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Abstract—In this paper, we consider the half-duplex relay channel. We seek to design multiple turbo codes to minimize the information outage in the block fading ...
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2010 proceedings

Design of Distributed Multiple Turbo Codes for Block-Fading Relay Channels Peng Hui TAN, Chin Keong HO, and Sumei SUN Institute for Infocomm Research, A*STAR 1 Fusionopolis Way, #21-01 Connexis, Singapore 138632 Email: {phtan, hock, sunsm}@i2r.a-star.edu.sg

Abstract—In this paper, we consider the half-duplex relay channel. We seek to design multiple turbo codes to minimize the information outage in the block fading channel. An analysis on the diversity order of the relay channel, which depends on the time sharing variable and the rate of the code, is given for practical modulations. We also give a code structure for the multiple turbo codes to achieve the full diversity when it is achievable. The codes are optimized using the extrinsic information transfer (EXIT) chart analysis, based on the convergence thresholds of the iterative decoding tailored for relay channel. Numerical examples shows that with our design technique, the achieved frame error rate is within 0.7dB of the information outage. To reduce complexity, a suboptimum code search approach is proposed, resulting codes which perform 1dB away from the information outage.

I. I NTRODUCTION Cooperative communication has been shown recently to increase the reliability, link quality, and data rate of the system. The most basic cooperative communication is the relay channel introduced in [1]. It involves a source transmitting to a relay and a destination in one phase and relay transmission to destination in another phase. We focus on practical implementations of the decode-andforward (DF) protocol, under the framework of coded cooperation [2]. The concept of coded cooperation was originally proposed using convolutional codes [2] in a relay channel, but is far from approaching the achievable information theoretical rate of the DF protocol. Various capacity-approaching coding schemes have since been considered in the relay channel to enhance the system performance. One of such schemes employs low density parity check (LDPC) codes [3–6] and turbo codes have also been successfully applied to the relay channel [7–10]. However, these works did not focus on the design of the constituent codes of the turbo codes, but rather on the description of the coding schemes or the optimization of the transmission parameters. In [7], a distributed turbo coding scheme was proposed. The source broadcasts a (punctured) recursive systematic convolutional code (RSC) to the destination and the relay. After decoding the message correctly, the relay interleaves the message and encodes it with another (punctured) RSC before transmitting to the destination. The destination receives both codes on different channels which can then be decoded using the turbo principle. To further improve the decoding capability at the relay, an enhanced turbo code scheme was

proposed in [9, 10], where a punctured turbo code is used at the source. The source node transmits a punctured sequence of parity bits from constituent codes C1 and C2 . After decoding, the relay then transmits some or all of the remaining parity bits to the destination. Thus, at the destination, the punctured turbo code is equipped with more parity bits. The distributed turbo coding schemes described previously can be extended to produce multiple parallel concatenation of component codes, as known as multiple turbo codes [11], at the destination [12]1 . For this distributed multiple turbo code scheme, the source also transmits a (punctured) turbo code. Instead of sending the punctured parity bits at the relay, the interleaved message is encoded with another (punctured) constituent code C3 . In [6], a code structure for full diversity LDPC was given, but it did not look into minimizing the gap between the outage probability and the error performance of the code. In this work, we design the multiple turbo codes on relay channel with block fading. Multiple turbo codes [11] are considered since they are capable of achieving good performance for all signal-to-noise ratio (SNR) and relatively easy to implement. Ideally, the outage boundary [13], which separates the outage region and non-outage region over all source-destination and relay-destination channels, of a good code on relay channel must be close to the information theoretic limit as possible for block fading channel. The distributed turbo coding schemes described in the literature might not lead to full diversity codes. We start with an analysis on the diversity order of the relay channel for practical modulation, which is dependent on the rate of the code and the time-sharing variable α, where α is the fraction of the first time slot in a two time slots protocol. The code structure of the multiple turbo codes for full diversity is also given. The extrinsic information transfer (EXIT) chart analysis [14] is generalized to analyze these codes and optimize the design of the component codes, which has not been commonly considered in the literature. A code search procedure is developed to minimize error probability at the destination, by searching over all available combination of component codes. A suboptimum search procedure is also proposed. With this systematic design method, the achieved frame error rate is within 0.5dB of the information outage, while a suboptimum code performs 1dB away from the 1 Although in [12] the multiple turbo codes is applied on a multiple-access relay channel, it can be applied to a relay channel easily.

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This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2010 proceedings

information outage. The rest of the paper is organized as follows. In Section II, we describe the system model for the relay channel. The diversity order is analyzed in Section III. We formulate and describe the code design problem on block fading channel in Section IV. Code examples obtained from code search and their performance are presented in Section V. Concluding remarks are given in Section VI. II. S YSTEM M ODEL We consider a time-division based half-duplex relay channel for single user, where S, R and D denote the source, relay and destination, respectively. The fading channel is assumed to be block fading. The channel coefficients are independent random variables, which remain constant over the whole duration of the codeword. The coefficients between the source-relay, source-destination and relay-destination nodes are denoted by gSR , gSD and gRD , respectively. The channel coefficient is −β/2 given by gi = hi di , where the variables hi , di and β represent channel gains and distances between the transmitter and receiver and attenuation exponent, respectively. Binary modulation is used in this paper and hence hi are real values. For Rayleigh fading channel, the distribution of hi is p(hi ) = 2hi exp(−h2i ). The relay operates in the non-orthogonal “ half-duplex mode, where the transmitting and receiving modes can not occur simultaneously. In the DF protocol, the transmission of a block of length n symbols is split into two time slots. S transmits in the first time slot of αn channel uses, where α is the time-sharing variable. R receives the coded bits sent by S and attempts to decode its message. If decoding of message is successful, it will be forwarded to D using Alamouti spacetime block code (STBC) by S and R. Otherwise, S will transmit its remaining coded bits. We assume D knows if R correctly decode the messages required for cooperation. This can be done by sending a short indicative message before the cooperation begins. Let Iij  I(γij ) be the mutual information of BPSK modulation as a function of the instantaneous SNR γij , where for example γSR  |gSR |2 γ and γ  Es /N0 . Es denotes the symbol energy and N0 /2 is the variance of the additive white Gaussian noise samples. The mutual information at R is thus R0 = αISR . Assuming the rate of DF protocol R ≤ αISR , which ensures R decodes the message from S successfully, an achievable rate region for R with Alamouti STBC coding is given by R



αISD + (1 − α)IST BC ,

(1)

where γSD  |gSD |2 γ and γST BC  (|gSD |2 + |gRD |2 )γ. However, if R does not decode the message correctly, i.e., R > αISR , then the destination uses the codewords received over SD channel in the first and second time slots for decoding. The achievable rate region is R ≤ ISD . More detailed derivation can be found in [15]. We focus on binary multiple turbo codes for generating the codewords. S broadcasts the first part of its encoded

message using a nonsystematic turbo code (C1 , C2 ) to R and the destination. C1 and C2 are the component convolutional codes of the turbo code. In the second time slot, S either cooperates with R or sends additional parity bits related to its own message using a convolutional code C3 . An additional component code can be added to C3 if relay transmits with a rate less than one. However, in this work, we will focus on one component code C3 at the relay. For both cooperation and non-cooperation modes, the destination sees the same multiple turbo code (C1 , C2 , C3 ). The codeword is transmitted only on the S − D channel for non-cooperative mode, while the codeword is transmitted on both S − D and R − D channels for cooperation mode. For comparison, we also consider the distributed turbo code scheme, where S broadcasts some punctured version of (C1 , C2 ) in the first time slot and the remaining code bits or repetition of partial codeword is sent in the second time slot. III. D IVERSITY O RDER OF O UTAGE P ROBABILITY The information theoretical limit for block fading channels is the outage probability Pout = P (I < R), where the random variable I is the instantaneous mutual information for a particular channel realization and R is the fixed transmission rate in bits per channel use. We assume that all the links has a common average SNR of γ¯ that grows to infinity. This assumption does not affect the analysis of the diversity order. For . Pr(E) = −r. an error event E, let Pr(E) = γ¯ −r if limγ→∞ loglog γ ¯ The outage probability can be written as Pout

=

  Pr(ESR ) Pr(ESD ) + Pr(ESR ) Pr(EST BC ), (2)

 is the event where ESR is the event {R > αISR }, ESD  {R > ISD } and EST BC is the event {R > αISD + (1 − α)IST BC }. For Rayleigh distributed single antenna channels, the maximum diversity order is one. If R ≥ αI ∗ , where I ∗  limγ→∞ I(γ) = 1 bit/use for the BPSK modulation, the relay cannot decode the message successfully. Cooperation is clearly not possible and the diversity order is at most one. Now, assume that R < αI ∗ . An upper bound for Pout is given by

Pout

  ≤ Pr(ESR ) Pr(ESD ) + Pr(EST BC ) . −2  = γ¯ + Pr(EST BC ) .  = Pr(EST BC ).

(3) (4) (5)

Here, (3) follows from that any probability is not greater . .  ) = γ¯ −1 , and than one, (4) follows from Pr(ESR ) = Pr(ESD (5) follows from the fact that the diversity cannot be greater than two since at most two distributed transmit antennaes are . present. Moreover, since Pr(ESR ) = 1, it follows that .   (6) Pout ≥ Pr(ESR ) Pr(EST BC ) = Pr(EST BC ). Combining (5) and (6), we have .  Pout = Pr(EST BC ),

(7)

which implies that the overall diversity is limited by the error  in the cooperation mode Pr(EST BC ). Further, it can be shown

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that if R ≤ (1−α), the cooperation mode has the full diversity [6]. To achieve full diversity, α and R have to be properly chosen such that R < α and R ≤ (1 − α). IV. C ODE D ESIGN M ETHOD The outage probability is the smallest achievable frame error probability for an infinitely large block length. Our objective is therefore to minimize the frame error probability by an appropriate selection of the component codes. The set component codes is constrained to be rate 1 non-systematic convolutional codes (with puncturing if necessary) of memory less than or equal to three. The EXIT chart analysis [14] is used to establish the relationship between the frame error probability Pe and the set of available component codes. Finally, we identify the best combination of component codes that minimizes Pe while meeting the required constraints. A. Frame Error Probability We start by analyzing the frame error probability Pe of the relay channel described in Section II from the coding point of view. The frame error probability has the same form as the outage probability in (2). The performance of turbo code (C1 , C2 ) on the S-R channel determines the error probability Pr(ESR ). For large block length, turbo codes have a threshold phenomenon with iterative decoding on AWGN channels [14, 16]. With block fading, the error probability can be written as Pr(ESR )

=

Pr (γSR < γ ∗ (C1 , C2 )) ,

(8)

where γ ∗ (C1 , C2 ) is the convergence threshold of (C1 , C2 ) under iterative decoding. This convergence threshold is obtained using EXIT chart analysis [14]. With this threshold phenomenon, Pr(ESR ) = 1 − Pr(ESR ). Without cooperation, the destination sees a multiple turbo codes (C1 , C2 , C3 ) from the S-D channel. Its performance can also be analyzed by its convergence threshold [17]. The error probability is therefore in the same form as the turbo codes.  ) for non-cooperation can Hence, the error probability Pr(ESD be written as  ) = Pr (γSD ≤ γ ∗ (C1 , C2 , C3 )) , Pr(ESD

(9)

where γ ∗ (C1 , C2 , C3 ) is the convergence threshold of multiple turbo code (C1 , C2 , C3 ). This convergence threshold is also obtained from the EXIT chart analysis [17]. The last remaining term in (2) is the error probability for  cooperation Pr(EST BC ). With cooperation, the destination also sees the same multiple turbo code, but the codeword is transmitted on the S-D and R-D channels. We obtain this error probability by looking at the outage region of the code [13] over all (γSD , γRD ) since they vary for fading channels. The SNR region of (γSD , γRD ) is partitioned into an outage region Do and a non-outage region for moderate and large fading values, separated by an outage boundary Bo (C1 , C2 , C3 ). With this boundary, the conditional error probability is given by    p(γSD , γRD ) dγSD dγ2D (10) Pr(EST BC ) = γSD

γRD

where (γSD , γ2D ) ∈ Do (C1 , C2 , C3 ) denotes the outage region of the multiple turbo code, and p(γSD , γRD ) denotes the joint distribution of the channel coefficients. Each point on the outage boundary of a code is obtained as follows. For each given point of γSD , we use the EXIT chart analysis to determine the value of γRD where the tunnel appears. The first point (in step of 0.1dB) where the decoding tunnel appears is defined as the convergence threshold. The line joining these convergence threshold points is the outage boundary of the code. In general, we want to design code such that Bo (C1 , C2 , C3 ) is as close to Bo∗ as possible for all channel coefficients γSD and γRD . The detail of this EXIT chart analysis to obtain these convergence thresholds is given in the next subsection. Fig. 1 illustrates the outage boundary of a multiple turbo code (4/7, 13/11, 2/3) with α = 2/3 and R = 1/3. The generator polynomials are given in octal form. Its outage boundary Bo (4/7, 13/11, 2/3) is above and to the right of the information theoretic outage boundary Bo∗ achievable by a real Gaussian channel with BPSK input. Bo∗ is the set of channel coefficients on the boundary of the set formed from (1). The information theoretic outage boundary Bo for the distributed non-systematic turbo code scheme is also included. For α = 2/3 and R = 1/3, the achievable rate is R ≤ ISD /3+I(2γSD +γRD )/3 because S and R repeat half of the codeword during the second time slot. Bo is given by the boundary points of this set. It can be seen that Bo lies further to the right than Bo (4/7,13/11,2/3). From (10), this implies that the multiple turbo code gives a lower error probability than the turbo code for this example. The diversity of code can also be determined from the shape of the outage boundary. According to (7), the diversity of the relay channel depends only on the cooperation mode. To have a diversity order of two (full diversity), we need to show that two outage events are necessary to lose the message. In Fig. 1, we also include the outage boundary Bo (DT C) of the same code (4/7, 13/11, 2/3) but transmitted using the distributed turbo code scheme. The whole codeword is divided into two, one for each time slot. The diversity order of this coding scheme is one since for any value of γRD , outage occurs when γSD is below certain threshold. The code (C1 , C2 , C3 ) needs to have certain structure in order to have full diversity. As SNR tends to infinity, the block fading channel can be modeled as a block-erasure channel, where gi ∈ {0, ∞}. It is sufficient to show that the code has full diversity on the block-erasure channel [18]. The necessary and sufficient condition is when codeword from either S-D or R-D is erased due to deep fade, the decoder at D must be able to retrieve the message. If codeword from C3 is erased due to deep fading in R-D channel, the codeword of C3 is still available from the S-D channel. The decoder at D is able to retrieve the message if (C1 , C2 , C3 ) has a finite convergence threshold on point-to-point channel. For this to be possible, two of the component codes must be recursive [17]. Hence, C1 and C2 must be recursive for the relay to decode the message. Note that this is a sufficient

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2010 proceedings

0 B*o B (4/7,13/11,2/3) o

Non−outage region

Bo(DTC) B"

o

γSD [dB]

−5

−10 Outage region

−15 −15

−10

−5

0

5

10

γRD [dB]

Fig. 1.

Outage boundaries for rate 1/3 relay channel.

but not necessary condition. C1 and C2 need not to be recursive if turbo code is not used. Similarly, when codeword from (C1 , C2 ) is erased due to deep fading in S-D channel, the decoder at D can only use C3 to retrieve the message. In our example in Fig 1, C3 is a rate 1 convolutional code. As SNR tends to infinity, the mutual information of the extrinsic output of the decoder for all convolutional codes used in this work tends to one for zero prior information [19]. This implies the ability to retrieve the message at D. Note that in this case, C3 needs not be recursive convolutional code. B. EXIT Chart Analysis Mutual information is used in the EXIT chart to predict the convergence behavior of iterative decoding. The computation of convergence threshold in (8) and (9) is based on the methods used in [14, 17]. We will now proceed to describe the steps to obtain the convergence thresholds in Fig. 1. These convergence thresholds can be determined by using the EXIT chart projection method [17]. On the vertical axis, we track the mutual information IE(U1 )  I(U1 ; E(U1 )) between the information bit sequence u1 and the corresponding sequence of extrinsic values E(u1 ), which is given by IE(U1 ) = TV (IE(U2 ) , γSD , γRD ). The SNR values γSD and γRD are stated explicitly to account for transmission over different channels during the two time slots. On the horizontal axis, we have IE(U2 )  I(U2 ; E(U2 )) = TH (IE(U1 ) , γSD , γRD ). A vertical step in the EXIT chart represents the activation of decoders for C1 and C3 until IE(U1 ) has converged. A horizontal step represents activations of decoders for C2 and C3 until IE(U2 ) has converged. The EXIT functions are generated from those of the component codes for (C1 , C2 , C3 ). For a fixed γSD and varying γRD , the convergence threshold can be easily determined if a tunnel exists between the two curves. C. Code Search As there can be a large number of possible combinations of component codes for (C1 , C2 , C3 ), we need to constrain the type of component codes available for selection. As

mentioned previously, we consider non-systematic rate 1 convolutional codes (CC) as our component codes. Puncturing is used to increase the rate. To reduce search space, we limit the memory of the component codes to less than or equal to three. According to [19], there are only 25 CCs that have unique EXIT functions: 4 feed-forward (type F), 4 feed-backward (type B) and 17 combined feed-forward/feed-backward (type C). The octal representations of their generator polynomials are {3, 7, 13, 17, 2/3, 4/7, 10/13, 10/17, 7/5, 7/6, 5/7, 6/7, 13/11, 13/12, 11/13, 12/13, 14/13, 15/13, 16/13, 17/13, 13/14, 13/16, 17/16, 13/17, 16/17}. To reduce the combinations of the component codes, we use a systematic sequential search for good codes, as follows. The search of all the convergence thresholds are done on grid of 0.1dB. First, we set the target rates R0 and R for the relay and the destination, respectively. We find a set of turbo codes (C1 , C2 ) whose convergence thresholds γ ∗ (C1 , C2 ) are within 1dB from the theoretical limit of channel for the first time slot. Puncturing patterns with longer periods have better convergence thresholds, but exhaustive searches are possible only when the period is small. To limit the search, we further restrict the two component codes to have the same periodic puncturing pattern p1 . Next, we consider the convergence thresholds γ ∗ (C1 , C2 , C3 ) of the available multiple turbo codes for non-cooperation mode on the S-D channel. (C1 , C2 ) is selected from the set of turbo codes found in the previous step and C3 is selected from the available component codes. The puncturing pattern of C3 is determined by the overall rate R of the multiple turbo code and we simply select an appropriate puncturing pattern p3 for each R. The last set of convergence thresholds we need to compute is the outage boundaries of the multiple turbo codes in the cooperation mode. This set of multiple turbo codes is formed as follows. (C1 , C2 ) is selected from the set of turbo codes. C3 is again selected from the set of available component codes. For each given (C1 , C2 , C3 ), we compute its outage boundary, as illustrated in Fig. 1, instead of just a convergence threshold for each code. With all the required sets of convergence thresholds, we can proceed to search for the code which has the minimum frame error probability for our design parameters. For a given set of design parameters (R, α, dSD , dRD and dSR ), we are interested in the target performance of the code at Pe = 10−2 (for example). Setting the outage probability to 10−2 , we obtain the corresponding γ value. With this γ value, we can generate the samples of channel parameters (γSD , γRD and γSR ). For each code (C1 , C2 , C3 ), we can then compute its Pe . The code which gives the minimum Pe is selected. Finally, we propose a simple but suboptimum code search method for the relay channel. For this simple method, we are only concerned in the code search for the cooperation mode. We consider three cases where γSD = 2γRD , γSD = γRD and 2γSD = γRD . For each of these cases, we search a multiple turbo code which has the smallest convergence threshold. Code

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TABLE I M ULTIPLE T URBO C ODES FROM CODE SEARCH METHODS . (A)

Outage Prob. Code (B): P e

(B)

R, α

5/11, 6/11

5/11, 6/11

C1 , C2 , C3

2/3, 10/13, 3

2/3, 7/6, 6/7

p1

[11100]

[11100]

p3

[1]

[1]

(A) listed in Table I is the result of this suboptimum search method and will be used in Section V. It has the best frame error probability for its corresponding rates.

−1

10

Code (B): n = 10000 Code (A): P e

Code (A): n = 10000 Outage/FER

Code

0

10

−2

10

−3

10

V. N UMERICAL E XAMPLES We use simulations to investigate the performance of the multiple turbo design codes for the relay channel. We consider the case, where dSD = 1, dSR = dRD = 0.3, and a path-loss exponent of β = 4. Using the code search method described in the previous section, we list one multiple turbo code (B) for relay channel which have the minimum Pe in Table I. For decoding, the decoders for component codes are activated sequentially and the maximum number of activation is limited to 100. The number of channel use n is set to 10000. The frame error rate (FER) and outage probability of the multiple turbo code (A) and (B), which have an overall rate of 5/11, are depicted in Fig. 2. Code (B) is designed specifically for relay channel and code (A) is the best code from the suboptimum search method. Their corresponding error probabilities Pe computed from EXIT chart analysis are also included. As we observe, the performance of the codes and outage probabilities have a diversity order of two. The FER of code (B) is about 0.7dB from the outage probability limit at 10−2 . It is also close to the error probability Pe . Code (A) is about 1.0dB from the outage probability limit. A gain of 0.3dB can be achieved if we design the code systematically, instead of the suboptimum approach. VI. C ONCLUSION We have considered the problem of designing multiple turbo codes for the relay channel with block fading. We show that the diversity order of the outage probability depends on the time-sharing variable and the rate of the code. When full diversity is achievable, the code structure for full diversity is also given. Using the EXIT chart analysis, we are able to obtain the frame error probability of the code for block fading channel. With proper code design method, the achieved frame error rate is within 0.7dB of the information outage, while suboptimum codes performs 1dB away from the information outage. R EFERENCES [1] T. M. Cover and A. A. E. Gammal, “Capacity theorems for the relay channel,” IEEE Trans. Inform. Theory, vol. 25, pp. 572–584, Sep. 1979. [2] T. E. Hunter and A. Nosratina, “Cooperative diversity through coding,” in Proc IEEE Int. Symp. Inform. Theory, Lausanne, Switzerland, Jul. 2002, p. 220. [3] P. Razaghi and W. Yu, “Bilayer low-density parity-check codes for decode-and-forward in relay channels,” IEEE Trans. Inform. Theory, vol. 53, pp. 3723–3739, Oct. 2007.

−4

10

−5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Es/N0 [dB]

Fig. 2. Outage and FER of code (A) and (B) at destination for relay channel. [4] J. Hu and T. M. Duman, “Low density parity check codes over wireless relay channels,” IEEE Trans. Wireless Commun., vol. 6, pp. 3384–3394, Sep. 2007. [5] A. Chakrabarti, A. Baynast, A. Sabharwal, and B. Aazhang, “Low density parity check codes for the relay channel,” IEEE J. Select. Areas Commun., vol. 25, pp. 280–291, Feb. 2007. [6] D. Duyck, J. J. Boutros, and M. Moeneclaey, “Low-density graph codes for slow fading relay channels,” [Online]. Available: http://arxiv.org/abs/0903.1502, Mar. 2009. [7] B. Zhao and M. Valenti, “Distributed turbo coded diversity for relay channel,” IEE Electronic Letters, vol. 39, pp. 786–787, Mar. 2003. [8] M. Janani, A. Hedayat, T. E. Hunter, and A. Nosratinia, “Coded cooperation in wireless communications: Space-Time transmission and iterative decoding,” IEEE Trans. Signal Processing, vol. 52, pp. 362– 371, Feb. 2004. [9] Z. Zhang and T. M. Duman, “Capacity-approaching turbo coding and iterative decoding for relay channels,” IEEE Trans. Commun., vol. 53, pp. 1895–1905, Nov. 2005. [10] ——, “Capacity approaching turbo coding for half duplex relaying,” in Proc. Int. Symp. Inform. Theory, Adelaide, Australia, Sep. 2005, pp. 1888 – 1892. [11] D. Divsalar and F. Pollara, “Multiple turbo codes,” in Proc. IEEE Military Communications Conference, San Diego, USA, Nov. 1995, pp. 279–285. [12] C. Hausl, “Joint network-channel coding for the multiple-access relay channel based on turbo codes,” Eur. Trans. Telcommun., vol. 20, pp. 175–181, Jan. 2009. [13] J. Boutros, A. G. I. Fabregas, and E. Calvanese, “Analysis of coding on non-ergodic block fading channels,” in Proc. Allertons Conf. on Comm. and Control, Illinois, USA, Sep. 2005. [14] S. ten Brink, “Convergence behavior of iteratively decoded parallel concatenated codes,” IEEE Trans. Commun., vol. 49, pp. 1727–1737, Oct. 2001. [15] A. Høst-Madsen and J. Zhang, “Capacity bounds and power allocation for wireless relay channels,” IEEE Trans. Inform. Theory, vol. 51, no. 6, pp. 2020–2040, Jun. 2005. [16] H. El Gammal and A. R. Hammons, “Analyzing the turbo decoder using the Gaussian approximation,” IEEE Trans. Inform. Theory, vol. 49, pp. 671–686, Feb. 2001. [17] F. Br¨annstr¨om, L. K. Rasmussen, and A. J. Grant, “Convergence analysis and optimal scheduling for multiple concatenated codes,” IEEE Trans. Inform. Theory, vol. 51, pp. 3354–3364, Sep. 2005. [18] J. J. Boutros, A. Guill´en i F`abregas, E. Biglieri, and G. Z´emor, “Low-density parity-check codes for nonergodic block-fading channels,” [Online]. Available: http://arxiv.org/abs/0710.1182v1, Oct. 2007. [19] F. Br¨annstr¨om, “Convergence analysis and design of multiple concatenated codes,” Ph.D. dissertation, Chalmers University of Technology, Gothenburg, Sweden, Mar. 2004.