Amplifying Matrix Design for Distributed Noncooperative SIMO Wireless Relay

Networks Under Jamming Environment

Zuojun Wang, Hyuck M. Kwon, Yanwu Ding, Kanghee Lee, and Yazan IbdahDepartment of Electrical Engineering and Computer Science, Wichita State University, Wichita, Kansas 67260

Emails: {zxwang3, hyuck.kwon, yanwu.ding, kxlee1, yxibdah}@wichita.edu

Abstract—The main contribution of this paper is the deriva-tion of an optimal diagonal amplifying relay matrix. This matrixis for a noncooperative distributed wireless relay network in ajamming environment. Minimum mean square error (MMSE)criterion was used in this study. Jamming location was alsostudied in this paper. Results show that diversity of a systemcan be degraded or even get lost depending on jamming location.A single-input multiple-output (SIMO) system strategy was im-plemented, using one-source, M-destination and N-relay nodes.System performance for the new derived matrix was evaluatedusing bit error rate (BER). Also, performance comparisonswith/without jamming were a focus in our study.

Index Terms—Broadband jamming, amplify-and-forward,minimum mean square error, single-input multiple-output.

I. INTRODUCTION

The amplify-and-forward (AF) relay protocol in wirelesssensor networks forwards an amplified version of its receivedsignals from a source node to a destination node. Comparedto decode-and-forward and compress-and-forward protocolsin [1], [2], the AF relay protocol has lower complexitybecause there is no required signal processing at the relaynodes in wireless networks for decoding and compressing. Inpractice, because of wireless communication features duringsymbol transmission, all communication nodes in wirelessnetworks are exposed under the threat of jamming signals.Therefore, partial-band and broadband noise jamming (BNJ)are considered in [3], [4], but for non-relay systems. Inaddition, the authors in [5], [6] considered the cooperativeAF single-input multiple-output (SIMO) system under no-jamming environment.

Noncooperative distributed minimum mean square error(MMSE)-based AF relay schemes under both jamming envi-ronment and node geometry in wireless networks with/withoutpower constraint for single-input single-output with 𝑁 relaynodes was studied in [7], [8], respectively. BNJ and channeluncertainty for noncooperative wireless relay networks underthe received power constraint was also studied in [9]. Effects

This work was partly sponsored by the Army Research Office under

DEPSCoR ARO Grant W911NF-08-1-0256, and by NASA under EPSCoR

CAN Grant NNX08AV84A.

Fig. 1. Wireless relay network with one-source 𝑁 -relay 𝑀 -destinationnodes under a BNJ environment.

of node geometry on noncooperative distributed SIMO wire-less relay networks was proposed in [10]. However, non-cooperative SIMO wireless relay networks under jammingenvironment was not studied in [10]. Therefore, this paperstudies the noncooperative distributed AF SIMO wirelessrelay system in a jamming environment using the MMSEcriteria. In particular, a BNJ for a jamming environment isused due to the fact a partial-band noise jamming affects morenegatively on bit error rate (BER) than a BNJ. For perfor-mance comparisons, no-jamming environment was simulated.In addition, this current paper presents how to determinethe optimal amplifying relay matrix for the noncooperativedistributed AF SIMO system under a jamming environment.Furthermore, the optimal amplifying relay matrix is selectedto be diagonal due to noncooperation relay protocol design.

The rest of this current paper is organized into four sec-tions. Section II describes the system model and symboltransmission strategy. Section III presents a noncooperativedistributed MMSE wireless relay scheme based on the AFstrategy. Section IV provides the simulation results. Finally,Section V concludes the paper.

Notation: Matrices and vectors are denoted, respectively,by uppercase and lowercase boldface characters (e.g., A anda). The transpose, complex conjugate, inverse, and Hermitianof A are denoted, respectively, by A𝑇 , A∗, A−1, and A𝐻 .An 𝑛 × 𝑛 identity matrix is denoted by I

𝑁. The expectation

2011 Fifth Asia Modelling Symposium

978-0-7695-4414-4/11 $26.00 © 2011 IEEE

DOI 10.1109/AMS.2011.46

186

2011 Fifth Asia Modelling Symposium

978-0-7695-4414-4/11 $26.00 © 2011 IEEE

DOI 10.1109/AMS.2011.46

211

operator is 𝐸[ ⋅ ]. Notations ∣𝑎∣, ∣∣a∣∣, and ∣∣A∣∣𝐹 denote theabsolute value of 𝑎 for any scalar, 2-norm of a, and Frobenius-norm of A, respectively.

II. SYSTEM MODEL

Figure 1 shows a wireless relay network with 𝑁 non-cooperative distributed relay nodes between a source nodeand 𝑀 destination nodes (𝑁 ≥ 𝑀). As depicted in Fig. 1,there are two stages for data transmission. The source nodetransmits a signal symbol 𝑠 in Stage I, while relay nodesretransmit their received signals to destination nodes in StageII, applying the optimal diagonal amplifying relay matrixfor relay noncooperation. It is assumed that all relay nodescannot communicate their received signal information fromthe source node to each other.

The BNJ is defined as a complex Gaussian noise signalwith zero-mean and power 𝑁𝐽 = 𝜎2

𝐽 . In other words, theBNJ column vector j𝑖 ∈ C𝑁×1, 𝑖 = 1, 2, 3, is a zero-meancomplex additive white Gaussian noise vector with covariancematrix 𝑁𝐽I

𝑁. It is assumed that 𝜎2

𝐽 = 𝜎2𝑗𝑖, 𝑖 = 1, 2, 3. As

shown in Fig. 1, the BNJ is added to the symbol flow inall node links. All relay nodes receive the transmitted signalwith the jamming signal in Stage I. They retransmit a scaledversion of the jammed signal to the destination node in ajamming environment. Then the destination nodes receivesthe jammed signal from all relay nodes.

It is assumed that all relay nodes have accurate informationof their local channel coefficients, both from the source nodeto the relay nodes and from the relay nodes to the destinationnodes. Let h𝑠 ∈ C𝑁×1 denote the perfect channel coefficientcomplex column vector from the source node to the relaynodes as

h𝑠 = [ℎ𝑠,1, ℎ𝑠,2, ⋅ ⋅ ⋅ , ℎ𝑠,𝑁 ]𝑇 (1)

where ℎ𝑠,𝑖, 𝑖 = 1, ⋅ ⋅ ⋅ , 𝑁 , is the 𝑖-th entry of h𝑠, representingthe channel coefficient from the source node to the 𝑖-th relaynode. The received signal complex column vector r ∈ C𝑁×1

at the relay nodes under a BNJ environment is written as

r = h𝑠𝑠+ v𝑠 + j1 (2)

where v𝑠 ∈ C𝑁×1 is a zero-mean complex additive whiteGaussian noise column vector with covariance matrix 𝜎2

𝑣𝑠I𝑁 .

Each channel ℎ𝑠,𝑖 is assumed to be independent identicallydistributed with a zero-mean circular complex Gaussian ofunit variance and quasi-static. Channel coefficients are fixedduring symbol data transmission. The amplified signal com-plex column vector x ∈ C𝑁×1 at the relay nodes under a BNJenvironment is written as

x = F1r + j2 (3)

where F1 ∈ C𝑁×𝑁 is called a diagonal amplifying relaymatrix at the relay nodes to enhance performance at thedestination nodes. Let H𝑦 ∈ C𝑀×𝑁 denote the perfectchannel coefficient complex matrix from the relay nodes tothe destination nodes as

H𝑦 = [h𝑦,1,h𝑦,2, ⋅ ⋅ ⋅ , h𝑦,𝑀 ]𝑇 (4)

where h𝑦,𝑚 = [ℎ𝑦,𝑚,1, ⋅ ⋅ ⋅ , ℎ𝑦,𝑚,𝑁 ], 𝑚 = 1, ⋅ ⋅ ⋅ ,𝑀 , is arow vector, representing the channel coefficient from all relaynodes to the 𝑚-th destination node. Each channel ℎ𝑦,𝑚,𝑁 isalso assumed to be independent identically distributed with azero-mean circular complex Gaussian of unit variance andquasi-static Rayleigh fading. The received complex signalcolumn vector y ∈ C𝑀×1 at the destination nodes under aBNJ environment can be represented as

y = H𝑦x + v𝑦 + j3 (5)

where v𝑦 ∈ C𝑀×1 is a zero-mean complex additive whiteGaussian noise column vector with covariance 𝜎2

𝑣𝑦I𝑁. Sub-

stituting (2) and (3) into (5), the received complex signalcolumn vector y ∈ C𝑀×1 at the destination nodes under aBNJ environment can be rewritten as

y = H𝑦F1h𝑠𝑠+ H𝑦F1v𝑠 + H𝑦F1j1 + H𝑦j2 + v𝑦 + j3. (6)

In the next section, the optimal diagonal amplifying relaymatrix F1 will be determined by using the MMSE criteriaunder a BNJ environment.

III. NONCOOPERATIVE DISTRIBUTED MMSE WIRELESS

RELAY SCHEME

An optimal diagonal amplifying relay matrix under a BNJenvironment is designed in this section. It is based on thenoncooperative distributed MMSE relay scheme in wirelessnetworks. The purpose is to minimize mean square errorbetween the signal component H𝑦x of the received signal atthe destination nodes and the transmitted signal 𝑠 in (5) canbe found from

F†1 = argmin

𝐹1

𝐽(F1) (7)

where the cost function 𝐽(F1) is written using (5) as

𝐽(F1) =𝑀∑

𝑚=1

𝐸[∣∣h𝑦,𝑚x − 𝑠

∣∣2]= 𝜎2

𝑠

𝑀∑𝑚=1

∥h𝑦,𝑚F1h𝑠∥22 + 𝜎2𝑣𝑠

𝑀∑𝑚=1

∥h𝑦,𝑚F1∥22

+ 𝜎2𝐽

𝑀∑𝑚=1

∥h𝑦,𝑚F1∥22 + 𝜎2𝐽

𝑀∑𝑚=1

∥h𝑦,𝑚∥22

−𝜎2𝑠

𝑀∑𝑚=1

h𝑦,𝑚F1h𝑠−𝜎2𝑠

𝑀∑𝑚=1

(h𝑦,𝑚F1h𝑠)𝐻+𝑀𝜎2

𝑠 .

(8)

187212

Differentiating 𝐽(F1) in terms of the complex conjugate of F1,i.e., F∗

1, using the properties of the complex derivative matrixin [11], [12], and employing the noncooperative strategydesign gives gives

𝑀∑𝑗=1

𝑁∑𝑖=1

ℎ∗𝑦,𝑗,𝑖ℎ𝑦,𝑗,𝑘𝑓𝑘 =

𝜎2𝑠ℎ

∗𝑠,𝑘

∑𝑀𝑗=1

∑𝑁𝑖=1 ℎ

∗𝑦,𝑗,𝑖

𝜎2𝑣𝑠

+ 𝜎2𝐽 + 𝜎2

𝑠

∑𝑁𝑖=1 ∣ℎ𝑠,𝑖∣2

(9)

where 𝑘 = 1, ⋅ ⋅ ⋅ , 𝑁 . Therefore, the optimum diagonalamplifying relay matrix F†

1 under a BNJ environment for allnode links in wireless networks can be written as

F†1 = diag

(𝑓1, . . . , 𝑓

𝑁

)(10)

where

𝑓𝑘=𝜎2𝑠ℎ

∗𝑠,𝑘

∑𝑀𝑗=1

∑𝑁𝑖=1 ℎ

∗𝑦,𝑗,𝑖

(∑𝑀

𝑗=1

∑𝑁𝑖=1 ℎ

∗𝑦,𝑗,𝑖ℎ𝑦,𝑗,𝑘)

(𝜎2𝑣𝑠+ 𝜎2

𝐽 +𝜎2𝑠

∑𝑁𝑖=1 ∣ℎ𝑠,𝑖∣2

)(11)

where 𝜎2𝐽 = 𝜎2

𝑗1.

From (8), the other three special cases for the SIMOwireless network system under a BNJ environment can bederived. Namely, a BNJ can be added only at the relayinputs in the source-relay links, only at the relay outputsnear the relay nodes in the relay-destination links, and onlynear the destination node. Therefore, the optimal diagonalrelay amplifying matrices F†

𝑖 , 𝑖 = 2, 3, 4, corresponding tothe locations of adding the BNJ in wireless networks can berepresented, respectively, as

F†2 = diag

(𝑓1, . . . , 𝑓

𝑁

)(12)

where

𝑓𝑘=𝜎2𝑠ℎ

∗𝑠,𝑘

∑𝑀𝑗=1

∑𝑁𝑖=1 ℎ

∗𝑦,𝑗,𝑖

(∑𝑀

𝑗=1

∑𝑁𝑖=1 ℎ

∗𝑦,𝑗,𝑖ℎ𝑦,𝑗,𝑘)

(𝜎2𝑣𝑠+ 𝜎2

𝐽 +𝜎2𝑠

∑𝑁𝑖=1 ∣ℎ𝑠,𝑖∣2

)(13)

under a BNJ environment at the relay inputs in the source-relay links, where 𝜎2

𝐽 = 𝜎2𝑗1

,

F†3 = diag

(𝑓1, . . . , 𝑓𝑁

)(14)

where

𝑓𝑘=𝜎2𝑠ℎ

∗𝑠,𝑘

∑𝑀𝑗=1

∑𝑁𝑖=1 ℎ

∗𝑦,𝑗,𝑖

(∑𝑀

𝑗=1

∑𝑁𝑖=1 ℎ

∗𝑦,𝑗,𝑖ℎ𝑦,𝑗,𝑘)

(𝜎2𝑣𝑠+𝜎2

𝑠

∑𝑁𝑖=1 ∣ℎ𝑠,𝑖∣2

) (15)

under a BNJ environment at the relay outputs near the relaynodes in the relay-destination links, and

F†4 = diag

(𝑓1, . . . , 𝑓

𝑁

)(16)

where

𝑓𝑘=𝜎2𝑠ℎ

∗𝑠,𝑘

∑𝑀𝑗=1

∑𝑁𝑖=1 ℎ

∗𝑦,𝑗,𝑖

(∑𝑀

𝑗=1

∑𝑁𝑖=1 ℎ

∗𝑦,𝑗,𝑖ℎ𝑦,𝑗,𝑘)

(𝜎2𝑣𝑠+𝜎2

𝑠

∑𝑁𝑖=1 ∣ℎ𝑠,𝑖∣2

) (17)

under a BNJ environment near the destination nodes. Asshown in (15) and (17), the optimal diagonal amplifying relaymatrices under a BNJ environment both at the relay outputsnear the relay nodes in the relay-destination links and near thedestination nodes are independent of the jamming signals.

IV. SIMULATION RESULTS

Monte-Carlo simulations were performed to evaluate BERperformance for Node Geometry. Noncooperative distributedAF SIMO system which uses MMSE relay schemes in a BNJjamming environment. In all simulations setup was as follow-ing one source node, 𝑁 =2, 3 noncooperative distributed relaynodes, and 𝑀 = 2 destination nodes. The perfect channelcoefficient row vector h𝑠 and matrix H𝑦 were generated fromindependent Gaussian random variables with zero mean andunity variance. All nodes have only one antenna and have thesame noise power, i.e., 𝜎2

𝑣𝑠=𝜎2

𝑣𝑦. The transmitted signal at the

source node is also modulated by 4-ary quadrature amplitudemodulation (4QAM).

A BNJ has one-side power spectral density 𝑁𝐽 , equal to5%, 10%, and 20% of the bit energy and generated as additivewhite Gaussian noise. Variances of jamming signal are chosento satisfy 10 log10(𝜎

2𝑠/𝜎

2𝐽)=13 dB, 10 dB, and 7 dB, where

𝜎2𝑠 = 𝐸𝑏 and 𝜎2

𝐽 = 𝑁𝐽 . For comparison, simulation resultsunder no-jamming environment are also included, i.e., 𝜎2

𝐽 =0.

Fig. 2. BER performance of 𝑁 = 2, 3 noncooperative distributed relay and𝑀 = 2 destination networks under a BNJ environment for all node links.

Figure 2 shows BER performance of 𝑁 = 2, 3 and 𝑀 = 2

network in a BNJ environment for all node links. Due to theeffect of the BNJ environment for all node links in entirenetwork, a worst BER is observed, and compared to the otherthree special cases. In addition, it is observed that the BNJcreates significantly negative effects on BER performancein wireless relay networks. In other words, increasing the

188213

variances of the BNJ, worse BER can be observed. It is alsofound that BER enhances with the increase in the number ofrelay nodes.

Fig. 3. BER performance of 𝑁 = 2 noncooperative distributed relay and𝑀 = 2 destination networks under a BNJ environment at the relay inputs inthe source-relay links.

Fig. 4. BER performance of 𝑁 = 2 noncooperative distributed relay and𝑀 = 2 destination networks under a BNJ environment at the relay outputsnear the relay nodes in the relay-destination links.

Figure 3 shows BER performance of 𝑁 = 2 and 𝑀 = 2

in a BNJ environment at the relay inputs in the source-relaylinks. Compared to other two special cases, it is observed thatthe BNJ did not have significant impact on BER performance.

Figure 4 provides BER performance of 𝑁 = 2 and 𝑀 = 2

network in a BNJ environment at the relay outputs near therelay nodes in the relay-destination links. While, Fig. 5 showsBER performance of 𝑁 = 2 and 𝑀 = 2 network in a BNJenvironment near the destination nodes. It is observed that theperformance degraded significantly if most BNJ location is atthe relay outputs near the relay nodes in the relay-destinationlinks.

V. CONCLUSION

This paper studied noncooperative distributed AF SIMO re-lay schemes in a BNJ environment at different locations. Theoptimal diagonal amplifying relay matrices based on MMSEcriterion were derived. Under a no-jamming environment,BER performance improves as the number of relay nodesincreases. While in a BNJ environment, it is observed thatincreasing the jamming power can cause a loss of diversityorder. In addition, it is observed that the most harmfuljamming location is at the relay outputs, near the relay nodesin the relay-destination links. Finally, it is also observed thatthe optimal diagonal amplifying relay matrix is independent ofthe jamming if a BNJ jamming is located at the relay outputsand near the destination node.

Fig. 5. BER performance of 𝑁 = 2 noncooperative distributed relay and𝑀 = 2 destination networks under a BNJ environment near the destinationnodes.

REFERENCES

[1] T. Wang, A. Cano, G. B. Giannakis, and J. N. Laneman, “High per-fromance cooperative demodulation with decode-and-forward relays,”IEEE Transactions on Communications, vol. 55, no. 7, pp. 1427-1438,July 2007.

[2] S. Lee and S. Chung, “When is compress-and-forward optimal?” IEEEITA 2010, San Diego, CA, Jan. 2010.

[3] J. S. Lee and L. E. Miller, “On the use of side information indiversity combining in frequency-hopping communications in partial-band jamming,” Proc. 1991 IEEE Int. Symp. Inform. Theory, June 1991,pp. 323.

[4] H. M. Kwon, L. E. Miller, and J. S. Lee, “Evaluation of a partial-band jammer with Gaussian-shaped spectrum against FH/MFSK,” IEEETransactions on Communications, vol. 38, no. 7, pp. 1045-1049, July1990.

[5] N. Krishnan and B. Natarajan, “Energy efficiency of cooperative SIMOschemes amplify forward and decode forward”, IEEE ICCCN 2009, SanFrancisco, CA, Aug. 2009.

[6] H. An and H. Song, “Cooperative communication in SIMO systemswith multiuser OFDM”, IEEE Transactions on Consumer Electronics,vol. 53, no. 2, pp. 339-343, May 2007.

189214

[7] K. Lee, H. M. Kwon, Y. Ding, Y. Ibdah, and Z. Wang, “Noncooperativedistributed MMSE relay schemes under jamming environment and nodegeometry in wireless relay network,” IEEE WTS 2011, New York, NY,April 13-15, 2011.

[8] K. Lee, H. M. Kwon, Y. Ding, Y. Ibdah, Z. Wang, and Y. Bi, “Nodegeometry and broadband jamming in noncooperative relay networksunder received power constraint,” IEEE SARNOFF 2011, Princeton,NJ, May 3-4, 2011.

[9] Y. Ibdah, H. M. Kwon, K. Lee, Z. Wang, Y. Bi, and M. Jo, “Broadbandjamming and channel uncertainty for noncooperative wireless relaynetworks under received power constraint,” IEEE AMS 2011, Manila,Philippines, May 23-27, 2011.

[10] K. Lee, H. M. Kwon, Y. Ding, Y. Ibdah, Z. Wang, “Effects of nodegeometry on noncooperative distributed SIMO wireless relay networks,”IEEE AMS 2011, Manila, Philippines, May 23-27, 2011.

[11] R. A. Horn and C. R. Johnson, 𝑀𝑎𝑡𝑟𝑖𝑥 𝐴𝑛𝑎𝑙𝑦𝑠𝑖𝑠. 1st ed., Cambridge,MA: Cambridge University Press, 1985.

[12] “The Matrix Cookbook,” Online at: http://matrixcook book.com.

190215