Labeling Diversity for 2x2 WLAN Coded-Cooperative Networks

470 S. EJAZ, FENGFAN YANG, HONGJUN XU, LABELING DIVERSITY FOR 2×2 WLAN CODED-COOPERATIVE NETWORKS Labeling Diversity for 2×2 WLAN Coded-Cooperative Networks Saqib EJAZ1, FengFan YANG1, HongJun XU2 1College of Electronic and Information Engineering, Nanjing University of Aeronautics and Astronautics, 29 Yudao Street, 210016, Nanjing, China 2School of Engineering, University of KwaZulu-Natal, King George V Avenue, Durban, 4041, South Africa saqib[email protected], yf[email protected], and [email protected] Abstract. Labeling diversity is an efficient technique recently proposed in the literature and aims to improve the Bit Error Rate (BER) performance of Wireless Local Area Network (WLAN) systems with two transmit and two receive antennas without increasing the transmit power and bandwidth requirements. In this paper, we employ labeling diversity with different space-time channel codes such as convolutional, turbo and Low Density Parity Check (LDPC) for both point-to-point and coded-cooperative communication scenarios. Joint iterative decoding schemes for distributed turbo and LDPC codes are also presented. BER performance bounds at an Error Floor (EF) region are derived and verified with the help of numerical simulations for both cooperative and non-cooperative schemes. Numerical simulations show that the coded-cooperative schemes with labeling diversity achieve better BER performances and use of labeling diversity at the source node significantly lowers relay outage probability and hence the overall BER performance of the coded-cooperative scheme is improved manifolds. Keywords Alamouti, boosted scheme, coded-cooperation, joint decoding, labeling diversity, LDPC, space-time, turbo, WLAN 1. Introduction The concept of cooperative diversity traces back to the work of Van der Meulen [1] and has proven to be an effective way to minimize the channel impairments on the transmission of a wireless signal. Usually, a typical cooperative system consists of one source S, one relay Rand a common destination D. This relay Rterminal has been used in different configurations such as Amplify and Forward (A-F), Estimate and Forward (E-F), and Decode and Forward (DF) [2]. In view of mobile communications, the relay can be another user (mobile station) and can share its antenna and resources with the source (mobile station) to make a virtual antenna array and provide spatial/path diversity to the destination (usually the base station) [3]. The concept of cooperative diversity opened a gateway for the mobile devices to exploit spatial diversity without using an actual MultipleInput and Multiple-Output (MIMO) [4]. Coded-cooperative diversity between the two users was introduced in [5]. It was the merger of existing channel coding techniques with the cooperative schemes. In a codedcooperation, the source and the relay terminal jointly construct more powerful (in terms of error correction) channel code at the destination. Joint decoding scheme is then established at the destination to recover the information transmitted at the source. Different channel codes such as convolution codes [6], turbo codes [7], [8], [9], Low Density Parity Check (LDPC) [10], [11], and recently polar codes [12], [13] have been utilized in the coded cooperative schemes. However, apart from many advantages offered by the coded-cooperative communication, one of the major drawbacks is the poor bandwidth efficiency as most of the proposed coded-cooperative schemes are designed for Binary Phase Shift Keying (BPSK) modulation. This issue can be resolved by using high order modulation schemes such as 4-QAM, 16-QAM or more higher order of Quadrature Amplitude Modulation (QAM) as in case of most practical scenarios [14]. However, as modulation order increases the BER performance significantly drops. Therefore, the concept Bit-Interleaved Coded-Modulation (BICM) evolved, which brings synergy between the code design and the modulation techniques [15]. It is a known fact that the Bit Error Rate (BER) performance of Bit-Interleaved Coded Modulation (BICM) depends significantly on the labeling map and is further improved with the help of iterative decoding [16], [17], [18]. However, the cost of improved performance is increased latency of the communication system, which may be one of the critical factors in some applications such as live telemetry and video broadcasts in satellite communication systems [19]. A map is a rule according to which the encoded binary bits are assigned symbols from the set of constellation χ. The idea of the optimal labeling for iteratively decoded bit-interleaved space-time coded modulation was presented in [20]. Recently, a novel diversity scheme known as Labeling Diversity has been introduced in the literature for Bit-Interleaved Space-Time Coded Modulation with ItDOI: 10.13164/re.2015.0470 APPLICATIONS OF WIRELESS COMMUNICATIONS RADIOENGINEERING, VOL. 24, NO. 2, JUNE 2015 471 erative Decoding (BI-StCM-ID) systems for two transmit and two receive antennas for WLAN applications [21], [22]. Until now, labeling diversity is proposed only for 16-QAM modulation, however the general idea of labeling diversity can be extended to other high order modulation schemes as well. Labeling diversity has shown promising BER performance gains over the systems without labeling diversity, and also lowers the Error Floor (EF) region by ensuring error free feedback during the iterative process [21]. The fundamental principle of labeling diversity is to apply two different labeling maps for the two adjacent spatial streams of the transmitter. In [20], the BI-StCM-ID was compared with the Boosted scheme [23] equipped with the labeling diversity and in this paper it is referred to as Bit-Interleaved Boosted Coded Modulation with Iterative Decoding (BI-BoCM-ID). Moreover, in [18], for the non-cooperative schemes, various analysis such as Extrinsic Information Transfer (EXIT) chart, distance spectrum, asymptotic gain and Monte Carlo simulations proved the supremacy of the BI-BoCM-ID scheme over BI-StCM-ID (without any labeling diversity). In this paper, we use the Boosted scheme (employing labeling diversity) for coded-cooperative communication systems, but without an iterative decoding to reduce the complexity of the receiver. To use the Boosted scheme in a coded-cooperative scenario, we consider two channel codes such as the turbo and LDPC codes as suitable candidates for the coded-cooperative diversity scheme [10]. The natural extension of the BI-StCM scheme, which uses only one convolution code is a Bit-Interleaved Space time TurboCoded Modulation (BI-StTCM). Ordinary Bit Interleaved Turbo Coded Modulation (BI-TCM) is a well-established bandwidth and power efficient coding technique [24], which has shown to achieve BER performance very near to the capacity limit over both Additive White Gaussian Noise (AWGN) and Rayleigh fading channels [25]. Its descendant for multi-antenna systems BI-StTCM is also well known and efficient approach [26]. BICM for LDPC codes is discussed in [27], [28], [29] for 16-QAM and even for high order of modulation. Space-time LDPC codes are suggested in [30] for orthogonal frequency-division multiplexing (OFDM) system with multiple transmitter and receiver antennas over correlated frequency-and time-selective fading channels. In the remainder of this paper, the Boosted scheme employing distributed turbo codes and labeling diversity is referred to as Bit-Interleaved Boosted Distributed Turbo Coded Modulation (BI-BoDTCM) and Boosted scheme employing distributed LDPC codes and labeling diversity is referred to as Bit-Interleaved Boosted Distributed LDPC Coded Modulation (BI-BoDLCM). For a benchmark reference to our proposed BI-BoDTCM and BIBoDLCM schemes, we consider Bit-Interleaved Space-time Distributed Turbo Coded Modulation (BI-StDTCM) and BitInterleaved Space-time Distributed LDPC Coded Modulation (BI-StDLCM) schemes, respectively. The receiver for BI-BoCM scheme and the BI-StCM are identical and so are the receivers for BI-BoDTCM/BI-BoDLCM and BIStDTCM/BI-StDLCM. The major contributions of this paper can be summarized as follows: •To the best of our knowledge, labeling diversity is used for the first time in conjunction with space-time turbo and LDPC codes in a non-iterative manner. •Performance analysis of the non-cooperative Boosted transmission scheme (BI-BoCM) is extended to the coded-cooperative Boosted transmission scheme (BIBoDTCM). •Derived theoretical BER performance bounds at Error Floor (EF) region are verified using numerical simulations. •Coded-cooperative scenarios are presented not only for ideal conditions but also for non-ideal (practical) scenarios. The remainder of this paper is organized as follows: Section 2 briefly explains the generalized three-terminal space-time coded-cooperation. Coded-cooperative encoding schemes are discussed in Section 3. Section 4 gives brief description of space-time diversity schemes considered in this paper. In Section 5, a brief overview and asymptotic gain due to the labeling diversity is discussed. Performance analysis for non-cooperative and coded-cooperative schemes is presented in Section 6. Joint iterative decoding schemes for space-time turbo and LDPC codes are discussed in Section 7. Monte-Carlo simulations are performed and different simulations scenarios are presented in Section 8. Finally, Section 9 concludes the article. 2. Generalized Space-time CodedCooperation The generalized three-terminal space-time codedcooperative communication system is shown in Fig. 1. The system topology is based on the famous idea of Van der Meulen three-terminal communication [1], where the three terminals are source S, relay R, and the destination D. The end-to-end transmission of information transmitted at the source takes two time slots and the communication takes place in half-duplex mode. Each communication node has two antennas to transmit and receive the data with an assumption that antennas have enough separation between them to avoid any mutual interference. During the first time slot, also referred to as broadcast phase, the information vector m1is encoded by a channel code C1, which may be any channel code such as convolutional, turbo or LDPC code. The encoded signal is then passed to the space-time diversity block as illustrated in Fig. 1, which modulates the binary encoded signal to space-time codeword XSand broadcasts over the radio frequency (RF) channel to the both relay and the destination. The details of this space-time diversity block are presented in Section 4. During the first time slot, 472 S. EJAZ, FENGFAN YANG, HONGJUN XU, LABELING DIVERSITY FOR 2×2 WLAN CODED-COOPERATIVE NETWORKS Fig. 1. Generalized block diagram of distributed space-time coded-cooperative scheme with reference to time. the signal received at the relay is modeled as, YS−R=XSHS−R+NS−R(1) where XSis a T×Ntmatrix, Tis the number of time slots for transmitting one space-time codeword XS,Ntand Nrare the number of transmit and receive antennas, respectively, HS−Ris Nt×Nrchannel matrix between the source and the relay nodes. In case of Rayleigh fading channel, the entries of the HS−Rmatrix are independent and identically distributed (i.i.d.) complex Gaussian random variables (RVs) with zero mean and variance 1/2 per dimension, and NS−R is the T×Nrnoise matrix whose entries are modeled as i.i.d. complex Gaussian RVs with zero mean and variance N0/2 per dimension. For a fast-fading channel, HS−Rremains constant during the transmission of one single space-time codeword XSand varies independently over the transmission of the next XS. The received signal YS−Ris then decoded to recover information bits ˜ m1transmitted at the source during first time slot, where ˜ m1may or may not be equal to m1 depending upon the S−Rlink condition. The received signal at the destination during the first time slot is given as, YS−D=XSHS−D+NS−D(2) where HS−Dand NS−Dare the channel and the Gaussian noise matrices, respectively, and defined similarly as HS−R and NS−R. During the second time slot, the relay re-encodes the decoded message vector ˜ m1using channel code C2. The resultant binary codeword is then passed to the space-time diversity block, which modulates and propagates the signal XRtowards the destination. The signal received at the destination during the second time slot is given as, YR−D=XRHR−D+NR−D(3) where HR−Dand NR−Dare the channel and the Gaussian noise matrices, respectively, and are defined similarly as HS−Rand NS−R. The received signals YS−Dand YR−Dduring each time slot are demodulated by a soft output demodulator, which generates log likelihood ratio (LLRs) for each received bit. These LLRs are then passed to the joint iterative decoder (in case turbo or LDPC codes are used), which then iterates for desired number of iterations and finally pass the updated LLRs to the slicer to recover the information vector as ˆ m1. The term joint refers to the fact that the received signals YS−Dand YR−Dare decoded jointly instead of decoding each received signal individually. This joint decoding results in significant bit error rate (BER) performance gains [9]. The code rate at the source is defined as Rc=K/N1, where Kis the information block size and N1is the codeword block size. The generalized overall code rate Ro cof a coded-cooperative scheme from the destination point of view is, Ro c=K/(N1+N2), where N2is the codeword block size generated at the relay. Let Xbe the set of all space-time codeword matrices generated at a transmitter (source or relay). The modulator and the space-time block code (STBC) jointly define overall one-to-one mapping rule M:{0,1}L→X.∀Xi∈X, where i={S,R}node, the corresponding L-tuple binary label is M−1(Xi). The symbol energy is normalized to unity i.e., Es=1 independent of transmit antennas Nt. The rate of the STBC is defined as RST BC =q/T, where q=Nw/log2(m),w=1,2 is the number of symbols in one transmitted frame, mis the modulation order and the overall information rate of the system is Roverall =mRo cRST BC, and the average energy per information bit is Eb=Es/Roverall =1/Roverall [16]. 3. Coded-Cooperative Encoding Schemes In this section, we briefly discuss encoding configurations for channel codes C1and C2as shown in Fig. 1. Two famous channel codes such as turbo and LDPC codes are considered in this paper and their performance is analysed in conjunction with the labeling diversity. 3.1 Distributed Turbo Code (DTC) In a typical distributed turbo code (DTC) scenario [9], the channel code C1used at the source shown in Fig. 1 is a recursive systematic convolutional (RSC) code, denoted as RSC1. During the first time slot, Kinformation bits are encoded to N1bits at the source and after modulation broadcasted to the relay and destination. If the decoding at the relay node is successful, then the decoded information bits are interleaved by S−random interleaver, where S≤pK/2 and further re-encoded by a second RSC encoder RSC2, assumed identical to RSC1. The encoded codeword is a binary vector of length 1 ×N2is then modulated and transmitted to the destination during the second time slot. The overall transmission rate from the destination point of view is Ro c=K/(N1+N2). RADIOENGINEERING, VOL. 24, NO. 2, JUNE 2015 473 3.2 Distributed LDPC Code (DLDPC) In a typical single relay LDPC based codedcooperation [31], at first, information vector m1is encoded by the C1=LDPCScode to a codeword given as, c1=hm1,m2,...,mN−M1,p(S) 1,p(S) 2,..., p(S) M1it (4) where parity-check matrix of LDPCSis given as, HS=AM1×(N−M1)DM1×M1).(5) The codeword c1is then modulated and transmitted to the both source and the relay channel. With an assumption of error free decoding at the relay, the recovered information bits are re-encoded by C2=LDPCR, with a parity check matrix given as, HR=BM2×(N−M1)DM2×M2).(6) From the encoded codeword c2= [m1,m2,...,mN−M1, p(R) 1,p(R) 2,..., p(R) M2]t, only the parity-check bits [p(R) 1,p(R) 2,..., p(R) M2]are further modulated and transmitted to the destination during the second time slot. From the destination point of view, the overall paritycheck matrix Hhas the following structure, H=AM1×(N−M1)DM1×M10M1×M2 BM2×(N−M1)0M2×M1DM2×M2(7) and the codeword cobtained at the destination is a vector of length N+M2given as, c=hm1,m2,...,mN−M1,p(S) 1,p(S) 2,..., p(S) M1,p(R) 1,p(R) 2,..., p(R) M2it (8) where Hc =0. The Bi-layer Tanner graph [32] of an overall parity-check matrix His shown in Fig. 2, where the first and second layers correspond to parity check matrices HSand HR, respectively. Fig. 2. Bi-layer Tanner graph, [32]. As shown in Fig. 2, vn(n=1,2,...,N−M1)are the common variable nodes of HSand HR, whereas, vn(n=N− M1+1,N−M1+2,...,N)are related to HSand vn(n= N+1,N+2,...,N+M2)correspond to HR. The check nodes c(S) m(m=1,2,...,M1)and c(R) m(m=1,2,...,M2)are related to HSand HR, respectively. The overall transmission rate from the destination point of view is Ro c= (N−M1)/(N+M2). 4. Space-time Diversity Schemes 4.1 Conventional Space-Time BICM Scheme The conventional transmitter of a space-time bit interleaved coded modulation scheme referred to as BI-StCM in the rest of the paper is shown in Fig. 3. Fig. 3. Transmitter of BI-StCM scheme. In this scheme, the information vector mis encoded by the channel code to the vector ¯ c, which is then bit-interleaved by a pseudo-random interleaver (PRI) and passed to the mapper ω, usually the Gray labeling mapper. After mapping of binary codewords to constellation points, they are further passed to the famous Alamouti space-time encoder before their upconversion to the radio frequency. The space-time codeword Xiis defined as, Xi=x2t(x2t+1)∗ x2t+1(−x2t)∗=ω(c2t)ω(c2t+1)∗ ω(c2t+1)−ω(c2t)∗(9) where * is a complex conjugate, the rows of matrix Xirepresent two spatial streams and columns represent time slots. 4.2 Boosted Scheme It is a known fact that the BI-StCM suffers from the significant BER performance loss particularly when Gray labeling is used in both spatial streams. The improvement to BI-StCM was suggested in [18] and the novelty was to use two different labeling maps in adjacent space streams at the transmitter, as shown in Fig. 4. Fig. 4. Transmitter of a Boosted scheme. 474 S. EJAZ, FENGFAN YANG, HONGJUN XU, LABELING DIVERSITY FOR 2×2 WLAN CODED-COOPERATIVE NETWORKS The information sequence mis encoded by the channel code (convolutional encoder in [18]) to a binary encoded vector ¯ c, which is then passed to PRI. After bit-interleaving, the binary codeword bit train is split in two short bit-trains of equal length and passed to the two different labeling mappers i.e., ω(1)and ω(2). In the lower branch, the outputs x(2) 2tand x(2) 2t+1 of mapper ω(2)are interleaved again and passed to the spacetime mapper. The space-time mapper takes two baseband signals as its input and generates a space-time codeword matrix Xidefined as, Xi="x(1) 2tx(1) 2t+1 x(2) 2t+1x(2) 2t#=ω(1)(c2t)ω(1)(c2t+1) ω(2)(c2t+1)ω(2)(c2t).(10) The non-iterative receiver common to both of the aforementioned transmission schemes is shown in Fig. 5. For original BI-StCM and Boosted schemes, there exists a feedback path between soft-input soft-output (SISO) decoder and the soft-demodulator. However, to keep the complexity of the receiver at minimum we assume that there is no feedback path between SISO decoder and the soft demodulator. Fig. 5. Non-iterative receiver for BI-StCM and Boosted schemes. The received signal Yjis passed to the soft-demodulator, which generates LLRs λdem corresponding to each received bit transmitted at the source. Next, λdem is de-interleaved by (De-PRI) block and the LLRs are passed to the SISO decoder, whose LLRs are further passed to the slicer for making hard decisions over LLRs. 5. Overview of Labeling Diversity The fundamental idea of the labeling diversity is to use two different labeling maps such as ω(1)and ω(2)for two spatial streams instead of using identical labeling maps in each spatial branch. However, search of such labeling maps or a labeling map pair is more complex and has been recently investigated in [21]. In this section, we present the coding gain achieved due to the labeling diversity. The following analysis is not new, however, it is necessary to present here to understand the magic behind the labeling diversity. Asymptotic coding gain is greatly affected by the distance between such space-time codewords Xiand Zithat the binary vectors assigned to them, i.e., w(Xi)and w(Zi)≡ ˜ wk(Xi), differ exactly at one position (denoted as k). The K bit length vector wconsists of two K/2-bit length vectors c and c0, which are mapped onto the constellation points according to the labeling maps ω(1)and ω(2), respectively, in two spatial streams as shown in Fig. 4. w= [c1×K/2|c01×K/2]1×K.(11) To determine the symbol cost, let ω(l)(c)be a constellation point onto which the binary vector cis mapped according to the labeling map ω(l), where l=1,2. Using the same notational analogy ˜ wk(Xi), vector ˜ ck, differing on position kfrom c, is introduced. Similarly, ˜ c0 kdiffers from c0. To compute the asymptotic gain denoted as ψ2, the product of the eigenvalues (or the determinant) of matrix ˜ Amust be evaluated, ˜ A= (Xi−Zi)H(Xi−Zi)(12) where Hrepresents the Hermitian operation. For k≤K/2 we get, Xi=ω(1)(c)ω(1)(c0) ω(2)(c0)ω(2)(c),Zi=ω(1)(˜ ck)ω(1)(c0) ω(2)(c0)ω(2)(˜ ck) (13) Substituting in (12) we get, ˜ A=|ω(1)(c)−ω(1)(˜ ck)|20 0|ω(2)(c)−ω(2)(˜ ck)|2.(14) For k>K/2 matrix Xiremains unchanged, however, matrix Ziis now re-defined as follows: Zi=ω(1)(c)ω(1)(˜ c0 k) ω(2)(˜ c0 k)ω(2)(c).(15) Consequently, the matrix ˜ Ais, ˜ A=|ω(2)(c0)−ω(2)(˜ c0 k)|20 0|ω(1)(c0)−ω(1)(˜ c0 k)|2.(16) To simplify the notations, we define a variable dl (c,˜ ck)to represent the Euclidean distance between the baseband signals ω(l)(c)and ω(l)(˜ ck), into which binary vectors cand ˜ ck are mapped according to the labeling rule ω(l): dl (c,˜ ck)=|ω(l)(c)−ω(l)(˜ ck)|2.(17) The asymptotic coding gain is then written as [18], ψ2="1 2KK∑ X K ∑ k=1det ˜ A−Nr#−1/ρNr (18) where d1 (c,˜ ck)=ω(1)(c)−ω(1)(˜ ck) 2 and d2 (c,˜ ck)= ω(2)(c)−ω(2)(˜ ck) 2 . The utility function κis mainly dependent on the Euclidean distances dl (c,˜ ck)between the signals x=ω(l)(c)and associated with the label ˜ ckdiffering from c exactly in the k-th position. It should be noted that shorter the distance, the worse is its impact on the asymptotic coding gain. The core idea is to provide individual mappings with a different set of constellation points for any c. This is what makes labeling diversity transmission scheme more robust as compared to the systems without labeling diversity. Each symbol (constellation point in χ) from different mapping has different distance spectrum and thus suffers from different fading when propagated in the channel. This fundamental rule for labeling diversity is illustrated in Fig. 6. RADIOENGINEERING, VOL. 24, NO. 2, JUNE 2015 475 (a) (b) Fig. 6. Optimal Labeling map pair (a) Labeling map ω(1)and (b) Labeling map ω(2)from [21]. It can be seem from Fig. 6 that labels associated with corner points according to the ω(1)are the most inner points according to the labeling map ω(2)and vice versa. Another useful inference was made in [21] that assigning a new labeling map is a necessary but not sufficient condition. The relation between labelings making the optimal pair is more complex, since (18) has a form of the sum over k, therefore, all the individual distance pairs (d1 (c,˜ ck)d2 (c,˜ ck))must be analyzed to achieve the optimal performance via labeling diversity. Six different optimal labeling pairs were found in [21] using the modified Binary Switching optimization Algorithm (BSA). The details of the BSA are beyond the scope of this paper, however, interested readers are referred to [21]. One of the optimal labeling pairs found in [21] is shown in Fig. 6 and in this paper, we consider this optimal labeling map pair to provide labeling diversity. 6. Asymptotic Performance Analysis In this section, we present the asymptotic performance analysis of space-time BICM schemes. Firstly, we consider non-cooperative scenario and later we extend our analysis to coded-cooperative scheme. 6.1 Non-Cooperative Labeling Diversity Scheme, BI-BoCM The following analysis is valid for convolutional as well as space-time distributed turbo codes. Assuming uniform random interleaving, the union bound estimate of the probability of bit error is given as, Pb≤1 k ∞ ∑ d=df WI(d)p(d,χ,M)(19) where dfis the minimum free distance of the turbo code, WI(d)is the total input weight of error events at Hamming distance d, and p(d,χ,M)is the pairwise error probability (PEP) and is a function of the Hamming distance d, the signal constellation χand the mapping rule M. At high SNR, the iterative turbo decoding shows a phenomena well known as error floor (EF) and the BER performance in this region is referred to as asymptotic BER performance. To determine the asymptotic BER performance, some of the error items in the p(d,χ,M)can be expurgated to achieve the error-free feedback performance and is given as [16], pUB(d,χ,M)≤  1 2KK K ∑ k=1 1 ∑ b=0 ∑ Xi∈χk b ∑ Zi∈χk ¯ b min sΦ∆(Xi,Zi)(s)  d . (20) For a non-cooperative scenario, i=S, and d=d1is the Hamming distance of the codeword generated at the source node. Then, (20) can be written as, pS−D UB (d1,χ,M)≤  1 2KK K ∑ k=1 1 ∑ b=0 ∑ XS∈χk b ∑ ZS∈χk ¯ b min sΦ∆(XS,ZS)(s)  d1 (21) where XSis a transmitted codeword at the source and ZSis the received codeword with errors at the destination during the first time slot, and Φ∆(Xi,Zi)(s)is the Laplace transform of the pdf of the difference metric ∆(XS,ZS)between the two space-time codewords defined as, ∆(XS,ZS) = ||YS−ZSHS−D||2 F−||YS−XSHS−D||2 F(22) where the subscript Frepresents the Frobenius norm. Further, the Chernoff bound for Φ∆(Xi,Zi)(s)is given as [16], min sΦ∆(XS,ZS)(s) = "ρ ∏ y=11+ΓS−Dλy 4#−Nr (23) where λyare the nonzero eigenvalues of the matrix ˜ Adefined in (12). At high SNR, using (21) and (23), we get the approximate upper bound for PEP as follows, pS−D UB (d1,χ,M)≈   1 2KK K ∑ k=1 1 ∑ b=0 ∑ XS∈χk b ∑ ZS∈χk ¯ b ρ ∏ y=11+ΓS−Dλy 4!−Nr  d1 =4 ΓS−DΛ2 SρNrd1 (24) 476 S. EJAZ, FENGFAN YANG, HONGJUN XU, LABELING DIVERSITY FOR 2×2 WLAN CODED-COOPERATIVE NETWORKS where, Λ2 S(χ,M,Nr),   1 2KK K ∑ k=1 1 ∑ b=0 ∑ XS∈χk b ∑ ZS∈χk ¯ b ρ ∏ y=1 (λy)!−Nr  −1/ρNr .(25) In logarithmic domain, the (25) can be written as, log10Pb≃ −ρNrdf 10 (RΛ2 S)dB +(ΓS−D)dB+c(26) where cis any constant. It should be noted that the difference between BI-BoCM-ID and BI-BoCM is only a horizontal shift at EF region, latter is assumed in this paper. For large values of Eb/N0defined as SNR per bit, the BER curve on a log-scale is a straight line with a slope proportional to the overall diversity order ρNrdf, which is a direct function of df, in other words, the diversity order significantly depends on the minimum distance of a binary code (convolutional, turbo or LDPC). The factor (RΛ2 S)dBin (26) shifts the BER curve horizontally towards the left, and shows the coding gain. 6.2 Coded-Cooperative Scheme with Labeling Diversity, (BI-BoDTCM) In case of coded-cooperative scheme with labeling diversity, i.e., BI-BoDTCM, the PEP in (20) becomes the joint PEP of the S−Dlink and the R−Dlink, i.e., pS−D UB (d1,χ,M) and pR−D UB (d2,χ,M), respectively. Mathematically, pUB(d,χ,M) = pS−D UB (d1,χ,M)pR−D UB (d2,χ,M)(27) where d=d1+d2is the Hamming distance of the codeword jointly constructed by the source and the relay nodes. Since, the transmitter at the relay is identical to the one used at the source, hence, all the approximations assumed for deriving (24) for non-cooperative scenario still hold true for the coded-cooperative scenario and (27) becomes, pUB(d,χ,M)≈pS−D UB (d1,χ,M)pR−D UB (d2,χ,M) =  1 2KK K ∑ k=1 1 ∑ b=0 ∑ XS∈χk b ∑ ZS∈χk ¯ b ρ ∏ y=11+ΓS−Dλy 4!−Nr  d1 ×  1 2KK K ∑ k=1 1 ∑ b=0 ∑ XR∈χk b ∑ ZR∈χk ¯ b ρ ∏ y=11+ΓR−Dλy 4!−Nr  d2 (28) where XRis a transmitted codeword at the relay and ZRis the received codeword with errors at the destination during the second time slot. (28) is further simplified using (24) as, pUB(d,χ,M)≈"4 ΓS−DΛ2 Sd14 ΓR−DΛ2 Rd2#ρNr .(29) The average error probability Pbof the BI-BoDTCM scheme can be determined by substituting pUB(d,χ,M)from (29) in (19). The diversity order of the coded-cooperative scheme is ρNr(d1+d2)and the full diversity is achieved when d16=0 and d26=0. It should be noted that for the fast fading, performance of the coded-cooperative scheme is identical to the performance of the non-cooperative scheme especially when the source and the relay have identical channel statistics, i.e., ΓR−D=ΓS−D. However, for dissimilar channel characteristics, i.e., ΓR−D6=ΓS−D,where ΓR−D> ΓS−Dthere is a definite BER performance improvement for a coded-cooperative scheme, also shown later in the simulation section. 7. Joint Iterative Decoding Schemes In this section, we briefly discuss joint iterative decoding schemes employed for bit-interleaved distributed spacetime turbo and LDPC codes with and without labeling diversity. 7.1 Joint Parallel Iterative Decoding for DTC The joint parallel iterative decoding scheme is shown in Fig. 7, [9], is primarily based on the decoding configuration shown in Fig. 5. The dashed line shows the reception of the data from relay node during the second time slot. The pseudo-random bit interleavers (PRI) used at the source and at the relay are represented as πsource,πrelay, respectively, also shown in Fig. 7. To improve the free distance of a distributed turbo code, S-random interleaver πSis used at the relay, where S≤pK/2. During the first time slot, the spacetime demapper outputs its extrinsic LLRs λe dem, which are soft decisions related to the systematic bits and the parity bits transmitted at the source. This λe dem is further sent to the deinterleaver π−1 source corresponding to the bit-interleaver used at the source πsource. These de-interleaved LLRs are then de-multiplexed into LLRs for the systematic and the parity bits such as λsand λp1. The SISO RSC decoder 1 takes these LLRs and generates its extrinsic information for the received parity and systematic bits µ[dec1,e] p1and µS12, respectively. The extrinsic information for the systematic bit µS12 is S-random interleaved and passed to the second SISO decoder. During the second time slot, the space-time demapper receives the signal from the relay and outputs its extrinsic LLRs for the second parity bits p2, which is de-interleaved (π−1 relay)corresponding to the bit-interleaver πrelay used at the relay. The de-interleaved parity bit LLRs λp2are then passed to the second SISO decoder. The second SISO decoder also generates the extrinsic LLRs for the second set of parity bits µ[dec2,e] p2and the systematic bits µS21, which are de-interleaved (π−1 S)and feedback to the first SISO decoder. The exchange of extrinsic information for the systematic bit between the two SISO decoders takes place for desired number of iterations and finally, the systematic bits from the sec- RADIOENGINEERING, VOL. 24, NO. 2, JUNE 2015 477 Fig. 7. Joint parallel iterative decoding for BI-BoDTCM with reference to time. ond SISO decoder are de-interleaved (π−1 S)and passed to the slicer for hard decisions on the measured LLRs. 7.2 Joint Min-Sum Iterative Decoding for Distributed Space-Time LDPC Codes The received signal is first passed to the soft demodulator, which generates the LLRs λe dem for each received bit. These LLRs are further de-interleaved by π−1 soruce and passed to the iterative joint min-sum iterative decoder that is primarily based on the bilayer Tanner graph as shown in Fig. 2 . During each iteration, the extrinsic information resulting from the variable and check nodes in the bilayer Tanner graph is exchanged. Furthermore, joint decoding of the two received signals from the source and the relay results in increased coding gains as compared to the individual decoding of the received signals, also shown in [31]. The details of this joint min-sum decoding are skipped here for brevity and interested reader is referred to [31]. 8. Numerical Results This section presents the numerical results for proposed encoding schemes in aforementioned sections. Three different types of space-time channel codes such as convolutional, turbo and LDPC codes are used in conjunction with labeling diversity. Therefore, this section is divided in three sub-sections to analyse the effect of labeling diversity for each channel code individually. The simulation parameters common to all the three channel codes are: perfect channel state information (CSI) is assumed at all receiving nodes. Each communication node has two transmit and two receive antennas, i.e., Nt=Nr=2, moreover, each node communicates in half-duplex mode. The optimal labeling maps used to provide labeling diversity are shown in Fig. 6, and for non-labeling diversity (NLD), Gray labeling is used as a benchmark comparison. Rayleigh fading channel is assumed among all nodes assuming that the fading coefficient remains constant for the duration of one space-time codeword and may change independently over the transmission of next space-time codeword. 5 10 15 20 10−6 10−4 10−2 100 SNR per bit, ΓS−D(dB) Bit error rate (BER) EF Bound BI−StCM BI−BoCM BI−BoCM−ID,iter=30 BI−StCM−ID, iter=30 Fig. 8. Comparison of space-time convolutional codes with and without labeling diversity, Rc=0.5. 8.1 Convolutional Codes At first, we present the BER performance of the convolutional codes as shown in Fig. 8. The generator matrix of the convolutional code is g= [1,5/7]8, in an octal form, information block size K=120 bits and Rc=0.5. Since, original BI-BoCM was presented with iterative decoding i.e., BI-BoCM-ID therefore we also present the performance of BI-BoCM-ID in comparion with BI-StCM-ID. It can be seen that both BI-BoCM and BI-BoCM-ID completely outperform the BI-StCM and BI-StCM-ID schemes with BER performance gains more than 10 dB at BER ≈10−4. The BER performance gain is more significant in case of iterative demodulator and soft-input and soft-output (SISO) decoder and approaches theoretical EF bound derived in (24). However, the penalty is the overall latency of the system, which may not be a feasible solution for most of the real time applications, hence, we restrict our analysis only to the BI-BoCM schemes. 8.2 Space-Time Turbo Codes In this subsection, we consider space-time turbo codes with and without labeling diversity in both codedcooperative and non-cooperative scenarios. The generator 478 S. EJAZ, FENGFAN YANG, HONGJUN XU, LABELING DIVERSITY FOR 2×2 WLAN CODED-COOPERATIVE NETWORKS matrix of both RSC encoders used at the source and at the relay is g= [1,5/7]8, in an octal form, and the size of S−random interleaver is 120 bits, S−constraint length equal to 7. The number of log-MAP decoder iterations are five. The turbo code is punctured to make the code rate Rc=0.5. The BER performances of BI-StTCM and BI-BoTCM are shown in Fig. 9, where, BI-BoTCM completely outperforms the BI-StTCM. Moreover, the theoretical EF bound derived in (24) is also validated by the help of numerical simulations. Next, we analyse coded-cooperative space-time turbo codes with and without labeling diversity. From Fig. 10, it is observed that BI-BoDTCM outperforms BI-StDTCM scheme again with a significant margin. It is assumed that relay has an additional gain of 2 dB as compared to the source i.e., ΓR−D=ΓS−D+2dB. Further, comparing Fig. 9 and Fig. 10, it can be seen that the BI-BoDTCM (coded-cooperative) also outperforms the BI-BoTCM (non-cooperative) scheme and the BER performance curve converges more rapidly to the EF performance bound derived in 29. Further in Fig. 11, we present the BER performance comparison of BI-BoDTCM and BI-StDTCM schemes for the most practical scenario, i.e., non-ideal S-R channel (ΓS−R6=∞). We assume that the link between the source and the relay node is at ΓS−R=10 dB, and from Fig. 11 it is observed that the BI-BoDTCM clearly outperforms BIStDTCM as the BER curve for BI-StDTCM becomes flat. This flattening of the BER curve is because the relay is in outage and there is no coded-cooperation between the source and the relay nodes. Usually in such scenarios, cyclic redundancy check (CRC) is employed at the relay to avoid error propagation at the relay which leads to erroneous decoding at the destination and results in BER curve flattening [6]. The BER performance of BI-BoDTCM with ΓS−R=10 dB is only 0.2 dB away from its BER performance for an ideal S−Rchannel, i.e., ΓS−R=∞dB, which shows the effectiveness of labeling diversity in the coded-cooperative scenarios. 8.3 Space-Time LDPC Codes In this subsection, we consider LDPC codes for the proposed encoding schemes. At first, we consider noncooperative scenario with and without labeling diversity. The parameters of LDPC code used are: dv=2 and dc=4, where dvis number of 1’s in columns and dcis number of 1’s in rows of a parity check matrix Hand the code rate is Rc=0.5. From Fig. 12, it can be seen that the BIBoLCM clearly outperforms BI-StLCM under identical conditions. Moreover from Fig. 12 it is observed that for longer length LDPC codes i.e., K=1000,N=2000 the coding gain for BI-BoLCM is further improved by approximately 7dB at BER≈10−6as compared to short length LPDC codes, which encourages use of longer length LDPC codes, obviously at increased decoding complexity. Furthermore, the longer length LDPC codes also lowers the EF region. 2 4 6 8 10 12 10−8 10−6 10−4 10−2 100 SNR per bit, ΓS−D(dB) Bit error rate (BER) BI−BoTCM, iterations=5 BI−StTCM, iterations=5 EF Bound, BI−BoTCM Fig. 9. Comparison of non-cooperative space-time turbo codes with and without labeling diversity, Rc=0.5. 2 4 6 8 10 10−8 10−6 10−4 10−2 100 SNR per bit, ΓS−D(dB) Bit error rate (BER) BI−BoDTCM, iterations=5 BI−StDTCM, iterations=5 EF Bound, LD Fig. 10. Comparison of coded-cooperative space-time turbo codes with and without labeling diversity, Ro c=0.5, ΓR−D=ΓS−D+2dB. 2 4 6 8 10 10−8 10−6 10−4 10−2 100 SNR per bit, ΓS−D(dB) Bit error rate (BER) BI−StDTCM, ΓR−D = ΓS−D+2dB, ΓS−R=10dB BI−BoDTCM, ΓR−D = ΓS−D+2dB, ΓS−R=∞ BI−BoDTCM, ΓR−D = ΓS−D+2dB, ΓS−R=10dB BI−BoTCM Fig. 11. BER performance comparison among different codedcooperative scenarios where S-R link condition varies as ΓS−R=∞,10 dB, and ΓR−D=ΓS−D+2 dB. The noncooperative scheme (with turbo code of Rc=0.5 and labeling diversity) acts as an upper bound on the BER performance of BI-BoDTCM. Moreover, code rate for coded-cooperative scheme is Ro c=0.5.