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TOPOLOGY-SELECTIVE JAMMING OFCERTAIN CDMA MONOHOP NETWORKS

Interim Technical Report

I

-xiomatix

DTIC

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DISTRUTION STATEMN A 827 16Approved f or public rol#,ame;

- - - ---- -------

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TOPOLOGY-SELECTIVE JAMMING OFCERTAIN CDMA MONOHOP NETWORKS

Interim Technical Report

a,, Andreas Polydoros and Unjeng Cheng

December 23, 1987

U. S. Army Research OfficeP.O. Box 12211 DTIC

Research Triangle Park, NC 27709-2211 D TECContract No. DAAL 03-87-C-0007 FEB 0 3 1988

Axiomatix9841 Airport Boulevard 4 H

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The view, opinions, and/or findings contained in this report are those of the author(s) andshould not be construed as an official department of the Army position, policy, or decision,

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Axiomatix Report No. R8712-5

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Approved for public release; dist!'ibution unlimited.

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IS. SUPPLEMENTARY NOTES

The view, opinions and/or findings contained in this report are those of theauthor(s) and should not be construed as an official Department of the Armyposition, policy, or decision, unless so designated by other documentation.

6 19. K(EY WORDS (Continue on, roer*e odd* If necessary and Identify by block number)

-, Communications,'A Code-Division-Multiple-Access (CDMA), Networks, Packet Radio, Spread Spectrum,'A Tenporal - Selective Jamming, Topology-Selective Jamm~ing.

20. ABSTRAcr rcm~oae ot m~re ads% if nocwooer m~ ideraify by block number)

The effects of Topology - selective jamming on local (monohop) Code-DivisionMultiple - Access networks is investigated. Topology -selective jamming attacks differentreceivers during different packet slot intervals. Probability density functions of thejamming strategy are defined, and analytical expressions for message throughput are

derived. Numnefical results for throughput as function of error correction capability of the

SEC5(URITY CLASSIFICArrOm or MSPG Wo Dae. Entered)

TABLE OF CONTENTS

Page

* List of Figures

List of Tables ii

I. Introduction 1

2. Network and Jamming Models 3

2a. Network Model 3

2b. Jamming Model 5

3. Performance Analysis 93a. Evaluation of pT(m) 10

3b. Evaluation of fMT(m) and Psjm 13

3c. Comparison with Temporal Jamming 16

4. Numerical Results 10

Appendix A 30

References 32

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LIST OF FIGURES

Page

Figure 1. Normalized throughput versus the spatial duty factor pp for 21jamming scenario 2.

Figure 2. Normalized throughput versus the spatial duty factor p~p for 22jamming scenario 2.

Figure 3. Normalized throughput versus the spatial duty factor Pp for 23jamming scenario 2.

Figure 4. Normalized throughput versus the spatial duty factor p for 24jamming scenario 2. The retransmission probability Pr is optimized.

Figure 5. Normalized throughput versus the spatial duty factor ps for 25jamming scenario 2. The retransmission probability Pr is optimized.


Figure 7. Normalized throughput versus the spatial duty factor pfor 27several error-correction codes for jamming scenario 2.


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LIST OF TABLES*

a-. Page

Table 1. Key Parameters 4

AL% a.%2

Topology-Selective Jamming

1. Introduction

The throughput/delay performance of Code-Division Multiple-Access (CDMA)

spread-spectrum networks has received wide attention in the past decade, particularly

because of the military interest in mobile packet radio [KaGrBuKu78], [Jubi85],

[ShTo851. Accordingly, the survivability of a Packet Radio (PR) network under jamming

attack is an important issue. Conceptually, a network could be attacked on three layers of

importance. namely the network, link, and physical layers. In such a jamming game, many

factors could affect the results. The communicators have the choices of routing algorithms.

channel quality monitoring schemes, and the network information exchange schemes. On

the other hand, jammer choices include temporal features (such as static, dynamic or

follower jamming) as well as topological features (i.e., selection of nodes or links to be

jammed).

The purpose of the present paper is to introduce certain concepts on topology-

selective jamming and further examine its impact on a certain class of "local" or "monohop"

networks, which we specify below. This particular class of networks has been selected

because there exists a convenient analytic vehicle which adequately describes their

performance [PoSi87]; thus, it provides a reasonable starting model upon which

topological jamming can be defined and assessed with a good degree of analytic ease. We

note, however, that the following definitions of topological selectivity are quite general and-p

can be applied to other models as well. We also note that topological selectivity can be

perceived as a complementary notion to temporal selectivity, which pertains to jamming

patterns with different time-domain profiles but homogeneous with regard to space;' the

reader is referred to [PrPo87] for a discussion of the latter case.

As mentioned, jamming with any particular selectivity feature manifests itself on all

three layers. Thus, it is not immediately clear how the "local" or "monohop" results of this

t in other words, the jammcr can be ON or OFF in a deterministic or stochastic way but, when ON. allnodes are jammed.

, % %

Topology-Selective Jamming ... 2

paper ought to be interpreted in a larger multihop environment, especially in view of the

fact that there does not seem to exist a unique, widely accepted analytic tool for

performance evaluation in this case. Here, we choose to focus on the monohop model

because (a) certain networks are indeed quite adequately described by this model and

(b) it allows for certain conclusions and assessment to be made (one has to crawl before

.walking).

The paper is organized as follows: In the next Section 2, we first review the basic

features of the network model (part 2a); subsequently, we introduce a general probabilistic

model for topology-selective jamming (part 2b). For illustrative purposes, we focus upon

two particular jamming scenarios which we consider in detail throughout the paper. The

O* corresponding throughput/delay expressions are provided in the following Section 3. An

efficient combinational algorithm is also presented for the recursive evaluation of certain

important probabilistic parameters, an enhancing feature of the theory in [PoSi87] in its

own merit. The paper is concluded with numerical results in Section 4.

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2. Network and Jamming Models

We identify in this section the particular monohop network and jamming models

which we employ in the present study. As mentioned, the monohop network environment

under consideration is the one analyzed in [PoSi87], while the probabilistic, topology-

selective jamming introduced here is quite general and can be used in conjunction with

other network models.

2a. Network Model

We consider an arbitrary topology involving NT transmitters and NR receivers

under a symmetry condition, namely, that each transmitter (TR) face the same probabilistic

circumstances in the channel; an analogous symmetry holds from each receiver's (RCVR's)

viewpoint. This local network is fully-connected, i.e., every TR can be heard from every

RCVR. Time is slotted. We shall assume that the number of receivers NR isfixed, and

that all RCVR's are always available (dedicated receivers). In terms of the notation of

Table I (repeated from [PoSi87], for convenience), this implies that MR = NR, i.e., all

RCVR's are always active. Such a scenario includes the full-duplex case as a special one

(NT = NR = U), but specifically excludes the half-duplex case whereby a modem is an

active RCVR whenever it is not transmitting (MR = U - MT). Note that, in all cases, the

number of active TR's MT is a random variable (0 < MT < NT) with an unconditional pdf

fMT(m). Let us note again that certain extensions to, say, the half-duplex model are

straightforward but will not be discussed here.

The access protocol is of the slotted ALOHA type with P0, Pr denoting the new-

packet transmission and backlogged-packet retransmission probabilities, respectively

[K1La75]. We shall assume a finite number of TR's, although extension to the infinite

Poisson model is again straightforward. The effect of buffering is not considered here;

thus, users are always busy (never idle with any empty buffer), meaning that they are either

active with a new packet or backlogged with an old one (this is the standard model of

.., , . , .. - . - . , . - - - . , -, "AC% %

Topology-Selective Jamming 4

,,

TABLE I

Key Parameters

U Total number of radio units in the local channel (fixed)

NT Maximum number of potential transmitters in a slot (fixed)

NR Maximum number of potential receivers in a slot (fixed)

Mr Number of active transmitters in a particular slot (r.v.)

. MR Number of active receivers in a particular slot (r.v.)

B Number of backlogged users at the beginning of a slot (r.v.)

MB Number of backlogged users retransmitting in a slot (r.v.)

M0 Number of new users transmitting in a slot (r.v.)

S Number of "channel successes" in a slot (r.v.)

' fMT(m) Prob (MT = m), unconditional pdf

A( , -, p) Pr{ successes in E trials), binomial with parameter p

J:.:

-NIN

- -. *e--,.

O'.

5-.

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Topology-Sclective Jamnming ... 5

V[KILa75]). For a network under jamming stress, the assumption of never idle users is

quite reasonable and corresponds to the interesting case of the jammer being sufficiently

strong to keep the network busy most of the time.

Finally, regarding the spreading code distribution, we assume either a common

code or TR-based code system, thus specifically excluding RCVR-based codes [SoSiS4],

[SuLi85], [Purs87]. The implication is that a TR's packet can be successfully received by

more than one RCVR, although it could potentially contribute only one unit to the total

success (or throughput) count.2 We have termed such as scenario competitive to account

for the TR's "effort" to secure some RCVR's attention. This is in contrast to the paired-off

scenario [PoSi871, [PrPo87] (which car, be thought of as a simultaneous TR and RCVR-

based system) where TR's and RCVR's form distinct pairs and only suffer from secondary

multi-user interference, while the RCVR attention is not an issue. A comparison between

the competitive and paired-off cases should provide an indication of the impact of the user

monohop topology upon the jamming game.

2b. Jamming Model

We adopt the following probabilistic model for the jamming action: the slotted

timing is perfectly known to the jammer, so that perfect slot synchronization exists (the cost

of removing such an assumption can be dealt with in a way similar to [PrPo87]). The

jammer performs an independent probabilistic action on a slot-by-slot basis. In each slot, a

Jrandom number of RCVR's, MR, is selected for jamming out of the NR total. The selection

of the specific jammed subset is independent from slot-to-slot, so that no memory can

be exploited by the users. Note that the jammer knows exactly the location of the dedicated

RCVR's. Also, the jamming pattern is assumed independent of the channel state (e.g., the

number of backlogged users), a realistic scenario for a rapidly changing environment.

7..

2 Alternatively, one can count as many successes as receptions regardless of origin; this would seem to be amore appropriate performance measure in a flooding-type of routing [PoSi87].

E

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However, extensions to a channel-depende'nt scheme would also be of interest, possibly

necessitating a different probabilistic jamming model than the one herein.JI. A particular topology-selective jamming strategy manifests itself in the way the MR

jammed RCVR's are selected. If we define the binary-valued r.v.'s A i=l.... N as

1 I, if RCVR i is jammedA. 0, otherwise

then we can quantify topological selectivity by the joint probability mass distribution

function

Pr[AJ=a] A Pr[A l = a1,.. A.= aN (2)

* where ai = 0 or 1; i 1 1,...,NR. Thus, the experiment performed by the jammer consists of

drawing a random vector A J in each slot, which then identifies the targeted RCVR's. The

assumption of independent trials (per slot) plus any given distribution (2) completely

determine the underlying probability measure.

Associated with the above are two important parameters, namely (a) what we shall

call the spatial duty factor, ps, defined as3

A. {sp}=(3)

PSp NR

• signifying the average fraction of jammed RCVR's and (b) the jamming power per attacked

RCVR per slot, JRCVR, assumed fixed throughout, as it relates to the average jamming

power Jav:

0.A av I~v - 4JRCVR(Psp) = = SP'-:: I {~ M J , NRIRp 4

3 Here, F(-} stands for expectation.

% -


In tile following analysis we shall assume that the jammer is average-power-limited,

i.e., J, is constant and given. Thus, varying the spatial duty factor presents the classical

tradeoff between percentage of jammed RCVR's versus the power JRCVR (p) directed to

each one of them. The functional dependence of equation (4) on Pp is meant to enhance

this point.

Definition (2) is quite general and can serve as a starting point for different

*" probabilistic jamming options. Here, we focus on two special cases which we shall call

scenario 1 and 2. respectively:

Scenario 1: Each RCVR is jammed with probability pj, independent of any other

RCVR. Then,

.., Pr[A'=a] = pj:a i (1 P pj)NR - (5)

where Yai = m (_) is the standard Hamming weight of the binary vector a. Note that

mJ (a) is simply the value of the r.v. M J in that slot. Consequently, E { M J } = PjNR,

which simply implies from (3) that

P = Pi (Scenario 1) . (6)

Scenario 2: The number of RCVR's jammed is fixed to NJ(a given constant), although

the specific subset changes randomly every slot. Then,

NR - ."."j , i f m R ( it ) = N JPr [AJ -- ] N R R NR

0 ,otherwise

regardless of the exact location of l's in a. Clearly,JNR

s N (Scenario 2) (8).NR

r0, " . " . " . " .,. . " , . % % % , ' . % % ' % % % % % % ' % " . " % % " . " .. " , . % % " '

I Topology-Selective Jamming 8

"" ~Note that 9sOf (8) is restricted to mnutipes of NR , while 9sof Scenario I can take on

-" any value in (0,11.

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Topology-Selective Jarnming ... 9

3. Performance Analysis

A network under the jamming model of the previous section can be analyzed by

generalizing the concepts in [PoSi87l. In particular, we shall assume that the probabilistic

symmetry conditions of the above model are still true and that the probability distribution of

the number of successes per slot can be uniquely determined, once the number of attempted

transmissions MT = m and the specific jamming pattern AJ = a are given for that slot. We

can evaluate the throughput f3 (packets/slot) as the expected number of successful packet

transmissions S, i.e.,

C Es) EMT, C {t S iM rA1 IZr'rIA' }

•Pr[A CM

. XPr[AJ a] I mpT(m,a) fMT(m) (9)rn S T

where feT (m) is the composite slot traffic (Table I) and pT(m,a) is the probability of

success from a typical TR's viewpoint, given another (m - 1) packets and a specific

jamming pattern a in the slot. In deriving (9), we have used the fact that the r.v. MT

(attempted transmissions) is independent of AJ, since there is no coupling between the

users' and jammer's actions within a slot. Upon interchanging the summations in (9) and

* defining the iamming-average probability of success from a TR's viewpoint, conditioned

on another (m - 1) packets, as

*T P[ a Tma

pS (m) F Pr[AS=al p (m,a) (10)we arrive at

= m pT(m) fMT(m) (11)

-m* which is a direct generalization of (42) in [PoSi87J. In order to proceed, we need to

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T, pl,)jgy-Selective Jamming ... 10

evaluate P(m) and fT(m) for any specitic jamming strategy as per (2).

3a Evaluation of P

TIn principle, ps (m) can be evaluated in accordance with Proposition I of [PoSi87]

(which is quite general and also holds in this jammed scenario) as

..

TIXp (m) m I S Psim (12a)

s=I

where"Pm I aPr [A'=]Psma (12b)

is the average (over the jamming strategy) of the probability

Ps'm.AJ = Pr[s successes, given m attempted transmissions

and jamming pattern A J] (13)

.PI For the nonjammed scenario, [PoSi87] provides examples of how to calculate psLm

in a variety of situations in terms of the number of RCVR's, their statistical dependence,

etc. This can be a very complicated procedure for arbitrary models; however, a

simplification occurs if one assumes that, conditioned on a specific jamming pattern in a

slot, each RCVR accepts packets in a statistically independent fashion from other RCVR's.

This assumption, which would be obviously valid if receiver thermal-noise were the only

deterrent, can be argued to be numerically satisfactory even in the presence of multi-user

noise because of random spreading patterns, fading, random distances, ground propagation

and formation, etc. Then, PsIm can be evaluated in a recursive way which we shall

present in the following section, since it is also required in the evaluation of fMT(m). Let us

Tjust note here that, once this evaluation is completed, pT(m) follows from (12).O.

Under the independence assumption outlined above, a shorter path for evaluating

rN -1.

Topology-Se'lctive Janming ...

p(m) is as follows: Let pR(m:J) and pR(m:Jc) denote the probabilities that a typical

RCVR will accept a packet in the presence of another (m - 1) contending ones, given that

this RCVR is jammed (event J) or not (event jc), respectively 4 Then, using (10) and an

immediate generalization of Proposition 2 of [PoSiX71, we get

P J

Psy (m) = Z Pr[A'=a] 1 - - pRm'__ -mRali RJC) fR R p TIJ

NR[ \ LI [I m) ( A m j

m

'1l

* (14)

'Zq- where M J is the Hamming-weight transformation (a r.v.) of the random vector A1 . Note

i the impact of the aforementioned symmetry assumption, which implies here that all jammed

,,.€' 'iRCVR's suffer the same interference level.

"-'" To illustrate, consider again the two scenarios of section 2b. We immediately have

""" that

Pr(MR =toR) = •~~N~P scenariolI (15a)

J J -

' " and

'{ 1, ifm NR

PrN = m)= 0, otewiescenario 2 (15b)

MR0

which, when substituted in (14). yield

* 4 Note that ofmth is identical to y(m , as intrduced m [PoSi7h. Here, because of th possible

-To jammng, c need to didgsh further and define these condional quantites.

-ht.

0r%'r' Am seai 1aR RR?,

and./fPV *~1~ *%P ~ ~%*

Topology-Selective Jamming 12

NRr p,~pPRj) + (I - P PR p(m:.J 11-1A- A . (scenario 1)m

TP..m) R R(nJl) P - Rp(MJ)I 1 PsP NR (16)

A1 - 1 I - P A J (scenario 2)

It should be kept in mind that pR(mJ) above depends on Pp via the jamming level

JRCvR(Psp). Note also that the result for scenario 1 depends only on the jamming-average

acceptance probability

PR A R R7 A(M) pR A(m;J) + (1 PP) pA(m;J ) (17)

which is an intuitively expected result, because RCVR's are jammed independently for this

case. Obviously, this is not true for scenario 2.

For the specific protocol choice p0 = p, = p (the "backlog-independent protocol"

of [PrPo87]) the composite traffic fMT(m) is simply the binomial distribution A(m,NT,p),

regardless of any jamming action. For this limited case, equations (11) and (16) suffice

to determine throughput. The more general and interesting case, however, is when

Po t Pr , which we examine below.

I%%W

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Topology -Selective Jamming ... 13

3b. Evaluation of fM (m) and PsimT PI

Following closely the steps in IPoSi87, Appendix Al, we can evaluate fMT(m) as

fNT(m) = n(b) fNTIBb(mlb) (18)b=0

where f (mib), the conditional composite traffic given b backlogged users at theM118I=b z'

be-inning of the slot, is evaluated from

fTIB=b(mlb) = A(m - n, b, pr) A(n , NT - b, p0 ) (19)

MT n

max(O, m - b) _ n 5 min(NT - b,m)

with (PO, Pr) the first transmission and retransmission probabilities, respectively. Note that

(19) is independent of the jamming action. Tn (18), B(b) is the appropriate eigenvector of

the jamming-average matrix P = { pi. }, i.e.,

.,B(b) = tl(b) P (20)

"S where5

2P =A(mO,NT - i, P0) " A(mb,i, P,) m (21)om0 =max(O~j-i) mb=max(OJ-.i)

q.

It is clear from (18), (20), (21) that knowledge of the set {"psim} is fundamental in the

evaluation of the composite traffic. Assuming, as in (13), that all patterns i with the same

I lamming weight m J.(a) result in the same PSm~a' we can rewrite (12b) as

5 Note that the eigenvector 7rri(b) of the comix)site system is no it self a jamming-average of eigenvectors,

hence we should not write itB(b )

"-

.

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'p.-... ..

h"i Top dogp v-Selective Jammitng ... 14

P•IM Pr[Nl' =*m' p (22)

In order to evaluate p , we shall assume that the MR receivers operateR

independently, conditioned on the jamming pattern. Let 9 { RCVRk, k=1 R.... mRK}

and " = {RCVRK' k=m+ .... NR } indicate that jammed and nonjanmned sets of

Rkreceivers, respectively. Let p generally stand for the probability of acceptance from the

AC

kth RCVR's viewpoint. Under the symmetry assumption for each of the sets " , we

. then have that;9 { R

." PArn;J), if RCVRke (k < mR)

AR C c= } PA(mJC), if RCVR k E (k > 2R)

Note that, in the present probabilistic framework, which individual RCVRs belong to the

sets is irrelevant, and their ordering can be arbitrary; it is the set size m = dim that

counts.

A recursive way to calculate p ,is the following: let P:k). denote the above. -[ s'm,m R

probability of s successes, given m attempts and 1 R jammed receivers, which is due to the

first k receivers, k = 0....N R Clearly, the sought probability is simply p = p(NR)*,. . m R K,~mJ" - We can first define

,'. 1 ,s = 0(0) (24a)

• SIMM1 , otherwise

Now consider the first receiver RCVR . Then

0

i,'%'S.

O~

• - , r . -,It . .. . |_ . .- -. -. - -. . -% ,, - -. .- ' -, -J -/ -,, ,, -, s• - v." :. ZV V- Z'?WV i - ,

.V. "

T pology-Selective Jamming 15

I 1 R s , md A_(1 = s-RIin

SI'm PA s m (24b)

s~~n~}0 s> 2 or mO

, here pA has been defined in (23) as a function of m and mR. Generalizing to the kth

RCVR's, we can use a standard combinatorial method to conclude that

-pA-) P(S- spk

(1 )Pk) + PRk s(k-1) +-(-. = L+ ] smm*R R

SIM -+ Is + Rk p(k-1) s ! inn(m,k)• A s-lIm 1 < <

0 "s > k or m

(24c)

The recursion stops when k= NR. Note that two different quantiies will be used in the

t%." R k

place of pA above, depending on whether k _< m or k > m , as per (23).

The above recursion can be used with any specific jamming number mJi to produce

'.-npm1 and then, via (22), Pslm . It is also useful for calculating PsIm in a nonjammed... " slin,m R

environment (thermal and multi-user noise only), thus augmenting the theory of lPoSi87].

Regarding our previous two scenarios, we note that

" , -- nNk)

PsIM sImSN; scenario 2 (25)

which is a direct result of (I 5b) and (24). For scenario 1, it can be shown (see Appendix

A) that PRM can be obtained directly from the recursion (24) (i.e., substitute p(k) for* sm,mJR

.1*

X-,

WW L~

Topology-Selective Jamming . 16

"(k) and identify p(NR) , -), so long as we use the jamming average PA(m) of (17)

sIm PsIm.Rmi PsIm

instead of PA This is also an intuitively appealing conclusion, in view of the fact that all

RCVRs are mutually independent and statistically identical* under scenario 1, each

described probabilistically by pR(m)

It is instructive to compare the above analysis for topology-selective jamming with

the dual concept of temporal selectivity. We take this issue up in the following section.

3c Comparison with Temporal Jamming

In our terminology, temporal implies a pulsed (blinking) two-level (ON-OFF)

-jamming pattern which, when ON, covers all local receivers under consideration with the

same power JRCVR. Let Pt indicates the temporal duty-factor of the jammer. It represents

the long-term fraction of time that the jammer is ON, as well as the probability that a

randomly observed slot will be found in the jamming state. As explained in IPrPo871, a

1 ': variety of jamming waveforms can be constructed which have the same Pt, but different

sample paths. In essence, one can vary the length (in slots) of the ON or OFF sessions by

different probabilistic mechanisms, yet keep Pt fixed.

The two extreme cases, from a temporal variability or slot-correlation viewpoint,

are (a) a long-term jammer, which stays in the same ON or OFF state for very long

(practically, infinite intervals of time and (b) a slot-by-slot independent jammer with

jamming probability pt. The mathematical model for (a) is that of initially chasing between

a "go-d" channel an( a "bad" channel with probability P, and (1 - P), respectively, and

staying there forever. In all cases, we can express total throughput asO

P= P3J + (l-Pt )i13 (26)

A. here the different temporal jamming strategies manife t themselves in the way we evaluate

' %--'S .- -..-. ". .-.'. .'.".-.-+ ." -..- " -- ,- - +--- -'? -- '. .-"""' -" . .

"% - r . ,i - -,- W U W -t W7. '.,Vi V 4 , - . .;- . . - - • •

ToPology-Selective Jamming.. 17

'S,

'.5-.. the conditional throu.hputs 1 and 1 For.the long-term jammer, it is immediate that

= ~mpT (m;j) fMT(mL.); =J orJ c (27)m

where the symbol I is used as a jamming index; thus, (26) is easily evaluated, virtually by

analvzing a multiple-access channel in two different interference levels.

It can be argued that (27) holds for all cases exactly as (26) does, except that

f.,,T(mill) should be with finite jamming block size, interpreted as the conditional

stationar' distribution of transmitting users, evaluated for the state-] slots only (I = J or

JC) In other words, fNT(mIJ) represents the probability of having MT - m transmissions

in a "typical" or randomly-chosen slot, assuming we only look at those slots of status .

Unfortunately, evaluation of these two conditional stationary distributions is not trivial,

necessitating the solution of a composite Markov chain of size which grows very quickly

x ,ith complexity:, the interested reader is referred to [PrPo87l. There is, however, one case

which is significantly simpler, namely the slot-by-slot independent jamming of case (b)

abwe. Then, a little thought will reveal that

fMT(mJ) = frT(mIJ c) = fMT(m) (28)

mcanin that any slot is a "typical" slot, regardless of whether it belongs to a jammed or

* nonjammed block. This is precisely so by virtue of the memoryless property of the

jamming mechanism from one slot to the next. Substituting property (28) into (26), (27)

(recall that the latter two equations hold for y scenario). We arrive at

13 = T m Psm) fMT(m) (29a)m

".

.5"

5'i

.. . - '. .. - ' ' . . - .' . .' ' ' ' . . -' .. "S . - .. ' . "'. " . .' . '' , • " .' % % " 5


where

T A T TP,(rm) A PtpT(mJ) +(I pt) p(m:jC) (29b)

is the temporal jamming-average probability of success.

Equation (29) is formally identical to (10), (11), the apparent difference being that

averaging here is performed over the temporal profile of the jamming process, as opposed

to the spatial jamming profile of Section 3. The root of the similarity is, of course, the fact

that the jamming decisions are independent from slot to slot. Furthermore, it is not hard to

see that this special case of memoryless temporal jamming can be reformulated in the

topology-selective framework by letting

I

P t if ai I for all i• -Pr [A' a] =(30).%A= 1- pt , if ai = 0 for all i

since a substitution of (30) into (10) will immediately yield (29b)

The last step for case (b) regards the evaluation of fMT(m) in (28a). If we

incorporate the above conceptual linkage, manifested by (30), into the procedure outlined in

(3b) for spatial jamming, we immediately conclude that fMT(m) can be obtained from

(18) - (21), with the temporal average

I

PSIm = lt PsmJ + (1 - PdslmJc (31)

'resulting from substituting (30) into (22). This is quite a convenient simplification which,I

as we mentioned, does not occur for jamming patterns with slot memory.

.1

W

4

.1 A

I.%I~...**.*.'

w '

--- . : -. • -- .- . - . -' '- . ."- '; ' .. .3. ; ' '. -.--.." ," , ,,. t,.- ". A- -4-. . ' ., ~ "- . '. * " '2 ,,

T()pohogy-Sehectiv-e Jammning-.. 19

4.0 Numerical Results

The main interest of this investigation is the network aspect of the jamrning thrcat.

Thus in the subsequent discussion, we assume the simplest s.igral format and jamming

format. nanely, the continuous tone jamming and the coded direct-sequence binary-phase-

R , Rmjcshift-key modulation. The probabilities of successtul reception PA (m,J) and p\(mj are

given bye

p( ) = , p(,A) (1- PCE(m,A))L - (32)

for the c-error correction code of block length L (it is also the packet length here). We also

let A =J if the receiver is jammed and let A =C if the receiver is not jammed. pcE(m,A) is

the channel symbol error rate, which is given by

.[? 4 2Es 1(33)

PcE(m,A) =Q Neq((mA)

where

2Ec5 Gbaud

N_- Ne(i, J) -+ JM)CNeq. Yin + (m - 1)cma +

A. '(34)

2EC Gbaud

eNq(mJC) Yin + (m -1)0tma

and

Q(x) e-x2/2 dx (35)

2J-x'C"

In the above equations, Gbh t( is the spreading ratio of the channel symbols, ygn is related to

*I the sinal-energy-to-thermal noise ratio per channel symbol by yin EC,(NOGbaud), and

"pL

i;

S . m. . , . ,. • . % , , . . .% . , % % % -,% . ., ,, ,


U.na is the multiple -access coefficient depending on the cross-correlation between the

particular multiple-access codes in use. The numerical results are computed for Oma = 1

and L = 1024. For the extended BCH code of block length 1024, the number of

correctable errors e and the coding rate can be related by

lIler (36)

1024

approximately.

In Figures 1 through 3, we show the normalized throughput versus the spatial duty

factor P with Po and Pr being the parameters for jamming scenario 2. The ten-error

correction code of rate 0.89 is used. We see that if Gbaud = 13 dB, the worst-case Psp

* approaches to 1 as P) and Pr approach 0.5, and if Gbaud = 20 dB, worst-case Ppp stays

about 0.5 for p0 and Pr less than 0.5. This phenomena can be explained as follows: Note

that the multiple-access self-interference noise is small if p0 and Pr are small for a small

GbauA, Or if Gbaud is large. If the self-interference noise is small, the worst-case Ppp is less

than 1 oecause the strong jamming power is needed to wipe out each jammed receiver. On

the other hand, if the self-interference noise is large, the worst-case p, is equal to 1

because the small jamming power is sufficient to wipe out each jammed receiver.

In Figures 4 through 6, we show the normalized throughput versus the spatial duty

factor P with p0 being the parameters for jamming scenario 2. The retransmission

probability pr is optimized in these three figures. We see that the effectiveness of the

worst-case Psp is slightly reduced for many cases by optimizing pr.

In Figure 7, we show the normalized throughput versus the jammer spatial duty

P factor with P0 = Pr = 0.3 for several error-correction codes for jamming scenario 2. In

the same figure, we also show the normalized throughput for the paired-off case. Notice

that the competitive scenario is more robust against the worst-case jamming than the paired-

4 off scenario.

4'

I%

3.0-Figure 1. Normnalized throughput versus the

spatial duty factor p, for jammingscenario 2

2.7

4 Yin 10 dB,( Gbaud 13 d B,S/J 3 3dB, L =1024,

V e = 10, r = 0.89

2.4 NT NR = 10

~2.1

PO = r 0.3

po p, = 0.4

N

EPo= =0.2

1.5- p0 =r =0.5

0.9 0

0.60 0.2 040.6 0.8 1.0

Spatial Duty FactorP4 S

V I~r %

3.9 -igIure 2. Normalized thiroughput versus thespatial duty factor p,, for Jammingscenario 2.

3.6-

7i-3 dB, GbadU(I 13 d B,S/J - 3dB, L = 1024,e = 10, r =0.89

3.3NT= NR = 10

S3.0- PO r 0.5

Pu = r= 0.4

- - .7

PO Pr =0.32.4-

2.1

1.8

PO p r =0.2

1.5 I

0 0.2 0.4 0.6 0.8 1.0

Spatial Duty Factor R

[-'ttire 3. Normalized through10put versus the3.9 Sp-,tial duty, factor pfor'arnm~ng 23

scenario 2. O ,0.

3.6-

3.3

PO = r 0.4

~3.0-

.7

EE

Po Pr =0.3

2.4-

2.1S/J 13 dB, L =1024,

e 0,10, r = 0.89

NT =NR = 10

II

0 0.2 0.4 0.6 0.8 1.0

I Spatial Duty Factor P

3. - 24

Fi~ure 4. Normalized throu01put versus thespatial duty factor jp for jammingscenario 2. The retransmissionprobability Pr is optimized.

2.7

2.4-

S2.1-PO r0.

I0.

Po Pr0.

1.5-

'Y. 3 dB,G baud 13 dB,S/J =-3 dB, L =1024.

e = 10, r = 0.89

1.2 NT NR =10

%:0.9p r0.

0.60 0.2 0.4 0.6 0.8 1.0

ISpatial Duty FactorPI

% % %

3.9 25

I- "ure 5 Normalized throughput -vcrsu,, tie

spatial duty factor p, !)r jammin gsc'nnirio 2. The rer: rni ,,Sronprohahility p,. i, vp iP n d.

3.6

' • P Pr = ."Po Pr = 0.4

A-

2.7

P p0.3

2.4-

"n= 10 dB, bd=3dBS/J 3 dB, L= 1024,

e = 10, r = 0.892.1-

NT NR =10

1.8po =P, =0.2

1.50 0.2 0.4 0.6 0.8 1.0* Spatial Duty Factor psp

-OIL

pt P, 0.5 26

3.

= ~=0.4

.5 pr

Figure 6. Normalized throughput versus the

-a spat IA duty factor p, for jammingsnrio 2. The ret-ransmissio

probability p, is optimized.

2.7

1.4

Yl= -3 dB, Gbaud = 20 dB,

SOJ= 13 dB, L =1024,

e = 10, r =0.89

62.1 NT N NR 10

1.8 p =p,- 0.2

S..

0 0.2 0.4 0.6 0.810

ISpatial Duty 17actorPI

... ~-5----' * ~s* 5 S'~ ~ -~% ~

2.4 -r 0 (.785 pair off 27

2.A

r 0.89

1.8

.~1.5 -Figure 7. Normalized throughput vessthespatial duty factor p~ for severalerror-cor-rection codes for jammingscenario 2.

S 1.2

Yi,= -3 dB, G baud 13 dB,

S/i-3 dB, L 1024,

0.9 P, PO0 .3

N NT N R -10

0.6-

0.3

pair off

0.0*0 0.2 0.4 0.6 0.8 1.0

ISpatial Duty Factor PI

~~~~~ ---, ' rp "*- .0p' .F .F.p' . -,.* . * * * S

Topologx -Stlective Jammin... 2

In Fi.zure 8. we show the normalized throughput versus the jammer spatial duty

factor P with p() being the parameters for jamming scenario 1. The retransmission

probability Pr is optimized in this figure. Compared with Figure 6, wke see that the worst-

case pp in scenario I is slig htlyv more effective than that in scenario 2.

p..

% %

,•

.

0

o

0,

3.9 PO) 0. 5 29)

3.6

PO =0.430

FiguLre 8. Normalized thrOug,-hput versus the.~ 3.0spatial duty factor p, for jammi ng

scenan o 1. Terransmissioniprobability Pr is optimized.

a'.4

'Yin .3 dB 2 B

p =N3

2.4 =1

a1..

0 ~~ 0. 10, r.6 0.89

Spta 2.1y NTto NR 1

a'% %

m10

Appendix A

As 11eC1nuonMed in Section h, ki~r the .imli ng , 'cnario 1, there are I"() wav% to

\I,\P' , ( o1.iite .",i 1'i euaton 22_, 1 here p, Imart: computCd by

t . 2 1, , f Fr each r

\ 1 . 2 (X:n::e p,,: dirc.tlv using recursion fonnula (24C) bv suhstituting-A k R

.Mi and p (m) tor P, and pA , respectively, and identifying

NR _ ANR

(NR;P ;IT [11R PA

In thi, appendix\, %e prove that these two methods are equivalent.

For the I s < min(m.k+! case, we have

1=0)______ k .1

(k= Pf - -Psp) k - (l-p) Pslmi + Psp li+(A

P P, +ir I S1 I (A-i1)

In the above equation, the first equality is due to equation (22), and the second equality is

obtained by applying the equality I (k + W thnapy herc sionL We then apply the recurs'o

(k+l) (k -) (k).formula (24c) to expand Ps1m+I and psim,i in terms of Psmi Note that, we useRi(k ) Rk p(k~i)

Rk+I (mJc) for pk I,) and p I(m'J) for p, 1 .,+i in the expansion. Thus. we end Ip with

.P1 •

# ~.

TopoZ j,'v-.S'hcuvtc jamming .. 3

ssr

m Pi j P,)k.P

+ nif s + I 1 k-1 k~ H p (

A

The last equality is obtained by applying equation (22) again and it is exactly Method 2.

Uising the similar technique, it is easy to verify the s =0 case.

-T, , -n.Sehctit-e .l ig... 32

References

[KaGrBuKu781 R. Kahn, S. Gronernever, J. Burchfiel, and R. Kunzeman,"Advances in Packet Radio Technology," Proc. IEEE, vol. 66,pp. 1468 - 1496, November 1978.

[Jubi85i J. Jubi. "Current Packet Radio Network Protocols," Proceedings ofINFOCOM '85, pp. 86 - 92, April 1985.

IShTo," 51 N. Shacham and J. D. Tornow, "Future Directions in Packet RadioTechnology," Proceedings of INFOCOM '85, pp. 93 - 98, April1985.

[PoSiM71 A. Polydoros and J. Silvester "Slotted Random Access Spread-Spectrum Networks," IEEE Journal on Selected Areas in Comm,,Vol. SAC-5, No. 6, July 1987.

* lPrPoS7] N. Pronios and A. Polydoros, "Spread-Spectrum, Slotted-ALOHA,Fully-Connected Networks in Jamming," submitted to IEEE Trans,Comm. ; see also preliminary version in MILCOM '87.

I K1La75] L. Kleinrock and S. Lam, "Packet Switching in a MultiaccessBroadcast Channel: Performance Evaluation," IEEE Trans,Comm., vol. COM-23, pp. 410 - 423, April 1975.

ISoSi841 E. S. Sousa and J. A. Silvester, "A Spreading Code Protocol for aDistributed Spread Spectrum Packet Radio Network," in Proc.GIOBECOM '84, Atlanta, GA, December 1984.

{SuLiS5I S. L. Su and V. 0. K. Li, "An Iterative Model to Analyze Multi-Hop Packet Radio Networks," in Proc. Allerton Conf., Oct. 1985,pp. 545-554.

[Purs871 M. B. Pursley, "The Role of Spread Spectrum in Packet RadioNetworks," Proceedings of the IEEE, vol. 75, pp. 116 - 134,

* January 1987.

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