A Novel Approach for using Extended LDPC codesin Cooperative Diversity

Hussain Ali and Maan KousaDepartment of Electrical Engineering

King Fahd University of Petroleum and MineralsDhahran, Saudi Arabia

Email: {hussainali, makousa}@kfupm.edu.sa

Abstract—Cooperative diversity or user cooperation achievesthe diversity gain without adding physical antennas to the usersor mobile stations. The users work in cooperative fashion usingtheir single antennas to create a virtual transmit diversity,called relay diversity or cooperative diversity. The diversity gainachieved by cooperative diversity can be further improved usingerror correction codes. Low-density parity-check (LDPC) codesare linear block codes with good error correction capabilities.We present a novel approach using extended LDPC codes toincrease the diversity gain in cooperative diversity. We alsocompare the extended LDPC codes with punctured LDPC codesin cooperative diversity and show that extended LDPC codes havelesser complexity than punctured LDPC codes in cooperativediversity.

I. INTRODUCTION

Wireless communications face the challenges of channelimpairments and fading that severely degrade the capacityof wireless channels. Numerous spatial diversity techniqueshave been in use to combat channel impairments and fading.One such technique is cooperative diversity in which the usersor mobile stations cooperate using their single transmittingantenna in a particular scenario to exploit the availabilityof good channels from users to base stations or destination.In cooperative diversity, generally, the destination receivesmultiple packets for the same data from independent chan-nels creating a virtual transmit diversity, called cooperativediversity. Cooperative diversity cannot guarantee error freetransmission, therefore, error control coding techniques areapplied in cooperative scenario.

User cooperation diversity has been used to achieve diver-sity gain using the partners transmitting antennas [1], [2]. Ifthe channel with one user to the destination is bad, then thechannels from other users, called partners, can be used tosend the packet to the destination. The destination receivestwo packets of the same data from two independent channelsthat may not be noisy or in deep fade at the same time. Thedestination provides decoding by maximal-ratio-combining(MRC) on both packets received and thus achieving spatialdiversity gain in simple repetition schemes. In a relay channel,each user acts only as relay, i.e. it only forwards the datawhich it receives by employing either detect-and-forward oramplify-and-forward or estimate-and-forward techniques [3].In cooperative communication, each user sends its own dataas well as relays the data of the partners in different time

slots. All the users have single transmitting antenna but theycooperatively create a virtual transmit diversity. In codedcooperative diversity or cooperation diversity through coding[4], rate-compatible convolutional (RCPC) codes [5] were usedjointly with cooperation which showed remarkable diversitygain.

Low-density parity-check (LDPC) codes were invented byGallager in his Ph.D. work [6] in 1960. LDPC codes belong tothe class of linear block codes. LDPC codes were ignored dueto lack of appropriate hardware in 1960s. A graphical repre-sentation of LDPC codes was proposed by Tanner [7] in 1981based on bipartite graphs. LDPC codes were rediscovered byMacKay [8] and others. LDPC codes have become more prac-tical due to the advancements in transistor technology leadingto high computational power of the hardware. LDPC codeshave gain attention due to their near-capacity performance.LDPC codes can be modified by puncturing and extensionto achieve rate-compatibility. Puncturing of LDPC codes wasdone to achieve higher code rate codes from lower code ratecodes [9]–[11]. Extended LDPC codes were introduced in[12]–[14] to achieve lower rate codes from high rate codes.A joint and efficient design for puncturing and extension isdiscussed in [13] which is preferred in cooperative scenariofor its rate adaptability.

Chuxiang Li et. al. used the rate-compatible design ofLDPC codes of [14] to extend the original codeword. Theyintroduced the half-duplex relay protocol in [15] for a singlerelay. In half-duplex relay protocol, the transmission is dividedin two time slots. In the first time slot, the destination andthe relay received the packets from the source. This modeof operation is called broadcast mode. In the second timeslot, the destination receives signals from both source and therelay. This mode of operation is called multiple access (MAC)mode. The code design for these operations becomes differentfrom the non-cooperative models. In the broadcast mode, thesource transmits an LDPC codeword to both the relay andthe destination. In MAC mode, the relay and the source sendsextra parity bits for the codeword to the destination. In MACmode, the destination receives two copies of the extra bitsthat can be combined optimally by MRC. The codeword isdecoded by successive decoding, first without extra parity bitsand second, if cyclic redundancy check (CRC) fails on thefirst transmission, then jointly decoded with extra parity bits.

978-1-4673-2054-2/12/$31.00 ©2012 IEEE 334

However, in this protocol the extra parity bits of the extendedcodeword does not constitute the complete codeword andcannot be decoded alone as a codeword. Factor graph decodingapproach of [16] was applied on two different relay protocolsfor cooperative relay systems in [17] where joint decoding atthe receiver was applied after receiving both packets in twoconsecutive time slots. The punctured LDPC codes for therelay channels were investigated in [18].

In this work, we present a novel approach using extendedLDPC codes for cooperative diversity. (This work is an ex-tension of the research paper accepted in [19].) We compareour work with punctured LDPC codes in cooperative diversityas well. The bit-error-rate (BER), encoding complexity anddecoding complexity are used as benchmarks for comparisonbetween the two approaches.

The paper is organized as follows: In section II, we in-troduce the system model for coded cooperative diversity. Insection III, we discuss punctured LDPC codes in cooperativediversity. These punctured LDPC codes will be used forcomparison with extended LDPC codes. In section IV, wepropose the novel approach for extended LDPC codes forcooperative diversity. The simulation results and discussionon complexity for the punctured LDPC codes and for theproposed design of extended LDPC codes for cooperativediversity are presented in section V. Section VI concludes thepaper.

II. CODED COOPERATIVE DIVERSITY

We assume a time-division based system with two terminalsT1 and T2 as users and one terminal T3 as destination. Thechannels for T1 and T2 transmission are assumed to be or-thogonal in time. The codeword N is divided into two weakercodewords denoted by N1 and N2. The frame transmissionfor N is divided into two time slots. The first time slot isreserved for each user’s own data. For the user T1, NT1

1 istransmitted to the destination T3 and to the partner T2 whereNT1

1 is the first codeword for user T1. Similarly, T2 sendsthe codeword NT2

1 to T1 and T3. Both T1 and T2 check theintegrity of data received by applying CRC. The transmissionin the second time slot is determined by the success or failureof decoding of these packets received in the first time slotfor each user. The four cases that arise after the first timeslot transmission are shown in Fig. 1. In case 1, both userssuccessfully decodes the packet received from their partners.Therefore, T1 will send NT2

2 for T2 and T2 will send NT12

for T1 in the second time slot. In case 2, both users fail todecode their partners transmission and continue to send theirown second codeword N2 in the second time slot. In case 3,T1 fails to decode the transmission from T2. In this case, bothusers will transmit NT1

2 for T1. In case 4, T2 fails to decodethe transmission from T1. In this case, both users will transmitthe codeword NT2

2 for T2.

��

��

��

��

��

��

��

��

��

��

��

��

������ ������

����� �����

��� �� ����������� �� ��������

Fig. 1: Four cooperative diversity cases based on transmission insecond time slot.

III. PUNCTURED LDPC CODES FOR COOPERATIVEDIVERSITY

LDPC codes are defined by their parity-check matrix H withG.HT = O where G is called the generator matrix and Ois an all zero matrix. The (wc, wr) regular LDPC codes haveconstant column weight wc and constant row weight wr. In thiswork, we will use regular LDPC codes with column weight3 and row weight 6, denoted as (3,6) regular code. The (3,6)regular code has the best error correction capabilities in theclass of regular LDPC codes. The LDPC codes are puncturedto generate two weaker codewords. In the cooperative diversitycontext, the codeword N is generated by an overall coderate k/n parity-check matrix and punctured periodically togenerate two weaker codewords N1 and N2. Both N1 andN2 can be decoded alone as a complete codeword to recoverthe information. At the decoder, erasures are inserted at thepunctured locations. At the destination, a three-step decodingis applied after receiving the transmission of both time slots.In the fist step, the information bits are decoded from N1

and checked by CRC for correctness of information. If theinformation received is not error free, then N2 is decoded andchecked for information integrity. If both decodings of N1

and N2 fail to recover the information, then N1 and N2 areconcatenated and decoded jointly. If the packet is error free,then it is accepted by the receiver otherwise an error is reportedto upper layers. The concatenation is similar to interleavingthe two codewords together. This interleaving effect increasesthe diversity gain by reducing the correlation between theconsecutive bits caused by the slow fading channel.

IV. EXTENDED LDPC CODES FOR COOPERATIVEDIVERSITY

Rate-compatible design is required to generate codewords ofdifferent lengths. The design of [13] is capable of embedding

Proceedings of the 2012 IEEE ICCS

335

higher rate codewords in lower rate codewords. The extendedLDPC codes keeps the integrity of information bits intact. Weexploit this design to be used in coded cooperative diversity.The extended parity-check matrix is defined according to thefollowing definitions of matrices

H2 =

[H1 OA B

]m′×n′

(1)

where m′ = n′ − k and H1 is the (3,6) regular parity-checkmatrix of dimensions (n− k)× n for the mother code of rateR = k/n. To extend the code rate to R′ = k/n′, the extraparity bits in the extended codewords will be ebits = n′ − nwhere n′ is the size of extended codeword for rate R′. The Omatrix is an all zero matrix of size k × ebits or k × (n′ − n).The A matrix is a very sparse matrix of size (n′ − n) × nwith at least one 1 in each row. The B matrix has dimensions(n′ − n)× (n′ − n) with column weight 3.

The parity-check matrix H1 is put in its systematic form byGauss-Jordan elimination over GF (2) and is represented as:

H1 =[PT

1 In−k

]m×n

(2)

where PT is the transpose of P. Similarly, the systematic formof H2 is given by

H2 =

[PT

1

PT2

In′−k

]m′×n′

(3)

The O matrix ensures that the higher rate codewords areembedded in extended lower rate codewords by keeping theintegrity of PT

1 . Therefore, the generator matrix for H1 andH2 becomes

G1 =[Ik P1

]k×n

(4)

and

G2 =[Ik P1 P2

]k×n′ (5)

respectively, where P1 has dimensions k × (n − k) and P2

has dimensions k × (n′ − n).We exploit this design of extended LDPC codes to the

cooperative diversity framework and modify the extendedLDPC codes to achieve decoding in three steps at the receiver.The codeword N1 is generated by using the generator matrixobtained from H1. Using (4), N1 takes the following form

N1 = [i p1]1×n (6)

where i is the information part and p is the parity part in thecodeword N1. The second codeword N ′

2 is generated by thegenerator matrix G2 mentioned in (5), in the following form

N ′2 = [i p1 p2]1×n′ (7)

0 2 4 6 8 10 12 14 16 18 2010−6

10−5

10−4

10−3

10−2

10−1

Average Received SNR at Base Station (both users equal) (dB)

BER

No cooperationPerfect interuser channel20dB interuser channel10dB interuser channel0dB interuser channel

Fig. 2: Bit-error-rate for cooperative diversity with punctured LDPCcodes, information bits = 512, 50% cooperation, both users equalSNR, very slow fading (block fading) channel.

where p1 is the same parity part as in N1 and p2 is the extendedparity part. This N ′

2 is modified to generate a codeword oflength n. The second codeword is transmitted in the followingformat

N2 = [i p2]1×n (8)

by eliminating the parity part p1. At the receiver, in thefirst step N1 is decoded using H1. If the codeword is notsuccessfully recovered in the first step, then the codeword isdecoded using H2 matrix with erasures inserted at p1 of N ′

2.If the decoding fails in the first two steps, then the codeword isconcatenated with i, p1 and p2 and jointly decoded using H2

to recover the codeword N . Whenever, we have the repetitionof i, the information part, we combine them optimally usingMRC.

V. SIMULATION RESULTS AND DISCUSSION

The decoding algorithm used in all simulations is thesum-product algorithm (SPA) [6] which has near optimalperformance. The maximum iterations for SPA decoder arekept at 100. All of these simulation results have been plottedas BER versus the channel SNR. The plots with BER versusinformation bit SNR will be identical with a shift of 10logRdB. We assume very slow fading (block fading) channelin which the channel coefficient remains constant for thetransmission of two time slots for each user. We also assumethat the channel coefficients are known at the receiver. TheBER for various values of inter user channel SNR have beenplotted versus channel SNR at the destination. We assume 50%cooperation between the two users which means both N1 andN2 are of equal length and users are cooperating equally foreach other.

Proceedings of the 2012 IEEE ICCS

336

The overall code rate for punctured LDPC codes is 1/4with information block of size 512. Fig. 2 shows the BERof punctured codes with cooperative diversity. The perfectinter user channel shows the diversity gain achieved throughcoded cooperation against the no cooperation case. At a BERof 10−3, a gain of 10 dB has been achieved with puncturedLDPC codes with perfect inter user channel versus poor interuser channel.

The codewords N1 and N2 that are transmitted over thechannel using extended LDPC codes also have same lengthof 1024 with 512 bits of information embedded in bothcodewords. For the extended LDPC codes in Fig. 3, the gainis also 10 dB for the perfect inter user channel as compared toworse inter user channel. However, there is a performance lossof 2 dB because the extended codes are decoded on relativelysmaller H matrix as compared to punctured LDPC codes.The information block size k = 512 is the same for bothpunctured and extended LDPC codes and the codewords N1

and N2 transmitted are also of same length equal to 1024 forfair comparison between the two approaches. The H matrixsize for all three steps of decoding of punctured LDPC codesis 1536× 2048 but the extended codes are decoded on H1 =512×1024 for the first step and H2 = 1024×1536 for the nexttwo steps. Therefore, the extended LDPC coded cooperativediversity has less complexity as compared to punctured LDPCcoded cooperative diversity. There is performance-complexitytradeoff between the punctured and extended LDPC codes inthe cooperative diversity framework.

Fig. 4 shows a comparison of punctured and extendedLDPC codes together for the 10 dB inter user channel. Theextended LDPC codes perform better at lower SNRs thanpunctured LDPC codes in cooperative diversity. This resultrequires some explanation. All codewords in extended LDPCcoded cooperative diversity always carries all the informationbits which are combined by MRC at the receiver providing areceive diversity gain. The punctured LDPC coded cooperativediversity has punctured or weaker codewords with some ofthe information bits punctured as well. These punctured bitscannot be recovered even with decoding on larger H matrix.Therefore, extended LDPC codes performs better at lowerchannel SNRs as compared to punctured LDPC codes incooperative diversity. However, on less noisy channels be-tween the users and destination, the punctured LDPC codesoutperforms extended LDPC codes in cooperative diversity.Extended LDPC codes can be useful when the users are atlarge distances from the destination or the transmit power forthe users is restricted within certain limits.

We assume an overall code rate of k/n for punctured codesas reference for complexity analysis. The punctured codes aredecoded on H matrix of size (n − k) × n. The approximatecomplexity in encoding for punctured and extended LDPCcodes in cooperative diversity is presented in Table I. Theencoding complexity is reduced by half for codeword N1

as compared to punctured LDPC codes case. The decodingcomplexity using SPA for both the punctured and extendedLDPC codes cases is mentioned in Table II. The products

TABLE I: Comparison of encoding complexity between puncturedand extended LDPC coded cooperative diversity.

AND XOR

Punctured LDPCN1 nk n(k − 1)

N2 nk n(k − 1)

Extended LDPCN1

n2k n

2(k − 1)

N23n4k 3n

4(k − 1)

TABLE II: Comparison of decoding complexity between puncturedand extended LDPC coded cooperative diversity (approximated forone decoder iteration.)

Products Additions

Punctured LDPC

N1 (n− k)(wr − 2) nwc

N2 (n− k)(wr − 2) nwc

N (n− k)(wr − 2) nwc

Extended LDPCN1 (n

2− k)(wr − 2) n

2wc

N2 ( 3n4

− k)(wr − 2) 3n4wc

N ( 3n4

− k)(wr − 2) 3n4wc

0 2 4 6 8 10 12 14 16 18 2010−6

10−5

10−4

10−3

10−2

10−1

Average Received SNR at Base Station (both users equal) (dB)

BER

No cooperationPerfect interuser channel20dB interuser channel10dB interuser channel0dB interuser channel

Fig. 3: Bit-error-rate for cooperative diversity with extended LDPCcodes, information bits k = 512, 50% cooperation, both users equalSNR, very slow fading (block fading) channel.

and additions are found approximately for one iteration. Thedecoding is done in 3 steps, therefore, all three cases for N1,N2 and N are shown. The decoding complexity for N2 andcombined codeword N is the same because N2 is decodedwith erasure insertion and both are decoded on H matrix ofsame size.

VI. CONCLUSION

We presented a novel approach for extended LDPC codedcooperative diversity framework. We compared the perfor-mance of punctured LDPC codes and extended LDPC codesin cooperative diversity. The two systems performs differentlyat lower and higher channel SNRs. The simulation results have

Proceedings of the 2012 IEEE ICCS

337

0 2 4 6 8 10 12 14 16 18 2010−4

10−3

10−2

10−1

Average Received SNR at Base Station (both users equal) (dB)

BER

Punctured LDPC codesExtended LDPC codes

Fig. 4: Bit-error-rate for cooperative diversity with punctured andextended LDPC codes, information bits k = 512, 50% cooperation,both users equal SNR, very slow fading (block fading) channel, Interuser channel 10 dB.

shown that there is a performance-complexity tradeoff betweenthe two techniques of LDPC codes for cooperative diversity.

REFERENCES

[1] A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity-part I: system description,” IEEE Transactions on Communications,vol. 51, no. 11, pp. 1927–1938, Nov. 2003.

[2] ——, “User cooperation diversity-part II: implementation aspects andperformance analysis,” IEEE Transactions on Communications, vol. 51,no. 11, pp. 1939–1948, Nov. 2003.

[3] A. Nosratinia, T. E. Hunter, and A. Hedayat, “Cooperativecommunication in wireless networks,” IEEE Communications Magazine,vol. 42, no. 10, pp. 74–80, Oct. 2004.

[4] T. E. Hunter and A. Nosratinia, “Diversity through coded cooperation,”IEEE Transactions on Wireless Communications, vol. 5, no. 2, pp.283–289, Feb. 2006.

[5] J. Hagenauer, “Rate-compatible punctured convolutional codes (RCPCcodes) and their applications,” IEEE Transactions on Communications,vol. 36, no. 4, pp. 389–400, Apr. 1988.

[6] R. Gallager, “Low-density parity-check codes,” IEEE Transactions onInformation Theory, vol. 8, no. 1, pp. 21–28, Jan. 1962.

[7] R. Tanner, “A recursive approach to low complexity codes,” IEEETransactions on Information Theory, vol. 27, no. 5, pp. 533–547, Sep.1981.

[8] D. J. C. MacKay, “Good error-correcting codes based on very sparsematrices,” IEEE Transactions on Information Theory, vol. 45, no. 2,pp. 399–431, Mar. 1999.

[9] D. Klinc and S. McLaughlin, “Rate-compatible punctured low-densityparity-check codes with short block lengths,” IEEE Transactions onInformation Theory, vol. 52, no. 2, pp. 728–738, Feb. 2006.

[10] S. McLaughlin, “Puncturing for finite length low-density parity-checkcodes,” in International Symposium onInformation Theory, 2004. ISIT2004. Proceedings., 2004, pp. 152–152.

[11] J. Ha, J. Kim, and S. McLaughlin, “Rate-Compatible Puncturing ofLow-Density Parity-Check Codes,” IEEE Transactions on InformationTheory, vol. 50, no. 11, pp. 2824–2836, Nov. 2004.

[12] D. Bi and L. Perez, “Rate-compatible low-density parity-check codeswith rate-compatible degree profiles,” Electronics Letters, vol. 42, no. 1,p. 41, 2006.

[13] J. Li and K. Narayanan, “Rate-compatible low density parity check codesfor capacity-approaching ARQ scheme in packet data communications,”in Int. Conf. on Comm., Internet, and Info. Tech.(CIIT), pp. 201–206.

[14] M. Yazdani and A. Banihashemi, “On construction of rate-compatiblelow-density parity-check codes,” Communications Letters, IEEE, vol. 8,no. 3, pp. 159–161, Mar. 2004.

[15] C. Li, G. Yue, X. Wang, and M. Khojastepour, “LDPC Code Designfor Half-Duplex Cooperative Relay,” IEEE Transactions on WirelessCommunications, vol. 7, no. 11, pp. 4558–4567, Nov. 2008.

[16] M. Khojastepour, N. Ahmed, and B. Aazhang, “Code design for therelay channel and factor graph decoding,” in Conference Record of theThirty-Eighth Asilomar Conference on Signals, Systems and Computers,2004., vol. 2, 2004, pp. 2000–2004.

[17] M. A. Khojastepour and M. Madihian, “LDPC-coded cooperative relaysystems: performance analysis and code design,” IEEE Transactions onCommunications, vol. 56, no. 3, pp. 485–496, Mar. 2008.

[18] J. Hu and T. Duman, “Low Density Parity Check Codes over WirelessRelay Channels,” IEEE Transactions on Wireless Communications,vol. 6, no. 9, pp. 3384–3394, Sep. 2007.

[19] H. Ali and M. A. Kousa, “Extended Low-Density Parity-Check Codesfor Cooperative Diversity,” in International Conference on WirelessInformation Networks and Systems, Rome, July 2012.

Proceedings of the 2012 IEEE ICCS

338