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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 6, JUNE 2006 1371

Collaborative Mitigation of

Partial-Time Jamming on Nonfading Channels

Jang-Wook Moon, John M. Shea, and Tan F. Wong

Abstract—We propose new collaborative reception techniques

for use in the presence of a partial-time Gaussian jammer. Under

the proposed techniques, a group of radios acts as a distributed

antenna array by exchanging information that is then used to

perform jamming mitigation. We propose two such jamming

mitigation techniques that offer a tradeoff between performance

and complexity. The results show that these techniques can

allow communications in much more severe jamming conditions

than when collaboration is not employed or when conventional

collaboration techniques based on maximal-ratio combining are

applied. Example scenarios with strong jamming show that three

collaborating radios can reduce the frame error rate by more

than two orders of magnitude over single-radio reception. In

another scenario it is shown that a jammer must jam at least

75% of the transmitted symbols to produce an unacceptable

frame error rate with three collaborating radios, but only 43%

of the transmitted symbols if there is no collaboration.

Index Terms—Partial-time jamming, jamming mitigation, col-

laborative communications, collaborative signal-processing.

I. INTRODUCTION

P

is especially true for military communication systems, which

may experience hostile jamming. We consider communication

in the presence of a partial-time Gaussian jammer. Such

a system also approximates the effects of a partial-band

jammer in a system that employs frequency-hopping spread

spectrum. In systems that do not employ jamming mitigation

techniques, the jammer can severely disrupt communications

by concentrating its power over just a few symbols of a

packet. If the jammer power becomes strong compared to

the signal power, then it becomes impossible to communicate

successfully. Therefore it is desirable to devise methods to

combat or even cancel the detrimental effects. In the absence

of multiple receive antennas, most previous research focused

on using time diversity via repetition and/or error-control

coding [1]–[14] to improve performance in the presence of

jamming.

ARTIAL-TIME jamming or interference can severely

affect the performance of communication systems. This

Manuscript received January 20, 2004; revised June 1, 2005 and June 8,

2005; accepted June 11, 2005. The associate editor coordinating the review

of this paper and approving it for publication was L. Hanzo. This work was

supported by the Office of Naval Research under Grant N00014-02-1-0554, by

the National Science Foundation under Grant ANI-0220287, and by the DoD

Multidisciplinary University Research Initiative administered by the Office of

Naval Research under Grant N00014-00-1-0565.

J.-W. Moon was with the Wireless Networking Group, Department of

Electrical Engineering, University of Florida and is now with VIA Telecom,

Inc., San Diego, California 92121 (e-mail: jmoon@via-telecom.com).

J. M. Shea and T. F. Wong are with the Wireless Networking Group,

Department of Electrical Engineering, University of Florida, Gainesville,

Florida 32608 (email: {jshea,twong}@ece.ufl.edu).

Digital Object Identifier 10.1109/TWC.2006.04015.

If multiple receive antennas are available, the jamming

signal can be reduced through nulling; see for example, [15]–

[17]. However, in many communication systems, multiple an-

tennas are not practical because of size constraints. Recently,

collaborative communication techniques [18]–[29] have been

proposed as a way to achieve spatial diversity without multiple

antennas on any single radio. In these techniques, a group of

radios cooperates by exchanging information to improve the

performance of one radio or the whole group. In particular, in

the collaborative reception techniques proposed in [25]–[27],

a cluster of radios act as a distributed antenna array to achieve

combining gain or diversity against fading.

Collaborative reception techniques can also be used to

provide diversity against a jamming signal. We propose two

such collaborative jamming mitigation techniques. In each of

the proposed techniques, copies of the demodulator outputs

for only the jammed symbols are exchanged among the col-

laborative radios. Iterative detection, estimation, and decoding

algorithms are applied to achieve jamming mitigation. We use

an error-correction code with maximum a posteriori (MAP)

decoding, and the two collaborative decoding techniques differ

in how the information from other radios is utilized and in their

complexity. We also propose techniques for determining the

set of jammed symbols, and we investigate performance for

both known and unknown jamming parameters.

The collaborative jamming mitigation schemes that are

proposed in this paper build upon previous research on co-

operative communication and on jamming mitigation through

error-control coding and adaptive antenna beamforming. Our

approach utilizes error-controlcoding and extends the previous

work [1]–[14], to take advantage of spatial diversity among

radios in a network. The techniques we propose use the radios

in a network to form a distributed antenna array. Unlike a

conventional antenna array, the relative locations of the radios

in a network are not carefully controlled, and thus the phase

of the jamming signal with respect to the message signal will

be random at each radio. These phases must be estimated,

and the performance may be reduced relative to a traditional

phased array. In addition, in conventional approaches [15]–

[17], the information from multiple receiver front ends is

exchanged over wires and there is no cost to exchange all

of the information. In our approach, we try to reduce the

required communication overhead by exchanging only the

jammed symbols. Each radio must individually estimate the

set of jammed symbols and the radios must reach a consensus

before exchanging the received values for these symbols.

Most previous work on cooperative communications [18]–

[27] assumes independent noise at the cooperating radios,

1536-1276/06$20.00 c ? 2006 IEEE

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1372 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 6, JUNE 2006

radio 0

radio 1

Transmitter

radio 2

radio 3

Jammer

Fig. 1.One transmitter is communicating with multiple radios.

TURBO

ENCODER

RECTANGULAR

INTERLEAVER

THERMAL

NOISE

DE-

INTER-

LEAVER

Message

Decoded

message

JAMMER

JAMMER

MITIGATION

& DECODING

JAMMER

STATE

ESTIMATION

θ je

DE-

INTER-

LEAVER

Decoded

message

JAMMER

MITIGATION

& DECODING

JAMMER

STATE

ESTIMATION

COLLABORATIVE INFORMATION

EXCHANGE

Radio 0

Radio 1

BPSK

MODULATOR

BPSK

DEMOD.

BPSK

DEMOD.

Fig. 2. The overall system model.

and thus uses combining schemes similar to maximal-ratio

combining (MRC). In the presence of correlated jamming, we

show that MRC does not provide adequate protection from

the jamming, and thus we propose more-sophisticated signal

processing techniques.

The rest of this paper is organized as follows. In Section II,

the system model that is used in this paper is introduced. In

Section III, two collaborative jamming mitigation techniques

are proposed. In Section IV, the overhead and complexity

of the two collaborative mitigation schemes are investigated.

In Section V, we first explain how the jammer state can

be estimated. Then, we propose a collaborative jamming

detection algorithm for the radios to reach consensus on the

set of jammed symbols. In Section VI, simulation results are

presented, and conclusions are made in Section VII.

II. SYSTEM MODEL

We consider a scenario in which a transmitter communicates

with N radios in the presence of a partial-time jammer, as

illustrated in Fig. 1. The overall system model for N = 2 is

illustrated in Fig. 2. Binary phase-shift keying (BPSK) is used

for modulation. A turbo code is used for channel coding, and

log-MAP decoding is performed using the decoding algorithm

proposed by Bahl, Cocke, Jelinek and Raviv [30], [31] (the

BCJR algorithm). Throughout this paper, we assume that the

turbo code has rate 1/3 and employs identical constituent

codes. However, our schemes easily generalize to other codes.

Let M denote the number of code symbols in a codeword. A

rectangular interleaver is used to break up jamming bursts at

the input to the decoder.

We model the jammer’s transmissions using a two-state

hidden Markov model (HMM). If at time k, the jammer is

in state 0, then the kth bit is not jammed. If it is in state 1,

then the bit is jammed, and the jamming signal is modeled as

white Gaussian noise with power spectral density NJ/(2ρ),

where ρ is the probability that a bit is jammed. Let E{TJ}

be the expected amount of time (in terms of number of bits)

spent in the jamming state before returning to the unjammed

state. Then the transition probability from state i to state j is

denoted by Pij and can be determined from ρ and E{TJ}.

If the kth bit is jammed, the demodulator output for the kth

received symbol at radio i can be modeled as

?

where Esis the symbol energy and ukis the modulated code

bit, which takes values from ±1. Here, n(i)

contributions from thermal noise and jamming, respectively,

and are zero-mean, circular-symmetric Gaussian random vari-

ables. The variances of n(i)

respectively. The total variance of the noise and jamming for

state 1 is Υ ? N0+ NJ/ρ. θ(i)is the relative phase of the

jamming signal at radio i with respect to the jamming signal

at radio 0 and is uniformly distributed on [0,2π). Without loss

of generality, we let θ(0)= 0. This does not imply that the

jamming signal at radio 0 is in-phase with the information

signal, as Jk itself is a complex random variable with phase

uniformly distributed on [0,2π). We assume that the value of

each θ(i)is fixed for the duration of each packet. Thus, as

shown in Fig. 2, the radios experience independent thermal

noise but a phase-shifted version of a common interference

from the jammer.

A group of radios acts as a distributed antenna array

by exchanging the set of symbols that are estimated to be

jammed and then applying collaborative jamming mitigation

techniques. These exchanges may occur over the same channel

as the original packet reception if the exchange will not

interfere with transmission of additional packets, for instance,

if the transmitter uses stop-and-wait ARQ. Alternatively, the

collaborative exchanges may take place over an orthogonal

communication channel, such as a wireless local area network.

We note that in the general case, it is possible that the

communications among the group of collaborating radios

may also be jammed. However, in this paper, we assume

that the receiving radios are clustered in a relatively small

area and thus have sufficiently high signal-to-noise ratio to

communicate in the presence of the jammer.

The radios use iterative estimation, information exchange,

and decoding algorithms. First the radios individually estimate

which symbols are jammed and then exchange messages

to reach a consensus on the set of jammed symbols. Then

the radios collaborate by exchanging information about the

jammed symbols. Finally, each radio uses all of the received

information to mitigate the jamming through one of two col-

laborative jamming mitigation techniques, which are described

in the next section.

y(i)

k=

Esuk+ n(i)

k+ Jkejθ(i),

(1)

k

and Jk are the

kand Jkare given by N0and NJ/ρ,

III. COLLABORATIVE JAMMING MITIGATION

The collaborative techniques used to mitigate jamming

should be designed to take advantage of the highly corre-

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MOON et al.: COLLABORATIVE MITIGATION OF PARTIAL-TIME JAMMING ON NONFADING CHANNELS1373

lated nature of the interference signal. In this section, we

present two techniques of differing complexity for jamming

mitigation. We begin by assuming that every radio has perfect

channel state information (CSI), which includes knowledge of

which bits are jammed as well as the statistics of the received

jamming signal. In Section V, we consider the problem of

estimating the jammer state and parameters of the jamming

signal.

The jamming mitigation techniques that we consider fall in

the class of collaborative decoding techniques [24]–[27]. As

in previous work, we base our jamming mitigation techniques

on the BCJR MAP decoding algorithm [30], [32]. Under the

BCJR algorithm, the a posteriori likelihood ratios for the

messages bits can be calculated as

p(uk= +1|y)

p(uk= −1|y)=

where U+and U−denote the sets of branches that corre-

spond to message bits with value +1 and −1, respectively.

Here, αk−1(s?) = P(s?,yk−1

probability, βk(s) = P(yM

probability, and γk(s?,s) = P(s,yk|s?) is the branch probabil-

ity. Here, yb

time indices a and b. It can be shown that αk−1(s?) and βk(s)

can be computed recursively using γk(s?,s) [30], [32]. The

rate 1/3 turbo codes considered in this paper are constructed

from two rate 1/2 constituent convolutional codes that are

decoded iteratively. In the absence of collaboration, the branch

metric for each of these convolutional codes is given by

?

U+αk−1(s?)γk(s?,s)βk(s)

U−αk−1(s?)γk(s?,s)βk(s),

?

(2)

1

) is the forward-looking state

k+1|s) is the backward-looking state

adenotes the vector of received symbols between

γ(i)

k(s?,s) = p(uk)p(y(i)

s,k|s?,s)p(y(i)

p,k|s?,s),

(3)

where y(i)

radio for the kth systematic and parity bit, respectively.

Since every radio receives copies of the same message in

the presence of phase-rotated versions of the same jamming

signal and independent thermal noise, it is desirable to utilize

the received symbols from every radio to discriminate against

the jamming signal and simultaneously achieve combining

gain. The greatest performance can be achieved if all of the

received symbols from every radio are employed in calcu-

lating the branch metrics. However, to do so would require

exchanging the soft decisions for every received symbol. As

the performance is often dominated by the jammed symbols,

we assume that only the jammed symbols are exchanged. We

propose two techniques to utilize the different copies of the

jammed symbols in the computation of the branch metrics.

The two techniques offer a tradeoff between complexity and

communication overhead, which we investigate in Section IV.

s,kand y(i)

p,kare the matched filter outputs at the ith

A. Joint Density For Jammed Symbols

In this section, we explain the first jamming mitigation

technique, which we call the joint density approach. In this

approach, the conditional densities for the received symbols

in the branch metric γ(i)

p(y(i)

sities for the symbols from every radio, p(ys,k|s?,s) and

p(yp,k|s?,s), for each symbol that is jammed. Here, ys,k=

?

k(s?,s) for the single receiver case,

p,k|s?,s) are replaced by the joint den-

s,k|s?,s) and p(y(i)

y(0)

s,k,y(1)

s,k,...,y(N−1)

s,k

?

, and yp,kis defined similarly.

Consider the conditional joint density function for the set of

received symbols representing a jammed information or parity

bit given the transmitted symbol uk. Under the assumption

that the jamming parameters NJ and ρ and phases θ(i)are

known, the conditional joint density function is Gaussian with

mean mkand covariance matrix Σk. The mean of y(i)

where μk=√Esuk. The variance of y(i)

yk = [y(0)

k

mk= [μk, μk, ..., μk]T. The covariance matrix is Σk=

E[(yk− mk)(yk− mk)H], where

Σk(l,m) = Σ∗

= E[(n(l)

= E[|Jk|2ej(θ(l)−θ(m))]

=NJ

ρej(θ(l)−θ(m)), l = 0,...,N − 1

k

is μk

k

is N0+NJ/ρ. Let

k, y(1)

k, ..., y(N−1)

]T. Then the mean of yk is

k(m,l)

k+ Jkejθ(l))(n(m)

k

+ Jkejθ(m))∗]

, m = l + 1,...,N − 1,

whereHdenotes complex conjugate transpose and∗denotes

complex conjugate.

In practice, the radios do not know a priori whether

a symbol is jammed. Neither do they know Σk, nor the

parameters needed to compute Σk, which are NJ, ρ, and

θ(i). The set of jammed symbols, as well as NJ and ρ can

be estimated using the Baum-Welch algorithm, which we

discuss in Section V. Then Σk can be estimated directly as

Σk =

|J|

set of jammed symbols. The mean mk is unknown because

the correct symbol value ukis unknown, so we calculate the

mean using the a posteriori estimate for ukfrom the previous

iteration of the BCJR algorithm.

1

?

k∈J(yk− mk)(yk− mk)H, where J is the

B. Jamming Signal Cancellation

Now we develop the second technique for jamming mit-

igation. We use an iterative estimation, cancellation, and

decoding process. In this technique, the jamming sig-

nals J

=

{Jk: k ∈ J} and the relative phases θ

?θ(0),θ(1),...,θ(N−1)?are estimated, and the phase-corrected

Then the symbols from different radios are combined to form

a new decision statistic

?

where ˆ Jk andˆθ(i)are the estimates for Jk and θ(i). The

new decision statistics are used in place of the y(i)

calculation of γk(s?,s) in the BCJR algorithm as

=

jamming signal is subtracted from the received symbols in J.

zk=

N−1

?

i=0

y(i)

k−ˆ Jkejˆθ(i)?

,

ks in the

γ(i)

k(s?,s) = p(uk)p(z(i)

s,k|s?,s)p(z(i)

p,k|s?,s).

(4)

We expect that this jamming cancellation scheme may suffer

from error propagation that is common to interference cancel-

lation schemes and thus will generally perform worse than the

joint density scheme. However, we show in Section IV that

the jamming cancellation scheme has a lower complexity than

the joint density scheme.

We consider the joint maximum-likelihood (ML) estimate

for the jamming signals J = {Jk: k ∈ J} and the relative

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1374IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 6, JUNE 2006

phases θ(i). We derive the estimators for θ(i)and J under the

assumption that u = {u(i)

is known. As in the previous section, since u is unknown,

we use the estimates from the previous iteration of the BCJR

algorithm. Given u, the ML estimator for θ and J is

[ˆJ,ˆ

θ(2),...] =argmax

[J,θ(1),θ(2),...]

k

: k ∈ J,i = 0,1,...,N − 1}

θ(1),ˆ

p

?

y(0),...,y(N−1)|J,θ,u

?

,

(5)

where y(i)is a vector of the received symbols in J at radio i.

Given θ, J, and u, the y(i)s are independent Gaussian random

variables, and thus the natural logarithm of (5) can be written

as

?

where C1and C2are constants that do not have any effect on

the maximization. Taking the gradient and setting to zero, it

is easy to see that the joint ML estimates for Jkand θ(i)must

satisfy

f = C2− C1

k∈J

N−1

?

i=0

???y(i)

k− μk− Jkejθ(i)???

2

,

(6)

ˆ Jk =

1

N

N−1

?

i=0

?

y(i)

k− μk

?

y(i)

k− μk

y(i)

k− μk

e−jθ(i), and (7)

ˆθ(i)= tan−1Im?

k∈J

??

J∗

k

Re?

k∈J

??

J∗

k

.

(8)

Thus, the ML estimate for θ(i)must satisfy

Im?

k∈J

sinθ(i)

cosθ(i)=

k∈J

?

y(i)

k− μk

?

??N−1

n=0

?

y(n)

k

?

− μk

− μk

?∗

ejθ(n)

ejθ(n).

(9)

Re?

y(i)

k− μk

??N−1

n=0

y(n)

k

?∗

We can similarly expand (7).

Consider first (9). It can be shown that an equivalent set of

N equations is given by

?N−1

n=0

where Am,n=?

ML= tan−1Im?

k∈J

and

ˆJk =y(0)

22

?

Similarly, for any number of radios, a solution to F0is given

by θ(i)= ∠Ai,0= tan−1Ai,0,∀i. This may not correspond

to the ML estimate for N > 2. Since the ML estimator of θ(i)

does not admit a simple form for general N, in what follows,

we use the estimate given by

ˆθ(i)= ∠Ai,0= tan−1?

Fi? Im

?

Ai,nej(θ(n)−θ(i))

?

= 0,i = 0,1,...,N−1,

(10)

k∈J(y(m)

k

−μk)(y(n)

?

y(0)

k

k−μk)∗. For two radios,

?∗?

− μk

the ML solution is given by

ˆθ(1)

k∈J

y(0)

k

− μk

y(1)

k

− μk

− μk

?

Re?

??∗?

y(1)

k

?

(11)

k

− μk

j tan−1Im?(y(0)

+y(1)

k

− μk

·exp

k

− μk)(y(1)

− μk)(y(1)

k

− μk)∗

− μk)∗

Re?(y(0)

kk

?

. (12)

k∈J

(y(i)

k− μk)(y(0)

k

− μk)∗,∀i. (13)

The value ofˆJkis then estimated using (7) with θ(i)replaced

by its estimate,ˆθ(i).

Note that the dependence of the estimators for each Jkon

the values ofˆθ(i), 0 < i ≤ N − 1 may makeˆJk sensitive to

the quality ofˆθ(i). The estimates forˆθ(i)are typically much

more accurate than for Jk. This is because more samples are

typically used forˆθ(i)than for Jk. Each θ(i)is constant over

the entire block, and the average is over time. The Jks change

from symbol to symbol, and the estimator is averaged over the

received values at different radios.

To calculate (7) and (13), every radio needs to broadcast

the received values for the jammed symbols. This is the

same information required for the joint density approach. This

information needs to be exchanged only one time after the

collaborators agree on the set of jammed symbols. However,

the estimate (7) is very sensitive to errors in ˆ μkfrom the output

of the BCJR decoder. We consider the following approach to

improving the estimate of ˆ μk for the jamming cancellation

technique. Each radio performs independent decoding and

generates separate a posteriori log-likelihood ratios (LLRs)

for each coded bit. For the jamming cancellation technique,

these LLRs are exchanged along with the received symbols.

The LLRs at different radios for the same bit will be correlated

because of the shared jamming samples. However, we use a

suboptimal combining technique to avoid the complexity of

optimally combining the correlated LLRs. The LLRs from

each radio are added together to generate a more-reliable LLR,

which improves the probability of correct decoding. As these

LLRs may change during the decoding, we investigate the

effects of different exchange strategies in Section VI-A.

The estimated codeword may contain errors, but we can

use the LLRs to estimate the reliability of the estimated code

bits and adapt the jamming cancellation accordingly, thereby

reducing problems from error propagation. The branch metric

is computed by conditioning on whether the bit decision is

correct and averaging over the two cases. To illustrate this,

consider the case of N = 2. In what follows, we assume

that θ(1)is known or accurately estimated. We then have the

following two cases:

1) The decision is correct. The probability that this happens

can be approximated as a maximum of a posteriori

probabilities (APPs), i.e. [33]

Prob(correct decision for kth bit at radio i| y(i))

≈ max

eL(i)(k)+ 1

?

eL(i)(k)

eL(i)(k)+ 1,

1

?

,

(14)

where y(i)is the total received vector at radio i and

L(i)(k) is the LLR for the kth message bit at radio i.

The reliability of L(i)(k) can be improved by adding all

L(i)(k), 0 ≤ i < N as previously explained. From (1)

and (12), the estimated Jk is a random variable given

by

ˆ Jk =n(0)

k

+ Jk+ n(1)

ke−jθ(1)+ Jk

2

+ n(1)

2

= Jk+n(0)

kke−jθ(1)

.

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MOON et al.: COLLABORATIVE MITIGATION OF PARTIAL-TIME JAMMING ON NONFADING CHANNELS1375

After cancellation using these estimates, we get the

new random variable z(i)

k

is Gaussian distributed with mean E{z(i)

variance Var{z(i)

2) The decision for uk is incorrect. For this case,ˆ Jk is

given by

= y(i)

k

−ˆ Jkejθ(i), which

k} = μk and

k} = N0(N − 1)/N.

ˆ Jk=2μk+ n(0)

k

+ Jk+ (2μk+ n(1)

k

+ Jkejθ(1))e−jθ(1)

2

.

The random variable z(i)

with

k

= y(i)

k−ˆJkejθ(i)is Gaussian

E{z(i)

k} = μk−2μk

N

N−1

?

?=0

ej(θ(i)−θ(?)),

and Var{z(i)

After jamming cancellation, the z(i)

prove the decoder performance. The z(i)

but in the interest of constraining the complexity of the

algorithm, we use zk =

statistic. Then zk and its reliability and moments are used

to calculate the branch metrics in the BCJR algorithm. The

terms of the form p(zk|s?,s) in (4) are calculated as

p(zk|s?,s) = p(zk|s?,s,correct decoding)P(k)

+p(zk|s?,s,incorrect decoding)[1 − P(k)], (15)

where P(k) is the probability of correctly decoding the kth

bit, which we approximate by (14).

k} = N0(N − 1)/N.

k

are combined to im-

are not independent,

k

?N−1

i=0z(i)

k

as our new decision

IV. OVERHEAD AND COMPLEXITY OF

JAMMING MITIGATION

The two proposed schemes offer a trade-off between over-

head and complexity, which we briefly investigate in this

section. We define the average overhead as the average

number of bits of collaborative information that is transmitted

by each radio for each symbol in the original packet. In Fig. 3

we provide a flowchart that illustrates the operation of the

two jamming mitigation algorithms along with the information

exchanges required. As the flowchart illustrates, the overhead

comes from several sources. First, every radio broadcasts the

indices for the set of symbols that are estimated to be jammed

in order to reach consensus on the set of jammed bits. If

L is the number of information bits in the block and R is

the rate of the code, then ?log2(L/R)? is the number of

bits required to index an arbitrary jammed symbol within the

block. Let Ji denote the set of symbols that are estimated

to be jammed at radio i, and the expected number of such

symbols is E [|Ji|]. The set of jammed symbols is highly

compressible because of the Markovian nature of the jammer.

In what follows we present analytical values for optimum

compression and for compression based on sending the indices

of the starting and stopping times of jamming bursts, both of

which are considered in detail in [34]. The analytical results

are based on exact knowledge of which symbols are jammed.

For the simulation results, the radios do not know the set of

jammed symbols, but select the set using the majority voting

scheme described in Section V-C.

The second set of information that is exchanged consists of

the complex received values for the consensus set of jammed

symbols, J. Let Q denote the number of bits needed to

represent a real number. Then the number of bits required

to exchange the received values for the jammed symbols is

N (2Q)E [|J|]. For the jamming cancellation technique, the

a posteriori LLRs are also exchanged for the jammed symbols.

In Fig. 3, we show only one exchange of these LLRs (which

has the best performance), but we also considered schemes in

which the LLRs are updated in several iterations. These are

real values, and thus each time the LLRs are exchanged, a

total of N (Q)E [|J|] bits are required.

The results in Table I show the average overhead required

for the two schemes with two or three collaborating radios and

four-bit quantization. For comparison, the overhead required

for MRC is also shown. The results are based on blocks of

1000 information bits that are encoded with rate 1/3 symmetric

turbo codes based on constituent convolutional codes with

feedforward polynomial 1 + D2and feedback polynomial

1+D+D2. For all of the results, ρ = 0.6. For the simulation

results, Eb/NJ = −6 dB and Eb/N0 = 6 dB. The joint

density scheme offers a significant overhead savings over

MRC, especially for N = 3. The jamming signal cancellation

technique requires more overhead, approximately equal to that

of MRC. These comparisons will depend on the parameters of

the jamming. For smaller ρ, the proposed schemes will require

much less overhead than MRC, while for larger ρ, it may be

more efficient to exchange all of the received symbols (as in

MRC).

The complexity of the two approaches can be estimated by

considering the number of multiplications required for each

jammed symbol in one iteration of the BCJR decoder. Except

where noted, all operations are assumed to be on complex

numbers. For the joint density approach, the complexity is

dominated by the matrix multiplications in the Gaussian den-

sity. The total number of complex multiplications is easily seen

to be N(N+1). For the jamming signal cancellation approach,

estimating θ(i)and Jk requires 1 and N multiplications

per jammed symbol, respectively. Computing (15) requires

2 complex and 2 real multiplications per symbol. Thus, the

equivalent number of complex multiplications required is

approximately N + 4. Thus, although the jamming signal

cancellation approach requires higher overhead than the joint

density approach, it is equally computationally efficient for

N = 2 and more computationally efficient than the joint

density approach for N > 2.

Since the jamming signal estimates can be updated over

multiple iterations, the jamming signal cancellation technique

will require the number of computations estimated above times

the number of iterations in which the jamming signal estimates

are updated. By comparison, the joint density approach does

not require the density to be updated in later iterations. Thus,

the actual trade-off in complexity depends not only on N but

also on the average number of iterations in which the jamming

estimates are updated for the jamming signal cancellation

technique. However, for large N, the O(N2) complexity of

the joint density scheme will generally be much larger than

the O(N) complexity of the jamming cancellation scheme.