differential
entropy of 3; given knowledge of 3. Since 3 and nare independent h(y|s) = h(n) (no
side information at transmitter) therefore, I (s; y) = h(y) — h(n). The input 3 is assumed

to be Gaussian as this maximizes the mutual information conditioned on H . Therefore,

E
Ryy : e{yy*} = 371:,— R83H* + NOIMR (28)

3.3.1. Capacity of MIMO channels
The capacity expression is obtained by the assumption of i.i.d. Gaussian H and the
memoryless property of the vector Gaussian channel upon conditioning onH and is

given in [3] as,

 

 

 

+HR H*
C = E H log ann 83 l (29)
|le
For the Special case of Rm, 2 N ()1 M R the capacity expression reduces to,
0:5 10 I. +LHH* 30
H 8 MR MT ( )

 

 

 

 

3

—0-; for fixed MR as MT gets large -A/Il—THH* “*IMR therefore

where p =

C —» M R log(l + p) . Therefore the capacity increases linearly in the number of receive

23

antennas. Also we note that as the number of receive antennas grow, the norm of H
grows which indicates the increase in the radio energy gleaned by the antennas.

It was shown in [8] that for MT 2 M R = M the capacity increase is linear in M as

M —> 00. This is possible only through joint optimal decoding of all the received Signals.

1 (31)

where r is the rank of the channel and A,- 's (2' = 1,2, ...r) are the positive eigenvalues of

Alternatively from [17] we note that,

r(H)

I(s;y) = 8 2 log 1+—B—x\¢
i=1 MT

 

 

HH*. This supports the intuition that multiple antennas at the transmitter and receiver
open up a large number of data pipes or communication modes through which
information can be tunneled. It was observed in [8], [18] (and references therein) that the

capacity as a function of SNR grows linearly in min(M R, M T) .

C(SNR)

=minM ,M l 32
SNR—+oolog(5NR) ( R T) ( )

3.3.1.1. Low SNR approximation of the ergodic capacity

2

 

C m 8 {log (33)

 

.0 2
1+— H F
MT” H

 

], IIHIII = ZIIHI.
2,]

We note that in the low SNR region the achievable capacity [17] depends on the energy

in the channel which is represented by p.

24

3.3.1.2. High SNR approximation of the ergodic capacity

Cars log

II

i=1

r(H)[

 

P .
Tat]

 

(34)
_p_

= r(H) log MT

 

 

r(H)
+ 5i 2 Iowa]
i=1

In the high SNR region the achievable capacity [17] depends on the channel rank

(number of modes) and the eigenvalue spread.

3.3.1.3. Outage capacity

In practice, coding is not feasible across large blocks and the channel is non-ergodic.

In such cases the notion of outage probability, Poutage% becomes necessary and is

defined as the probability with which a given rate, R is achievable. Outage in this context
refers to the fact that in non-ergodic channels a given rate R may not be achievable for
some channel realizations. Outage capacity is the capacity or rate that can be achieved for

the remaining (100 - Poumge)% realizations [l8] (and references therein).

Poutage (R, 17(3)) = P{H 3 I(siylHU‘) = H) < R} (35)

Here p(s) is assumed to be the transmission policy. Assuming a Gaussian (white)

codebook the outage probability for a given rate, R can be written as,

Poutage (Ra 17(3)) = P 10g

 

 

IMR +MP?HH*

 

< R] (36)

25

3.3.1.4. Effective diversity order

We note that Spatial multiplexing performance which determines the rate and spectral
efficiency depends on the rank of the channel H and the diversity gain depends on the
norm "H F "2 of the channel. These two measures become key factors in Space-time code

design. In practice the probability of error serves as a reliable performance measure. The

pair wise error probability (PEP) between two code word sequences 3 = [s(0),s(l),..]

and e = [e(0),e(1),....] for the Rayleigh flat fading channel is bounded [6][18] as,

 

 

 

 

P(s —-> e) < 1 (37)
II (1 + PAT)
_ i=1
This is further reduced to the following form,
1- "MR —1‘MR
P(s—WK HA.- E3 (38)
1:1 4N0

 

 

 

Definition: A coding scheme [16] which has an average error probability PC(SNR) as a

function of SNR that behaves as,

m log(Pe(SNR))
SNR—+00 10g(SNR)

 

= ‘d (39)

is said to have diversity order of d .

In other words the probability of error, Fe m SNR—d implies that the diversity order

1‘ I 7‘
is bounded as d g rM R and the coding gain is approximately “(A0 based on

i=1

26

equation(38).
Definition: A coding scheme [16] that has a transmission rate of R(SNR) is said to have
a multiplexing gain q if,

1m R(SNR)
SNR—>00 108(SNR)

= q (40)
Also to be noted iS the tension between the achievable diversity gain and multiplexing
gain, i.e. high multiplexing gain comes at the cost of lesser diversity gain and vice-versa
[16]. This is a manifestation of the classical error probability versus rate tradeoff [18].
Based on this discussion the two important criteria that emerge in the design of Space-

time codes are as follows

I Rank criterion: In order to achieve the maximum diversity MTMR, the

codeword difference2 matrix has to be full rank over all pairs of distinct
codewords. If the rank of the codeword difference matrix is r overall then a

diversity order of rMR is achieved.

T 1/1‘
I Determinant criterion: Maximize H (A,) for a given rover all pairs of
i=1

distinct codewords in order to maximize the coding gain.

 

2 The codeword difference matrix is defined as the difference matrix of two distinct codeword sequences.

27

3.4. Introduction to Space-Time Codes

AS discussed earlier, spatial diversity techniques deal with incorporating transmit
and/or receive diversity. In general, receive diversity is easier to implement at the base
station and utilizes the channel state information. Transmit diversity techniques are more
challenging due to absence of channel state information and the need to incorporate
correlation in data streams. Moreover transmit diversity techniques need to be
complemented with effective receiver processing in order to make full use of the design.
Multiple antenna systems follow the law of diminishing returns [18] with respect to error
probability as a function of increasing number of antennas, i.c. the SNR gain margin
decreases.

Space-time coding is primarily an open-loop transmit diversity technique. Some of
the advantages include the fact that multiple transmit antennas are easier to implement in
the downlink channels. Good design can incorporate coding gain along with diversity
gain and with the additional benefit that channel state information is not required at the
transmitter [5] [4] [18] [21]. STC techniques also have the effect of stabilizing the link

which leads to effective power control.

3.4.1. Space-Time Block Codes (STBCs)

Several methods [20] of constructing space-time codes exist such as trellis based
codes; linear dispersion codes etc. but of specific interest are the Space-time Block Codes
(STBC) over quasi-static fading channels. Some of these code structures will be applied

in the later part of this work.

28

3.4.1.1. Orthogonal space-time block codes (O-STBC)

The most popular space-time block code was discovered by Alamouti [2] and can be
defined by a 2 x 2 space-time block structure which is equivalent to having two transmit
antennas and the codeword is transmitted over two time slots assuming there is a single

antenna at the receiver.

3.4.1.2. The Alamouti space-time block code

Let s = (30,31) represent the data—tuple to be transmitted. The space-time codeword

S is defined below where the rows represent the transmit antennas and the columns

represent the time slots,

30 81

S = (41)

* *
‘91 30
For the signal model under consideration let y be the received signal vector, S the

transmitted space-time codeword, h = [ho hl]T the channel vector which is assumed to

be quasi-static over the transmission interval and n = [n0 n1 ]T is the additive Gaussian

noise vector with variance NO. The vector representation of the received signal for the

code block, S is given by,

y = Sh + n (42)
The received Signals at both the time instants are captured through the vector

representation3, y .

 

3 Bold small letters represent vectors and bold capitalized letters represent matrices in all the equations.

29

81,80

. TX‘ ' ho

369 —Sf , 7V
RX
:7 01 ‘

. TX2

- p.

t1,t0

Figure 4. Representation of the Alamouti space-time block coding scheme

Due to the orthogonal structure of the space-time codeword, the received signal can

be re-arranged into the following form.

 

 

 

 

 

 

 

yO h0 h] 80 n0
-yi — -hi" hi -3i -ni‘ (43)
‘w-J \.-_
y H s n.
y 2 H3 + n (44)

This form is the “equivalent channel” form of the received vector where the channel
matrix, H reduces to an orthogonal structure which greatly simplifies detection as well
as estimation. The equivalent channel form converts the quasi-static channel vectors to a
matrix form and in turn converts the codeword matrix to a vector form. Note that not all
Space-time codewords have an equivalent channel representation.

Assuming channel state information is available at the receiver, the Maximum

likelihood decoding reduces to scalar processing at the decoder as follows [25],

30

H*y = ||h||2 s + n', where n' = H*n (45)

Note that the noise vector n’is still white Since the transformation H * is orthogonal.
The structure S corresponds to Bi-quaternions which were invented by Hamilton [15].

Bi-quatemions may be viewed as pairs of complex numbers [25] where the product of

two quatemions (30,31) and (52,33) is given by (3032 — 3133,3033 + 3135) and can be

SO 31
obtained by associating the pair to the matrix form [ * *]. It follows that the
"31 S0

representation of the Alamouti STBC in (41) is the quatemion form of the complex pair

(30,31). This structure corresponds to a complex orthogonal design of size two.

3.4.1.3. Real orthogonal Space-time block code for four transmit
antennas

The full rate real orthogonal design for a 4 x 4 STBC can be determined based on the
Alamouti complex design, by representing complex numbers as 2 x 2 matrix algebra over

real numbers [25]. The complex numbers so = 2:0 + i271 and 31: {1:2 + i233 E C

30 $1 $2 $3
correspond to the matrices a
—:L‘1 $0 —$3 1132

] over IR respectively. Therefore

the real orthogonal design of Size four can be obtained by substituting for 30 & 31 with

equivalent real matrices in terms of 270, $1,352 & 3:3 in equation (41).

’ $0 $1 $2 $3 '
$1 $0 173 $2
(46)

R4“ = “172 1‘3 *930 $1

 

 

—-:L'3 —:L‘2 —.’L‘1 -.’)30

31

3.4.1.4. Complex orthogonal space-time block code for four transmit

antennas

The complex orthogonal 4 x 4 design is based on Octonions [25] or the Cayley numbers
[15]; they can be represented as four—tuples of complex numbers. It must be noted that
full rate complex orthogonal designs do not exist for block sizes greater than two [18]
[25] and the complex 4 x 4 orthogonal design does not lead to the full rate real 8 x 8
design as seen in the earlier instance. The right multiplication of an octonion
a = (a0,a1,a2,a3)with another of the form b = (b0,b1,b2,0) is represented as
ab = aM (b) where abbz- E C. This represents the code structure (Octonion) of a rate -%

complex orthogonal design of size four,

 

FbO I’1 92 0i

M(b0vb19b2a0)= .4): b0 (: b2 (47)
“b2 0 bi) "bl
.0 —s bi“ bod

 

3.4.1.5.Quasi-orthogonal designs

Since full rate orthogonal designs are scarce, in order to achieve higher rates another
good approach [25] [6], is the adoption of quasi orthogonal designs an example of which
is the following 4x4 full rate design which can be interpreted as an adaptation of (47).

l “0 “1 a2 a3 1

* * *

—a1 “0 ‘03 ‘02
Q : :1: at * * (48)
-(12 —a3 0'0 al

a3 a2 —a1 “0

 

 

32

3.4.1.6. Diversity embedding in space-time codes

An interesting approach that was proposed in [20] was to provide multiple levels of
Quality of Service (QOS) by providing different levels of reliability for data streams. This
is a form of multi-layer Space-time coding. This is achieved by embedding a high
diversity code (low rate) within a high rate code which opportunistically makes use of the
channel conditions (SNR) depending on the reliability level [20]. As an example consider
the 4x4 rate 3A complex design based on the Octonion structure in (47) which can be

improved in rate by embedding another message set in the code as below,

 

 

'b0 b1 b2 00'

M(b0.b1,b2.0)=_b1: b0 0 b2 (49)
‘92 0 50 ‘91
I 0 —b2‘ blk b0

The two sets of messages are A,B with messages {a0} 6 X and

{b0,b1,b2} E X where X represents the constellation that is used for transmission of the
. . 1 3 .
messages. We note that In this case Ra = ZlogIX | and H), = Zlog|X| leading to a total

rate oflog|X|.

The field of space-time code design and that of MIMO systems is a very vast and
active one. Although we have covered the salient features of the necessary background,
this is by no means complete. For an end to end overview and discussion of issues and
open research problems in this area an interested reader is referred to [26] and references

therein.

33

4. CHANNEL AWARE DATA FUSION AND
DISTRIBUTED DETECTION

Wireless Sensor Networks are being deployed for applications such as habitat
monitoring, environmental sensing, target detection, localization and tracking
applications. As discussed in Chapter 1 , the crucial factor in the design of these networks
is to maximize the network lifetime while ensuring data reliability and minimum
distortion. The data reliability problem itself can be sub-classified into the categories of
coverage and deployment, information accuracy and reliable data transport. As discussed
heretofore, dense sensor deployments in WSNS lead to a high degree of redundancy in
measurements. Also one must note that these nodes are geographically distributed
therefore the overall system has in-built diversity. The focus of this work is to jointly
address the information accuracy and the reliable data transport problem in an energy
efficient manner by exploiting the correlation and the inherent diversity. Here our
approach to “exploiting” correlation in not compressive based on Slepian-wolf coding
framework [1]. In a sense is dual to Slepian-wolf framework since we consider multiple
joint transmissions at reduced power. This can also be interpreted as a form of multiuser
diversity. Although space-time block structures have been used for energy efficiency they
have been applied entirely from a data-centric perspective, i.c. the ultimate goal is in
using co-operative diversity purely for recovering the transmitted bit/data streams.
Considering that the ultimate goal is to discern a hypothesis these approaches tend to be
based on separation, which may not be optimal given data correlation. In this work we

apply Space-time coding without co-operation between the nodes where our approach is

34

completely inference-centric, i.c. the end is objective to discern the presence or absence
of a target or phenomenon of interest and not to reconstruct the transmitted data streams.
This chapter is bi-partite i.c. as a first step to analyze the feasibility of the proposition
(i.c. of applying space-time coding in a completely distributed manner for detection) we
model the sensor noise as a simple binary symmetric channel (BSC) and apply space-
time techniques for transmission and data fusion. Throughout this work we assume that
the wireless links between the sensors and the fusion center undergo attenuation via
channel fading and are affected by additive noise at the receiver. In the latter half we
adopt the more general Gaussian density function for modeling the sensor noise, perform
local detection and apply Space-time coding techniques for the dual purpose of
transmission and data fusion in wireless fading channels. The aim is to be energy and
bandwidth efficient while maintaining low latency and maximizing the probability of

detection.

35

Source Uncemlnty Transmission Channel

 

 

 

 

 

 

 

 

 

Bscm.) m
.@ ,, 850(32) F—{Saf ------------ I. I I)
: 1.11:6
O .. _, hN ~/
. , " Fusion Center
“ BSC(B~)' SN

 

 

 

Figure 5. System representation of the Parallel fusion network with the Binary Symmetric
Channel (BSC) modeling the source uncertainty

4.1. Channel Aware Data Fusion I - Uncertainty via
Binary Symmetric Channels

In this preliminary section we model the hypothesis testing problem in a region of
interest as a binary random source and where the uncertainty in the sensor decisions is
modeled Via an independent binary symmetric channel between the source and an

individual sensor.

4.1.1. System representation and problem formulation

4.1.1.1. Source uncertainty — binary source
We consider a set of NT sensors distributed sensors in a localized random field

I:

measuring an observable arbitrary scalar parameter 0 in noise. Here :13,- iS the lath

observation at the ith sensor and I)? is the measurement noise.

36

Based on the observations individual sensors perform local detection based on the
observable features and the decisions are transmitted to the fusion center. The distortion
in sensor measurements may arise due to several factors such as ambient noise, sensor
calibration, quantization effects etc. and the noise statistics may or may not be known. In
order to keep the formulation general we abstract the hypothesis in effect as binary
random source that is Bernoulli distributed with parameters or equivalently prior

probabilities 7r0 and 7T1 corresponding to H0 and H1 respectively.

(50)

We use indicator functions to represent the decisions transmitted by the sensors to the

fusion center i.c. u,- 6 {0,1}. In general the approach to detection is using sufficient

statistic and comparing it with a threshold. Each sensor’s local threshold is chosen in
order to minimize the probability of detection error. In order to keep the formulation
general we model the source uncertainty aS a binary symmetric channel between the

source (binary) and the local detector. The conditional probability distribution of the

decision a,- at the ith

sensor given that a hypothesis is in effect is also Bernoulli
distributed with parameter I3) (also synonymous with the crossover probability of the
BSC),

Pi(ui = 0|H1)= P1011 =1|H0) =fiz'

(51)
Pi(ui = 01110): 101% = 1|H1) =1 - fit

37

4.1.1.2. Transmission channel

The transmission channel between the nodes and the fusion center is assumed to be a
discrete time quasi-static block fading Multiple Input Single Output channel (assuming
that the Signaling rate is much higher than that of the environmental change). The local

sensor decisions are transmitted to the fusion center using the Binary Phase Shift Keying

(BPSK). L61 8 = 81 82 SNT E CNTXI represent the transmit vector consisting

of the BPSK modulated sensor decisions. Since we can have upto NT joint

transmissions this represents a system with multiuser diversity and if multiple antennas
are deployed at the fusion center this would be a Multiple Input Multiple Output (MIMO)

cNRXI.

system, where the received vector y E can be represented as,

y = H3 + 19 (52)
where H E CNRXNT is the channel matrix of the fading coefficients, the elements of
H are modeled as zero mean complex circular symmetric complex Gaussian
(ZMCSCG) random variables with unit variance and I9 6 (C‘Nli’x1 is the additive noise

vector that is also ZMCSCG with variance 0,29 and E S is the peak power constraint.

4.1.2. Data fusion based on space-time coding for two sensors

Considering that the active sensors are co-located with the phenomenon of interest it
might be assumed that the local decisions are highly correlated. In the case of complete

correlation the multiuser diversity order N T is equivalent to the transmit diversity order

of the system. In a query based sensor network, co-located sensors on receiving a request

38

to send (RTS) from the fusion center can begin Simultaneous transmission of local
decisions according to a pre—specified block structure. The transmissions at each sensor
correspond to independent columns in the Space-time blOck and can be assigned offline
based on specific criteria. Since the nodes are assumed to be co-located the inter-node
delays in receiving the RTS from the fusion center are assumed to be negligible. AS a
first step we present the Space-time fusion for the two sensor case in which delay [19] and
multiuser diversity are combined for distributed space-time coding and for data fusion at
the receiver. The space-time block is based on the Alamouti orthogonal code structure as
discussed in 3.4.1.2.

In this scheme two sensors performing local detection buffer their consecutive
decisions subject to protocol specific requirements (this method can be extended to a
sequence). The key variation from classical space-time coding is that there is no co-
operation between the transmit antennas in this case the participating sensors,
independently encode and transmit local decisions based on the IO-STBC structure.
Therefore the scheme is entirely distributed in that there is no exchange of information
except for the requirement of synchronization with the fusion center.

kth

The linear algebraic formulation is as follows, let uf represent the sensor

decision at the ith node. Let [sn1(k) Sn,2 (10] and [31110: +1) 31,209 + 1)] represent the

consecutive observations at two sensors indexed by III and ha. The distributed space-

time block transmission is modeled by the codeword S and the received vector is

modeled by y . The columns represent the time slots and the rows represent the antennas

or in this case the path diversity of the data streams from the two sensors.

39

 

 

872.1 (k) 3721 (k + 1)
== * I (53)
—sn,2 (k + 1) s",2 (k)
3,10.) 3n1(k +1) «9.- T
= HS + 19 = h h 54
y i k k+1]_3;2(k+1) 522(k) 797.41 ( )

 

 

 

 

In this scheme we have NT = 2 and N R = 1. In effect four sensor decisions are

transmitted to the fusion center in two time slots which results in a latency reduction of
50% when compared to a round robin transmission approach. And better utilize the
available spectrum.

The fusion center obtains linear combinations of the independent transmissions from
the sensor nodes as represented in (54). Assuming that the fusion center has full CSI
(through training symbols), the Maximum Likelihood (ML) decision rule for the
Alamouti structure can be obtained by scalar processing due to the Quaternion sstructure.
This can be applied to the distributed structure in case of perfect correlation. As discussed

in Chapter 3.4.1.2 upon re-arranging the terms the received vector can be expressed as,

A

311: hr hit-+1 3k 19k

+

(55)
-r/i+1

 

 

 

 

 

 

 

 

* * all! *
-h1.~+1 his —Sk+1 ”91.41
Therefore the estimate 3 can be obtained by scalar processing of the received vector due

to the orthogonal form of the channel matrix in equation (55) as follows,
* 2 2 A r
H y = (lhkl +|hk+1| )s + 19 (56)
where the modified noise term 19' is Still white due to the orthogonal transformation H * .

It must be noted that when the sensor noise is very high and sensors decisions do not

coincide the transformation may no longer be orthogonal since the Alamouti code

40

structure may not be preserved. Some of these effects will be discussed in the sequel. In
the case of BPSK modulation the vector S can be decoded by implementing a hard
limiter in order to yield the global decisions for the two time slots. The above decoding

rule (56) implicitly performs data fusion to yield global decisions based on consecutive

local sensor decisions 14:1, 2151+ 1, 1152, 215,2“ as,
uglobal=1
. >
sgn(sk) < 0 (57)

”globalzo

k+1 k+2 k+3

Assume {aspunl ,unl ,unl } be a sequence of buffered binary decisions at a

sensor node 721 for k = 0,1,2-u. Let {ng Si” s32. S32. ...} be the corresponding

modulated waveforms at the nfh sensor and let {agruki , n.2,, , 2732. ...} be the sequence of

global decisions (post fusion) at corresponding time instances. AS discussed earlier as
compared to classical round robin transmission schemes, the space-time based fusion via
joint transmission and data fusion achieves a latency reduction of fifty percent which
makes it very applicable for active modes in WSNS where high transmission rates are

required and bandwidth is limited. Table I illustrates the NT = 2 space-time fusion

scheme’s transmission and decision times in comparison to round robin based schemes.

41

Table I. Comparison of Transmission and Decision times for Space-time fusion vs.
Round Robin schemes for two sensors

Transmission Sequence for the Space-time fusion scheme for N T = 2

 

 

 

 

 

 

 

 

 

 

Time (II) 0 1 2 ' 3 ......
Sensor II) 312109) 31:10“ ‘1' 1) 3n1(k + 2) 31:10“ ‘1‘ 3) ........
Sensor n2 —8;,2 (k + 1) 3:2 (It) —s;';,2 (k + 3) — :2 (k + 2)

F133;; ’2’“ 22k+1 fik+2I 11k+3

 

 

Transmission sequence for a Round Robin Scheme for NT = 2

 

 

 

 

Time (II) 0 1 2 3 .......
Sensor n. 87110:) 81.106 '1" 1) .......
Sensor mg 3112 (k) 3n1(k ‘1' 1) .......
Data 111;, “ll/H1 .......
Fusion

 

 

 

 

 

 

 

 

4.1.3. Simulation results and performance analysis

The performance of the distributed Space-time fusion scheme relies heavily on the
degree of spatial and temporal correlation in the sensor measurements. Since we use a
discrete representation for modeling both the source uncertainty; a measure that captures
the degree of correlation (or the lack of it) in the sensor decisions is needed. Since the

source uncertainty is Via a binary symmetric channel, the error parameter )6 indicates the

noise level. We use information as a measure to capture the degree of correlation in the
sensor decisions. Mutual information when appropriately normalized serves as a measure
of correlation. It generalizes the classical measures of linear correlation. The usefulness
of this method of capturing degree of data correlation was recognized early on, see [38].
We note that, “information could serve as a rational basis for a universal measure of the

42

degree of stochastic dependence between two random elements".

In the results to follow we use information to capture the data correlation and the
noise level given binary symmetric channels with different error parameters. As depicted
in Figure 6 when the sensor measurements are highly correlated the mutual information

approaches 1(i.e. S,- —> 0). As [3, the crossover probability increases, the mutual

information approaches the value of 0.5 as expected for Bernoulli random variables.

 

 

Iogtpg-Bscz —8 -8 Iog(p,)-BSC1
Figure 6. Plot of Mutual Information vs. crossover probabilities of Bernoulli distributed
local sensor decisions
We now address the channel uncertainty aspect, i.c. parameters that affect the
performance due to transmission of the local sensor decision through a noisy physical
channel. To make the problem more tractable we assume that the channel state

information is available at the fusion center. In Figure 6 the probability of error, Pe is

characterized both as a function of the channel SNR as well the degree of correlation in

43

 

log (P)

 

 

SNR (db) Mutual Information

Figure 7. Probability of error (P,3 ) as a function of Channel SNR and Mutual Information

sensors’ decisions, i.c. in terms of mutual information. We observe that when the degree

of correlation in the measurements is high, (i.c. mutual information —'+ 1) and the SNR is

in the medium to high range [16], the P8 is of the order of SNR—d as discussed in
section 3.3.1 where d is the diversity order.

The performance of distributed space-time fusion approaches the co-operative
Alamouti [2] scheme as the sensor correlation increases (or equivalently as the reliability
of sensor measurements improves) as seen in Figure 8. We also infer that if the sensor
decisions are not reliable, even good transmission conditions (channel SNR in the
medium to high range) will not help the overall system performance. Therefore diversity
aids the performance only given a minimum degree of accuracy (and correlation) in

sensor measurements. In Figure 8 it must be noted that the all the sensors are assumed to

44

r,=r,=s

 

‘

‘

i

.

.1

fl

- I \\ 7
I

10.- "' 3

-3
10 5‘ -—e— lee-Ol \

: “-9-“ 5:10-02

 

 

Probability of Error

’ ———.——. B=1e03 :
10' + 8:16-04 \' ‘=
E + 8:10-05

’ + lee-OG
"IR-"Alamouti Oo-op
5 10

‘1

10‘

 

 

 

 

I l

15 20
SNR (db)

 

* MAJLAA 1.111

Figure 8. Probability of error vs. channel SNR for two sensors with equal source
uncertainty (cross over probabilities of the BSCS eqUal)

be affected by the same degree of noise and therefore the BSC crossover probabilities are
assumed to be identical for both the sensors. Therefore as the crossover probability
decreases the performance of the distributed space-time fusion approaches the Alamouti
scheme with complete co-operation between the transmit antennas.

In conclusion we note that when the sensor decisions are in the medium to high
reliability range (low sensor noise), the performance of the space-time fusion scheme
without co-operation approaches that of the Alamouti scheme with complete co-

operation.

45

4.2. Channel Aware Data Fusion 11 - Source
Uncertainty via Gaussian Noise

In this section we consider a more general form of the source uncertainty i.c. the
sensor noise is modeled as additive white Gaussian. The problem formulation and the
system model are the same as in 4.1 except for the nature of the source and observation
noise at the sensors. The new overall formulation with additive Gaussian sensor noise in

depicted in Figure 9.

m
6.1
Whisk.
® >\$flb®j “22(5)

INT - - > " " . Fusion Center

Source Uncertainty Transmission Channel

Figure 9.System representation of a parallel fusion network observing a parameter 0 in
the presence of Gaussian noise at the sensors

4.2.1. Problem formulation — Gaussian observation noise

We consider a set of distributed sensors in a region of interest, measuring an
observable, arbitrary scalar parameter or Signal 0 , in noise. The measurements are
assumed to be corrupted by an additive white Gaussian noise (AWGN) with zero mean,
whose variance is a function of effects such as atmospheric changes, sensor calibration

errors, proximity to the phenomenon etc. This can be modeled as,

46

161(k) = 6’ + 771(k), i=1,2,---NT (58)

where I)? isW (0, 0,27) distributed and i is the sensor index and k is the time index of the

sensor observation If . The noise at the sensors is assumed to independent and identically
distributed. For an arbitrary 6 (the parameter can be estimated) the detection problem is

that of a binary hypotheses test,

H0 i$i1ki = milk) (59)
H1 = 151(k) = 9 + 7710'“)

Using the Neyman Pearson criteria, the likelihood ratio test, [If , the likelihood ratio at

individual sensors is given by,

 

AI,- _ p($t(k);H1) > _ (60)
z _

where 7',- is the individual sensor threshold. The probability of . false alarm at the
individual detector is given by, Q(7i) the complementary cumulative distribution

function (CCDF) of the standard normal distribution. Therefore the detection
performance depends upon the noise characteristics i.c. the noise variance and the
thresholds at the individual sensors. We assume identical sensor thresholds since the

transmission channel noise is assumed to be the identical for all the wireless links4.

 

" We note that independent identically distributed Gaussian additive noise with the same variance
at all the sensors does not necessarily imply identical sensor thresholds since the thresholds are
chosen based on maximizing the global probability of detection which is jointly optimized based
on the transmission channel characteristics, local and global thresholds [23] [10].

47

ROC at Individual Sensors for 1“r=1’“I=0} vs. Observation Noise(oz)

1,L_A A7; L.A.~ --H ..L__T_f~-___

 

 

Pdetection

 

 

 

7 0.2 7 7 0.4 V 0.6 I A 0.8— " 1
Figure 10.Receiver operating characteristic of local sensors as function of noise variance

The receiver operating characteristic of the local sensors as a function of varying
level of sensor noise is shown in Figure 10. As expected the best performance is achieved

2 2

when the noise variance 0 is minimum. Note that we assume a = 0,271. i.e. noise level

identical for co-located sensors.
The transmission channel between the sensors and the fusion center is the same as
4.1.1.2 where the channel is assumed to be discrete time flat fading Multiple Input

Multiple Output channel. The received vector is represented as,

E3
= /—Hs + a
5’ NT (61)

where the transmit vector 3 E CNTX1 represents the vector of BPSK modulated local

48

cNRXI

sensor decisions. The received Signal vector at the fusion center y E and the

vector of local sensor decisions, 3 are related by (61). Here H E CNRXNT is the

channel matrix of fading coefficients that are zero mean circularly symmetric complex
Gaussian distributed (ZMCSCG) with unit variance, I9 6 (CNRX1 is the additive

ZMCSCG distributed noise vector with variance ogI and E5 is the peak power

constraint. We note that the transmit power per time slot is normalized over the number

of users, NT .

4.2.2. Data Fusion using space-time block codes

We discuss Space-time fusion schemes based on the complex orthogonal block codes
in this section. AS motivated earlier the aim of the work is to make implicit use of Spatial
correlation in sensor measurements for the dual advantage of reliable data transport as
well as decision fusion. The data fusion scheme for NT 2 2 with Gaussian sensor noise
is identical to the scheme covered in section 4.1.2 from a sensor-to-fusion center

communication perspective. Figure 11 represents the schematic of the fusion scheme for

a quick recap. We now discuss Space-time fusion for four sensors.

 

 

 

 

I i V h
i °°°7372,1(3):8n1(2)73n1(1)78n1(0) - TXI 0 ér—7
<7 711 R’“

__ 7 ._.L
O

at,(2>.—s:,<3).s:,(o>,—s:;,<1II ..

 

Figure 11.Space-time transmission and fusion for two sensors without co-operation and
single receive antenna at the fusion center.

49

4.2.3. Space-time fusion for four sensors

In this section we extend the space-time fusion scheme to the four sensor scenario.
The complex orthogonal space time block code for the fOur transmit one receive antenna
case was discussed in 3.4.1.4 and is related to Octonions or Cayley numbers [25] [15].
For Space—time fusion each element in the space-time block is substituted by distributed
local decisions from four co-located sensors. As discussed earlier [25] Since this structure
is rate 3A this translates to an overall transmission of three consecutive observations per

sensor spread out over a transmission time of four time Slots.

As in the earlier case the symbol sf corresponds to the BPSK modulated waveform

,th
uj’ decision. The four participating sensors transmit their consecutive measurements

based on the following O—STBC structure (62) that combines multiuser and delay

diversity. At each time Slot tk three simultaneous transmissions occur from any three
sensors, for a total of four consecutive intervals, i.e. {t0,t1,t2,t3}. The overall

transmission block is depicted by the matrix S representing the sensors and

corresponding transmission times.

 

 

 

 

30”1 SI"1 92’“ 0 i
”2* ”2* ”2 sensorl 7‘
a 50 0 32 sensor2 ,
S: _52713* 0 383* _Sl”3 sensor3 > (62)
4
“4* ”4* sensor >
0 -32 31 30

 

 

50

The received vector at the fusion center can be modeled for the four node space-time

scheme based on the MIMO system form in (61) and is represented in (63).

 

 

 

 

 

 

 

’ n1 n1 711 I
. ' T % éI n I) . T
WT In "P %* In %
yr _ hr ‘31 50 0 32 + 191 63
W — h? _s;3* 0 883* _S{l3 192 ( )
93 h. I9
. . . 3‘ 0 _S;I4* 8114* 88.4 i i 3‘

 

We also assume that the channel is quasi-Static, over the four time slots. As in the earlier
case, every participating sensor transmits its respective decisions regarding the local
event based on a pre-determined column (or row Since orthogonal structure) in the space-
time transmission block. In (62) the rows of transmission block represent the local
decisions at individual sensors and the columns represent the respective transmission
times. We also note that all the sensors do not transmit in all the four time slots since the
code is not full rate. There is a delay of four time slots before the global data fusion can
occur at the fusion center as indicated in Table H. Since the noise is additive Gaussian
and we assume a non-informative prior, the Maximum Likelihood detector is optimal and
the fusion-detection rule is given by,

§=arg in Ily—HS(x)||

vzegMTXI (64)
where {:13} is the set of all possible transmit vectors corresponding to vectors of local

sensor decisions, {It} . The joint detection approach once again serves the dual purpose

of detection as well as fusion of the multiple sensors’ observations.
As noted earlier, based on the assumption of identical noise at the sensors and

identical sensor thresholds we infer that the sensor performance indices are identical, i.c.

51

Table H. Comparison of transmission times for Space-time fusion scheme vs. fusion
usiflRound Robin data collection scheme for four sensors

 

Transmission Sequence for the space-time fusion scheme (4 Sensors)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Time (k) 0 1 2 3' 4 5 6
Sensor(n1) 531 sill 331 sgl 321 357,11
Sensor(n2) —s{i2* 582* 332 321 331
Sensor(n3) —s; 3 33 3 * —s{7' 3 3:1 331
SensOr(n4) —s;4 * sf“ * 384 3:1 821

Fusion Center 120,121, 1’12
Round robin transmission sequence for optimal fusion (4 Sensors)
Sensor(n1) 3611
Sensor(n2) 389
Sensor(n3) 33 3
Sensor(n4) 534
Fusion Center 120
Sensor(n 1) sill
Sensor(n2) 3179
Tutsi/III." E, E, E, E, E, E, E,

 

 

 

 

 

 

 

 

the probabilities of false alarm, miss and detection will be the same for all the sensors
therefore equal weights can be assigned to all the elements in the received vector and the
sensor decisions in the search space. Therefore the fusion rule assigns equal weight to all
the sensor decisions. We will shortly analyze the detection performance of the proposed
schemes for the two and four sensor space-time fusion schemes.

We note the significant reduction in latency that is possible through such a scheme;
this is highlighted in Table H where we compare the four sensor space-time fusion

scheme’s time to decision with that of an optimal fusion scheme that utilizes round robin

52

 

transmissions to the fusion center. In a round robin scheme, the latency both in data
aggregation as well as fusion is directly proportional to the number of sensors and the
sensor sampling rate. From a latency reduction perspective notice that the four sensor
Space-time fusion scheme results in a latency reduction of approximately 67% per
transmission block while operating under the identical energy constraints as round robin
transmissions deploying optimum fusion. This has potential benefits to sensor networks
when they operate in active modes or when data is sampled at higher rate i.c. when more
nodes in a region of interest are active due to event triggering. This particularly useful in

dense deployments which are not delay tolerant and are bandwidth limited.

4.2.4. Detection performance and comparison with optimal
fusion schemes

The performance of the space-time fusion schemes depends on the correlation in the
local sensor decisions, which in turn is influenced by the sensor variance and the
selection of the local sensor thresholds. The performance analysis as a function of sensor
thresholds and noise variance is presented in this section. The results give an insight into
conditions under which space-time fusion performs as well as optimal fusion schemes, so
that other potential benefits of the scheme such as reduced latency and scheduling
complexity, increased throughput and overall energy and bandwidth efficiency can be
utilized. We analyze the Receiver Operating Characteristic (ROC) of the global system as
a function of local sensor thresholds, noise variance at the sensors and channel SNR. The
performance of the Space-time fusion schemes is compared to optimal fusion schemes
that deploy round robin transmissions for aggregation.

In this section we present a brief review of the optimal fusion scheme(s) that will be

53

used for comparative analysis. We then present the detection performance as a function
of global probability of detection and false alarm in variable sensor thresholds and noise.
These results are presented for the two and four sensor space—time fusion schemes. Since
the Receiver operating characteristics are presented for a Specific channel SNR, in order
to complete the picture we present a probability of detection VS. channel SNR
perspective. We also present comparative results for equal time or equivalently energy
for Space-time fusion schemes with the optimal fusion, the particulars of which will be

discussed in the sequel.

4.2.5. Review of comparative optimal fusion rules in fading
channels

In this section we present a short review of the data fusion schemes that will be used
for benchmarking the performance of the proposed schemes. Classical data fusion
schemes for various sensor topologies [10] were formulated under the assumption of an
ideal transmission channel between the local sensor and the fusion node. For the parallel
sensor network topology, the fusion rule is usually. a weighted Boolean function of
individual sensor decisions, where the weights are determined based on the sensor
performance metrics as discussed in 2.5. Recent work [22], [23], [24], extends classical
fusion schemes to wireless channels, in particular the fading channels in case of wireless
links. The global decision is determined by formulating a fusion rule i.c. based on a
Likelihood Ratio (LR) Test which is a weighted combination of the individual sensor
decisions, sensor metrics and the channel conditions,

f()’/H1)
f(y/H0)

A] opt = log

 

 

(65)

54

The optimum Likelihood Ratio (LR) based fusion rule given full channel state
information and individual sensor performance indices, i.c. probability of detection and
false alarm was discussed in the background chapter. It is presented here for

completeness in (66) where NT , corresponds to the number of co—located sensors.

Therefore the global decision (or the fusion rule) at a given instant is based on assigning
weights to individual sensor decisions in terms of the sensor performance metrics as well

as the channel conditions.

' T 6 20v +(1_ p])e 20.0
A] = log pd d (66)
Z III-III? _||yj+hr"2

. 2 . 2
2 ., 2
The 0‘ + (1 - Pfak 0‘"

 

 

 

 

A sub-optimal LR based fusion rule which is a high SNR approximation to the optimal

LR rule [23], [24] is given by (67),
NT
N_sz:( (ch Pit) )hjyj (67)

This reduces to the following analogous form of the Maximal Ratio Combiner (MRC) as
in (68) for local sensors with identical performance indices, i.c. identical probabilities of
detection and false alarm at the sensors. Henceforth, we refer to (68) as the MRC fusion
rule, which is the optimal fusion rule when the sensor indices are identical. In this work
we apply the MRC fusion rule where the channel fading coefficients are complex

(assume in-phase and quadrature components in the received signal).

AMRC = N— Eighty (68)
Tj= —1

55

We compare space-time fusion to schemes employing the MRC fusion rule (68) in
conjunction with round robin transmissions for data collection. These schemes are
equivalent in the sense that both the metrics require channel state information and assume

identical local sensor performance indices. Therefore for NT sensor data fusion with

round robin data collection, given identical sensor performance indices, the MRC fusion
rule yields the optimal global decision based on NT sensor transmissions over NT

symbol intervals.

4.2.6. Detection performance of space-time data fusion

The performance of space-time based fusion schemes relies heavily on the degree of
spatial correlation in the sensor measurements (which directly translates to low
observation noise and dense distribution). This is because there is no information
exchange between the sensors in order to be energy efficient (in this case the available
resources could be channeled to facilitate event capture at a higher resolution). For
Simulation purposes we assume a deterministic Signal/parameter of interest, 0 which
takes the form (69) depending on the hypothesis in effect.

1 if H1 in effect

= 0 H0 otherwise (69)

Assuming that the sensor cluster is co-located with the event of interest, we restrict

2

the variance of i.i.d. sensor noise 77,; to the range 0.1 S 077

_<_ 0.5. In general however 0

and 0727:. can be chosen or scaled according to on topology Specific parameters, which can

be determined offline or prior to deployment. The selection of the local sensor thresholds

is dependent on the local sensor noise, transmission channel conditions, density

56

of active sensors and tolerable level of the system-wide false alarm. Analysis of the
selection of the local sensor thresholds as a function of various system level parameters is
a very important aspect of the distributed data fusion prOblem. It is beyond the scope of
the current work and an interested reader is referred to [24] for a complete discussion on
optimizing thresholds. We analyze the global performance in terms of the Receiver
operating characteristics, over a range of local sensor thresholds. We characterize the
system performance metrics, i.c. the global probability of detection and false alarm rates
as a function of the local sensors’ noise, thresholds and the transmission channel Signal-

to-noise ratio.

4.2.6.1. Results for two sensor space-time data fusion

We analyze the system level performance as a function of the sensor noise, local
sensor thresholds and the transmission channel. The Receiver Operating Characteristics

(ROC) in terms of the global probability of detection and false alarm for the sensor noise
range 0.13 0,27 3 0.5 are presented for a fixed channel SNR in order to reduce the

dimensionality of the analysis. The SNR is fixed at 15db, which is a typical operating
point and offers a good system probability of detection without trading off the false alarm
rate. We compare the performance of the space-time fusion scheme based on the
Quaternion structure with the Maximum Ratio Combining (MRC) data fusion statistic for
the two sensor case.

We note here that the total energy expenditure of the both schemes is identical,
however the bandwidth or time required by the MRC based fusion scheme is higher due

to round robin data collection. The comparison is not entirely fair to the Space-time

57

fusion scheme since the MRC fusion does not experience any co-channel interference.
Space-time fusion on the other hand tries to exploit the multiuser diversity by
constructively utilizing the multiple correlated transmissions. However, if the sensor
noise is high and the decisions are uncorrelated, the multiuser diversity may degrade the
performance due to destructive interference. The following ROCS compare space-time
fusion schemes with optimum fusion schemes under identical transmission power

constraints.

58

ROCs for Dan Mimi with NT-2,o:-0.l, era-net SNR-Isa

 

 

 

 

 

 

 

 

I W fii; v n W
0.8 ' ~
0611‘ -
3‘ 1: 4;
94
0.4if .
I;-
0.2 ’ .
—e— Space-time fusion (Qtnternion) for NT-Z
—*— Maxirmrm Ratio Combining fusion rule
0 1 f i J
0 0.2 0.4 0.6 0.8 I
system
Pf:

Figure 12. Receiver Operating Characteristics of Space-time fusion vs. Optimum fusion
based on MRC rule with 0,2) = 0.1 for the two sensor scenario

ROCs for Data mm with NT-2,o:-O.2, MI SNRclSO

l 1 r v i.— .-— J.
.

 

 

0.8 .

0.6 » "

system

 

P

0.4 i'

0.2

 

—+— Madman Rdio Combining fusion rule
—e— Space-time fusion (Qutemion) for NT-Z
0.2 0.4 0.6 0.8 1

system
Pfa

 

 

 

 

 

 

Figure 13. Receiver Operating Characteristics of Space-time fusion VS. Optimum fusion
based on MRC rule with 0,27 = 0.2 for the two sensor scenario

59

ROC! for Data MIN with Nlepi-Ofi, chine] SNR-l

I w . ____-,-=—_ . —— .

- 0

 

   
 

 

 

’ —e— Space-time fusion (Mon) for FIT-2
—*— Murmur: Rtio Combir'g fusion rule
0.2 0.4 0.6 0.8 l

syn-m
Pf:

Figure 14. Receiver Operating Characteristics of Space-time fusion vs. Optimum fusion

 

 

 

based on MRC rule with 0,2, = 0.3 for the two sensor scenario

 

ROCs ror Du Mic: with RT-z,o:-0.4, channel SNR-15.

 
 

043/
08» Ar .
g 0.6 ’- ..
K‘v
“I.

0.4. '

 

0.2

 

—*— Mnimm Rnio Combining fusion rule
—e— Space-time fusion (Quiemion) for lit-2

 

 

 

 

 

 

0.2 0.4 0.6 0.8 I

'6

Figure 15. Receiver Operating Characteristics of Space-time fusion vs. Optimum fusion
based on MRC rule with 0,2, = 0.4 for the two sensor scenario

60

A. o;-0.1 B. off-0.2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0-95 KLEg—e—s 0.95. ' ' '
b
(19 / . 0.9 . .
0.05 . 0.052;: :f' “i ‘ <
0.8 - - 0.8 I .
0.75 . —G— Space-time firsion . 0,75 I —e—— Space-time fusion .
—I-— MRC fusion —I— MRC fusion
0.7 -. 1 ‘ 0.7 ‘ . 1
5 10 15 20 25 5 10 15 20 25
system
Pd 0. o:=0.4 D. 0:205
0.95 . 7 ' ' 0.95 ’ ' I
—9— Space-limo fusion —€>— Space-time fusion
0.9 ~ —-I— MRC fusion . 0.9 . —#— MRC fusion
0.85 ~ - 0% >
0.8 ~ I 0.8 [
M—H—W—fi
6,147 . -II-—'—‘— * é“ @— F
0.75 075wa—
0.7 . - . a] I . .
5 10 15 20 25 5 10 15 20 25

8NB(db) —-——->
Figure 16. Probability of detection of the system as a function of Channel SNR for
different levels of sensor noise for two sensor data fusion
Based on the Receiver operating characteristics for the noise range under consideration it
becomes clear from Figure 12, Figure 13, Figure 14 and Figure 15 that space-time fusion
without co-operation performs at par with an optimum MRC fusion scheme utilizing
round robin data collection. Another View that is presented in Figure 16 is that of the
system probability of detection as a function of the transmission channel SNR for various
levels of sensor noise. We note that in this case also, in the medium to high SNR range
the system performance of the non-cooperative scheme closely matches that of the

optimum fusion scheme.

61

These results clearly indicate the conditions under which space-time block structures
can be deployed in a completely distributed manner without data exchange between
participating nodes. Such an approach serves the dual purpose of channel aware signal
processing as well data fusion. The proposed fusion scheme for the two sensor case
yields a 50% percent reduction in the latency during data transmission and fusion phases,
in low to medium sensor noise ranges, this can be inferred from Table I. In cases where
there is very high decision correlation the performance is at par with that of co-operative
multiple-antenna communications, thereby contributing to reliable data transport by

improving link/data reliability through diversity.

4.2.6.2. Results for four sensor space-time fusion

In this section we present performance results for the four sensor fusion schemes i.c.
the Octonion based space-time fusion discussed in 4.2.3 vs. MRC fusion based on round
robin transmissions [2.5.2.2] for varying levels of sensor noise. we note that in the low
sensor noise range, the performance closely matches the optimum (MRC) fusion as seen
from Figure 17 and Figure 18. Low sensor noise leads to highly correlated sensor
decisions within the co-located nodes. In high sensor noise, i.c. Figure 19 and Figure 20
there is a marginal degradation in performance which can be attributed to the perturbation
of the code structure due to high noise. In the high sensor variance region the
performance deviation from the optimum seems higher compared to that of the two node
Space-time fusion because of higher interference. This can be thought of in terms of the
Signal-to-Interference noise ratio which is lower in the four sensor fusion case as

compared to the case of two sensor space-time fusion.

62

noes m: on ram with lit-4, Iii-oi, err-net m-rso

 

 

 

1 .
0.8 , -
g 0.6 -
tau

0.4 .

0.2 ' . . .

. —e— Space-tune fusron (Octomon) for FIT-4

—9— MRC fusion rule for NT-t

0 0 0.2 0.4 0.6 0.8 1
PM
to

Figure 17. Receiver Operating Characteristics of Space-time fusion VS. Optimum
fusion based on MRC rule with 0,2, = 0.1 for four sensor data fusion

ROCs for Dan Fusion with NT-4, oi-oz, chm] SNR-15d:
1 - 3 _ ,' _ - _ O a: 1. .p

 

 
 

0.8 7 1.! '

 

 

 

g 0.6 I .
has
0.4 ..
0.2" . . .
—e— Space-tune fusron(0ctomon) for NT-It
—e—MRC fusion rule for NT-t
0 0.2 0.4 0.6 0.8 1
system
Pr.

Figure 18. Receiver Operating Characteristics of Space-time fusion vs. Optimum
fusion based on MRC rule with 0,27 = 0.2 for four sensor data fusion

63

)-

 

ROCs for on Fusion with HT-I, Iii-0.8, m1 arm-1

I _"‘,‘=-—iv;— ;: '\,-" ‘4
,‘.f" - ‘

    

0.8

 

 

—e— Space-time fusion (Octonion) for NT-4
+MRC fusion rule for LIT-=4

 

 

 

 

 

0.2 0.4 0.6 0.8 1
system
P50
Figure l9.Receiver Operating Characteristics of Space-time fusion vs. Optimum

fusion based on MRC rule with 0,2, = 0.3 for four sensor data fusion

ROCs for Data Fusion qu: N194, 11:30.5, charmer SNR-15¢!)
1 v T _ :5 1.. _

  
 

- no.0.

 

 

 

—e—— Space-time fusion (Octonion) for N14
+ MRC fusion rule for NT-t

 

 

 

 

l I I

0.4 0.6 0.8 1
system
Pf:

Figure 20.Receiver Operating Characteristics of Space-time fusion VS. Optimum
fusion based on MRC rule with 0% = 0.5 for four sensor data fusion

I. 03-01 b. o’aoz
'1 n

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.982;!‘8 ‘3 7 it 0.98 » . 7
0.0 . - 00%:
c
0.85 I . 0.85 I -
0.8 I —e— Space-time fusion < 0.8 . —€I— Space-time fusion
—‘1'_ MRC fusion —*_ MRC fusion
0.70 r . . . 0.75 - . .
5 10 15 20 25 5 10 15 20 25
system
Pd 0. o:-0.3 (1.01:0.5

 

 

v V Y *7

0.95 I —e— Space-time fusion ' 0.90 I —e— Space-time fusion
0.9» +0433 fUSIOI'l . 0.9 . +MRC 0.151011

 

 

 

 

 

 

(D -II-
(>41-

 

 

 

 

 

 

 

 

0.857— 0.85 . .
(V’s,— , .. ..
0.8 I I 0.8% . . . -p
0.75 1 . . . 0.75 W
5 10 15 20 25 5 10 15 20 25
Channel SNR (db)-——->

Figure 21. System probability of error vs. channel SNR for four Sensor data fusion

In Figure 21 we present the channel SNR VS. probability of detection perspective for
the four node space-time fusion and compare it with the optimum fusion scheme, i.c.
maximum ratio combining rule. We note that the performance of the four sensor space-
time fusion scheme closely follows that of the MRC fusion rule. The energy constraints
in both the schemes are identical for these results. Note however the deviation from the
optimal performance increases as the sensor variance increases (or equivalently decision
correlation decreases) which again stresses the need for a threshold level of correlation
required for the distributed scheme to yield benefits. It also indicates that joint

transmissions from co-located sensors at lower transmission power can be very

65

beneficial. It is interesting to note that while operating high sensor noise, the overall
system has to tolerate a high false alarm rate to achieve a given probability of detection

and this hold true for both the fusion schemes under consideration.

4.2.7. Performance results of two sensor fusion under equal
energy and time constraints

The space-time based fusion schemes, by virtue of the joint transmissions benefit in
terms of the overall throughput. An interesting perspective would be to look at the system
performance from an equal energy and “time” perspective, i.e. have identical time/delay
constraints for global decisions at the fusion center. In the two sensor fusion case we note
that data transmission only takes half the time required by a round robin transmission
scheme (in which case there is at least a four symbol delay for in order to make a global
decision). The total time can be equalized by repeating the space-time codeword, with the
energy per block reduced by half such that the transmit power constraint is unaltered. In
this case however the time to global decision i.c. four time slots would be the same for
both the schemes under comparison. The equivalent space-time block is a repetition of
the transmission block S as in equation(53), at half the transmit power per symbol and

can be represented by the formulation in equation(70),

 

 

 

 

 

 

 

 

’ 5.10)) 3,10.- + 1) 0 0 ‘
—s;';2 (1- + 1) 3:2, <1) 0
S, 2 0 0 9.101) s..., (1- +1) (70)
0 0 —s;;, (1 +1) st, (1)
._ n a a a _, a :1; a ._ :9 _.
K 2 2 2 2 )

 

 

66

Equal time ST-fusion waveforms
(per information/decision symbol)

1..

 

2 7 .
\JNJT— #X/Akr—fi
< To/2 . < To/2 »

Round robin transmission waveform
(per information/decision symbol)

E. A

U \f“

4 To >

//\
r/ \
__4_/

Figure 22.The transmission time and energy per decision for space-time fusion vs. round
robin transmissions for the two sensor fusion for identical energy and delay constraints

Figure 22 illustrates the transmitted waveforms for equal energy and identical delay
constraints. The results for the equal energy equal time ST fusion scheme for the 2 sensor
case are presented in Figure 23, Figure 24, Figure 25 and Figure 26.

It becomes clear that performance of the equal time ST fusion is identical to the
optimal MRC fusion scheme for the entire range of sensor noise. We also observe that the
system probability of detection as a function of the channel SNR is higher than of the
round robin transmission scheme in low to medium SNR ranges, which is not quite
discernible in the ROCS. In the low sensor noise case the path diversity provides error
resilience under noisy transmission conditions but as the channel becomes reliable (high
SNR) we observe that the performance of MRC fusion approaches that of space-time

fusion.

67

noes (equal atom) with ai-on, channel SNR-15d:

 

 

 

 

 

 

 

 

 

 

1 M3 u w ' u u ' u 7-3 SCH
0.8 ~ .
g 0.6 .
I“
9..
0.4 -. -
0.2 . .
. —o—MRC fusion - 4 time slots
:..i —9— ST fusion - 4 time slots

0 .‘I. J 1 1 I

0.2 0.4 0.6 0.8 l
PSysm

in
Figure 23. Receiver Operating Characteristics for data fusion schemes for two sensors

with identical energy and delay constraints with sensor variance, 0,27 = 0.1

Roe: («pa energy/um) with «Ii-0.2, channel sun-1m

 

 

    
  

 

 

 

 

 

 

 

0.9 I .1” . .
0.8 . i3 .
0.7 I .
0.6 . .
g‘ 0 5 d
m i

0.4 9’ .
0.3 .
0W2 d
0.1 —B—MRC ftlsion-‘ttimeslotsT

+ sr fusion - 4 time slots
0 0.2 0.4 0.6 0.8 l

Psm
b

Figure 24. Receiver Operating Characteristics for data fusion schemes for two sensors
with identical energy and delay constraints with sensor variance, 0,2, = 0.2

68

ROC: (cqml mam/time) with 01-04,ch sun-15a.

. . - . .. “
1 I, ... :I d: 3. .m n . - “I"

 

 

 

—B—MRC fusion-4time slots
+S‘I‘t‘usion-4time slots

0.4 0.6 0.8 1
System
Pf:

 

 

 

 

 

Figure 25. Receiver Operating Characteristics for data fusion schemes for two sensors

with identical energy and delay constraints with sensor variance, 0% = 0.4

   
 

  

ROC: (equal energy/um) with nines, chm! sun-1

. . 'N-‘A‘ -
."m -'v"

 

  

--
It

 

 

—E+—MRC fusion-4mm slots
+Sl'fusion-4timeslots

0.4 0.6 0.8 1
System
pf.

 

 

 

 

 

Figure 26. Receiver Operating Characteristics for data fusion schemes for two sensors
with identical energy and delay constraints with sensor variance, 0,2, = 0.5

69

In conclusion as depicted in Table H, for four sensors space-time fusion the overall
data transport latency is reduced by approximately 67% percent without any significant
deviation from optimal performance, while operating within identical energy constraints.
This enables event capture and data sampling at higher granularity and while reducing the
time spent in waiting for data transmission as well as global decisions. The key strength
of the space-time based schemes lies in the ability to significantly improve the throughput
of the network during the event triggering modes. These features become particularly
important in dense sensor networks which are pre—dominantly designed with low

transmission rates and bandwidth.

70

5. DISTRIBUTED ESTINIATION - IMPACT OF
RECEIVE DIVERSITY AND CORRELATION

5.1. Background and Overview

Wireless Sensor Networks are the key to ubiquitous sensing, a paradigm that is
widely applicable to industrial and scientific applications. They impact communication,
instrumentation and control mechanisms by enabling distributed sensing, higher data
granularity and information acquisition. The design of such networks contends with
several issues [9], [27] such as sensor placement, topology changes, environmental
effects, reliability etc. while operating under stringent energy and bandwidth constraints.
In applications such as target tracking, habitat and environment monitoring [39] [40] etc.
where there is dense deployment of sensors over a geographic spread, the sensor
measurements are spatially and temporally correlated. There has been significant research
in exploiting correlation for reliability and energy efficiency; where majority of the
approaches [28], [37], [4]] have been compressive i.e. based on eliminating the
redundancy given data correlation and distortion constraints. Reference [46] discusses the
influence of node density and spatio-temporal correlation on the total distortion in field
estimation, in the context of hop by hop routing.

Diversity based techniques in broadband wireless access with their ability to provide
higher data rates, spectral efficiency and reliability, have spurred their application and
extension to equivalent scenarios in sensor networks. This has mainly been in the area of
co-operative relay based routing schemes. The methodologies proposed in [29] [30] [31]

[42] [47] [51], incorporate the benefits of diversity and joint transmission via relays.

71

 

 

 

 

 

(a) Dense Mode (b) Sparse Mode
Figure 27. Representation of operating modes in a sensor network

Cooperative source-channel coding schemes [43] [48] present a cooperation strategy
where neighboring nodes act as virtual antennas. However, the design and analysis was
focused towards coding with little emphasis on the effects of node density, separation or
correlation models. In [49] the concept of a Wireless Information Retriever (WIR)
equipped with an antenna array for querying sensor ensembles using wideband
waveforms was proposed. The approach highlights the benefits of receive diversity and
joint transmission albeit only for the extreme cases of identical or completely
independent sensor measurements and the analysis does not extend to underlying
correlation structures.

Emerging approaches to energy efficiency include the advocacy of active sensor
networks [44], which introduce in-network processing and dynamic computation into
sensor networks. While dense sensor deployments do exhibit redundancy in
measurements there is the necessity of maximizing the network life—time which entails

low sensor utilization. Sensor networks are heading towards reactive approaches where

72

the distribution of active sensors in regions of interest is adaptive. This is particularly
relevant to applications such as [27] [39] [40] where sensor clusters are deployed over a
large area. Typically the network operates in a sparse mode, Figure 27(b), when there are
very few active sensors per cluster. Upon event triggering, (for instance increased seismic
activity), the data fidelity and resolution has to be much finer. Therefore the number of
active sensors in a region of interest needs to be higher; thereby, the network segments
adaptively switch to a dense mode, Figure 27(a). In the dense mode the sensors are
required to sample at higher data rates.

The data in such scenarios which is inherently correlated can be suitably modeled by
spatially correlated jointly Gaussian random variables. Also, as pointed out in [1], we
note that “in order to maximize the capacity of a Gaussian multiple access channel, one
should preserve the correlation between the inputs of the channel. Slepian-Wolf encoding
on the other hand gets rid of the correlation”.

Therefore we consider joint transmissions of the sensor measurements to the sink or
the base station without coding. Also, since we consider multiple antennas at the base
station or virtual antenna configurations for reception, the system is equivalent to a
MIMO system. On enumerating various parameters from this perspective, the following
emerge as key factors in benchmarking the performance of the estimator,

I Spatial Correlation in measurements i.e. the degree of correlation.

' Multiuser Diversity which in the context of joint transmissions corresponds to the

number of independent links between users spread out over a geographic area;

- Receive Diversity which is equivalent to the number of cooperating neighbors (in

case of virtual antenna configurations) or the number of receive antennas

73

deployed at the fusion center or base station.

The aim of this work is to present a holistic analysis of integrating spatial correlation,
joint transmissions and receive diversity and its impact on the achievable estimation
accuracy. Although a wealth of related work exists, each of the above parameters is
addressed in a disparate manner that may obscure the complete picture. In this work we
formulate a system abstraction that integrates the key factors in order to characterize the
achievable performance limits. The abstraction is high level in the sense that we do not
focus on implementation specific details in order to be consistent with the scope of the
work. We demonstrate the performance gain in WSNS by integrating spatial correlation,
multiuser and receive diversity without sacrificing bandwidth.

Such analyses are applicable to scenarios as in [39] [40] in order to adaptively
determine:

I The number of sensors that need to be activated for required estimation accuracy.

I The required instantaneous receive diversity for minimizing the distortion

measure given fixed node density, spatial variance of events or energy constraints.
This knowledge could be used to create an adaptive virtual array for reception
using neighboring cluster nodes.

I Impact of node density on estimation accuracy given fixed constraints on number

of available nodes or receive diversity.

In the sections to follow the overall system model used to capture the physical
process and the associated correlation model of the measurements is described. This is
followed by the description of transmission channel model, derivation of the optimum

estimator and the analytical expressions for theoretical MSE as a function of correlation,

74

node density, receive diversity and the channel conditions. Finally the simulation results

and corresponding analyses are presented in the final section.

5.2. The Physical Model and Data Correlation
5.2.1. Modeling the physical phenomenon

The objective of sensor deployment is to be able to assess, accurately the field under
observation which can be characterized in terms of its spatio-temporal behavior.
However, the degree of spatial and/or temporal correlation is entirely a function of the
underlying physical process. Consider a Region of Interest (ROI), the sensor observations
could be noisy samples of a point source, S (e.g. target tracking applications [27]),
where the correlation in sensor measurements depends on the distance between the
location of source and sensor. In other cases there may be multiple spatially correlated
sources that evolve in time, for e. g. seismic activity. Whether the objective is to estimate
an event source based on distributed sensor observations or that Of independent sensor
samples of a spatially correlated random processes, joint Gaussian random fields are
widely used in modeling such processes [46] [41].

We model the spatially correlated physical process, f(a:,-,y2-,t) as a joint Gaussian

random process that evolves in space and time. The observations of the process at the

NT sensors are modeled by a random vector X formed by joint Gaussian random

variables, where Xz-(t) is the instantaneous sample of the correlated random process

f(a:z-,yz-,t) or equivalently the sample of an event source, S' at the ith sensor. For

simplicity we assume that the quantization is performed with negligible distortion of

source characteristics and that the sensors are fairly reliable. Therefore we have

75

X ~ W(yX,CX) and if we consider a point source S to model the information in the
region of interest we assume p5 = px & GS = CX. Further, individual sensor

observations are assumed to be identically distributed and independent in time. Therefore
the PDF of X is given by:

I
(2W)N% ICX|N%

where E(X,) = ii,- = 0;Var(Xz-) = 0% Vi.

 

p(X)=

 

1 T —1
exp --2-(X flux) CX (X '“X) (71)

5.2.2. The correlation model
The correlation of the observations is modeled by the widely used [46] [41] power

exponential correlation model which is a function of the separation between the nodes.

Let dij = “if, — fj" denote distance between the nodes i, j located at positions 5?,- & 5?]-

respectively. The correlation between any two nodes i, j is given by ogpz-j , where pij is

the correlation coefficient. Therefore the covariance of the observations for p X = 0 is

given by,
f 2 2)
Oz 912035 plN T 0:1:
2
@101; i Z
CX = . . 2 . <72)

: : 0x :

(pNT 10:1: 0:1: )

 

 

(73)

 

[in
p.j=e 91 ;01>o.626(1,21

76

7(dz'j) = — (74)

The correlation coefficient is given by (73) and the value 727- represented in (74) captures

the dependence on the spatial separation between the nodes.

5.3. System Model and Optimal Estimation

5.3.1. The optimum estimator

In wireless sensor networks it is reasonable to assume that the signaling rate is
significantly higher than the rate of change of the propagation parameters [27], therefore
we use a frequency non-selective fading channel with additive noise, where the same
channel is used multiple times to transmit observation vectors. We consider uncoded but

joint transmissions of correlated observations from N T nodes in a region of interest via

the transmission channel. The received vector Y at the virtual antenna or sink, with

receive diversity N R is modeled by

I C
Y = —HX N
NT + (75)

where H E (CNRXNT , is the matrix of channel fading coefficients that are assumed to be

zero mean circularly symmetric complex Gaussian (ZMCSCG) with unit variance. The

power constraint is given by E(XHX(/NT) = C assuming 03 = 1.Therefore the total

power per channel use is normalized in (75) by the factor C/NT . The received vector

at the sink or the base-station is represented by Y E (CNR X1 where X E CNT XI denotes

the transmitted vector and N E (CNRX1 ~ W (0, 0,2, ) is the receiver noise which is

77

additive and is also assumed to be zero mean circularly symmetric complex Gaussian

(ZMCSCG).

X1
ié H, Sensor1

Sensor 2

 

 

 

 

 

 

 

 

 

 

SNT XNT

M ’ Sensor NT

Spatially Correlated Gaussran
Random Field
Y .5
loioeuog ie uoueinfiguoo
euueiuv lenuwameoaa

 

 

 

 

 

 

Source Transmission Channel

............

 

Figure 28. Multiple Input Multiple Output System representation of a Gaussian random
field with multiuser and receive antenna diversity

In cases where prior information is available for parameter estimation, the Maximum
A Posteriori (MAP) Estimator is known to be optimal whereas the typically used

Maximum Likelihood Estimator (MLE) assumes a non-informative prior. For accurate
estimation we need to determine If that maximizes the a posteriori PDF given by,

X = argmfix 19(X/Y) (76)

Using the Bayes Rule,

p(Y/X)p(X)
100’)

(77)

 

arg max p(X/Y) = arg max

The optimum estimator, i.e. the MAP estimator is given by (77) which for the present

case reduces to (79) as follows.

78

A

X = arg my p(Y/ X) p(X) (73)

Y— —C—HX
liNT

Since the model represented by (75) fits that of a Bayesian Linear Model for

2
XMAP =min +XTCX(p)X
VX

(79)

 

 

 

 

 

 

deterministic H we conclude that the posterior PDF is also Gaussian [50]. Therefore in
this case, the MMSE estimator is equivalent to the MAP estimator, and is also simple in

its implementation. The MMSE estimator for the system defined by (75) is given by,

—1
XMMSE = 7VLCX(p)H* [7:— HCX (p)H* + 0'12vl Y (80)
V T T

We represent the source covariance as C X (p) to underline the dependence of the
estimator on the correlation coefficient, which is related to the node density (or spatial

separation of the nodes). It emerges as an important factor 'in characterizing the

performance of the estimator.

5.3.2. The performance of the estimator

The performance of the MMSE estimator for the model under consideration is well
characterized and can be measured in terms of the error a = X — X , the variance of the

estimator [50] is given by equation (81) where E{} indicates the expectation operator,

—1
E(ee*) = CE = C}1(p) + NLH‘ICXIIH (81)
T

Therefore, the theoretical Mean Square Error (MSE) is given by 05, where the

diagonal elements yield the individual nodes’ MSE.

79

MSE(X,-,-) = of (82)
Based on equation (81) we note that the MSE depends directly on the following factors:
I Correlation in the sensor measurements via C X (p) (the source covariance)
I Diversity (Multiuser and Receive) the effect of which is captured via the quadratic
form of H (NR x NT).
I Signal to Noise Ratio (SNR), the channel conditions, impact the MSE in terms of
the noise power through C N , the noise covariance.

In the sequel, we analyze the achievable performance gain in terms of minimum MSE.

80

5.4. Simulation Results and Analysis

We re-iterate the focus of the work, which is to analyze the impact of a general
framework that integrates multi-user diversity and receive diversity with sensor
correlation on the overall field estimation accuracy. In the previous section it was shown
that the variance of the optimum estimator is impacted by the parameters based on the
MIMO perspective. In this section we quantitatively analyze the effect of these factors,
i.e. node density, spatial separation between co-located nodes, multiuser diversity and
receive diversity on the minimum MSE of the system. The theoretical MSE was
calculated by averaging the individual nodes’ MSE over several channel realizations. We
use the power exponential model as defined in (73) with 02 = 1. As noted in [41] , 01
controls degree of correlation between nodes as a function of distance. It can be
interpreted as a topology related scaling coefficient. We use the measure 7 outlined in

(74) as a measure of the node density (number of nodes per cluster for a given topology)

or the spatial separation. For example, the value of 'y = 0. 001 may correspond to a value

of 01 = 10 and a node separation of d = 0. 01 or in a different instance could

correspond to a topology specific value of 01 = 1 but with node density much higher, i.c.

with separation d = 0. 001. Also we note that the power constraint was set to C = 1.

MSEovemu = 11111 ‘27. c}; (83)

n—Ioo n
The overall MSE value was calculated based on (83), where Cf} is individual sensor

MSE.

81

5.4.1. MSE as a function of spatial correlation (node
separation)

The correlation model under consideration is a' function of the spacing between
individual nodes. We consider uniformly spaced nodes in a fixed topology; therefore 01
is set to be constant. The parameter 7 is varied over a range of values, which is
equivalent to varying the spatial separation of the sensors. Note that the value of gamma
is inversely proportional to the degree of correlation in the corresponding observations.
Since the covariance function is entirely parameterized by the separation between nodes,
the same approach can be extended to randomly placed nodes.

Based on Figure 29 and Figure 30, we note that beyond a given threshold of 'y, the
achievable MSE saturates for fixed NT,N R and channel SNR, for the case under
consideration, if 7Threshold 6 (1,00). This is apparent from the logarithmic MSE vs. '7
characteristic see Figure 30 when ’7 6 (0,1] the MSE is significantly lower and remains
unchanged beyond this range. In further analysis we consider the values within 7 6 (0,1]
only. Figure 29(b) highlights the significant reduction in MSE that is possible by
increasing the receive antenna diversity from NR=5 to NR=IO for a fixed number of
sensors, NT=10. This is particularly useful in the “sparse mode” of operation in sensor
networks where the measurements of sensors tend to be more uncorrelated due to fewer
active sensors per cluster. In such modes of operation event detection accuracy can be
improved by varying the receive diversity order (at the base station or in a virtual beam

forming configuration) at the same time ensuring low sensor utilization.

82

(I) lit-'10, 3.16 (3) 31-10, N‘IIIO
0 7 v w

 

 

 

 

 

 

 

 

 

 

0.7 l i _
-G— SNR-10 & —9— SNR-10 h
+ SNR-20 d) —I— SNR-20 ®
0.6 r + SNR-30 d) j
05 > d

 

 

 

 

 

1 J

t 2 3 0 l 2 3
In." Nod: Win/Didacmfl/Q

 

 

Figure 29. Mean square error of the estimator as a function of sensors’ spatial
separation, 7 (inversely proportional to degree of correlation)

 

 

(a) NT'IO, 3'6 (5)1‘r-10, 17.-10
. :, l f ‘
W” *
x91 2'87. - I E e e
,67 flw D
l ,«3/6

10'

A ALL
7 fi vv‘
RKQ‘
A \Al
'I'

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

—e~—smi-loao —e——SNR-loa>‘
—I—8NR-20 do i l —I—SNR-20 ch i
+smz-3o d) _, +SNR-3o a
L. * 1 to t: J
0 0.5 l O 0.5 1
In." Node WMfimp

Figure 30. Logarithm of MSE as a function of spatial separation of sensors
highlighting the threshold range for high data correlation

83

5.4.2. MSE as a function of multiuser diversity (joint
transmissions)

We now consider the effect of joint transmissions of the co—located sensors with

correlated data. The parameters 0,7 are fixed. Similarly the parameters N R and the

channel SNR are held fixed. We assume joint synchronized transmissions from the co-

located nodes to the base station or the virtual array. The peak power constraint is met by

normalizing the per-node transmission energy by the factor K / N T .

Presented in Figure 31 is the theoretical MSE of joint correlated transmissions. It is
interesting to note that under identical transmission energy constraints, correlated
transmissions significantly reduce the estimation error vis-a-vis independent uncorrelated
transmissions. This can be attributed in part to the transmit diversity that is inherent in
the system. This has several implications in “dense” modes of operation mentioned early
on, since this approach achieves the dual purpose of improving estimation accuracy while
remaining energy efficient. In applications requiring high data fidelity on event triggering
[40], the active sensors per cluster can be adaptively increased for higher fidelity. In a
sense the approach of sending more correlated measurements at lower power is the dual
to sending fewer compressed measurements at higher power, i.e. the duality between the
Slepian wolf coding and Multiple Access Channel (MAC) problems. Space-time coding
schemes in MIMO systems strive to incorporate this correlation into otherwise
independent data streams, whereas in sensor networks this correlation and diversity is
inherently present. In Figure 31(b) we consider the combined effect of increasing receive
diversity given multiuser diversity which further improves the estimation accuracy. This

will be discussed in greater length in the section to follow.

84

MSE re number of simultaneous ueere (multi- ueer diversity, N1" 7:0.1)

 

 

I v T - r fl

“fluorine!“
Trmnleslone

I T
i
. Uncorrelded

 

1’er

M
O
O
0
¢
0
a

10' .~* . 10"

\
s"
\
AAA I

1030183)
beam!)

10'2 , 3 10'

 

 

 

* —— SNR-30 db 1
-8 f O 4
10 SNR—10 db 10
—e— SNR-20 db
Nflo —-Ii— SNR-30 db

1 1 J 4

10 20 30 40 50 10 20 30 40 50
(e) Multi- ueer diversity, NT (b)

 

LLIILKr

 

 

 

 

 

 

 

 

 

Figure 31. Performance of the estimator as a function of Multiuser Diversity in the
presence of correlation vs. independent transmissions

5.4.3. MSE as a function of receive diversity

In analyzing the effect of receive diversity on the system MSE, the number of nodes
per cluster NT , the correlation and node spacing relating to 7 and the additive noise
power are fixed. Although various aspects of data gathering in WSNS have been explored
in the past, the area of multiple antennas or diversity reception is still nascent. In order to
gain a better insight into to this aspect, the MSE for varying degrees of sensor data
correlation as a function of receive antennas is presented in Figure 32 and Figure 33. In
MIMO systems the gains due to multiple transmit and or receive antennas is due to the
improvement of the detection SNR due to multiple data paths.

85

(a) run-20 (bu-0.1.8?»

   
   

 

 

 

 

 

 

 

 

 

 

0.9 —e—smz:-10(|)L 0.9- —O—SNR=-10d> ‘
+ SNR=20 d) —a— SNR=20 a)

0.8 - +SNR=30 (1,5 0.8 i ’ —I—SNR-30d>r

0.7 I ~ 0.71

0.6 . - 0.6
a 0.5 I ,3 « a 0.5 .

‘12
° 4 F ii. an ‘ 0'4 ’
0.3 I h “a . 0.3 .
.5 a
. us
0 2 I 91.. 1" o 2
11.1.
“m

 

 

 

 

 

 

5 10 15 20
Nunber d Receive Antennae (1")

Figure 32. Mean Square Error as a function of receive diversity for varying inter-node
spacing i.c. 7 E {1, 0.1}

 

 

 

 

 

 

 

 

 

 

(eh-0.01.1119” (”finalist-20
0.3 +SNR-IOd) . 0.3» —e— SNR-10d) ,
l ——i— SNR-20 a) -—¢— SNR-20 a
l -B— SNR-30 ch -Ei— SNR-30 (b
0.25 .| . 0.25 . 1
7 .
0.2

E3015

0]-

005-

 

 

 

 

 

 

Figure 33. Mean Square Error as a function of receive diversity for varying inter-node
spacing i.c. 7 E {0.01, 0.001}

86

 

in) H9131!”

Y

 

  
  
  

 

 

+smi-loih ‘
+SNR=20<I>
+smi-3oei

 

 

 

 

 

 

 

lo' I

 

 

 

l A J

 

 

 

5 10 15 20

Barber of Receive AIM.)

Figure 34. Logarithm of the Mean Square Error as a function of receive diversity,
demonstrating the diminishing return on MSE as antennas increase

The performance improvement is through array gain since the multiple paths are
combined such that the SNR is maximized. For MIMO systems,iit directly depends on
the norm of the channel matrix H which indicates the radio energy gleaned by the
antennas, therefore as the number of antennas increases the norm increases. The same
holds true for transmit diversity (which could in theory be equated to multiuser diversity
in case of very high correlation). In Gaussian models the gain in SNR directly affects the
MSE of the estimator. Therefore the decrease in MSE as a function of receive antenna
diversity can be attributed to improvement in detection SNR at the base station due to
maximal ratio combining. As expected the MSE drops significantly as the number of
receive antennas approach the number of simultaneous sensor transmissions as seen in

Figure 32 and Figure 33.

87

Since deploying multiple antennas at the base station or collector yields significant
improvements in detection SNR this has implications to increasing the overall network
life time. Joint transmissions allow the nodes to communicate at lower transmit power,
resulting in fewer re-transmissions and scheduling costs; thereby improving the overall
lifetime of the network. Adaptive beam-forming can be achieved via a several methods,
in sensor networks it would correspond to virtual antenna configurations [51] [49] and
would require changes to the MAC layers for achieving full advantage [52]. Another
advantage of exploiting receive diversity is in alleviating the reach back problem. This
approach can address the reach back problem by reducing the required transmit power
levels of the already power-limited nodes that are nearer to the collector since the
detection SNR can be improved by increasing the receive antenna diversity.

An effect of increasing the diversity order of the system is the law of diminishing
returns in multiple antenna systems and can be noticed in Figure 34. In traditional MIMO
systems it was noted that “an attribute of receive (or transmit) diversity is that they
experience diminishing returns as the number of antennas increases, i.e., with respect to
the SNR gain for a given error probability criterion”. An equivalent effect for the system
under consideration is that of diminishing returns as the number of antennas increase with
respect to MSE of the estimator. Based on Figure 34 it can be observed that the
improvement in MSE begins to decrease for the same increments in the number of
receive antennas for a fixed SNR.

Therefore while the benefits of receive diversity and joint transmissions are doubtless
very encouraging one has to bear in mind effects such as these while designing systems

that utilize diversity gains. They can be enumerated as,

88

I The decoding complexity which increases as the multiuser diversity increases.
I The diminishing returns on the increasing the number of antennas
Therefore for optimal design, the factors should be kept in consideration and the order of

NT and N R optimally chosen based on the application specific resources and

requirements.

In summary this chapter an integrated framework that brings together multiuser,
receive diversity and data correlation for accurate field reconstruction. The framework
provides not only a significant reduction in estimation error but also bandwidth efficiency
and improved network lifetime. The optimum estimator and the associated bounds on the
Mean Square Error (MSE) of the framework were presented. The achievable performance
gains in terms of the MIMO system parameters were presented in the simulation results

and performance analysis.

89

6. CONCLUSIONS AND FUTURE WORK

The performance results support the conclusion the system throughput can be
improved significantly while ensuring energy efficient by using the distributed space-
time fusion schemes. The performance of the space-time fusion schemes is at par with
optimum fusion schemes deploying round robin transmission schemes under identical
energy constraints. Space-time fusion schemes mitigate the latency by a huge margin, i.c.
50% and 67% for the two and four node schemes during the data collection and fusion
stages. Latency is critical to sensor network applications requiring high rate event data
collection, i.e. when the density of active nodes per cluster is high. Larger gains can be
achieved as the node diversity increases. This also enables the sensors to sample data at
much higher granularity or sampling rates despite limited on chip buffer size or storage
capacity. We harness multi-sensor diversity in a new approach to address not just the
reliable data transport issue but also reliable event detection and data fusion for co-
located sensors, an aspect that has not been explored heretofore. We also show that not
only is it feasible to apply space-time coding in a ‘truly’ distributed manner for certain
operating ranges but in a manner such that the global event detection performance of
system approaches that of optimal fusion schemes. This is achieved through channel
aware receiver processing. In fact, when the sensor measurements are reliable the
performance approaches the co-operative Alamouti scheme (source identical at all the
antennas).

The work presented in relation to distributed detection represent preliminary results

of an effort to realize distributed schemes that draw upon classical space-time techniques.

90

This was achieved by abstracting arbitrary sensor networks as MIMO systems for joint
data fusion and data transport reliability in order to be energy efficient. There remain
several open challenges such as synchronization issues and adapting network protocols to
facilitate multi-user diversity and smart antenna techniques for sensor network
applications. In the generalized approach we intend to extend the proposed work to an
arbitrary number of parameters and sensors, wherein sensors in the vicinity of a
parameter of interest act like multiple transmit antennas. The current fusion schemes take
into account only the spatial correlation in the measurements, exploiting the temporal
correlation aspect also might potentially lead to gains in the overall system performance
which will be addressed in our future work.

Addressing the distributed estimation problem, the analytical as well as the simulation
results indicate the strong dependence of estimation accuracy on spatial diversity. The
results also demonstrate significant gains in accuracy and energy efficiency that can be
achieved by integrating multiuser diversity with spatial correlation. The benefits of
incorporating receive diversity into Wireless Sensor Network (WSN) applications that
require high data fidelity and resolution upon event triggering were presented. Such an
approach is particularly applicable to active sensing scenarios, where the distribution of
active sensors across the network varies or can be adaptively varied. By integrating
spatial correlation in wireless sensor networks with receive diversity, the field estimation
accuracy can be significantly improved while ensuring energy and spectrum efficiency.

We have abstracted the overall system using a novel framework that captures the
multiple effects of correlation, joint transmissions and receive diversity into one model.

The uniqueness of such an approach is that the interaction between the various factors is

91

more clearly understood as opposed to a disparate treatment of factors. The combined
effects of the data correlation, node density, and multi-user and receive diversity on the
overall estimation accuracy are investigated. The proposed framework is highly relevant
to sensor network applications that require variable levels of data fidelity, bandwidth and

energy efficiency and perform adaptive network resource allocation.

92

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

REFERENCES

T. M. Cover and J. A. Thomas, Elements of Information Theory. New York: Wiley
1991.

S. Alamouti, “A simple transmit diversity technique for wireless communications”,
IEEE J. Select. Areas in Communications, vol. 16, pp.1451-1458, Oct. 1998.

I. E. Telatar, “Capacity of multiple antenna Gaussian channels,” Eur. Trans.
Telecommunications, vol. 10, no. 6, pp. 585-595, Nov./Dec. 1999.

V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block codes from
orthogonal designs,” IEEE Trans. Information Theory,vol. 45, pp. 1456-1467, July
1999.

V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block coding for
wireless communications: Performance results,” IEEE J. Select. Areas in Commun.,
vol. 17, no. 3, March 1999. -

V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space-time codes for high data rate
wireless communications: Performance criterion and code construction,” IEEE
Trans. Information Theory, vol. 44, pp. 744-765, Mar 1998.

E. Biglieri, J. Proakis, and S. Shamai, “Fading channels: Information-theoretic and
communication aspects,” IEEE Trans. Information Theory, vol. 44, pp. 2619-2692,
Oct 1998.

G. Foschini, “Layered space-time architecture for wireless communication in a
fading environment when using multiple antenna elements,” Bell Labs Tech.
Journal, pp.41-59, Sept 1996.

Eds. S. S. Iyengar and RR. Brooks, Distributed Sensor Networks. Chapman &
Hall/CRC 2005.

[10] P.K. Varshney, Distributed Detection and Data Fusion. New York: Springer-

93

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18]

[19]

[20]

[21]

[22]

[23]

[24]

Verlag 1997

P. Gupta and P. R. Kumar, “The capacity of Wireless Networks,” IEEE
Transactions on Information Theory, vol. 46, pp. 338-404, Mar. 2000.

B. M. Hochwald and T. L. Marzetta, “Capacity of mobile multiple-antenna
communication link in Rayleigh flat-fading,” IEEE Trans. Information Theory, vol.
46. pp. 139-157, Jan. 1999.

J. G. Proakis, Digital Communications, 4‘h ed. New Yorkch-Graw Hill 2000.
S. Verdu, Multiuser Detection. Cambridge, U.K.: Cambridge Univ. Press, 1998.

J. H. Conway and D. A. Smith, 0n Quatemions and Octonions: Their geometry,
arithmetic and symmetry. Natick, Massachusetts: A. K. Peters, 2003.

L. Zheng and D. N. C. Tse, “Diversity and Multiplexing: A fundamental tradeoff in
multiple antenna channels,” IEEE Trans. Information Theory, vol. 49, pp. 1073-
1096, May 2003.

A. Paulraj, R. Nabar, and D. Gore, Introduction to space-time wireless
communications. Cambridge, U.K.: Cambridge University Press, 2003.

S. N. Diggavi, N. Al- Dhahir, A. Stamoulis and A. R. Calderbank, “Great
Expectations. The value of spatial diversity in wireless networks,” Proceedings of
the IEEE, vol. 92, No. 2, Feb 2004.

A. Wittneben, “A new bandwidth efficient transmit antenna modulation diversity
scheme for linear digital modulation,” in Proc. ICC, 1993, pp. 1630-1634.

S. N. Diggavi, N. Al-Dhahir, and A. R. Calderbank, “Diversity Embedding in
Multiple Antenna Communications,” DIMACS workshop on Network Information
Theory 2004.

N. Al-Dhahir et. al, “Space-time processing for broadband wireless access,” IEEE
Communications Mag, pp. 136-142, Sept 2002.

B. Chen, R. Jiang, T. Kasetkasem, and P. K. Varshney, “Channel aware decision
fusion in wireless sensor networks, ”IEEE Trans. on Signal Proc., vol. 52, no. 12,
pp. 3454-3458, Dec 2004.

R. Niu, B. Chen and, P. K. Varshney, “Fusion of decisions transmitted over
Rayleigh fading channels in wireless sensor networks,” IEEE Trans. on Signal
Proc., vol. 54, no. 3, pp. 1018-1027, March 2006.

B. Chen, L. Tong and P. K. Varshney, “Channel-aware distributed detection in
94

[25]

[26]

[27]

[28]

[29]

[30]

[31]

[32]

[33]

[34]

wireless sensor networks,” IEEE Signal Proc. Magazine, pp. 16-26, July 2006.

AR. Calderbank and A. F. Naguib, “Introduction to Space-time Codes”, in Space-
Time Wireless Systems. Cambridge Univeristy Press, July 2006.

Special Issue on Gigabit Wireless, Proceedings of the IEEE, vol.92, no. 2, 2004.

H. Karl and A. Willig, Protocols and Architectures for Wireless Sensor Networks.
Wiley, May 2005.

S. S. Pradhan, J. Kusuma and K. Ramchandran "Distributed compression in a dense
sensor network," IEEE Signal Processing Magazine, March, 2002.

S. Cui and A.J.Goldsmith, “Cross-layer Design in Energy—constrained Networks
Using Cooperative MIMO Techniques,” EURASIP's Signal Processing Journal,
Special Issue on Advances in Signal Processing-based C ross-layer Designs, 2005.

B. Sirkeci-Mergen, A. Scaglione, “Randomized Distributed Space-Time coding for
Cooperative Communication in Self-organized Networks,” IEEE Workshop on
Signal Processing Advances in Wireless Communications, 2005.

J. Geng and U.Mitra, “Distributed Space-Time Processing in Sensor Networks,”
Asilomar 2004.

M. L. Chebolu and S. K. Jayaweera, “Integrated Design of STBC-based Virtual-
MIMO and Distributed Compression in Energy-limited Wireless Sensor
Networks,” 2”“ European Workshop on Wireless Sensor Networks, 2005.

R. Yeung, A First Course in Information Theory. Kluwer Academic Publishers,
2002.

A. Ribeiro and G. B. Giannakis, “Bandwidth-Constrained Distributed Estimation
for Wireless Sensor Networks, Part 1: Gaussian PDF," IEEE Transactions on

' Signal Processing, vol. 54, no. 3, pp. 1131-1143, March 2006.

[35]

[36]

[37]

A. Ribeiro and G. B. Giannakis, “Bandwidth-Constrained Distributed Estimation
for Wireless Sensor Networks, Part 11: Unknown PDF,” IEEE Transactions on
Signal Processing, vol..54, no. 7, pp. 2784-2796, July 2006.

Z.-Q. Luo, “Universal decentralized estimation in a bandwidth constrained sensor
network,” IEEE Trans. Inf. Theory, vol. 51, no. 6, pp. 2210—2219, Jun. 2005.

15.0. B. Akan and I.F.Akyildiz, “Event-to-sink reliable transport in wireless sensor

networks,” IEEE/ACM Transactions on Networking (TON), vol.13 no.5, pp.lOO3-
1016, October 2005.

95

[38]

[39]

[40]

[41]

[42]

[43]

[44]

[45]

[46]

[47]

[48]

G. A. Darbellay, “The mutual information as a measure of statistical dependence,”
ISIT Germany, 1997.

A. Mainwaring, et. al, “Wireless Sensor Networks for Habitat Monitoring,” ACM
Workshop on Sensor Networks and Applications, September 2002.

G. Allen, et.al, “Deploying a Wireless Sensor Network on an Active Volcano,”

IEEE Internet Computing, Special Issue on Data-Driven Applications in Sensor
Networks, March/April 2006.

M. C. Vuran, O. B. Akarl, and I. F. Akyildiz, “Spatio-Temporal Correlation:
Theory and Applications for Wireless Sensor Networks,” Computer Networks
Journal (Elsevier), vol. 45, no 3, pp. 245 -259, June 2004.

A. Scaglione, S. Kirti and BS. Mergen, “Error Propagation in Dense Wireless
Networks with Cooperation,” Proc. of the International Symposium on Information
Processing in Sensor Networks (IPSN '06), April 2006.

AD. Murugan, P.K.Gopala and H. E. Gama], “Correlated Sources over Wireless
Channels: Cooperative Source-Channel Coding,” in IEEE Journal on Selected
Areas in Comm, vol. 22, no. 6, pp. 988-998, August 2004.

P. Levis, D. Gay and D. Culler, “Active Sensor Networks”, Second Symposium on
Networked Systems Design and Implementation, May 2005.

LE. Akyildiz, W. Su, Y. Sankarasubramaniam, and E. ‘Cayirci, “A Survey on
Sensor Networks”, IEEE Communications Magazine, vol. 40, no. 8, pp. 102-116,
August 2002.

Eds. Franceschetti, Stomelli, Joint Source-Channel Coding on Graphs and
Applications in Sensor Networks, Wireless Networks. Academic Press, 2006 (to

appear)

S. Cui, A.J. Goldsmith and A. Bahai, “Energy Efficiency of MIMO and Co-
operative MIMO in Sensor Networks”, IEEE Joum. Select. Areas in Commun.,
August 2004. ’

G. Kramer, M. Gastpar and P. Gupta, “Co-operative Strategies and Capacity
Theorems for Relay Networks”, IEEE Transactions on Information Theory, vol.

’ 51, no. 9, pp: 3037-3063, September 2005.

[49]

[50]

T. Sivanadyan and A. Sayeed, “Active Wireless Sensing for Rapid Information
Retrieval in Sensor Networks”, Proceedings of the International Symposium on
Information Processing in Sensor Networks (IPSN '06), April 2006.

SM. Kay, Fundamentals of Statistical Signal Processing: Volume I and II. New

96

[51]

[52]

[53]

[54]

[55]

Jersey: Prentice Hall, 1993.

A. Scaglione and Y.W. Hong, “Opportunistic Large Arrays: Cooperative
Transmission in Wireless Multihop Ad Hoc Networks to Reach Far Distances”,
IEEE Transactions on Signal Processing, vol. 51, no. 8, pp: 2082—2092, August
2003.

J.H. Winters, “Smart Antenna Techniques and their Application to Wireless Ad
Hoc Networks”, IEEE Wireless Communications Magazine, August 2006.

S. Devarakonda and H. Radha, “The impact of receive diversity in the presence of
correlation in wireless sensor networks”, Proc. of Conference on Information
Systems and Sciences, March 2007.

K. Misra, S. Karande and H. Radha, “Multi-hypothesis distributed video coding
using LDPC codes”, Proc. of Allerton Conference on Communications, Control

and Computing, Sept. 2005.

T. Rappaport, Wireless Communications: Principles and Practice, 2nd Edition.
Prentice Hall, 2001.

97

 

lllllll‘ll‘lM il Iii ||i llll‘” illill'il

 

293 028 459