Distributed Turbo Coding with Hybrid Relaying ... - Semantic Scholar

Distributed Turbo Coding with Hybrid Relaying Protocols Yonghui Li and Branka Vucetic

Jinhong Yuan

School of Electrical and Information Engineering University of Sydney, Australia Email: [email protected]

School of Electrical Engineering University of New South Australia Email:[email protected]

Abstract— Distributed turbo coding (DTC) has been shown to be an effective coding scheme to approach the capacity of a wireless relay network. However, most of existing DTC schemes only consider a relay network with single relay node and assume that relay can perform an error free decoding, which we refer to as a perfect DTC scheme. In this paper, we consider a general 2hop relay network with an arbitrary number of relays and design the DTC for such a network when taking into account imperfect decoding at each relay. We propose a generalized distributed turbo coding (GDTC) scheme with hybrid relaying protocol for such relay networks. In each transmission, based on whether relays can decode correctly or not, each relay is included into one of two relay groups, referred to as a decode and forward (DAF) relay group and an amplify and forward (AAF) relay group. Each relay in the DAF relay group decodes the received signals from the source, interleaves, re-encodes and forwards it to the destination, while each relay in the AAF relay group amplifies the received signals and forwards it to the destination. At the destination, all signals transmitted from the relays in the DAF relay group are combined into one signal and that in the AAF relay group are combined into another signal. These two signals form a generalized DTC codeword. Theoretical analysis and simulation results show that the proposed GDTC scheme benefits from a significant coding gain contributed from the DTC relay group compared to the distributed coding with pure AAF relaying and simultaneously overcome the detrimental effects of error propagation due to the imperfect decoding at relays in the conventional DTC schemes. It also approaches the perfect DTC as the signal to noise ratio (SNR) increases.

I. I NTRODUCTION In wireless communication systems, diversity has been an effective technique in combating detrimental effects of channel fading caused by multi-path propagation and Doppler shift. Specifically, a spatial diversity can be achieved by a multipleinput-multiple-output (MIMO) system by exploiting space time coding. However, in a cellular network or wireless sensor networks, due to the limited size, cost and hardware limitation, it may not be possible for a mobile terminal or wireless sensor nodes to equip with multiple transmit antennas. To overcome such limitation, a new form of diversity technique, called user cooperation diversity or distributed spatial diversity [1]- [3], has been proposed recently, for cooperative cellular networks or wireless sensor networks. The relayed transmission can be viewed as a good example of distributed diversity techniques. Several relaying protocols for employing relays have been proposed in the literature, including an amplify and forward (AAF) approach and a decode

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and forward (DAF) approach. Recently, some cooperative or distributed coding schemes employing AAF and DAF relaying protocols have been proposed to explore the cooperative spatial diversity and cooperative coding gains in a wireless relay network, such as distributed space time block codes [11] [12] and distributed space time trellis codes [13] have been proposed and analyzed for both synchronous and asynchronous relay networks. To further improve the performance of a relay network, a distributed turbo coding (DTC) scheme by using the principle of turbo codes has been developed in [5] [6]. It has been shown that such a coding strategy performs close to the theoretic outage probability bound of a relay channel [5]. However, in most existing DTC schemes, it is assumed that the relay can perform error-free decoding, which we refer to as a perfect DTC. In [10], a distributed turbo coding scheme with soft information relaying (DTC-SIR) has been developed for a system when imperfect decoding occurs at the relay. Results shown that the soft information relaying can effective mitigate the error propagation due to the imperfect decoding, thus improve the performance of multi-hop relay transmission. However almost all the existing DTC schemes were based on the AAF and DAF relaying protocols, which suffer from a disadvantage of either noise amplification or error propagation. Furthermore, most DTC schemes are only designed for a relay network with single relay node. It is very important to develop a DTC scheme for a general relay network with any number of relays. In this paper, we propose a generalized DTC scheme with a hybrid relaying protocol (GDTC) for a two hop relay network with an arbitrary number of relays. For the proposed scheme, in each transmission, based on whether relays can make correct decoding or not, all relays are divided into two groups, which we called a DAF relay group, and an AAF relay group. All relays, which can correctly decode the signals transmitted from the source, are included in the DAF relay group and the rest of relays, which fail to make a correct decoding, are included in the AAF relay group. Each relay in the AAF relay group will amplify the received signals from the source and forward it to the destination, while each relay in the DAF relay group will decode the received signals, interleave, re-encode the interleaved symbols and forward them to the destination. All signals forwarded from the relays in the AAF relay group are combined into one signal at destination and that from

the DAF relay group are combined together into another signal. After combination, the overall codeword consists of combined coded information symbols transmitted from the relays in the AAF relay group and combined coded interleaved information symbols transmitted from the relays in the DAF relay group. These two signals form a generalized distributed turbo coding. Simulation results show that the proposed DTC scheme considerably outperforms the distributed coding with AAF (DC-AAF) scheme and circumvent the error propagation due to imperfect decoding at relays in the conventional DTC schemes. It also approaches the perfect DTC scheme at high signal to noise ratio (SNR). In a practical system, in order to determine whether a relay can decode correctly or not, some CRC bits can be appended to each information block (frame). After decoding each frame, the relay can examine the CRC checks to determine if the received signals are decoded correctly or not. For practical implementation, the destination also needs to know which of DAF or AAF relay group each relay belongs to. This can be done by simply transmitting one bit flag by each relay. II. S YSTEM M ODEL For simplicity, in this paper we consider a general twohop relay network, consisting of one source, n relays and one destination. Without loss of generality, we also assume that there is a direct link from the source to the destination. The transmitted source binary stream, denoted by B, is represented by B = (b1 , . . . , bk , . . . , bl )

(1)

where bk is the kth binary symbol and l is the information block length. The information sequence B is first encoded by a channel encoder. For simplicity, we consider a recursive systematic convolutional code (RSCC) with a code rate of 1/2. Let C represent the corresponding codeword, given by C = (C1 , . . . , Ck , . . . , Cl )

ysr,i (k, j) =

ysd (k, j) =

Psr,i hsr,i sk,j + nsr,i (k, j)

Psd hsd sk,j + nsd (k, j)

(4) (5)

where Psr,i = Ps ·(Gsr,i )2 , Psd = Ps ·(Gsd )2 are the received signal power at the relay i and destination, respectively, Ps is the source transmit power, Gsr,i and Gsd are the channel gains between the source and relay i and that between the source and destination, respectively. Also hsr,i and hsd are the fading coefficients between the source and the relay i and between the source and destination, respectively. In this paper, we consider a quasi-static fading channel, for which the fading coefficients are constant within one frame and change independently from one frame to another. Furthermore, nsr,i (k, j) and nsd (k, j) are zero mean complex Gaussian random variables with two sided power spectral density of N0 /2 per dimension. The relays then process the received signals and send them to the destination. Let xr,i (k, j) represent the signal transmitted from the relay i at time 2(k − 1) + j. It satisfies the following transmit power constraint, E(|xr,i (k, j)|2 ) ≤ Pr,i

(6)

where Pr,i is the transmitted power limit at the relay i. The corresponding received signal at the destination at time 2(k − 1) + j, denoted by yrd,i (k, j), can be written as yrd,i (k, j) = Grd,i hrd,i xr,i (k, j) + nrd,i (k, j)

(7)

where Grd,i is the channel gain between the relay i and destination, hrd,i is the fading coefficient between the relay i and destination and nrd,i (k, j) is a destination noise with two sided power spectral density of N0 /2 per dimension.

(2)

where Ck = (bk , ck ) is the codeword of bk , ck ∈ {0, 1}, bk is the information symbol and ck is the corresponding parity symbol. The binary symbol stream C is then mapped into a modulated signal stream S. For simplicity, we consider a BPSK modulation. The modulated signals S is given by S = (S1 , . . . , Sk , . . . , Sl )

destination, at time 2(k−1)+j, j = 1, 2, denoted by ysr,i (k, j) and ysd (k, j), respectively, can be expressed as

(3)

where Sk = (sk,1 , sk,2 ), sk,1 , sk,2 ∈ {−1, 1} are the modulated information and parity signals transmitted by the source at time 2k − 1 and 2k, respectively. We assume that the source and relays transmit data through orthogonal channels. For simplicity, we will concentrate on a time division multiplex transmission scheme [3], for which the source and relays transmit in the separate time slots. The source first broadcasts the information to both the destination and relays. The received signals at the relay i and

III. G ENERALIZED D ISTRIBUTED T URBO C ODING (GDTC) WITH A H YBRID R ELAYING P ROTOCOL In this section, we propose a generalized distributed turbo coding (GDTC) scheme. For the proposed scheme, in each transmission, based on whether relays can make correct decoding or not, each relay is included into either an AAF or a DAF relay group, denoted by ΩAAF and ΩDAF , respectively. Each relay in the AAF relay group amplifies the received signals from the source and forwards it to the destination, while each relay in the DAF relay group decodes the received signals, interleaves, re-encodes the interleaved symbols and forward them to the destination. All signals forwarded from the relays in the AAF relay group are combined into one signal at destination and that from the DAF relay group are combined together into another signal. These two combined signals form a distributed turbo coding. Fig. 1 shows a block diagram of the GDTC.

B. DAF Relay Group

Fig. 1.

The DAF relay group consists of all the relays, which can make an error-free decoding. Upon receiving signals, each relay in the DAF relay group will decode the received signals from the source, interleave the decoded information symbols, re-encode and send them to the destination. Since all the relays in the DAF relay group can decode correctly, each relay in the DAF relay group can accordingly recover the binary information stream B. B is then interleaved ˜ encoded and modulated into S, ˜ where into B,

Block diagram of a generalized distributed turbo coding

A. AAF Relay Group An AAF relay group, denoted by ΩAAF , consists of all the relays, which could not decode correctly. Upon receiving signals from the source, each relay in the AAF relay group simply amplifies the received signals from the source. Let xr,i (k, j), i ∈ ΩAAF , represent the signal transmitted from the relay i at time 2(k − 1) + j, then it can be expressed as xr,i (k, j) = μi ysr,i (k, j), i ∈ ΩAAF

(8)

where μi is an amplification factor such that xr,i (k, j) satisfies the power constraint in (6) and it can be calculated as Pr,i μi ≤ (9) |hsr,i |2 Psr,i + N0

˜k , . . . , S ˜l ) ˜ = (S ˜1 , . . . , S S

(13)

˜ k = (˜ where S sk,1 , s˜k,2 ), sk,j is the modulated signal transmitted by the relay at time 2(k − 1) + j. The relay i in the DAF relay group will then forward the ˜ with power Pr,i , to the destination, modulated symbols S (14) xr,i (k, j) = Pr,i s˜(k, j), i ∈ ΩDAF The received signals at the destination, transmitted from the relay i in the DAF relay group, become yrd,i (k, j) = Grd,i hrd,i Pr,i s˜(k, j) + nrd,i (k, j), i ∈ ΩDAF

At destination, all signals forwarded from the DAF relay group, at time 2(k − 1) + j, are then combined together. Let yrd−DAF (k, j) represent the combined signal, and it is given By substituting (8) and (4) into (7), the received signal at the by destination, transmitted from i-th relay, become yrd−DAF (k, j) = wr,i yrd,i (k, j) (15) i∈ΩDAF Psr,i hsr,i s(k, j) + nsr,i (k, j) yrd,i (k, j) = Grd,i hrd,i μi + nrd,i (k, j), i ∈ ΩAAF

(10)

At destination, all signals forwarded from the AAF relay group, at time 2(k −1)+j, are then combined with the signals directly transmitted from the source as follows, wr,i yrd,i (k, j) (11) yrd−AAF (k, j) = wsd ysd (k, j) + i∈ΩAAF

where yrd−AAF (k, j) is the combined signal at time 2(k − 1) + j, and wr,i , i ∈ ΩAAF are the combining coefficients. The optimal values of wsd and wr,i are given by [9] √ μi Grd,i Psr,i h∗rd,i h∗sr,i Psd h∗sd , wr,i = 2 , i ∈ ΩAAF wsd = N0 (μi |Grd,i hrd,i |2 + 1)N0 The corresponding destination SNR for the combined signals in the AAF relay group, denoted by γAAF , can be approximated by 1 γAAF = γsd |hsd |2 + H2i (12) 2 i∈ΩAAF

P

rd,i where γsd = PNsd0 , γrd,i = N , Prd,i = Pr,i Grd,i , γsr,i = 0 Psr,i 2 1 1 −1 i is called the Harmonic Mean p=1 λp,i ) N0 , H2 = ( 2 of variables λp,i , p = 1, 2, λi,1 = |hsr,i |2 γsr,i , and λi,2 = |hrd,i |2 γrd,i .

Similar to the calculation in the AAF group, the destination SNR of the combined signal in DAF is given by γrd,i |hrd,i |2 (16) γDAF = i∈ΩDAF

C. Iterative Decoding of GDTC From Eqs. (11) and (15), we can observe that an overall codeword for a generalized distributed turbo coding, consists of the combined coded information symbols transmitted from the AAF relay group, given in Eq. (11), and the combined coded symbols of the interleaved information sequence sent from the DAF relay group, shown in Eq. (15). These two signals at the destination are denoted by yrd−AAF and yrd−DAF , respectively. Since the information part of yrd−AAF and yrd−DAF carry the same information and they should be properly combined before decoding. Let the combined signal of yrd−AAF and yrd−DAF be yAAF and yDAF , respectively. A turbo iterative decoding algorithm is performed between these two decoders associated with yAAF and yDAF . The decoder is based on a BCJR MAP decoding algorithm. Two MAP decoders with input symbols yAAF and yDAF calculates the a posteriori probability (APP) of the transmitted information symbols and the interleaved information symbols and respective extrinsic information, respectively. The extrinsic information of one decoder is used to update the a priori probability of the other decoder in the next iteration. After

several iterations, the decision is made based on the APPs of the first decoder. There is one possible scenario that for some transmission blocks there are no relays, which can make correct decoding. This could happen at the low signal to noise region. In this case, there are no relays in the DAF relay group and all relays are in the AAF relay group, so we only need to decode yrd−AAF , from which we get the information symbol estimates.

where d1 and d2 are the Hamming weights of the erroneous codewords with Hamming weight d, transmitted from the AAF and DAF group, respectively, such that d = d1 + d2 . The probability that the AAF relay group consists of any q relays and the DAF relay group consists of the rest (n − q) relays is given by

IV. P ERFORMANCE A NALYSIS

Due to the uniform distribution of relays and assumption of γsr,i = γsr for all i = 1, . . . , n, the average PEP at high SNR, denoted by P GDT C (d), can be calculated as

In this section, we analyze the performance of the GDTC and compare it with other relaying protocols. For simplicity of calculation, we assume that γsr,i = γsr and γrd,i = γrd for all i = 1, · · · , n. A. Error probability of the GDTC Let us first calculate the PEP for a scenario where the AAF relay group consists of q relays numbered from 1 to q and the DAF relay group consists of (n − q) relays numbered from (q + 1) to n. Let γAAF,(q) and γDAF,(n−q) represent the instantaneous received SNR of the combined signals in the AAF and DAF relay groups, then we have from Eqs. (12) and (16) that γAAF,(q) = γsd |hsd |2 + γDAF,(n−q) = γrd

1 i H 2 i=1 2

(17)

|hrd,i |2

(18)

n

q

i=q+1 i (dsr , γsr,i |hsr,i ) be the conditional pair-wise error Let PF,sr probability (PEP) of incorrectly decoding a codeword into another codeword with Hamming distance of dsr in the channel from the source to the relay i. Since we assume that γsr,i = γsr for all i=1, 2,. . ., n, then we have i 2 2dsr γsr |hsr,i | PF,sr (dsr , γsr,i |hsr,i ) = Q i Let PF,sr (γsr |hsr,i ) represent the conditional word error probability in the channel from the source to the i-th relay, i PF,sr (γsr |hsr,i ) =

2l

i A(dsr )PF,sr (dsr , γsr |hsr,i )

dsr =dsr,min

the code minimum Hamming distance, where dsr,min is

l

A(dsr ) = i=1 il p(dsr |i), il is the number of words with Hamming weight i and p(dsr |i) is the probability that an input word with Hamming weight i produces a codeword with Hamming weight dsr . Then the conditional PEP at high SNR for this scenario, GDT C (d|hsd , hsr , hrd ), can be calculated as denoted by P(q)

q

(i1 ,...,in )∈(1,...,n) k=1

P

GDT C

n

n

i

k PF,sr (γsr |hsr,ik )

ik 1 − PF,sr (γsr |hsr,ik )

k=q+1

q n i (d) ≤ PF,sr (γsr |hsr,i ) E q q=0 i=1 n

i 1 − PF,sr (γsr |hsr,i ) Q 2d1 γAAF,(q) + 2d2 γDAF,(n−q)

i=q+1

q n

n i i 1 − PF,sr = PF,sr (γsr |hsr,i ) (γsr |hsr,i ) E q q=0 i=1 i=q+1

2d2 γDAF,(n−q) Q 2d1 γAAF,(q) Q n−q n 1 − PF,sr n −1 ≤ (d1 γsd ) (f (d1 ))q d2 γrd q q=0 n 1 − PF,sr −1 = (d1 γsd ) f (d1 ) + d2 γrd n

i where f (d1 ) = E PF,sr (γsr |hsr,i )Q( d1 H2i ) and 2l −1 −1 PF,sr = γsr dsr =dsr,min A(dsr )dsr . The exact closed form expression of f (d1 ) is too complex to be presented here. At the high SNR, it can be approximated as f (d1 ) ≤

2l

−1 A(dsr )γsr

dsr =dsr,min

1 1 + dsr + d1 dsr (d1 γrd + dsr γsr )

By substituting (19) into P GDT C (d), we have n d2 γrd P GDT C (d) ≤ (d1 γsd )−1 (γrd d2 )−n 1 + d1 γsr

(19)

(20)

Let PbGDT C be the BER upper bound for the GDTC. At the high SNR, it can be approximated as PbGDT C ≤

4l d=dmin

A(d)P GDT C (d)

(21)

l where A(d) = j=1 jl jl p(d|j), jl is the number of words with Hamming weight j and p(d|j) is the probability that an q input word with Hamming weigth j produces a codeword with i GDT C P(q) (d|hsd , hsr , hrd ) ≤ PF,sr (γsr |hsr,i ) Hamming weight d. i=1 From Eq. (20), we can observe that a diversity order of n

i (n + 1) can be achieved for the GDTC scheme in a relay 1 − PF,sr (γsr |hsr,i ) Q 2d1 γAAF,(q) + 2d2 γDAF,(n−q) −1 γrd → 0. network with n relays, when γsr i=q+1

Fig. 2.

FER performance for 1 relays

Fig. 4.


Fig. 3.


Fig. 5.


Similarly, for a perfect DTC, in which all relay are assumed to decode correctly, the average PEP of incorrectly decoding P erf ect (d), can be to a codeword with weight d, denoted by PDT C calculated as P erf ect PDT (d) ≤ (d1 γsd )−1 (d2 γrd )−n C

(22)

The average BER upper bound of perfect DTC, at high SNR, denoted by PbP erf ect , can be approximated as PbP erf ect ≤

4l

P erf ect A(d)PDT (d) C

(23)

d=dmin

Eq. (21) can be further expressed as n 4l d2 γrd P erf ect PbGDT C ≤ A(d)PDT (d) 1 + C d1 γsr

(24)

d=dmin

−1 γrd → 0, It can be noted from Eq. (24) that as γsr P erf ect → Pb and the performance of the GDTC approaches the perfect DTC. .

PbGDT C

V. S IMULATION R ESULTS AND D ISCUSSIONS In this section, we provide simulation results comparisons for various relaying schemes with various numbers of relays. All simulations are performed for a BPSK modulation and a frame size of 130 symbols over quasi-static fading channels. We use a 4-state recursive systematic convolutional code

(RSC) with the code rate of 1/2 and the generator matrix of (1, 5/7). For simplicity, we assume that γsr,i = γsr and γrd,i = γrd for all i = 1, · · · , n, and γrd and γsd are the same. It has been observed by earlier research that the conventional DTC schemes require the relays to fully decode the source information and this limits the performance of DTC to that of direct transmission between the source and relays, so it does not offer a diversity gain. By contrary, other distributed coding schemes, including the DC-AAF, GDTC, can all achieve a full diversity order for large SNR. Considering this, in this paper, we will not compare to the conventional DTCs, simply because that it cannot achieve a full diversity. Instead, we compare to the perfect DTC, where relays are always assumed to decode correctly. Figs. 2-5 compare the performance of the GDTC, distributed coding with AAF (DC-AAF) and the perfect DTC for various numbers of relays. It can be noted that as the number of relays increases, the GDTC significantly outperforms the DC-AAF in all SNR regions, and perform very close to the perfect DTC as γsr increases. This conclusion is consistent with the analysis in Section IV. It can be noted from the above results that for the low γsr and high γrd values, the GDTC and the DC-AAF have the similar performance in this case. However, as n increases, such as n = 4, 8, the GDTC can achieve considerable performance

gain compared to the DC-AAF. This can be easily explained in the following way. For low γsr values, the channel from the source to the relays is very noisy and probabilities of decoding errors at each relay are very high, so most of relays cannot correctly decode the received signals. In this case, as the number of relays is small, most of relays are included in the AAF relay group at a very high chance and the DAF relay group only occasionally includes few relays. However even such a limited contribution of coding gain from the DAF relay group is significant if the channel from the relay to the destination is poor (corresponding to the low γrd values), because in this case the relays in the DAF relay group can considerably improve the overall channel quality. As the number of relays increases, the probability that the DAF relay group contains at least one relays also increases and the contribution of coding gain from the DAF relay group become significant. This explain the reason why the GDTC can provide a significant coding gain over DC-AAF, even at low γsr values, as the number of relays increases. The above simulation results confirm that the GDTC can provide a significant coding gain in all SNR regions and perform very closely to the perfect DTC as γsr increases. VI. C ONCLUSION In this paper, we propose a generalized distributed turbo coding (GDTC) scheme for a general two hop relay networks consisting of any number of relays when imperfect decoding occurs at relays. The proposed DTC scheme employs a hybrid AAF/DAF relaying protocol. Based on whether relays can correctly decode or not, we include all the relays into two groups, referred to as an AAF relay group and a DAF relay group. Our results reveal that the proposed scheme can provide a significant SNR gain compared to the DC-AAF schemes due to the contribution of coding gain from the DAF relay group. This gain increases as the number of relays increases. The proposed scheme can also circumvent the error propagation due to imperfect decoding at relays, which usually occurs in the conventional DTC schemes, and approach the perfect DTC scheme at a high SNR region. R EFERENCES [1] A. Sendonaris, E. Erkip, and B. Aazhang, ”User cooperation diversity -Part I: system description,” IEEE Trans. Commun., vol. 51, pp. 19271938, Nov. 2003. [2] J. N. Laneman and G. W. Wornell, ”Distributed space-time-coded protocols for exploiting cooperative diversity in wireless networks,” IEEE Trans. Inform. Theory, vol. 49, pp. 2415-2425, Oct. 2003. [3] J. N. Laneman, D. N. C. Tse and G. W. Wornell, ”Cooperative diversity in wireless networks: Efficient protocols and outage behavior,” IEEE Trans. Inform. Theory, vol. 50, no. 12, Dec. 2004, pp. 3062-2080. [4] M. O. Hasna and M. S. Alouini, ”Harmonic mean and end-to-end performance of transmission systems with relays,” IEEE Trans. Commun., vol. 52, Jan. 2004, pp.130 - 135. [5] B. Zhao and M. C. Valenti, ”Distributed turbo codes: towards the capacity of the relay channel,” VTC 2003-Fall, vol. 1 , 6-9 Oct. 2003, pp. 322 - 326. [6] M. Janani, et al., ”Coded cooperation in wireless communications: space-time transmission and iterative decoding,” IEEE Trans. Signal Processing, vol. 52, Feb. 2004, pp.362 - 371.

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