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IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 31, NO. 8, AUGUST 2013
LDPC Codes for Soft Decode-and-Forward in Half-Duplex Relay Channels Marwan H. Azmi, Student Member, IEEE, Jun Li, Member, IEEE, Jinhong Yuan, Senior Member, IEEE, and Robert Malaney, Member, IEEE
Abstract—We investigate the use of rate-compatible lowdensity parity-check (RC-LDPC) codes as part of a soft decodeand-forward (SDF) protocol over the half-duplex relay channel. We propose a new methodology to design the degree distribution of the RC-LDPC codes with a lower triangular parity-check matrix, enabling the additional parity bits to be linearly and systematically encoded at the relay. Our proposed methodology introduces the concept of a K-layer doping matrix to represent the structure of a lower triangular parity-check matrix. As a result of our methodology, the asymptotic performance of RC-LDPC codes can be analyzed and predicted using the multi-edge-type density evolution. Then, we derive the soft-re-encoding of the additional parity symbols at the relay using our designed RCLDPC codes. Moreover, we propose a novel method, which we refer to as soft fading, to compute the log-likelihood ratio (LLR) of the received signal at the destination for the SDF protocol. We demonstrate that our proposed soft fading method outperforms the best known method in the literature by up to 0.7 dB in terms of BER performance. Finally, we derive a new bound for the power multiplication factor at the relay, which limits the amount of soft-errors forwarded by the relay to the destination. The BER performance of our new RC-LDPC codes improves significantly once the power multiplication factor at the relay satisfies this bound. Index Terms—Rate-compatible low-density parity-check codes, multi-edge-type ensemble, K-layer doping matrix, soft decodeand-forward, soft fading, soft noise, relay channel.
I. I NTRODUCTION OOPERATIVE communication [1] for wireless relay networks promises improved transmit diversity and increased spectral efficiency. In the context of integrating cooperation with coding, the use of a decode-and-forward (DF) strategy instead of the simple amplify-and-forward (AF) strategy in a three-node relay channel has proven to boost system performance and provide additional coding gain [2], [3], especially when the source-to-relay link is strong. In [4], a special DF strategy called coded cooperation, where the relay decodes the message received from the source and then
C
Manuscript received 14 August 2011; revised 1 March 2012. Part of the material in this paper was presented at the 2011 IEEE International Symposium on Information Theory (ISIT), Saint Petersburg, Russia. The research leading to these results has been partially supported by the Australian Research Council (ARC) under Discovery Projects DP110104995 and DP120102607. M. H. Azmi is with the Faculty of Electrical Engineering, Universiti Teknologi Malaysia. He is currently on a study leave and working towards his Ph.D. at the School of Electrical Engineering and Telecommunications, University of New South Wales, Australia (e-mail:
[email protected]). J. Li, J. Yuan, and R. Malaney are with the School of Electrical Engineering and Telecommunications, University of New South Wales, Sydney, NSW 2052, Australia (e-mail: jun.li, j.yuan,
[email protected]). Digital Object Identifier 10.1109/JSAC.2013.130805
transmits additional parity bits of the message to the destination, has been shown to achieve better system performance (i.e. smaller outage probability) compared to the repetitionbased DF strategy. Subsequent to [4], the use of capacityapproaching codes like the distributed turbo codes [5], [6] and the LDPC codes [7]–[9] were shown to achieve the theoretical limit performance of the DF strategy in the relay channels. While the application of coding using a conventional DF strategy requires an error free source-to-relay channel, this requirement cannot always be guaranteed in practice. In order to take into account this factor, a more advanced relay protocol known as the soft DF (SDF) scheme was independently proposed, using the recursive systematic convolutional (RSC) codes [10] and the distributed turbo codes (DTC) [11], to mitigate the effect of error propagation due to erroneous decoding at the relay. In the context of DTC, the work in [11] proposed soft-re-encoding for the parity symbols when only the posteriori probabilities of the information symbols are known at the relay. Furthermore, they modeled the soft-errors introduced by performing the soft-re-encoding at the relay as Gaussian noise, which was then added at the destination when computing the received log-likehood ratio (LLR) for the BCJR decoder. However, the work in [12] subsequently showed that the LLR distribution at the destination for the turbo code used in [11] is not purely Gaussian. Even though the Gaussian noise assumption proposed in [11] is not completely accurate, this assumption has been repeatedly employed in a more complex relay system such as the MIMO relay system [13] and the cognitive radio relay system [14]. This is due to the fact that modeling the soft-errors as Gaussian noise is currently the best known method available in the literature. In another class of work [15], distributed turbo product codes (DTPC) with soft information relaying (SIR) over cooperative networks were proposed. The main purpose of this work is to investigate the use of SIR under the SDF scheme with rate-compatible (RC) LDPC codes in the half-duplex relay channels. More specifically, we address two main challenges that arise when adopting the RCLDPC codes under the SDF scheme. The primary advantage of RC-LDPC codes is that their degree distributions can be explicitly optimized to closely achieve specific achievable rates (theoretical limit) of the DF strategy in the half-duplex Gaussian relay channels [9]. However, the first challenge of performing SIR using RC-LDPC codes is the soft-re-encoding of the additional parity bits at the relay. This is due to the fact that LDPC codes in general have high encoding complexity.
c 2013 IEEE 0733-8716/13/$31.00
AZMI et al.: LDPC CODES FOR SOFT DECODE-AND-FORWARD IN HALF-DUPLEX RELAY CHANNELS
While the work in [9] only focused on the optimization of the RC-LDPC code structure, the encoding of RC-LDPC codewords (especially for the additional parity bits) at the relay may not be feasible unless special structure is introduced to the Tanner graph of RC-LDPC codes, e.g. accumulator [17] or lower triangular parity-check matrix [18]. This is because the RC-LDPC codes form a nested sequence of code bits where the parity bits of higher rate codes for the sourceto-relay channel are embedded in those of lower rate codes for the source-to-destination. The second and more general challenge that arises when employing any code (not only RCLDPC codes) under the SDF scheme is how to precisely compute the LLR of the received signal at the destination. This paper addresses these challenges and proposes four main contributions. We start by constructing RC-LDPC codes with a lower triangular parity-check matrix, enabling the additional parity bits to be linearly and systematically encoded at the relay. Even though the structure of the lower triangular parity-check matrix is straightforward to accomplish simple encoder, the optimization of its degree distribution is not trivial. This is because the optimized RC-LDPC codes in [9] and the capacityachieving irregular LDPC codes [19] in general disallow the parity-check matrix to have this lower triangular structure. To overcome this problem, our first contribution is a proposal of a new methodology to design the degree distribution for the RCLDPC codes with a lower triangular parity-check matrix based on the framework of multi-edge-type (MET) ensemble [20]. The MET framework has been shown to be a very powerful tool in designing the degree distribution for different LDPC codes in cooperative channels, such as the bilayer LDPC codes [21], [22] and the root-check RC-LDPC codes [23]. Our methodology introduces the concept of a K-layer doping matrix to represent the structure of the lower triangular paritycheck matrix. By representing different layers of the dopingmatrix with different edge-types, the asymptotic performance of the RC-LDPC codes with a lower triangular parity-check matrix can be analyzed and predicted using the MET density evolution [20]. Note that our proposed design methodology can also be used for designing any LDPC code with a lower triangular parity-check matrix. In [24], an idea similar to K-layer doping matrix called the control doping has been introduced to explain the concept of coding gain evolution in root-check LDPC codes. The key difference between our work and the work in [24] is that we use the K-layer doping matrix as a means of allowing systematic and linear encoding complexity at the relay. Furthermore, we propose a method to specifically optimize the degree distribution of the K-layer doping matrix in order to maximize the coding gain of the RC-LDPC codes. The second contribution of this work is we derive the soft-re-encoding for the additional parity symbols at the relay using our designed RC-LDPC codes. Then, our third contribution is a novel method to compute the LLR of the received signal at the destination under the SDF scheme, where the soft-errors introduced at the relay are described via fading coefficients. We refer to our new method as soft fading. We demonstrate that our soft fading method outperforms the best known method in [11] by up to 0.7 dB in terms of BER performance. Finally, our fourth contribution is that we derive
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Broadcast Mode
Multiple-access Mode
R
R
W2
V1 S
Y1
X1
D
S
X2
Y2
D
1-t
t Broadcast Time Fraction
Multiple-access Time Fraction
Fig. 1. The half-duplex relay channels.
a new bound for the power multiplication factor at the relay, which controls the amount of soft-errors forwarded by the relay to the destination. The BER performance of our new RC-LDPC codes are shown to improve tremendously when the power multiplication factor at the relay satisfies the derived bound. II. S YSTEM M ODEL AND LDPC C ODING S CHEME A. Half-Duplex Relay Model The half-duplex relay channel is illustrated in Fig. 1, where one source (S) sends its messages to the destination (D) aided by one half-duplex relay (R). Given a time window L, the total of N = N1 + N2 symbols are received at D. In a fraction of time tL (the broadcast (BC) mode), S broadcasts N1 symbols to both R and D. In the remaining fraction of time t L = (1 − t)L (the multiple-access (MAC) mode), both S and R transmit N2 symbols to D simultaneously. We denote the source transmitted signal, relay received signal, relay transmitted signal, and destination received signal by using X, V , W and Y (see Fig. 1), respectively. To distinguish between the transmission modes, we denote the BC mode using subscript 1 and the MAC mode using subscript 2. With the above notations, the half-duplex relay channel is defined by V1 Y1
= hSR X1 + MR1 = hSD X1 + MD1
(1) (2)
Y2
= hSD X2 + hRD W2 + MD2 ,
(3)
where hij is the channel realization between node i and node j, MD1 and MD2 are the noise realizations at D in BC and MAC modes, respectively, and MR1 is the noise realization at R in BC mode. All MR1 , MD1 and MD2 noises are Gaussian with zero mean and unit variance (σ 2 = 1), i.e. N (0, 1). In this work, S and R transmit with equal power P and BPSK modulation is employed. The nodes S, R and D are assumed to lie on a straight line1 , where we normalized the distance between S and D to unity, and d denotes the position of R relative to S with 0 ≤ d ≤ 1. With this setup, the channel gains are hSD = 1, hSR = √1dα and hRD = √ 1 α , where (1−d)
1 Even though we assume all the nodes lie on a straight line, the extension to any topology of relay channels is straightforward.
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α is the path-loss exponent. In addition, perfect global channel knowledge is assumed at all received nodes. B. LDPC Coding for Half-Duplex Relay Systems From the achievable rate point of view, the relay coding strategy is encompassed by two fundamental extremes. In one extreme, R and S simultaneously send completely independent information, which interfere with each other. In the other extreme, R and S simultaneously send identical information (i.e. X2 = W2 ), which forms a coherent transmission. It is this latter strategy that corresponds to the relay coding strategy adopted in this paper. In this strategy, S transmits a binary LDPC code w 1 with length N1 and dimension K1 in the BC mode, i.e. with rate R1 = K1 /N1 . The code w1 is defined by a (N1 − K1 ) × N1 parity-check matrix H1 . Assuming perfect decoding at R2 , both S and R transmit another codeword segment w 2 with length N2 that only contains additional parity-check bits in the MAC mode. Thus, D will receive the full LDPC code w = [w 1 w 2 ] with rate Rc = K1 /N . This full codeword is defined by a (N − K1 ) × N parity-check matrix H, which includes H1 , given by [9], [30] (w 1 ) (w 2 ) ⎤ H1 0 H =⎣ − − − − −− ⎦ H2 ⎡
(4)
We call the code a rate-compatible LDPC code since the code has to achieve rate R1 = K1 /N1 and rate Rc = K1 /N for the decoding of the LDPC code w1 and the full LDPC code w = [w 1 w 2 ] at R and D, respectively. The transmission model using the proposed RC-LDPC codes in the half-duplex relay channels is √ V1 = P · hSR · w 1 + M R1 (5) √ Y1 = P · hSD · w1 + M D1 (6) √ P · (hSD + hRD ) · w 2 + M D2 , (7) Y2 = where the codeword vectors transmitted in the BC and MAC modes can be expressed as w 1 = [w1,1 , w1,2 , . . . , w1,N1 ]T and w2 = [w2,1 , w2,2 , . . . , w2,N2 ]T , respectively. Similarly, the received vectors can be expressed as V 1 = [v1,1 , v1,2 , . . . , v1,N1 ]T , Y 1 = [y1,1 , y1,2 , . . . , y1,N1 ]T and Y 2 = [y2,1 , y2,2 , . . . , y2,N2 ]T . The parity-check matrix H in (4) represents the decoder structure for the full codeword w = [w 1 w 2 ] of the RC-LDPC codes at D, which can be optimized so that the RC-LDPC codes achieve the theoretical limit of the DF scheme in the half-duplex relay channel [16]. Here, the encoding of the RCLDPC codes would be prohibitively complex unless special structural properties are imposed on H since the sub-matrix H1 by itself must be in the ensemble of LDPC codes of rate R1 = K1 /N1 and the position of K1 information bits from S must only be in the LDPC code w1 . So, we cannot simply perform the Gaussian elimination to the parity-check matrix H in order to obtain the generator matrix G [25]. This is because there is a possibility that a systematic generator G can only be generated if a column permutation is performed 2 Code
w1 is designed to achieve the capacity of the S-R channel.
between the two columns of the sub-matrix H1 and the zero padded matrix, as a consequence, it changes the structure of parity-check matrix H in (4). We introduce a special structure to H so that the encoding of the full codeword w = [w1 w2 ] is possible. We set H2 to be a lower triangular parity-check matrix as depicted in Fig. 2(a). By having this lower triangular structure, S firstly encodes the LDPC code w 1 using a systematic form of generator matrix G1 . The generator matrix G1 can be obtained by performing the Gaussian elimination to the parity-check matrix H1 [25]. Any column permutation operation when performing the Gaussian elimination to H1 must be recorded so that the same column permutation can be performed to the sub-matrix H2 . This ensures the degree distribution of the original parity-check matrix H remains unchanged. Note that restricting the Gaussian elimination to only H1 prevents the possibility of performing column permutation between the two columns of the sub-matrix H1 and the zero padded matrix. Secondly, once the LDPC code w 1 is encoded, the additional parity bits w2 can be encoded using the sub-matrix H2 . This linear and systematic encoding can be performed using backsubstitution as H2 is a lower triangular parity-check matrix. The encoding and decoding of the proposed RC-LDPC codes for the half-duplex relay channel are summarized in Fig. 2(b). Note that the parity bits w 2 can also be encoded using the generator matrix G2 , where G2 · H2T = 0, if the right most N2 columns of H2 are full rank. This full rank condition is necessary in order to avoid the column permutation operation when generating G2 from H2 using the Gaussian elimination. Here, the proposed H2 with a lower triangular parity-check matrix satisfies this full rank condition. III. C ODE D ESIGN Now, we propose the methodology to optimize the degree distribution of the RC-LDPC codes in Fig. 2(a). Note that the predefined structure of the sub-matrix H1 and the lower triangular matrix of H2 disallows the parity-check matrix H to be represented using the standard degree distribution polynomials [19]. The structure of RC-LDPC codes has two special features. The first special feature is that there is a statistical distinction between the edges in the parity-check matrix H. This is because the sub-matrix H1 by itself must be in the ensemble of LDPC codes of rate R1 = K1 /N1 , optimized to operate for the S-R channel, and the connections of the edges in the sub-matrix H2 must also follow the predefined structure of the lower triangular matrix. The second special feature is that different segments of the received symbols at D are transmitted through different channels and experience different SNRs. Using (6) and (7), the received SNRs for the decoding of w 1 and w 2 at D can be written, respectively, as γ1 γ2
= =
P · (hSD )2 P · (hSD + hRD )2 ,
(8) (9)
since MD1 and MD2 noises are Gaussian with zero mean and unit variance (σ 2 = 1). The two special features of RC-LDPC codes can be represented using the MET-LDPC ensemble. The MET-LDPC ensemble can be specified through the following
AZMI et al.: LDPC CODES FOR SOFT DECODE-AND-FORWARD IN HALF-DUPLEX RELAY CHANNELS
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EϮ
Eϭ
,ϭ
Ϭ
, с
EϭͲ0
μ[i,0,0,0] xi1 +
dc,2
μ[0,j,0,1] xj2 x4
j>0
dc,3
+
μ[0,0,k,1] xk3 x4 .
(12)
k>0
where r1 and r2 denote the N1 and N2 received variables associated with w 1 and w 2 , respectively. The EDV d = [i, j, k, l] represents four types of edge degree, with i, j, k and l denoting the variable or check node’s degrees (sockets) of type 1, type 2, type 3 and type 4, respectively. Note that the diagonal element of the identity matrix in the 2-layer doping matrix is represented using type 4 variables x4 . Here, the edge degree of type 4 is only one, i.e. l = 1, which corresponds to one nonzero element in each column and row of the identity matrix (refer to (11) and (12)). Now, the design problem is to find the optimum ensemble of MET-LDPC codes given in (11) and (12) such that the submatrix H1 is optimized to operate for the S-R channel at rate R1 = K1 /N1 , while the parity-check matrix H of RC-LDPC codes is optimized at rate Rc = K1 /N for the decoding of w = [w 1 w 2 ] at D. This optimization is executed in two steps4 . In the first step, the code design begins with optimizing the ensemble of H1 at rate R1 for the S-R channel. H1 can be designed using a standard irregular LDPC code’s optimization method [26] as it is only represented via one edge type (edge degree of type 1). The node perspective degree distribution of a standard irregular LDPC code consists the polynomials of dv,1 ˚ i−1 dv,1 ˚ λ(x) = x and ˚ ρ (x) = ρi xi−1 , where λ i i>0 i>0 ˚ 3 It
is straightforward to generalize the multinomials of RC-LDPC codes when a K-layer doping matrix of H2 is used. 4 We do not jointly optimize H and H because this optimization is 1 2 more complex. In addition, joint optimization puts additional constraints on optimizing H1 for the S-R link, which may lead to a sub-optimal H1 when compared to optimizing H1 alone for the S-R link. Note that the decoding of w at D depends on the successful decoding of w1 at R. Hence, in our design, we first optimize H1 for the S-R link to minimize the error propagation effect from the relay, and then based on a fixed H1 , we optimize H2 to design H.
˚ the coefficient ρiare non-negative real numbers dv,1of˚ λi and ˚ dc,1 ˚ ρi = 1, respectively. Since satisfying i>0 λi = 1 and i>0 N1 variable nodes represent a subcode LDPC code w 1 inside H, the optimized ˚ λ(x) and ˚ ρ(x) polynomials of H1 can be represented using the MET-LDPC code’s multinomials in (11) and (12) as υ(r, x) =
r1
dv,1
υ[0,1,0],[i,0,0,0]xi1
(13)
i>0
μ(x) =
dc,1
μ[i,0,0,0] xi1 ,
(14)
i>0
dv,1 = L1 and respectively, with i>0 υ[0,1,0],[i,0,0,0] dc,1 μ = L · (1 − R ). Each non-negative real 1 1 i>0 [i,0,0,0] number of υ[0,1,0],[i,0,0,0] and μ[i,0,0,0] can be computed λi and μ[i,0] = L1 · ˚ ρi · (1 − R1 ), using υ[0,1,0],[i,0] = L1 · ˚ respectively, where L1 = N1 /N is the percentage of N1 variables in H. The second step is to design H so that it achieves the rate Rc . The design of H is performed by optimizing the ensemble of sub-matrix H2,1 and H2,2 of the 2-layer doping matrix simultaneously5, while fixing the optimized ensemble of sub-matrix H1 obtained from the previous optimization and the identity matrices in the 2-layer doping matrix, i.e. the search for the optimum degree distribution of H in (11) and (12) is only performed within the edge degree of type 2 and type 3. The optimization setting for H is different from the standard optimization of LDPC codes in point-to-point channels, as N1 and N2 variables are transmitted and received through different channels and experience two different SNRs. Here, we define σN1 and σN2 as the noise standard deviation (thresholds) for the N1 and N2 variables, respectively. So, the 2 corresponding received SNRs for w 1 and w2 are γ1 = 1/σN 1 2 and γ2 = 1/σN2 , respectively. With this definition, the average noise threshold for the overall N1 + N2 codeword is given by
1
σavg = , (15) 1 L1 · σ2 + L2 · σ21 N1
N2
where L2 = N2 /N is the percentage of N2 variables in H. We find the optimum parity-check matrix H at Rc = K/N by maximizing (15) under the MET density evolution [20], subject to σN1 ≤ σBClim and σN2 ≤ σMAClim . Here, σBClim and σMAClim denote the theoretical limits for the BC and MAC Gaussian channels at D, respectively. Note that the optimization is only performed to the K sub-matrices in the K-layer doping matrix of H2 , using the differential evolution [27]. IV. S OFT D ECODE AND F ORWARD Now, we will explain how the SIR under the SDF scheme can be performed, when the relay fails to decode LDPC code 1 = w1 (refer to Fig. 2(b)), using the proposed w 1 , i.e. w RC-LDPC code structure of the previous section. 5 For general setup of K-layer doping matrix, there are K sub-matrices need to be optimized simultaneously.
AZMI et al.: LDPC CODES FOR SOFT DECODE-AND-FORWARD IN HALF-DUPLEX RELAY CHANNELS
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A. Calculation of Soft Code Bits at the Relay
B. Calculation of LLR at the Destination
The first step required for SIR is the calculation of the LLR L(w 1 /V 1 ) of w1 from the received sequence V 1 at the relay. Since w 1 is an LDPC codeword, this computation can be easily performed using the Belief Propagation (BP) decoder of H1 . We denote Lp (w 1 /V 1 ) as the LLR of w1 after p iterations of the BP decoder. From the computed LLR values, the soft bits of w 1 are given by
In order to perform the BP iterative decoding using the H matrix, D has to compute the input LLRs for the codeword vectors w1 and w2 based on the received signals Y 1 and Y 2 , respectively. Here, we start by discussing the LLR computation for the DF scheme, i.e. for the case where R can correctly decode w1 . Under the DF scheme, the received LLRs L(y1,i /w1,i ) and L(y2,i /w2,i ) at D for each bit w1,i and w2,i , respectively, can be computed using √ P · hSD L(y1,i /w1,i ) = 2 · · y1,i (21) 2 √ σ P · (hSD + hRD ) · y2,i . (22) L(y2,i /w2,i ) = 2 · σ2 Note that L(y2,i /w2,i ) is obtained because S and R coherently transmit the same value of the parity-check bit w2,i . For the SDF scheme, R will transmit the soft bits β w
2 instead of the hard decision bit of w 2 if it fails to correctly decode w1 in the BC mode. Under the SIR of the SDF scheme, the received LLR L(y1,i /w1,i ) for w1,i remains the same as in (21), since there are no changes in the transmission of LDPC code w 1 in the BC mode. The only difference is the computation of L(y2,i /w2,i ) because S and R are transmitting two different vectors w 2 and β w
2 , respectively, in the MAC mode. Hence, the accurate and precise LLR computation of L(y2,i /w2,i ) requires the knowledge of the error probability P r(β w
2,i = w2,i ). In [11], the knowledge of P r(β w
2,i = w2,i ) was modeled as w
2,i = w2,i (1 − ns ), (23)
w
1 = tanh(Lp (w 1 /V 1 )/2).
(16)
The second step required for SIR is the soft-re-encoding, or calculating the soft bit of parity-check w2 from the reliability of soft bits w
1 . The soft parity-check bits w2 can be encoded linearly using back-substitution since H2 has a lower triangular structure. The value of the first soft bit w
2,1 in vector w
2 can be calculated using w
2,1 = w
1,k , (17) k∈B1
while the remaining soft bits in vector w
2 are given by w
2,j = w
1,k w
2,k for j =2, . . . , N2 , (18) k∈B1
k∈B2 ,k=j
where B1 (B2 ) is the set of message nodes incident to check node j that carries the soft information from vector w
1 (w
2 ), i.e. the message nodes that involve in the parity-check equation j. After all soft bits in w
2 have been computed, the signal transmitted from the relay can be written as √ w 2 = P · β · w
2, (19) where the power multiplication factor per transmission β can be calculated by satisfying the transmitted power constraint at the relay, given by 1 β= . (20) 1 N2
2,i |2 i=1 |w N2 Note that β > 1 when the transmitted power constraint at the relay is satisfied under the SDF scheme, since the value of soft parity-check bits at the relay is always 0 < |w
2,i |2 < 1. Remark 1: For the case where relay can successfully decode w 1 using the BP decoder of H1 , the signs of L(w1 /V 1 ) satisfy all parity-check equations spanned by H1 . When this happens, the soft bit value of w
1,i is either 1 or −1, i.e. the hard decision for a 0 or 1 binary bit w ˘ 1,i . As a consequence, the encoding of a binary bit w ˘ 2,i using back-substitution can be written as [18] N i−1 1 w ˘2,i = ⊕(H2 (i, j)w ˘1,j ) ⊕ ⊕(H2 (i, j+N1 )w˘2,j ) j=1
j=1
where the Σ⊕ and ⊕ symbols denote the summation and addition operators in the binary field, respectively. The power multiplication factor for the hard decision encoding is β = 1 because w
2,i ∈ {−1, +1}. Note that this is the standard DF scheme.
where ns ≥ 0 is the equivalent noise introduced at R. Note that the soft bit w
2,i can be seen as the received signal of w2,i through a special noisy channel of −w2,i · ns . Here, ns corresponds to the magnitude of the equivalent noise variable, while −w2,i corresponds to the sign of the equivalent noise variable, i.e. +1 or -1. This equivalent noise can then be added to the noise realization MD2 when computing the received LLR at D. In this paper, we refer to ns as soft noise, where ns is assumed to be Gaussian with the mean value of μns = E{1−w2,i w
2,i } and variance σn2 s = E{(1−w2,i w
2,i −μns )2 } [11]. Now, we will briefly explain how the statistics6 of ns can be incorporated in the computation of L(y2,i /w2,i ) when soft bits w
2 are transmitted by R. Using the soft noise method, the signal transmitted from R in (19) can be written as √ w 2,i = P · βsn · w2,i (1 − ns ), (24) where the power multiplication factor βsn can be computed by using7 1 βsn = . (25) (1 − μns )2 + σn2 s 6 All statistical values of the soft noise (i.e. μ 2 ns and σns ) are computed offline numerically, and then stored at D for the computation of received LLR during real time transmission. Hence, R does not need to provide any extra information to D. 7β sn is the average power multiplication factor calculated on the assump2 . Note that in tion that ns is Gaussian with mean μns and variance σn s general βsn is not equal to β in (20), where β is the power multiplication factor for every codeword w 2 during real time transmission.
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At D, the received signal corresponding to the transmissions from S and R in the MAC mode is given by √ y2,i = P · hE · w2,i + nE , (26) where hE = hSD + hRD βsn b(1 − μns ) and nE is the equivalent Gaussian noise at D, with zero mean and vari2 = σ 2 + P |hRD βsn |2 σn2 s . The approximated LLR ance σE Lsn (y2,i /w2,i ) can then be computed using √ P · hE y2,i . (27) Lsn (y2,i /w2,i ) = 2 2 σE Here, we propose a new method called the soft fading to incorporate the knowledge of P r(β w
2,i = w2,i ) when computing the received LLR at D. Our proposed method represents the relationship between the soft bit w
2,i and the correct bit w2,i in the following manner w
2,i = hs · w2,i ,
(28)
where hs can be seen as the fast fading coefficient, i.e. the soft bit w
2,i is the received signal of w2,i through a channel with multiplicative fast fading of hs . Here, we call hs the soft fading variable. When comparing (23) and (28), we can see that hs = 1 − ns . (29) The key difference between our soft fading method and the method of [11] lies on how the soft information w
2,i is used for the detection of the received signal at D, in particular, the computation of the LLR values of the received signal at D in MAC mode. Using the soft fading method, the coherent received signal at D in MAC mode can be written as √ P · (hSD · w2,i + hRD · βsf · w
2,i ) + MD2 y2,i = √ = P · (hSD + hRD · βsf · hs ) · w2,i + MD2 = hF · w2,i + MD2 ,
(30)
where hF is referred to as the coherent fading variable8 , and the power multiplication factor βsf can be computed by using9 1 (31) βsf = 2 . E(hs )
depends on the pdf of the soft fading variable hs , denoted by fH s (hs ). Even though the full knowledge of fH s (hs ) is unknown, there are still some important properties about fH s (hs ) that can be obtained by studying the characterization of the soft bits wˆ2,i . One of the most important properties of hs is given by; Property 1: The soft fading variable hs is a continuous random variable with value between −1 and 1, i.e. −1 ≤ hs ≤ 1. From equation (28), the value of hs depends on the value of the soft bit w
2,i , given by hs = w
2,i /w2,i . There are two extreme cases for the value of the soft bit w
2,i . The first extreme case occurs when the value of soft bit w
2,i is equal to the correct bit w2,i , given by w
2,i = w2,i . For this case, the value of hs is 1. The second extreme case occurs when the value of soft bit w
2,i gives a hard decision error, represented by w
2,i = −w2,i . For this case, the value of hs is −1. Since the range of hs is small (i.e. −1 ≤ hs ≤ 1), we can approximate the continuous random variable hs using ∗ a discrete random variable hs without losing much information. This approximation can be done by quantizing the continuous value of hs into Q + 1 discrete values. Let S = {−1, −1 + Δ, −1 + 2Δ, . . ., −1 + (Q − 1)Δ, +1} be the ∗ set of the sample space for hs , where Δ is the quantization size10 . The probability mass function (pmf) of the discrete ∗ random variable hs can be written as ∗
∗
fH s (hs ) = P rH s (hs = i)
for i ∈ S.
(33)
∗
Now, the challenge is to obtain the pmf of hs given in (33). ∗ Here, we obtain the pmf of hs by generating the histogram ∗ of hs numerically, which can be performed offline before the real time transmission take place11 . Once the pmf (histogram) is generated, the power multiplication factor βsf in (31) can be computed. Finally, the LLR L(y2,i /w2,i ) in (32) can be easily approximated using ∗
log
Σi∈S fY2 |W2 ,H F (y2,i |w2,i = +1, hF )P rH s (hs = i) ∗
Σi∈S fY2 |W2 ,H F (y2,i |w2,i = −1, hF )P rH s (hs = i)
(34) ∗
where the discrete coherent fading variable is given by hF = √ P · (hSD + hRD · βsf · i). The main advantage of using the soft fading method to compute the received LLR values at D is that it removes the required Gaussian assumption of the soft noise method. Since ∗ the soft fading method uses the entire pmf (histogram) of hs , where this method achieves better BER performance as we will see later in Section V. In addition, such knowledge of pdf hs 2 (y2,i − hF · w2,i ) 1 ∗ fY2 |W2 ,H F (y2,i |w2,i , hF ) = √ exp − or pmf hs allows one to understand the characteristic of the 2 2σ 2πσ soft bit w
2,i and the soft errors. For example, three different decision regions can be classified when R forwards a soft bit and fH F (hF ) is the pdf of hF . Note that the only random to D by using the soft fading method. These three decision w
2,i variable in hF is hs (refer to (30)). So, fH F (hF ) solely regions are given by the following three properties; If we know the probability density function (pdf) of hF , the received LLR of L(y2,i /w2,i ) can be written as [28] +∞ −∞ fY |W ,H (y2,i |w2,i = +1, hF )fH F (hF )dhF log +∞ 2 2 F (32) f (y2,i |w2,i = −1, hF )fH F (hF )dhF −∞ Y2 |W2 ,H F
8 If we consider a relay system with no coherent transmission during the √ MAC mode (i.e. only R transmits in the MAC mode), hF = P · hRD · βsf · hs . 9β sf is an averaged power multiplication factor computed using the distribution of hs . Note that in general βsf is not equal to β in (20), where β is the power multiplication factor for every codeword w2 during real time transmission. If N2 → ∞, then β → βsf .
10 Note that the smaller the quantization size Δ, the more accurate the discrete distribution to the original continuous distribution. 11 This offline computation is similar to the case of soft noise modeling. The numerical pmf (histogram) can then be stored at D for the LLR computation during real time transmission. Hence, the transmission complexity of the soft fading model is similar to the soft noise model.
AZMI et al.: LDPC CODES FOR SOFT DECODE-AND-FORWARD IN HALF-DUPLEX RELAY CHANNELS
Property 2: The relay forwards the correct soft information bit w
2,i to the destination if the value of the soft fading variable is positive, i.e. hs > 0. Property 3: The relay forwards the wrong soft information bit w
2,i to the destination if the value of the soft fading variable is negative, i.e. hs < 0. Property 4: The relay does not forward any information to the destination if the value of the soft fading variable is zero, i.e. hs = 0. Note that the value of hs = 0 is achieved when the soft-reencoding in (17) or (18) produces the soft bit w
2,i = 0. This scenario is also similar to the case where R does not transmit any signal to D in the MAC mode, i.e. R remains silent. Now, we will explain how the three decision regions can be used to improve the performance of SIR under the SDF scheme. From Property 2, R forwards a wrong soft information bit w
2,i if hs < 0. Using the transmission model of the soft fading method in (30), the value of hs has a direct impact on hF in the coherent transmission of the MAC mode. Beside the Gaussian noise MD2 at D, the soft bit w
2,i forwarded from R during the coherent transmission in the MAC mode, can also cause an error at D if the value of the soft fading variable hs changes the value of the coherent fading variable hF to negative. Mathematically, this can be written as hF = hSD + hRD · β · hs < 0.
(35)
From the above equation, the only parameter that R can control in order to reduce the impact of error propagation when forwarding the soft bit w
2,i to D is the power multiplication factor per transmission β. The following proposition gives the bound for β in order to avoid the coherent transmission in the MAC mode being in error. Proposition 1: The coherent transmission in the MAC mode will not be in error if R forwards the soft bit w
2,i with a power multiplication factor per transmission β that satisfies β
0 even though R forwards the worst soft error, which can be represented by hs = −1. Mathematically this can be written as hF = hSD − hRD · β > 0.
(37)
The above equation can be satisfied with a power multiplicaSD , which completes the tion factor per transmission of β < hhRD proof. Finally, the new value of β at R according to Proposition 1, which prevents an error occurring during the coherent transmission in the MAC mode is always β < 1. This is
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because the S-D channel gain hSD is always smaller than the R-D channel gain hRD , i.e. the distance between S and D is normalized to 1, while the distance between R and D is 1 − d. V. N UMERICAL R ESULTS We evaluate the performance of the proposed SIR under the SDF scheme using the RC-LDPC codes for the following half-duplex relay channel; d = 0.29; t = 0.5; α = 2; and w 1 and w = [w 1 w 2 ] with rate R1 = 0.9 and Rc = 0.45, respectively12. Using the results in [16], the capacity bound for the above half-duplex relay channel can be achieved with a transmission power of P = −4.9 dB. Note that the transmission power P is equal to the SNR of the S-D channel because the distance between S and D is normalized to unity, and the noise at D is Gaussian with zero mean and unit variance. Before the SDF scheme can be performed for the chosen half-duplex relay channel, we search the optimum multinomials for the RC-LDPC codes that operate at rate R1 = 0.9 and Rc = 0.45. This searching is performed using differential evolution under the MET density evolution algorithm with the maximum degree of the variable nodes in H set to 25. We first design Code 1, shown in Table I, where w 1 and w = [w1 w2 ] perform asymptotically within 0.08556 dB and 0.29036 dB, respectively, from the theoretical limits. Note that Code 1 is designed by assigning the sub-matrix H1 with edge-type 1 and the sub-matrix H2 with edge-type 2, i.e. we are not representing the sub-matrix H2 with K-layer doping matrix. As a result, Code 1 represents the parity-check matrix in (4), i.e. H2 is not a lower triangular matrix. For a similar relay channel setting, we design Code 2, shown in Table I, using a 4-layer doping matrix, i.e. there are six different edge-types in the code’s structure. For a fair comparison, we used the same optimized sub-matrix H1 as in Code 1, which performs asymptotically at 0.08556 dB for the S-R channel at rate R1 = 0.9. After optimizing the four sub-matrices H2,1 , H2,2 , H2,3 and H2,4 (the edge degree of type 2, type 3, type 4 and type 5) in the 4-layer doping matrix, the full LDPC code w = [w 1 w 2 ] of Code 2 performs asymptotically within 0.32783 dB of the theoretical limit. Next, we simulate the finite-length performances for both codes, given in Fig. 5, with block length of N1 + N2 = 20000 bits and with the assumption that the S-R channel is error free, i.e. R can always decode the subcode w 1 . Here, the BER curve for Code 1 is simulated using the all zero codeword13. This is because that it is very difficult to satisfy the optimized degree distribution and the full-rank constraint simultaneously in the construction of H in (4)14 . While both Code 1 and Code 2 have almost similar asymptotic and BER performances at the error 12 The parameters for the half-duplex relay channel in this work represent an example of an RC-LDPC code design at R1 = 0.9 and Rc = 0.45. Note that, other parameters can also be chosen. 13 We assume that all K information bits from S are all zero. This generates all zero codeword of length N1 for w1 and also all zero parity-check bits of length N2 for w 2 . 14 We generate H based on the optimized degree distribution of Code 1 in Table I using the Progressive-Edge-Growth (PEG) algorithm [29]. The PEG algorithm only maximizes the girth of the LDPC codes but it cannot guarantee the full rank condition as explained in Section II-B.
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TABLE I T HE OPTIMIZED MULTINOMIALS FOR H DESIGNED AT (R1 = 0.9 , Rc = 0.45). B OTH CODES HAVE THE SAME OPTIMIZED SUB - MATRIX H1 , BUT C ODE 2 IS DESIGNED WITH A 4- LAYER DOPING MATRIX . CODE 1 υb,d 0.00883 0.05062 0.11447 0.03720 0.03053 0.12645 0.04093 0.03799 0.00455 0.04589 0.00252 0.49826 0.00174
Variable b (RDV) 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1
d (EDV) 2 0 2 1 2 4 3 0 3 15 3 12 6 4 7 5 7 4 15 7 15 6 0 2 0 20
R1 , Rc ∗ ∗ σSR , σavg ∗ ∗ σBC , σM AC gapSR , gapavg
CODE 2 Constraint d μd (EDV) 0.05 44 0 0.041 0 5 0.459 0 6
0.9 , 0.45 0.510738 , 0.9731 1.412 , 0.788 0.08556 dB , 0.29036 dB
Variable b υb,d (RDV) 0.08236 0 1 0 2 0.09157 0 1 0 2 0.02370 0 1 0 3 0.17048 0 1 0 3 0.03612 0 1 0 6 0.00481 0 1 0 6 0.01350 0 1 0 7 0.02904 0 1 0 7 0.02341 0 1 0 15 0.02501 0 1 0 15 0.03066 0 0 1 0 0.18878 0 0 1 0 0.18617 0 0 1 0 0.09440 0 0 1 0 R1 , Rc ∗ ∗ σSR , σavg ∗ ∗ σBC , σM AC gapSR , gapavg
0
10
Code 1 Code 2 The theoretical limit -1
10
Bit Error Rate (BER)
10
-3
10
-4
10
-5
10
-6
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-7
-5
0 0 0 0 1 1 0 0 1 0 1 2 1 0
0 0 0 0 0 0 0 0 0 0 1 1 1 1
μd 0.05 0.02937 0.00129 0.13829 0.05049 0.12326 0.06290 0.00212 0.09229
44 0 0 0 0 0 0 0 0
d (EDV) 0 0 0 6 0 0 7 0 0 0 4 0 0 5 0 0 0 6 0 0 7 0 0 0 0 0 0
0 0 0 0 0 0 0 6 7
0 1 1 1 1 1 1 1 1
0.9 , 0.45 0.510738 , 0.969 1.41, 0.784 0.08556 dB , 0.32783 dB
of a codeword that is spanned by the parity-check matrix H. Therefore, the result that we show here is remarkable because our proposed code design can now be applied to the optimization of a lower triangular parity-check matrix, which can simplifies the encoding structure. Now we evaluate the performance of SIR under the SDF scheme using Code 2. Before the real time transmission takes place, all statistics of the soft noise and soft fading methods are computed numerically. Fig. 6 illustrates the numerical pmf ∗ of soft fading variables hs using Code 2 for different SNRs of the S-R channel. Note that the pmf is purely dependent on the quality of S-R channel. While the quality of the S-R channel ∗ improves, the pmf of hs becomes closer to the case where the S-R channel is error free (i.e. E(h2s ) = 1).
-2
10
Constraint d (EDV) 0 0 1 0 1 4 0 0 0 0 1 2 0 0 0 2 7 5 5 0 0 2 6 4 0 4 0 2 6 0 0 3 2 0 0 1 0 0 0 0 0 0
-4.5
-4
-3.5
-3
-2.5
Transmission Power, P (dB)
Fig. 5. The BER performances of the overall RC-LDPC codes w = [w1 w2 ] (i.e. H) with rate Rc = 0.45 at the destination for Code 1 and Code 2. The dotted line represents the theoretical limit of the half-duplex relay channel for the corresponding design rate.
probability of 10−5 , Code 2 has a simpler encoding structure for the additional parity-check bits w 2 because H2 is a lower triangular matrix. This allows Code 2 to be encoded using the method explained in Section II-B. As a result, the BER curve for Code 2 is simulated with all possible combinations
Fig. 7 depicts the simulation results at D using Code 2. Firstly, we show the performance of two extreme cases of perfect S-R channels and direct transmission. Their BER curves are represented by two distinct waterfall regions. The first waterfall region at low SNR corresponds to the code’s performance under the assumption that the S-R channel is error free (the similar BER curve for Code 2 in Fig. 5). The second waterfall region at higher SNR corresponds to the code’s performance when there is no cooperation from R in the MAC mode, i.e. R is always silent in the MAC mode. The latter case can also be seen as the code’s performance for direct link transmission from S to D. There is an improvement of about 4 dB between these two ideal cases, which shows the benefit of using a relay. Secondly, we evaluate the code’s performance at D when there are transmission errors at R. In the simulations, we fixed the SNR of the S-R channel at 6.25 dB (the Shannon limit is at 5.75 dB) to demonstrate the impact of transmission errors at the relay, and how SIR under the SDF scheme can be applied to mitigate these errors. We start by evaluating a state-of-the-art scheme called Selective DF, where R will only transmit if it can decode w 1 in the BC mode. Otherwise, R will just be silent. For the Selective
AZMI et al.: LDPC CODES FOR SOFT DECODE-AND-FORWARD IN HALF-DUPLEX RELAY CHANNELS
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-2
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-3
10 Bit Error Rate (BER)
Probability
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-8
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-0.8
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Fig. 6. The numerical pmf with 21 discrete values (i.e. Q = 20) of the soft ∗ fading variable hs using the optimized RC-LDPC code of Code 2, which operates at Rc = 0.45 and R1 = 0.9 for different SNR of the S-R channel. The code’s block length is 20000 bits and 1000 samples of block errors are collected at the relay to generate the pmf. Here, we found that choosing Q bigger than 20, i.e. Q = 200, does not improve the BER curves of the SIR using the soft fading method depicted in Figure 7.
DF scheme, the error in the S-R channel causes significant error floor performance at D for the SNR region between the two ideal cases. The BER curve then decreases sharply once the SNR is equal or greater than the capacity of the direct transmission since D can decode the full codeword w = [w 1 w 2 ] by only using the received signal from S. The high error floor performance at the low SNR region for the Selective DF scheme can be significantly improved when SIR is applied at R. Unfortunately, performing SIR at the high SNR, when error exists in the S-R channel, does not greatly reduce the BER curves as in the case of the Selective DF scheme, i.e. the error floor still exists at the high SNR region. This error floor, even at the high SNR region, is caused by the error propagation from the relay to the destination. When a decoding error occurs at R, some soft values of w
2 are in error. These error soft values are multiplied with β in (19), before they are forwarded through the stronger R-D link. At the same time the correct values of w2 from S are sent through the weaker S-D link. The soft values of the wrong parity check bits in w
2 from R may cause the coherent transmission at D to remain in error. Depending on the channel coefficient hSD and hRD , the errors in the received signal at D may be too large to be corrected by the iterative decoder. These errors introduced by the relay cannot be removed by increasing the system SNR because the increased transmission power at R will also amplify the soft values of the error parity bits in w
2 . One of the methods to mitigate the error propagation
Perfect S-R channel Direct transmission Selective DF protocol SIR (soft noise) SIR (soft fading) SIR (soft noise + power control at relay) SIR (soft fading + power control at relay)
-9
10
-5
-4
-3 -2 -1 0 Source Transmission Power , P (dB)
1
2
Fig. 7. The BER performances for different cases using Code 2, which operates at rate Rc = 0.45 at the destination.
is by reducing the transmitted power at R by choosing β appropriately. For example, we can choose β such that the worst soft error from the relay can be corrected by the coherent transmission signal from the source. That is, when the value of β satisfies Proposition 1, the received coherent transmission at D will be error free even though R fails to decode w1 in the BC mode. In Fig. 7, the error floor of the SIR improves significantly as the SNR improves at the high SNR region when a controlled β = 0.9 · hSD /hRD is used. Furthermore, our scheme has an improvement at about 0.5 dB compared to the Selective DF scheme at the BER of 10−6 . Fig. 7 also demonstrates the performances of SIR using two different LLR computations at D. We can see that our proposed soft fading method achieves better BER performance when compared to the soft noise method introduced in [11]. The BER curve of the soft fading method is better than the BER curve of the soft noise method for β = 0.9 · hSD /hRD , where an improvement at about 0.7 dB is achieved at the BER of 10−6 . We simulate the performance of Code 2 using different values of β and plot the results in Fig. 8. From Fig. 8, we can see the effects of using a lower transmitted power at R when the values of β satisfies Proposition 1. Firstly, we see that the BER performances at the small SNR region (i.e. SNR < −1 dB) start to decrease when a smaller β is used. It is also shown that the error floors at the high SNR region (i.e. SNR ≥ −1 dB) start to improve when a smaller β is used, and the error floor has been completely removed when β ≤ 0.7 · hSD /hRD . At the high SNR region, we found that β = 0.5 · hSD /hRD gives the best BER performance and it has the improvement of 0.6 dB when compared to the
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method outperforms the best known method in the literature by up to 0.7 dB in terms of BER performance. Finally, we derived a new bound for the power multiplication factor at the relay, which limits the amount of soft-errors forwarded by the relay to the destination. The BER performance of the designed RC-LDPC codes improved significantly once the power multiplication factor at the relay satisfies this bound.
0
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-1
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-2
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-3
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R EFERENCES -4
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-6
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0.9 hsd/hrd -7
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-3 -2 -1 Source Transmission Power, P (dB)
0
1
Fig. 8. The BER performances of Code 2 for different values of β that satisfy the bound given by Proposition 1.
selective DF scheme at the BER of 10−8 . In addition, from Fig. 8 we see that the BER curves at the high SNR region become closer to the BER curves of the Selective DF when the values of β are further reduced, i.e. β = 0.3 · hSD /hRD and β = 0.1 · hSD /hRD . Based on these observations, we would point out that by choosing β = 0.7 · hSD /hRD , we can significantly reduce the error floor or error propagation from the relay, and slightly degrade the error performance at the waterfall region. Even with the reduced transmission power, the error performance of the proposed scheme with β = 0.7 · hSD /hRD is consistently better than the Selective DF scheme for the SNR regions of practical use. VI. C ONCLUSION We developed a new SIR under the SDF scheme using LDPC codes in the half-duplex relay channels. We introduced a structured RC-LDPC code with a lower triangular structure so that the encoding of the additional parity-check symbols at the relay is linear and systematic. Then, we proposed a new methodology to design the degree distributions for the RCLDPC codes, where the concept of a K-layer doping matrix is introduced to represent the structure of a lower triangular parity-check matrix. As a result of our methodology, the asymptotic performance of RC-LDPC codes can be analyzed and predicted using the MET density evolution. Next, we developed the soft-decoding and soft-re-encoding algorithms for the proposed RC-LDPC codes, which allows the relay to forward soft messages to the destination when the relay fails to decode the source’s message. Furthermore, we proposed a novel method, which we refer to as soft fading, to compute the LLR of the received signal at the destination for the SDF scheme. We demonstrated that our proposed soft fading
[1] A. Nosratinia, T. E. Hunter, and A. Hedayat, “Cooperative communication in wireless networks,” IEEE Trans. Inf. Theory, vol. 42, no. 10, pp. 74–80, 2004. [2] T. M. Cover and A. El Gamal, “Capacity theorems for the relay channel,” IEEE Trans. Inf. Theory, vol. 25, no. 5, pp. 572–584, Sep. 1979. [3] G. Kramer, M. Gastpar, and P. Gupta, “Cooperative strategies and capacity theorem for relay networks,” IEEE Trans. Inf. Theory, vol. 51, no. 9, pp. 572–584, Sep. 2005. [4] T. E. Hunter, “Coded cooperation: A new framework for user cooperation in wireless systems,” Ph.D. dissertation, University of Texas at Dallas, Texas, USA, 2004. [5] B. Zhao and M. C. Valenti, “Distributed turbo coded diversity for relay channel,” Electron. Lett., vol. 39, pp. 786–787, May 2003. [6] Z. Zheng and T. M. Duman, “Capacity-approaching turbo coding and iterative decoding for relay channels,” IEEE Trans. Commun., vol. 53, no. 11, pp. 1895–1905, Nov. 2005. [7] J. Hu and T. M. Duman, “Low density parity-check codes over wireless relay channels,” IEEE Trans. Wireless Commun., vol. 6, no. 9, pp. 3384– 3394, Sep. 2007. [8] P. Razaghi and W. Yu, “Bilayer LDPC codes for decode-and-forward in relay channels,” IEEE Trans. Inf. Theory, vol. 53, no. 10, pp. 3723–3739, Oct. 2007. [9] C. Li, G. Yue, M. A. Khojastepour, X. Wang, and M. Madihian, “LDPC code design for half-duplex cooperative relay,” IEEE Trans. Wireless Commun., vol. 7, no. 11, pp. 4558–4567, Nov. 2008. [10] H. H. Sneessens and L. Vandendorpe, “Soft decode and forward improves cooperative communications,” in Proc. 1st IEEE Int. Workshop Computational Advances Multi-Sensor Adaptive Process., Dec. 2005, pp. 157–160. [11] Y. Li, B. Vucetic, T. F. Wong, and M. Dohler, “Distributed turbo coding with soft information relaying in multihop relay networks,” IEEE J. Sel. Areas Commun., vol. 24, no. 11, pp. 2040–2050, Nov. 2006. [12] P. Weitkemper, D. Wübben, V. Kühn, and K.-D. Kammeyer, “Soft information relaying for wireless networks with error-prone source-relay link,” in Proc. Int. ITG Conf. Source and Channel Coding, Jan. 2008. [13] K. Lee and L. Hanzo, “MIMO-assisted hard versus soft decoding-andforwarding for network coding aided relaying systems,” IEEE Trans. Wireless Commun., vol. 8, no. 1, pp. 376–385, Jan. 2009. [14] R. Thobaben and E. G. Larsson, “Sensor-network-aided cognitive radio: On the optimal receiver for estimate-and-forward protocols applied to the relay channel,” in Proc. 41st Asilomar Conf. Signals, Syst. Computers (ACSSC), Nov. 2007, pp. 777–781. [15] E. A. Obiedat and L. Cao, “Soft information relaying for distributed turbo product codes,” IEEE Signal Process. Lett., vol. 17, no. 4, pp. 363–366, Apr. 2010. [16] M. A. Khojastepour, A. Sabharwal, and B. Aazhang, “On capacity of Gaussian ’cheap’ relay channel,” in Proc. IEEE Global Telecommun. Conf. (GLOBECOM), Dec. 2003, pp. 1776–1780. [17] H. Jin, A. Khandekar, and R. McEliece, “Irregular repeat-accumulate codes,” in Proc. 2nd Int. Symp. Turbo Codes Related Topics, Sep. 2000, pp. 1–8. [18] T. J. Richardson and R. L. Urbanke, “Efficient encoding of low-density parity-check codes,” IEEE Trans. Inf. Theory, vol. 47, no. 2, pp. 638– 656, Feb. 2001. [19] T. J. Richardson, A. Shokrollahi, and R. L. Urbanke, “Design of capacity-approaching irregular low-density parity-check codes,” IEEE Trans. Inf. Theory, vol. 47, no. 2, pp. 619–637, Feb. 2001. [20] T. J. Richardson and R. L. Urbanke, Modern Coding Theory, 1st ed. Cambridge University Press, 2008. [21] M. H. Azmi and J. Yuan, “Design of multi-edge type bilayer-expurgated LDPC codes,” in Proc. IEEE Int. Symp. Inf. Theory, 2009, pp. 1988– 1992. [22] M. H. Azmi, J. Yuan, G. Lechner, and L. K. Rasmussen, “Design of multi-edge-type bilayer-expurgated LDPC codes for decode-and-forward in relay channels,” IEEE Trans. Commun., vol. 59, no. 11, pp. 2993– 3006, Nov. 2011.
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[23] D. Duyck, M. Moeneclaey, M. H. Azmi, J. Yuan, and J. J. Boutros, “Universal LDPC codes for cooperative communications,” in Proc. 6th Int. Symp. Turbo Codes Iterative Inf. Process. (ISTC), Oct. 2010, pp. 73–77. [24] J. J. Boutros,“Diversity and coding gain evolution in graph codes,” in Proc. Inf. Theory Applications Workshop (ITA), Feb. 2009, pp. 34–43. [25] W. E. Ryan and S. Lin, Channel Codes: Classical and Modern, 1st ed. Cambridge University Press, 2009. [26] “A fast and accurate degree distribution optimizer for LDPC code ensembles.” [Online]. Available: http://ipgdemos.epfl.ch/ldpcopt/ [27] K. Price and R. Storn, “Differential evolution - A simple and efficient heuristic for global optimization over continuous spaces,” J. Global Optimiz., vol. 11, pp. 341–359, 1997. [28] R. Yazdani and M. Ardakani, “Linear LLR approximation for iterative decoding on wireless channels,” IEEE Trans. Commun., vol. 57, no. 11, pp. 3278–3287, Nov. 2009. [29] X. Y. Hu, E. Eleftheriou, and D. M. Arnold, “Regular and irregular progressive edge-growth tanner graphs,” IEEE Trans. Inf. Theory, vol. 51, no. 1, pp. 386–398, Jan. 2005. [30] D. Duyck, J. J. Boutros, and M. Moeneclaey, “Low-density graph codes for coded cooperation on slow fading relay channels,” IEEE Trans. Inf. Theory, vol. 57, no. 7, pp. 4202–4218, June 2011.
Marwan Hadri Azmi received the B.Eng. degree in electrical and telecommunications engineering from Universiti Teknologi Malaysia in 2003 and the M.Sc. degree in communications and signal processing from the Imperial College of Science, Technology and Medicine, University of London, in 2005. He is now a lecturer in Universiti Teknologi Malaysia. He is currently on a study leave and working towards his Ph.D. at the University of New South Wales, Australia. His research interests include communication, information and coding theory focusing on cooperative communications, and LDPC coding.
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Jun Li (M’09) received the Ph.D. degree in electronic engineering from Shanghai Jiaotong University, Shanghai, P. R. China, in 2009. From January 2009 to June 2009, he worked in the Department of Research and Innovation, Alcatel Lucent Shanghai Bell, as a Research Scientist. Since June 2009, he has been a Research Fellow in the School of Electrical Engineering and Telecommunications, University of New South Wales, Australia. He served as a Technical Program Committee member for APCC’2009, APCC’2010, VT’C2011 (Spring), and ICC’2011. His research interests include network information theory, channel coding theory, wireless network coding, and cooperative communications.
Jinhong Yuan received the B.E. and Ph.D. degrees in electronics engineering from the Beijing Institute of Technology, Beijing, China, in 1991 and 1997, respectively. From 1997 to 1999, he was a Research Fellow at the School of Electrical Engineering, the University of Sydney, Sydney, Australia. In 2000, he joined the School of Electrical Engineering and Telecommunications, the University of New South Wales, Sydney, Australia, where he is currently a Professor for Telecommunications of the school. He is currently serving as Chair of the NSW Joint Chapters of the IEEE Communication Society and Signal Processing Society. He has published two books, two book chapters, and over 150 papers in telecommunications journals and conference proceedings. He co-authored 3 Best Paper Awards and 1 Best Poster Award, including a Best Paper Award of the IEEE Wireless Communications and Networking Conference (WCNC), Cancun, Mexico, in 2011, and a Best paper award of the IEEE International Symposium on Wireless Communications Systems (ISWCS), Trondheim, Norway, in 2007. His publication is available from http://www2.ee.unsw.edu.au/wcl/JYuan.html. His current research interests include wireless communication, communication theory, error control coding, and digital modulation.
Robert Malaney is currently an Associate Professor in the School of Electrical Engineering and Telecommunications at the University of New South Wales, Australia. He holds a Bachelor of Science in physics from the University of Glasgow, and a Ph.D. in physics from the University of St. Andrews, Scotland. He has over 100 publications. He has previously held research positions at Caltech, UC Berkeley - National Labs, and the University of Toronto. He is a former Principal Research Scientist at CSIRO.
