O N O P T I M U M S Y S T E M D E S I G N F O R
W IR E L E S S C O M M U N IC A T IO N S
by
BO W U
M.Sc., T siaghua University, Beijing, 1992 B.E. & B.Sc., T singhua University, Beijing, 1990
A D issertation S u b m itted in P artial Fulfillm ent of the R equirem ents for the Degree of
D O C T O R O F P H IL O S O P H Y
in th e D epartm ent of Electrical and C om puter E ngineering
We accept this dissertation as conforming to th e required stan d a rd
Dr. Q iangW ang, Supervisor (D ep artm en t of EC E )
Dr. Vijay K. Bhargava, D e p a r tm e ^ d M em ber (D ep artm en t o f ECE)
Dr. W u-Sheng Lu, D ep artm en tal M em ber (D ep artm en t of ECE)
Dr. R. Nigel Horspool. O utside M em ber (D ep artm en t of CS)
. .J. Han Vinck. External M ember (U niversity o f Essen)
© B O WU. 1996 U niversity of V ictoria
All rights reserved. Dissertation may not be reproduced in whole or in part, by photocopying or other means, without the permission o f the author.
11
S u p e r v is o r : Dr. Q iang Wang
A B S T R A C T
This dissertatio n addresses the issue o f o p tim u m system design to achieve reli able com m unication in th e presence of various types of interference. M ultiobjective form ulation is used with noncooperative and cooperative approaches owing to the natu re of the problem s under consideration.
Since intentional Jam m ing is one of th e m ost severe kinds of interference, a n ti ja m techniques are crucial for com m unications in a hostile environm ent. T h e ja m
and anti-jam problem is modeled as a tw o-person zero-sum gam e in which th e com m unicator and th e ja m m e r have antagonistic objectives and are viewed as th e two players. The concept of Nash equilibrium is introduced and its characterizations such cis existence, uniqueness, stability, robustness, and sensitivity are investigated. This model is th en applied to a frequency-hop spread sp ectru m M -ary frequency- shift-keying system where ratio-threshold diversity is used to com bat p artial-b an d noise and m u ltito n e jam m ing. E quilibrium perform ance in term s of cutoff ra te and bit error ra te is shown to be superior to th a t pred icted by worst-case analysis.
W hen m u tu al interference caused by sim ultaneous transm issions is the m a jo r concern in a heterogeneous packet netw ork, a m ultiobjective framework is proposed in this dissertation w ith the objectives a n d constraints of the individual users taken into consideration. Near-far effect and Rayleigh fading m ay occasion packet cap tu re and therefore create unfairness in favor of closer users. Thus, m ultiobjective o ptim ality is introduced, in which criterion of fairness is em bedded. O p tim u m strategies controlling transm ission pro b ab ility a n d /o r power are exam ined to yield the Pareto o p tim al solution in a slotted .4 L 0 H A network. Then, the sam e control strategies are stu d ied with the channel u tilizatio n being th e m axim ization objec tive. O ptim ization results are obtained in various situations, and effectiveness of different strategies is compared.
I l l
A m ultim edia direct-sequence spread sp ectru m system may su p p o rt m ultiple services with different transm ission rates an d diverse quality-of-service require m ents. To facilitate m ultim edia applications and maximize the sy stem capacity, average power control, erro r correction coding, and tim e diversity are incorporated into th e system . The capacity of such a system is evaluated in m u ltip a th Rayleigh fading channels. Average bit error rate, outage probability, and corresponding in form ation theoretic bounds are discussed. C oncatenation of Reed-Solomon codes and convolutional codes is considered for erro r correction to account for different q uality and delay constraints. It is shown through a numerical exam ple th a t th e system capacity can be increased significantly by an appropriate sy stem design.
Exam iners:
---Dr. Q iang^V ang, Supervisor (D epartm ent of ECE)
________________
Dr. Vijay K. B hargava, D epartm ental M em ber (D epartm ent o f ECE)
Dr. Wu-Sheng Lu, D epartm ental M em ber (D epartm ent of EC E )
Dr. R. Nigel Horspool, O utside M em ber (D epartm ent of CS)
IV
Table o f C on ten ts
A b stra ct ü
T able o f C on ten ts iv
L ist o f T ables v ii
L ist o f F igu res v iii
L ist o f A b b rev ia tio n s ix
A ck n ow led gm en ts x ii D e d ic a tio n x iii 1 In trod u ction 1 1.1 N oncooperative A p p ro a c h e s ... 2 1.2 Cooperative A p p ro a c h e s ... 3 1.3 C ontributions of th e D is s e r ta tio n ... 5
2 A G am e T h eo retic M o d el for A n ti-J a m C om m u n ication s 6 2.1 I n t r o d u c t i o n ... 6
2.2 G am e-Theoretic M odel ... 6
2.3 Existence and Uniqueness of E quilibrium ... 9
2.3.1 Existence of E q u i l i b r i u m ... 9
2.3.2 Uniqueness o f E q u ilib riu m ... 13
2.4 C haracterizations of E q u ilib r iu m ... 15
2.4.1 Stability and Robustness of E q u ilib r iu m ... 15
Table o f C ontents v
2.5 C o n c lu sio n s ... 20
3 G am e T h e o r e tic S tu d y for an A n ti-J a m F F H /F S K S y s te m U sin g R a tio -T h r e sh o ld D iv ersity 21 3.1 I n t r o d u c t i o n ... 21
3.2 R atio-T hreshold D iversity C o m b i n i n g ... 21
3.3 N um erical R e s u l t s ... 27
3.3.1 B ER Perform ance of Uncoded S y ste m s ... 27
3.3.2 C utoff R ate Perform ance of Coded S y s t e m s ... 29
3.4 C o n c lu sio n s ... 31
4 A M u ltio b je c tiv e A n a ly tic Fram ew ork for W ir e le ss H ete r o g e n e o u s N etw ork s 37 4.1 I n t r o d u c t i o n ... 37
4.2 M ultiobjective O p tim ality and Fairness ... 38
4.3 N etw ork D e s c rip tio n ... 41
4.4 .A.pplication E x a m p le s ... 44
4.5 C o n c lu sio n s ... 47
5 M a x im iz a tio n o f C hannel U tiliz a tio n in W ireless H ete r o g e n e o u s N etw o rk s 51 5.1 I n t r o d u c t i o n ... 51
5.2 P roblem Form ulation ... 51
5.3 Strategies to M aximize C hannel U t i l i z a t i o n ... 53
5.3.1 Strategy of Transm ission P o w e r... 53
5.3.2 Strategy of Transm ission P r o b a b ility ... 54
5.3.3 .Joint S t r a t e g y ... 58
5.4 C o n c lu s io n s ... 64
6 C a p a city o f a M u ltim ed ia D S /C D M A S y ste m in M u ltip a th Fading C h an n els 65 6.1 I n t r o d u c t i o n ... 65
Table o f Contents v i
6.2 S ystem and Channel M o d e l ... 6 6 6.3 Inform ation Theoretical C onsiderations ... 70 6.4 Perform ance M e a s u re s ... 74 6.-5 E rro r Correction Scheme ... 7-5 6 . 6 N um erical E x a m p l e ... SO 6.7 C o n c lu sio n s ... SI
7 C o n clu sio n s 86
7.1 S u m m ary of the D i s s e r t a t i o n ... S6 7.2 Suggestions for Future R esearch ... S7
A p p en d ix A M eth o d s to Find P a r e to O p tim a l S o lu tion s 89 A p p en d ix B Pow er C apture M o d e l 91 A p p en d ix C D erivation o f E q u a tio n s (5 .1 6 )-(5 .2 1 ) 93
V II
List o f Tables
3.1 BER with E b l N j = 10 dB an d M = 2 28
4.1 C apture Probabilities in Rayleigh Fading C h a n n e l s ... 45
5.1 C apture Probabilities in Rayleigh Fading C h a n n e l s ... 57
5.2 M aximization Solution w ith Equal Individual Throughput ... 57
5.3 M aximization Solution w ith = 0.3 and S2 = S z ... 57
6.1 Required Eb!Iq (dB) for voice users using B P S K ... S3 6.2 Required Eb!Iq (dB) for voice users using B D P S K ... S3 6.3 Required Eb/Io (dB) for d a ta users using B P S K ... S4 6.4 Required Eb!Iq (dB) for d a ta users using B D P S K ... S4
Vlll
List o f Figures
3.1 The FFH noncoherent A /FSK spread spectrum receiver... 22 3.2 BER versus Q at various p w ith (M , m , Eb/ Nj) = (2 ,3 ,1 6 dB) under
PBN jam m in g ... 32 3.3 The worst case BER versus E b / N j a t various 0 w ith M = 2 and
m = 3 under PBN ja m m in g ... 32 3.4 Saddle point BER perform ances a t various m w ith M = 2 .4 , S. and
16 under PBN jam m in g ... 33 3.5 Saddle point BER perform ances a t various m w ith M = 2 ,4 .8 . and
16 under M T jam m in g... 34 3.6 M inimum E b j N j needed to achieve saddle point cutoff ra te at vari
ous m w ith M = 2 ,4 .8 , and 16 under PBN ja m m in g ... 35 3.7 M inimum E b fN j needed to achieve saddle point cutoff rate a t vari
ous m w ith M = 2 .4 .8 . and 16 under M T ja m m in g ... 36 4.1 .A.dmissible set for transm ission probability of f/g, and tran sm is
sion probability of i/3, pa... 49 4.2 .Admissible objective space for th ro u g h p u t of i/g, 5a, and the recip
rocal of average delay of i/3, l / r a ... 49 4.3 .Admissible objective space for Si and So with c = 20... 50 4.4 .Admissible objective spaces for and S2 with various c... 50
5.1 M aximum channel utilizatio n when all users are in one group: a) with near-far effect only: b) in th e presence of Rayleigh fading. . . 55
List o f Figures ix
5.2 M axim um channel u tilizatio n w ith infinite p o p u latio n using group preem ptive stra teg y in a near-far environm ent, a) w ithout individual th ro u g h p u t co n strain ts, b) w ith equai individual th ro u g h p u t am ong users... 61 5.3 M axim um channel u tilizatio n versus P t i / P t z in Rayleigh fading
channels under equal individual throughput req u irem en t, a) PT2I P t z =
2. b) P t 2 /P t3 = 1. c) P t 2 /P t3 = 10, d) PT2I P t z = 0.5... 64
6.1 Ebllo required to achieve th e average Shannon capacity versus pro cessing gain Gb w ith various L in Rayleigh fading channels... 71 6.2 Eb/Io required to achieve th e cutoff rate versus code rate R w ith
various L in Rayleigh fading channels... 73 6.3 O utage probability versus ratio A = 7 ^ / 7 5 w ith various L in Rayleigh
fading channels... 76 6.4 System capacity using BPSK in Rayleigh fading channels: a) Z, = 3
and hard decision; b) L = 5 and hard decision; c) L = 3 and soft decision: d) £ = 5 a n d soft decision: e) Shannon lim it w ith L = 3; f) Shannon lim it w ith £ = 5... S5 6.5 System capacity using B D PSK in Rayleigh fading channels: a) £ =
3 and hard decision; b) £ = 5 and hard decision; c) £ = 3 and soft decision: d) £ = 5 a n d soft decision... 85
X
L ist o f A b b rev ia tio n s
A BR available b it rate
ARQ a u to m a tic repeat request AVVGN ad d itiv e w hite Gaussian noise
BD PSK binary differential phase shift keying BER bit error rate
BFSK bin ary frequency shift keying BPSK b in ary phase shift keying BSC bin ary sym m etric channel GBR co n stan t bit rate
CDMA code division m ultiple access DS direct sequence
ECO erro r correction coding
FDM A frequency division m ultiple access F EC forward erro r correction
FFH fast frequency-hop FH frequency-hop MAC m edia access control
MAI m ultiple access interference MDS m axim um distance separable :V/FSK M -ary frequency shift keying MRC m axim um ratio combining . u s e iV/-ary sym m etric channel MT m u ltito n e
PBN p a rtia l band noise
PCS personal com m unication services pdf probability density function PD F probability distribution function
List o f Abbreviations x i
PN pseudonoise
QOS
quality-of-service R-T ratio-threshold SFH slow frequency-hop SNR signal to noise ratio SS spread spectrumTD M A tim e division m ultiple access VBR variable bit rate
X ll
A ck n ow led gem en ts
I would like to express my deep appreciation to m y supervisor, Dr. Q iang Wang, for his invaluable guidance, inspiration, and encouragem ent throughout my Ph.D . study at the U niversity of V ictoria.
I wish to acknowledge N atu ral Science and Engineering Research Council and D epartm ent of C om m unications for th eir financial su p p o rt through a Strategic G rant and a research co n tract aw arded to Drs. Vijay K. B hargava and Qiang Wang.
I am indebted to Mr. Xin W ang for his collaborative effort on th e work in C hapter 2 and Dr. Gang Li for his assistance on the work in C h ap ter 3.
The friendly atm osphere in th e com m unications group a t th e U niversity of Vic toria, the fruitful discussions, and various helps from m y fellow g ra d u a te students benefit me significantly.
I am grateful to Drs. V ijay K. Bhargava, Wu-Sheng Lu, and R. Nigel Horspool for serving on my supervisory com m ittee, and Dr. A. .J. Han Vinck for serving as the external exam iner in m y Ph.D . oral exam ination.
Finally, my special th an k s go to my wife. Qing, for her su p p o rt, consideration, and endurance throughout this work.
Xlll
C h ap ter 1
In tro d u ctio n
In wireless com m unication system s, in addition to background therm al noise, there exist o th er forms of d istu rb an ce to the tran sm itted signals. The d istu rb an ce is referred to as interference and m ay be characterized as an y com bination o f th e fol lowing: intentional or unintentional interference from o th e r users: m ultiple access interference due to sp ectru m sharing by coordinated an d non-coordinated users: and m u ltip ath interference or self-jam ming by delayed signal. This dissertatio n is concerned w ith th e issue of optim um system design for wireless com m unica tions where suppression of various types of interference is crucial to achieve reli able transm issions. M ultiobjective optim ization problem s arise in these situ atio n s where several objectives are to be satisfied [1. 2] and gam e theoretic approaches are often called for.
G am e theory is th e m ath em atical study of conflict an d cooperation between intelligent rational decision-m akers [3]-[5]. Although it was originated in econom ics and behavior science, nowadays we can find its sp ectacu lar applications in m any o th e r branches of social science as well as in engineering areas. Basically, gam e theory is the study o f equilibrium, which provides the evaluation of an optimal situ a tio n where th ere are a num ber of individual objectives and independent a n d /o r dependent constraints. In a typical game, there are a num ber of players with individual strategy sets and payoffs. Depending on w hether the players are able to m ake binding agreem ents, games are divided into tw o classes: noncooperative gam es and cooperative gam es, and this categorizes the problem s under study.
Chapter I. Introduction 2
1.1
N o n c o o p e r a tiv e A p p ro a ch es
In noncooperative situations, each player acts individually to o p tim ize his objec tive w ith o ut concern for th e others’ objectives. T h e outcom e o b tain ed in this way is th e Nash equilibrium [6]. T he main application o f th e noncooperative approach in wireless com m unications is in anti-jam com m unications. It is known th a t in ten tional jam m in g m ay degrade th e system perform ance significantly if no protective m easures are taken by th e com m unicator. T hus, various kinds of anti-jam tech niques have been studied [7, S] where an intelligent jam m er is often assum ed and th e w orst case perform ance is considered. In reality, however, th e com m unicator m ay also adopt intelligent counterm easures in order to obtain a perform ance b e t ter th a n th a t predicted by the worst case analysis. Since the com m unicator and th e ja m m e r are in antagonistic positions and can be viewed as two players in a gam e. Blachm an [9] and Dobrushin [1 0] modeled th e jam and an ti-jam problem as a tw o-person zero-sum gam e. In recent years, th ere have been m any a tte m p ts to ap p ly this model to anti-jam com munications [11]-[16] w here an exact m a th e m atical model is im plicitly assumed. In practice, approxim ation in m odeling and p e rtu rb a tio n due to noise are inevitable, and ch aracterizations of equilibrium require further investigation, to which C hapter 2 is devoted.
In [12]. Chang analyzed the ratio-threshold (R -T ) anti-jam technic[ue in a frequency-hop spread spectrum (FH /SS) binary frec[uency-shift-keying (B FSK ) system from th e inform ation theoretic point of view. She used gam e theory ap proach in the analysis and showed th e perform ance im provem ent o f a tern ary o u tp u t channel generated by the R-T technique over a binary sy m m etric channel (BSC) resulted from hard decision reception. By using a R -T test, an estim a tion o f the channel condition can be obtained. T his estim ation can then be used to form a soft decision decoding m etric. By applying the gam e theoretic m odel stu d ied in C h ap ter 2. we extend Chang's work to an .V/-ary FSK schem e in C hap ter 3 w here the R-T test technique is used in diversity com bining with q u a tern ary
Chapter I. Introduction 3
o u tp u t. We assum e th a t th e ja m m e r knows everything about th e system except th e pseudonoise (PN ) code which controls the carrier frequency-hop sequence. We also assume th a t the to ta l jam m in g power of th e jam m er is finite and fixed. Two types of intelligent jam m in g are considered, partial-band noise (P B N ) and mul ti tone (M T) jam m ing. PB N jam m in g concentrates the total ja m m in g power in a fraction of th e spread spectrum signal bandw idth and injects jam m in g power into a receiver in the form of additive G aussian noise. MT jam m in g injects the to tal jam m ing power into a finite num ber of tones, which coincide w ith som e of th e FSK signal tones used by the com m unicator. .According to th e d istrib u tio n of th e jam m ing tones. M T jam m in g can be divided into two classes: band m u ltito n e jam m ing and independent m ultitone jam m in g . It is shown th a t the worst case MT jam m ing usually occurs when there is only one tone in a jam m ed hopping channel.
1.2
C o o p e r a tiv e A p p r o a c h e s
In cooperative situations where binding agreem ents are possible, each player tends to choose a cooperative equilibrium strategy, which is applicable to m ultiple ac cess networks where users are coordinated in a distributed way or are com pliant to central control. In fact, th e cooperative approach has already been applied to admission control, flow control, routing, and resource sharing in w ireline networks [17]-[20]. In wireless netw orks, th e em erging personal com m unication services (PCS) will support a variety of services, e.g., voice, video, image, and d a ta , with different rates and diverse quality-of-service (QOS) requirem ents. Therefore, m ax imizing the spectrum efficiency while m eeting th e QOS requirem ents, w hich is a m ultiobjective problem in n atu re, rem ains a challenge in the design of a PCS sys tem . To address this problem , we present a m athem atical framework in C h ap ter 4 using cooperative approaches.
In m ultiple access networks, slotted .ALOH.A is a simple random access tech nique [21] and is often a com ponent of more complex protocols [22. 23]. In such
Chapter I. Introduction 4
a network, th e overall perform ance suffers severely from the collision betw een two or more packets th a t arrive at a base statio n sim ultaneously. Classically, all the packets involved in a collision are assum ed to be destroyed, which plagues the system w ith a low throughput and a high average delay. In a wireless scenario, however, th e near-far effect and th e channel fading result in significantly different power levels am ong received packets, and th e strongest packet m ay c ap tu re the base statio n in th e presence of a n u m b er of colliding packets. This is known as th e packet c a p tu re [24, 25]. .A.lthough th e cap tu re effect increases th e network throughput considerably [26], it m ay also create unfairness am ong rem ote users in a heterogeneous network where th e y have different requirem ents a n d /o r loca tions. For instance, if the transm ission power level is identical for every user in a network, a user closer to the base statio n m ay have a higher probability of cap turing the bcise statio n and therefore receive a b e tte r service in term s of higher th ro u g h p u t, lower average delay, and lower packet loss probability. T he unfairness is highly undesirable and may even m ake th e whole network drift to an unstable state under a high load. To rectify this problem , certain control strategies have to be em ployed to ensure th at each user has an optim al and fair access to the base statio n . T h is is usually done through a netw ork controller at th e base sta tion by ad ju stin g each user's transm ission power, transm ission (or retransm ission) probability, erro r correction capability, or o th er controllable param eters. Though capture effect in mobile radio channels has been widely studied, few contributions have addressed th e optim ization problem especially for a heterogeneous network, which is th e focus of C hapters 4 and 5 where strategies of controlling transm ission probability an d power are considered.
Direct sequence (DS) spread sp ectru m is an o th er technique for m ultiple ac cess in PCS and has drawn much a tte n tio n due to its inherent advantages such as statistical m ultiplexing gain, an ti-m u ltip ath and an ti-jam capability, and privacy [27]-[30]. In C h a p te r 6. user num bers of different services are used as th e optim iza tion objectives, and the effect of erro r correction in m u ltip ath Rayleigh fading is
Chapter I. Introduction 5
studied in conjunction w ith average power control and tim e diversity in th e form of RAKE reception.
1.3
C o n tr ib u tio n s o f th e D is se r ta tio n
T h e m ain contributions of this dissertatio n are as follows:
• Study of the characterizations of equilibrium for a two-person zero-sum gam e th at is used to model an an ti-jam com m unication system.
• Game theoretic study of ratio-threshold diversity for a fast frequency-hopped iV/FSK system in the presence of partial-b an d or m ultitone jam m ing.
• C onstruction of a m ultiobjective framework for cooperative heterogeneous networks with optim ality an d fairness addressed.
• O ptim um strategies by controlling transm ission probability and power to maximize the channel utilizatio n in a slo tted ALOHA heterogeneous network. • Evaluation of the system capacity for a m ultim edia CDMA network in m ul tip ath Rayleigh fading channels with m ultiple services having different rates and quality requirem ents.
C h ap ter 2
A G am e T h eoretic M od el for
A n ti-J a m C om m unications
2.1
In tr o d u c tio n
In a hostile com m unication environm ent, the com m unicator and th e ja m m e r are in antagonistic positions and each tries to use his best strategies to defeat th e o th er's purpose. For instance, the ja m m e r can select different types of jam m ing , such as p artial band jam m ing, single or m ulti-tone jam m ing, follower jam m in g , rep eater jam m in g , and predictive jam m ing, and adjust corresponding jam m in g param eters to achieve th e best jam m ing effect. On the oth er hand, in order to com bat ja m m ing, th e com m unicator may choose from various system considerations, including m odulation schemes, channel codes, diversity combining, interleaving, and signal energy. Since jam m ing may degrade the system perform ance considerably, it has to be effectively counteracted by th e com m unicator using certain kinds of anti-jam techniques [7]. The ja m and anti-jam problem is often form ulated as a two-person zero-sum gam e with the com m unicator and th e ja m m e r being the two players. The payoff o f the gam e may be one o f the figures of m erit, such as channel capacity, cut-off rate, bit error rate, and th ro u gh p u t.
2.2
G a m e -T h e o r e tic M o d e l
.Assume com plete information for both sides and consider the single-period s itu a tion in which the com m unicator and the jam m er determ ine their optim al strategics
Chapter 2. .4 Game Theoretic Model fo r A nti-Jam Communications 7
sim ultaneously and continue to use th e m thereafter. Let a A:-dimensional vector ÛC € .4c be th e com m unicator’s stra teg y and an /-dimensional vector Uj 6 A j be the ja m m e r’s strateg y , where .4c and A j are th eir respective strateg y sets. T he C artesian p ro d u ct .4 = .4c x A j is the stra teg y set of th e game. Assume th a t .4c.
Aj, and .4 are a p p ro p riate m etric spaces. A strateg y com bination a = {ac, a_,) G .4 is known as a pure strategy. Denote the com m u n icator’s payoff as u (a), which is a function defined from .4 to the real set R . T hen the payoff of th e ja m m e r is —u(a) in this zero-sum gam e. Therefore, the gam e can be represented by a strategic form (.4. u). In gam e theory, the most im p o rtan t concept is Nash equilibrium, a vector of strategies th a t neither player would d ev iate from given the o th e r’s strategy. D e fin itio n 2 .1 For a game (.4, u), a N ash equilibrium is a strategy combination
a~ = (a “. a “) € .4 such that u(a') > u(ac,aj) fo r all Uc G .4c and u{a“) < u{a~.aj) fo r all Uj G A j .
u(a‘ ) is known as the value of the gam e with a‘ and a" being the respective
optim al strategies for the two players. T he value of a game and the optim al strategies are regarded as the solution o f the game.
If the ja m m e r knows th e com m unicator’s strategy, he can choose a strateg y to achieve th e best jam m in g effect, or m inim ize the com m unicator’s payoff, yield ing the worst case jam m ing. T he com m unicator may then choose a strateg y to minimize th e w orst case jam m ing effect, resulting in a payoff of u,n,n,
U j n i n = m a x m i n u( a), (2.1)
dcSA-c
which is the com m unicator's value or security level. If is chosen in this way. the com m unicator's payoff can be g uaranteed to be no lower than Umin against any jam m er.
On the o th e r hand, if the com m unicator knows the jam m er's strategy, he may choose a stra teg y to maximize his payoff, which corresponds to the best anti-jam
Chapter 2. .4 Game Theoretic Model fo r Anti-Jam Communications 8
system . T he ja m m e r may th e n choose an a p p ro p riate aj to m axim ize his jam m in g effect for th e best anti-jam sy stem , resulting in a payoff of Umax>
Umax = min m ax u (a), (2.2 )
which is th e ja m m e r's value or security level. If aj is chosen in this way. the com m unicator's payoff cannot be higher th an
Umax-It is easy to see th a t Umm < Umax- If th ere exists a° = (a^,aj) such th a t
tl(^a ) — Um(71 — Umax? (2.-I)
this point is a saddle-point equilibrium . In this case, the two sides tend to use th e optim um pure strategies and u^, respectively. If one side does not use the optim um strategy, the opponent m ay gain advantage. Thus, neither player would have incentives to deviate from th e saddle-point strategy.
In general. Umm and Umax are unequal; th a t is, equilibrium cannot be achieved using pure strategies. In this case, both sides m ay use random ly varying param eters instead of fixed param eters w ith th e payoff function replaced by the expected value. .•\ mixed strategy is obtained which is determ ined by a probability distrib u tio n over pure strategies as follows. T h e com m unicator selects an ap p ro p riate probability distrib utio n function (P D F ) o f F(uc) and uses his strategy accordingly, whereas th e jam m er selects his stra teg y according to a PD F of G { a j ) . T he payoff is then th e expectation of u(u) over Oc and Uj. Let A (.4 c ) and A (.4 j) denote the sets of PD Fs on Borel subsets of .4c an d A j , respectively. A. mixed strateg y equilibrium results if the following equation holds:
Uq = max
F(ae)€A(.4c) G{ m in )€^(.4y) JAjJAc/ / u{ac.a,) dF{ac) dG(a.)
= m in m ax / / uiuc.a,) dF[ac) dG(a,). (2.4) G(a,)&MAj)Ftar_)eMAc)J.\jJAc
If a.; and aj have discrete values of and with corresponding probabilities of /3(«,-„, ) € A '(.4 c I and p { a , „) € A''(.4v) where A"'(D) is defined as the set of
Chapter 2. .4 Game Theoretic Model fo r A nti-Jam Communications 9
p ro b ab ility distributions over the discrete set D\ th a t is,
= {p : D -J- R I ^ p ( / ) = 1, and p{l) > 0, V/ 6 D}. (2.3)
IÇ.D
T h en , we have the discrete form of (2.4)
In sum m ary, from the com m unicator’s viewpoint, if th e re exists a saddle point, th e o p tim u m can be obtained by m axim izing the worst case u[a). This is o p ti m um in th e sense th at using any o th er strategy may result in a worse perform ance if th e ja m m e r employs his optim um strategy. If a saddle point does not exist, m ixed strategies can be em ployed, and equilibrium perform ance of this system is b e tte r th a n th a t of a system using pure strategies.
2 .3
E x is te n c e a n d U n iq u e n e ss o f E q u ilib r iu m
T he finite two-person zero-sum games have been extensively studied, and the key result is th e well-known von Neumann Minimax Theorem [3], which guarantees th e existence of an equilibrium . However, further analysis is needed for the above gam e theoretic model, which is usually infinite in practice.
2 .3 .1 E xisten ce o f Equilibrium
E xistence of equilibrium is usually addressed by applying m ath em atical fixed point theorem s. In th e following, we present existence theorem s for the gam e (.4. u) under study.
F irst, we assume th a t the payoff function u(a) is continuous on .4 and only pure strategies are used. T he first existence theorem is given as follows.
Chapter 2. .1 Game Theoretic Model fo r Anti-Jam Com munications 10
T h eo re m 2.1 [f fo r any aj Ç A j , maXa^^Ac “ (“ ) a unique solution; f o r any d-c E A c , u{a) has a unique solution; u{a) is continuous on .4." and either
4 c or Aj is a compact convex set, then there exist a* E 4 c and a ’ € A j such that
a" = {a‘ ,(Lj) is an equilibrium, i.e.,
min max u(a) = m ax m in u(a) = u (a ”). (2.7) aj6.Aja^€.4c ' a„€.4c
P roof. Suppose first th a t ,4.j is a com pact convex set. Since maXa.-e.4c u(w) has a unique solution for any Uj E .4j, denote this solution by / ( c j ) : .Aj -4- .4c,
f i a j ) = argjna^x u{a),
where /(U j) is the best response of th e com m unicator if the ja m m e r takes the strateg y Oj.
Sim ilarly, let g{ac) be the best response of the jam m er if th e com m unicator's stra teg y is Oc- i.e., g{ac) : 4 c -4- .4y.
g{ac) = arg m in u (a).
a^6.4j
By assum ption. / is a function from .4y to .4c, and ^ is a function from .4c to A j . Since u(a) is continuous on .4. by Maximum Theorem [31] we have th a t / and
g are continuous. Now. define h: A j - 4 A j as
h{aj) = g{f{ai)).
If the ja m m e r takes strategy aj and the com m unicator takes his best response
f { a j) . th e n h(aj) = g{f(aj)) is the best response th a t the ja m m e r can make to the
com m unicator's f{ aj) . Hence, we know th a t if there exists àj such th a t h{cij) = cij. then th e ja m m e r will choose a_,.
Since g and / are continuous, h is continuous. By the Brouwer Fixed Point
Chapter 2. .4 Game Theoretic Model fo r A nti-Jam Com munications 11
a'j € Aj such th a t h{a’ ) = a'-. Let a “ = f { a j ) . T hen we have u{a') > u(ac.aj) for any Oc € -4c and u(a’ ) < u ( a ' , a j) for any aj E A j . Thus, a* = (a*, a ”) is an equilibrium .
We know th a t u( a‘ ) > u(ac-,aj), Voc E Ac- Therefore,
u(a") = m ax u(ûc. O;)- ac6.4c
and
u (a ') > m in m ax u(a).
aj6Ajace.-ic
Similarly. u(a") < max^^e.^^ miria^gAy u (a). Hence,
ti(a“) = m in m ax u(a) = m ax m in u(a). ( ^ j € A j a c € A c f t c € A c <^ j € A y
W hen -4c is a com pact convex set instead of .4c, the proof is sim ilar. I
T he result of Theorem 2 . 1 is rem arkably strong and reveals some im p o rta n t properties. F irst, we do not need to random ize th e com m unicator's and ja m m e r's strategy sets to guarantee the existence of an equilibrium . Second, we do not need both .4c and .4y to be com pact convex sets in a finite dim ensional space. In contrast, the proof of the existence of the Nash equilibrium in an n-player gam e requires th a t each player's set of (m ixed) strategies be a com pact convex set in a finite dim ensional space.
W ith the conditions of Theorem 2 . 1 changed slightly, the following existence theorem s result.
T h eorem 2.2 / / / o r any aj E .4y. max^^g.^g. u(a) has a unique solution: / o r any
«C 6 .4 c .u (a ) ^-5 quasi-convex on aj: u(a) is continuous on .4; and .Aj is a compact convex set. then there exists an equilibrium.
P r o o f. Define / . g. and h in the sam e way as in the proof of Theorem 2.1. We know th a t / is a continuous function from . \ j to .4c by .Maximum Theorem, and
Chapter 2. .1 Game Theoretic Model fo r Anti-Jam Com m unications 12
th a t g is a convex-valued u p p er sem i-continuous correspondence from .4c to A j by th e fact th a t u{a) is quasi-convex on aj. Because A j is a co m p act set and u(a) is continuous on .4, g(ac) is a com pact set in A j for any
Oc-We also know th a t A is a convex-valued correspondence from A j to A j and has a closed graph. Then, by th e Kakutani Fixed Point Theorem [31], there exists a* G Aj such th a t a* 6 h{a~). Let a~ = / ( a J ) . Hence, we know th a t ( a '. a j ) is an
equilibrium . I
T h eo re m 2.3 I f fo r any Oc G .4c. miua^g.^^ u{a) has a unique solution: f o r any cij G A j . u{a) is quasi-concave on 0^: u{a) is continuous on .4," and A c is a compact convex set, then there exists an equilibrium.
T h e proof is analogous to th a t of Theorem 2.2.
T h eo re m 2 .4 I f fo r any Uc G .4c, u(a) is quasi-convex on aj; f o r any aj G 4y.
u(a) is quasi-concave on a^; u{a) is continuous on .4," and .4c and .Aj are compact convex sets, then there exists an equilibrium.
P roof. Define r: .4 —)■ .4 such th a t
r(â ) = (arg m ax u ( a c ,â j) ,arg min u(âc,aj) ), (2.8)
ac€Ac ajBAj
where â = (â ^ .â j), and r is th e best response correspondence of th e game. By assum ptions, r is a convex-valued function having a closed g rap h . By applying th e
Kakutani Fixed Point Theorem, we know th a t there exists a fixed point, i.e.. an
eciuilibrium exists. H
T h e above theorem s are concerned w ith the existence of saddle points for a continuous payoff function w here only pure strategies are used. W hen a saddle point does not exist, we need to apply m ixed strategies and reform ulate the ex istence problem as follows. T he strategies Oc and Uj are replaced by PDFs F(f/c) and C{aj). and th e strategy sets .4c and A j are replaced by A (.4 c ) and A (.4 j).
Chapter 2. .1 Game Theoretic Model fo r Anti-Jam Com munications 13
respectively. T h e payoff function is then the expected value o f u (a ), as expressed in (2.4) an d (2.6) at an equilibrium . If A c and A j are non em pty com pact sets, th en th e re exists an equilibrium for the mixed strategy (see T heorem 1.3 in [5]).
2 .3 .2
U n iqueness o f Equilibrium
In general, equilibrium is nonunique because there may exist m ore th an one fi.xed point in th e equation a = r(a ). .A.n equilibrium may or m ay no t be achieved when m u ltip le equilibria exist in a noncooperative game, since even if each player selects a s tra te g y associated w ith an equilibrium , th e resulting com bination may not be an eq u ilib riu m . Hence, th e issue of uniqueness for equilibrium arises. Paralleling to T heorem s 3.4 and 3.5 in [32]. we present two uniqueness theorem s below for the gam e (.4, u).
.Assume th a t the conditions of Theorem 2.1 are satisfied. T h en , equilibria exist and th e best response r{a) defined in (2.8) is a single-valued function. The first uniqueness theorem requires th a t r(a) be a contraction, which m eans th at there exists a positive scalar F < 1 such th a t for any x, «/ 6 A,
d(r{x).r{y)) < Fd{x.y).
w here d { x . y ) is the distance from x to y on .4.
T h e o r e m 2 .5 I f the conditions of Theorem 2.1 are satisfied and r(a) is a con
traction, then there exists a unique equilibrium.
P r o o f . From Theorem 2.1, we know th a t there exists a t least one equilib rium . Suppose th at Ui € .4 and a-2 G .4 are equilibria, i.e., ai = r(a ,) and
Ü2 = r ( ü2). T hen. d (a i,a g ) = d{ r( ai) ,r (a2)). Since r(a) is a contraction, we have
chat d ( r ( a i ) . r( ü2)) < F t/lo i.a o ) for F < 1 . This can only hold for «i = 0 2- which
Chapter 2. .4 Game Theoretic Model fo r A nti-Jam Communications 14
The second uniqueness theorem does not require r(a) to be a contraction, but needs the differentiability of th e payoff function.
Let . 4 be the interior of .4 and be the set of aJI the real functions continuously differentiable to th e n th order on an open set Denote u'{a) as
u'(a) = dujdac —d u jd ü j
which is a function from 4 to and its .Jacobian as J[u).
T h e o r e m 2 .6 Assume that the conditions of Theorem 2.1 are satisfied and u G
O 0 0
C '(4 ). I f negative quasidefinite fo r all a G. 4 . and r{a) 6 4 fo r any a G .4,
then there exists a unique equilibrium.
O
P r o o f . By assum ption, we know th a t equilibria exist. Since r{a) G.4, th e re is no eciuilibrium on the boundary of .4. Thus, all equilibria satisfy th e first-order condition:
d u /d ü c = d u / d a j = 0.
O .4.n equilibrium a' corresponds to u'{a') = 0. Since u'{a) is a function from 4 to
and its .Jacobian is negative quasidefinite, by th e Gale-Nikaido Univalence
Theorem [32], u'{a) is univalent and achieves th e value u'(a) = 0 only once. This
establishes the uniqueness of th e equilibrium . B
0
By extending the differentiability and quasidehnitness conditions from 4 to .4 and applying the Rosen Uniqueness Theorem [33]. we can obtain a stronger global uniqueness result and elim inate th e restriction in Theorem 2.6 th at all ec[uilibria
O m ust be in .4
-Chapter 2. .4 Game Theoretic Model fo r A nti-Jam Com munications 15
2 .4
C h a r a c te r iz a tio n s o f E q u ilib r iu m
T h e above model is assum ed to give an exact description o f th e system ’s behavior. However, approxim ation m ay occur in constructing th e m odel, and sm all p e rtu r b atio n caused by system error or noise is inevitable in practice. For a g am e of (.4. u), deviation or p e rtu rb a tio n m ay exist in the o ptim al strategies and payoff function. We thus seek to evaluate the effectiveness of changing p aram eters a t an equilibrium for the following two cases: stability and robustness of eciuilibrium w ith respect to deviations from optim al strategies, and sensitivity of eq u ilib riu m w ith respect to p aram eter p ertu rb atio n .
2 .4 .1 Stability and R ob u stn ess o f E quilibrium
W hen eith er of the two players deviates his strategy slightly from an equilibrium a* = ( a ', a j ) intentionally or unintentionally, the opponent m ight adjust his s tr a t egy by choosing the best response to the deviated strategy. C ontinuing to ite ra te th e process results in a dynam ic process. .Assuming sim ultaneous moves w ith / . g, and r defined in the sam e way as before, we have the process as:
= r ( o '- ') (2.9)
w ith = (a°,a°) 6 .4 being the initial point. If the process converges to a~ for any initial point, the equilibrium is said to be stable. If th e convergence is valid only under small initial deviations, the equilibrium is said to be locally stable. O therw ise, it is unstable.
From th e results of dynam ic analysis, we have the following stability th eo rem
[5]-T h e o r e m 2 .7 .4 fixed point a' o f r is locally stable in the process o f (2.9) i f all
the eigenvalues o f have real parts whose absolute vahies are less than 1.
Chapter 2. .4 Game Theoretic Model fo r A n ti-Ja m Com munications 16
L e m m a 2.1 f f X and Y are open, xp{x,y) : .V x V' —>• R satisfies tp € C ^ { X x V').
and min^gx ( o r m ax^gx ) é [ x ^ y ) has a unique solution denoted by o(y), i . e . . L' iy) = arg min^gx ( or a r g max^gx ) tp{x,y). then
dv \ d \ ' ] - 1
dy dx"^ d x d y (2.1 0)
provided that [ | ^ ] exists.
P r o o f . Since .V a n d Y are open and é € C ^ { X x V'). then if for any y G V'. x ‘ =
L' i y) is the solution of min^gx (or m ax^gx) Ç {x ,y ), by th e first-order condition.
d p = 0. for any y G K {■r'.y) (2.11) which is dip
^ ( o ( i / ) , I/) = 0, for any t / G V', D ifferentiating (2.12) with respect to y yields
(2.1 2) d ^ p dv d^ih d x - dy ^ d x d y (2.13) Hence, if [ | ÿ - 1 exists. dv dy - 1 r dH ' d x d y (2.14)
From Theorem 2.7 and Lem m a 2.1. we can o b tain th e following corollary. C o r o lla r y 2.1 When both a^ and Oj are one-dimensional, a sufficient condition
C hapter 2. .4 G a m e Theoretic Model f o r A n ti-J a m C o m m u n ica tio n s 17
c o n d i t i o n on the s l o p e s o f the best respon se f u n c t i o n s ,
< 1 (2.15) d f dg duj doc o r d'^u < d'^u \ dücdaj
in an open neighborhood o f the equilibrium.
d^u
(2.16)
N ext, we address briefly the issue of robustness of equilibrium w ith respect to the payoff" function p ertu rb atio n . It is our concern w hether an equilibrium of the original game (.4, u) is an approxim ate equilibrium of the game {A. ù) w ith a p e rtu rb e d payoff ù. A n equilibrium of a game is robust if there exists a nearby equilibrium for any nearby game.
D e f in itio n 2.2 .An equilibrium o f the game ( A ,u ) . a ', is robust i f fo r any s > 0
there exists r/ > 0 such that fo r any ù satisfying |u — ù| < q. there exists an
equilibrium à of the game (.4.Û) such that d(a". ù) < c. .4 game is robust i f all its equilibria are robust.
For sim ultaneous-m ove finite games, we have the following robust theorem [5]. T h e o r e m 2 .8 .Almost all finite strategic form games are robust.
For th e game (.4, u), "alm ost all" means th a t the set of finite games w ith two players' strategy sets fixed is open and dense in the Euclidean space of dim ension
2.A*.Aj where the com m unicator and the jam m er have .A* strategies and .4*
strategies, respectively. Those games th a t are not robust are said to be exceptional. Thus, we can roughly say th a t finite games are robust in general.
Chapter 2. .4 Game Theoretic Model fo r A nti-Jam Com munications 18
2.4 .2
S en sitivity A nalysis o f E quilibrium
T h e m easure of the sensitivity of an equilibrium to a param eter is th e sensitivity function w ith a form of derivatives of th e o p tim al strategies w ith respect to th a t p aram eter. In the proof o f th e sensitivity theorem , th e following lem m a is required in addition to Lemma 2.1.
L e m m a 2 . 2 I f cp : X x V* -4- X satisfies 0 G C ^ ( X x V'). and X is a compact
convex set in R^'. then by the Brouwer Fixed P oint Theorem , there exists x ' such that
0 ( x ',y ) = z ‘ . (2.17)
Furthermore, if x “ is in the interior o f X , and Y' is open, then
(2.1 8)
'%/ (...w (/y (T..„
provided that — ij exists where I is the identity matrix.
P r o o f . We know th a t
o { x '{ y ) ,y ) = x '{ y ) . for any y 6 V'. (2.19)
D ifferentiating (2.19) w ith respect to y. we have
d x ' d é , d y d i ~ . d y (x’.y) d é d x ' d é à x ' d x d y d y d y (2.2 0) Thus. d x '
W
- I d p dy provided th a t — l | exists. (2.2 1).\ssu m e the conditions of Theorem 2 . 1 are satisfied where . \ j is a com pact con vex set and / . y. and h are defined in the sam e way as in the proof of T heorem 2.1.
Chapter 2. .4 Game Theoretic Model fo r Anti-Jam C om m unications 19
We stu d y th e sensitivity of an equilibrium w ith respect to th e p e rtu rb a tio n of a pa ram eter s € (a, 6). D enote th e payoff as u (a ,s ). We have th e following sensitivity theorem .
T h e o r e m 2 .9 I f fo r any s € (a, 6), u (a ,s ) G C ^[A x {a,b)) satisfies all the con
ditions o f Theorem 2.1 with an equilibrium a ' = (a ’ ,a j ) E.4, then
d a ' ds (a*.s) dh duj - I dh ds (2.2 2) and where with da; d s da j ds ds dh _ dg d f daj da^ daj
dh _ dg d f dg ds dac ds d s ' (2.23) dg dg -I dac' ds d (a j,s )^ d{aj,s)dac \ d f d f ] ' -I
d a j ' ds d{ac.s)'^ d[ac, s)daj
P r o o f . T he proof follows from Lemmas 2.1 and 2.2.
T h e equilibrium th a t is stable, robust, and insensitive w ith respect to per tu rb a tio n is of more interest for a practical problem , since otherw ise equilibrium perform ance is m eaningless due to inevitable approxim ation and p ertu rb atio n . C om p u tatio n algorithm s for equilibria can be found in [34. 35].
Chapter 2. A Game Theoretic Model for Anti-.Jam Com m unications 20
2.5
C o n c lu sio n s
In this chapter, a single-period com plete-inform ation gam e theoretic m odel has been presented for th e com m unication-jam m ing problem . T he concept o f ec[ui- librium has been em phasized, and related fundam ental issues, such as existence, uniqueness, stability, robustness, and sensitivity, have been discussed w ith corre sponding theorem s presented. T he next chapter considers an ap p licatio n of the gam e theoretic m odel to th e design of a practical anti-jam com m unication system .
2 1
C h ap ter 3
G am e T h eoretic S tu d y for an
A n ti-J a m F F H /F S K S y stem
U sin g R atio-T h resh old D iv ersity
3.1
I n tr o d u c tio n
In this c h ap te r, we apply th e two-person zero-sum model discussed in th e previous chapter to a fast frequency-hopped (FFH ) M -ary frequency shift keying (A /FSK ) system under partial-b an d noise (PBN ) or band m ulti-tone (M T) jam m in g . The stru c tu re of th e receiver is shown in Figure 3.1. R atio-threshold (R -T ) diversity com bining technique [12. 36] with or w ithout error correction coding (ECC ) is used to com bat jam m ing. T he com m unicator chooses th e ratio -th resh o ld and th e diversity o rd er as his strategies. The jam m er selects th e fraction of ja m m e d band w idth in the to ta l bandw idth in PBN jam m ing or th e power ratio of signal to one jam m in g tone in M T jam m in g as his strategies. Bit error rate (B E R ) is used as th e payoff function when only diversity combining is employed, and cutoff rate is considered w hen ECC is also used.
3.2
R a tio -T h r e sh o ld D iv e r sity C o m b in in g
W hen a R-T diversity com biner is used in a F F H /F S K system , we denote the diversity ord er as m and the ratio threshold as 0. .-Assume th a t th e o u tp u t of
Chapter s. Game Theoretic Study fo r an A nti-Jam F F H /F SK System 22 D ecision O utput Matched Filter /m-i Matched Filter FH DEMOD Envelope Detector Envelope Detector Matched Filter Envelope D etector Combining Diversity Samples at 7^
F ig u r e 3 .1 . The F F H noncoherent M F S K spread spectm m receiver.
t = 0 .1 iV/ — 1. and I = 1 . 2 . . . . , m. The hop decision is m ade based on which non-coherent m atched filter o u tp u t is the largest. Suppose th a t, for th e /th hop.
(3.1)
where / = i. 2 , . . . , m. and the quality bit q is set according to [36]
_ f 0 (good) if §[[- > g for all i 7^ j . 1 1 (poor) otherwise.
(3.2)
The o u tp u ts of all hops are accum ulated with good quality bits and poor quality bits, respectively. If there is at least one hop decision with a good cjuality bit and there is a m ajo rity decision w ithout a tie, then th e o u tp u t of the com biner is the m ajority decision with a good o u tp u t quality bit attach ed . If there is a tie betw een 0 and 1 with good quality bits, or if there is no hop decision with a good q u ality bit.
Chapter 3. Game Theoretic Study fo r an Anti-Jam F F H /F S I\ System 23
a decision is m ade based on hop decisions w ith poor quality b its (if th e re is a tie, flip a coin), and this o u tp u t of the com biner is attached w ith a poor o u tp u t quality bit. W hen th e logg M binary inform ation bits associated w ith each tra n sm itte d
M -ary sym bol are ideally interleaved, the result is a channel w ith binary inputs
and q u a tern ary o u tp u ts for each hop.
T h e tran sitio n probabilities of the channel are known to be [36]
Pc = Fc { e ) + ( ^ ^ - l ^ F e { 9 ) . P c x = F c ( I ) - F c (0) + ( y - i ) [ F £ (1) - F £ (0)]. M Pe x = — [F e (1) - F£:(0 )], M Pe = - F s (0).
where Pc and Pq x are the probabilities o f correct reception w ith good and poor quality, Pe and Pe x are the probabilities of error reception w ith good and poor quality, and
Fc[9) = Pr{.Vj7 > GXa for all / # j | j sent}.
Fe[9) = Pr{A'„/ > OXii for a speciflc n ^ j and all i ^ n \ j sent}.
which can be derived from the distributions of M m atched filter o u tp u ts .Vo/. . \ 't / , . ... .V(,v/_i)/ using the average com putation model for P B N ja m m in g [37] and band M T jam m in g [38], respectively.
.4. P B X Jam m ing
\ PBN ja m m e r is supposed to ja m a fraction p of the transm ission band with
noise power density X j / p . We have [37]
/ V/ — I
Fc{9) = ^ ( - 1 ) ^
A: = 0 0 2 f t
p e x p fc log , M Eb 0- + k m (:\o + X j Ip)
Chapter S. Game Theoretic Study fo r an A nti-Jam F F H /F S K System 24 + ( 1 - p )e x p k log 2 M E b ' 9--{-k mNo ^ r 1 \ d^ + k logoM E b \ ^
^
j 9 ^ - - { - k e ^ - ^ k - \ - l 9 ^ - ^ k - \ - l m { N o + N j / p ) Jwhere Eb is th e energy per inform ation bit and No is the power sp ectral density of th e therm al noise m odeled as additive white G aussian noise (AVVGN).
B. M T Jam ming
W hen band M T ja m m in g is assumed where th e ja m m e r has a single ja m m in g tone per jam m ed band w ith equal power, Fc { 9 ) and F e { 9 ) can be co m p u ted as [38]
E c ( 0 ) = [ i - - g ) E c o ( 9 ) + / j . Fc i { 9 ) .
F e { &) = - f ) F e q { 9 ) + i-lF e \.[9 ),
where p. is the pro b ability of one M -ary band being ja m m e d
F =
a r u M logo M E b / N j
w ith Q being th e ratio of signal power to the power of one ja m m in g tone. T he expressions for Fco{ 9) , F c i { 9 ) , Feo{9), and Fe i { 9 ) can be found in [38].
Let o be the o u tp u t of th e diversity com biner before tie-breaking. T h ere are five possible values for o,
0 0 decision is 0, good quality. 0 1 decision is 0, poor quality, o = < I tie.
II decision is 1, poor quality. 1 0 decision is 1. good quality.
Chapter:]. Game Theoretic Study fo r an Anti-.Jam F F H /F S K System 25
T h e conditional distribution of o given th a t “0” is sent is Pr{o = 0 0 |0 s e n t} = ^ n.ki >k2 Pr{o = 0 1 |0 s e n t} = ^ n , t ‘ i = t 2 , A : 3 > f c | P r { o = l l |O s e n t } = ^ n.ki =k2,kî<k4 Pv{o = 10 I 0 sent} = g pk, pk,^ pk,^ , n.ki <k2
Pr{o = z | O s e n t } =
^
n . i ' l = ^ 2 , ^ 3 = ^ 4 w here Q. = {/l'i, A'2, /l3. A:4 I 0 < k1. k2. k z , k^ < m . k i + k2 + k j + A.4 = m }an d jg m ultinom ial coefficient.
W hen ECC is not used, the o utputs of diversity com biner are also th e o u tp u ts of th e receiver. Suppose the two binary symbols before interleaving are equiprobable. T h e B E R at the diversity combiner o u tp u t is
Pb = Pr{o = II I 0 sent} + Pr{o = 10 | 0 sent} + ^ P r {o = x | 0 sent}. (3.3)
W hen ECC is used and the cutoff rate of the coding channel is considered, the e n tire diversity combining channel can be viewed as an equivalent binary inputs q u a tern ary o u tp u ts channel with the corresponding transition probabilities being
P c = Pr{o = 0 0 I 0 sent},
P c x = Pr{o = 0 1 1 0 sent} + ^ P r (o = X 10 sent},
Pe x = Pr{o = 1 110 sent} + - P r {o = X 10 sent}.
Chapter 3. Game Theoretic Study fo r an Anti-Jam F F H /F S K System 26
T he norm alized cutoff rate Rq of this channel is related to th e tra n sitio n probabil ities by
/?o = 1 - log-2 (^1 + 'l\J PePc + 2 \ / Pe xPc x^ (b its/b in a ry sym bol). (3.4)
In the gam e theoretic model for the system under study, th e com m unicator can adjust 9 an d m to minimize the B ER or maximize the cutoff rate, while the ja m m e r can ad ju st p or q under different types of jam m ing to m axim ize th e BER or m inimize th e cutoff rate. Then, = [O^m] is the c o m m u n icato r’s strategy, and aj, which is p under PBN jam m ing or q under MT ja m m in g , is th e ja m m e r's strategy. The payoff function a is Pb as in (3.3) or Rq as in (3.4).
Then, we have th e strategy set for the com m unicator
OC-,4c = {(0,p )| 1 < 0 < oo; p = ( p i , . . . , p , , . . . ) . Pi > 0. = 1} » = 1 = {é>| 1 < ^ < oo} X { p |p = ( p i , P i ,...) , Pi > 0. Y ^pi = 1} 1 = 1 = Ac{9 ) X .4c(p).
where pi = Pr{m = T he strategy set for the jam m er is
-4j(p) = { p | 0 < p < 1} (3.5)
under PBN jam m in g , or
-4y(o) — {n I OCjjxtTi ^ ^ O^mar} (3-6) under MT jam m in g where a-min is the lower bound small enough to include the lowest point of th e worst o , and oimax is such an a th a t causes p = 1. It is obvious th a t Aj is a com pact convex set.
Chapter:]. Game Theoretic Study fo r an .Anti-Jam F F H /F S K System 27
3.3
N u m e r ic a l R e su lts
In this section, vve apply th e gam e-theoretic m odel to the a n ti-ja m problem and study the system perform ance through num erical m ethod w here Eb/No is fixed at 13 dB which corresponds to a BER of 2.3 x 10“ ^ for a system w ith m = 1, M = 2. and no jam m in g .
3.3.1 B E R Perform ance o f U n co d ed S ystem s
W hen R-T diversity is used without ECC, we evaluate the B E R as given in (3.3). .4. PBIV J a m m in g
We assum e first th a t th e com m unicator ad just 0 a t a fixed m . In this case, we can verify th a t th e conditions of Theorem 2.1 are satisfied; thu s, th e saddle point perform ance can be achieved. In Figure 3.2, BER versus 6 w ith various p
at {.\[.m . Eb f Nj ) = (2 ,3 , 16 dB) is shown. T h e value of 9 a t which th e m inim um BER is achieved can th en be selected by th e com m unicator to o b tain the best anti-jam result a t different p. Similarly, the value of p a t which th e m axim um BER is achieved can be adopted by the ja m m e r to obtain th e worst case result at different 9. Figure 3.3 shows the worst case BER versus E b / N j w ith various
0 a t .\I = 2 an d m = 3. It is observed th a t th e optim um 9 is alm ost the sam e
at different Eb/I^'j; th a t is, there is alm ost no need to ad ju st 9 when an optim um value of 9 is found at a fixed m. T he com m unicator’s value can be obtained by selecting the m inim um point a t each E b /N j which corresponds to th e saddle point perform ance. In Figure 3.4, the saddle point perform ances are shown for various
m with .\I = 2 .4 .8 . and 16. from which it can be concluded th a t if th e diversity
order is g reater th a n a value of rn (e.g., m = 3 at M — 2 and 4. and rn = h at M = S and 16). then noncoherent com bining loss becomes do m in an t and the system equilibrium perform ance is uniformly worse than th a t w ith a sm aller ni.
It is obvious th a t th e com m unicator can gain more if it changes m as well as 9. In this case, a saddle point equilibrium does not exist. T he com m u n icato r's value
Chapter 3. Game Theoretic Study fo r an A nti-Jam F F H /F S K System 28
can be o b ta in e d by choosing m and 6 associated w ith th e sm allest BER. This
corresponds to th e lower envelopes of th e B ER curves in F igure 3.4 a t different A/. It is ev id en t th a t th e com m unicator tends to use th e largest M to achieve th e uniform ly b est B E R perform ance. A m ixed strateg y can th en be used to obtain th e (m ixed) eq uilib riu m performance. For exam ple, suppose m € {1.3}. M = 2. and a saddle point equilibrium is achieved a t {6i , pi ) for m = 1 or {6 2, P2) for m = 3.
T he co m m u n icato r can select m = 1 and a corresponding 8 1 w ith a probability
of pc and select m = 3 and a corresponding 8 2 with a p ro b ab ility of 1 — pc- On the o th e r han d , the jam m er can choose pi w ith a p ro b ab ility o f pj and choose p2
w ith a p ro bab ility of i — pj. Then we get a gam e m a trix w hich can be solved to o b ta in a m ixed strategy equilibrium . For instance, w hen E b / N j = 10 dB and
\ [ = 2. th e gam e m atrix is shown in Table 3.1, where th e value is the BER when
corresponding param eters in th a t row or colum n are ad o p ted .
T a b le 3 .1 . B E R with E b / N j = 10 dB and M = 2 Pi = 0.26 /? 2 = 1 - 0
m = 1 , 0 1 = 1 . 0 0.0412 0.0231 ^ = 3, 0 3 = 1.7 0.0179 0.0590
Using th e Shapely-Snow procedure for games [39]. we can find th a t th e solution for this gam e m atrix is 0.0341 with p^ = 0.6943 and Pj = 0.6064. In general, we can ob tain a m ixed equilibrium by solving a game w ith an m x n gam e m atrix T formed by m aking p into n discrete points as
T = [cq]mxn (3.7)
where £,j = Pb{{8i. m ,).p^), 1 < / < m and I < j < n.
B. M T J a m m in g
Sim ilarly, we can apply the above analysis to M T ja m m in g . T h e worst case a is found through searching in the interval [omm.Qmax] num erically. W hen the
Chapter 3. Game Theoretic Study fo r an Anti-.Jam F F H /F S K System 29
com m unicator's strategy set is Ac{9) w ith a fixed m , th e conditions of Theorem 2 . 1 are also satisfied and the saddle point perform ance can be achieved. The corre sponding results are shown in Figure 3.5 at various m w ith M = 2 ,4 ,8 . and 16.
It is observed again that there is no need to have a diversity order greater th an a value of m (e.g., m = 4 at M = 2 and 4, and m = 3 at M = 8 and 16). W hen th e com m unicator further changes m together w ith 9 as his strategies w ith the stra teg y set enlarged to .\c{9) x .4c(p), a saddle point does not exist and m ixed strategies m ay be used. The perform ances at the co m m u n icato r’s value correspond to the lower envelopes in Figure 3.5 for different M .
By com paring corresponding lower envelopes of th e curves w ith the same M in Figure 3.4 and Figure 3.5, it is clear th a t the ja m m e r will alm ost always use MT ja m m in g (except for a small region of E b /N j w hen M — 2 ) to achieve a b e tte r ja m m in g effect, while M = 8 is most desirable for th e com m unicator.
3 .3 .2
C u toff R ate Perform ance o f C o d ed S ystem s
N ext, we discuss the equilibrium perform ance in th e presence of ECC. T he pay off function is now the cutoff rate as given in (3.4). T he results are interpreted through the m inim um value of E b / N j th a t is needed for practical and reliable com m unications. It is determ ined from the cutoff ra te in th e following m anner. Suppose th a t th e cutoff rate is a function of the sym bol signal-to-jam m ing noise ratio: R = Ro{Es/ Nj ) . If we use codes of the ra te /?, th en E(,/Ay is given by
E , / N j =
R log? M
If /? < R o i E , / N j ) , then we have
