New Multi-User OFDM Scheme: Braided Code Division Multiple Access

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1 New Mlti-User Scheme: Braided Code Division Mltiple Access Marcos B.S. Tavares, Michael Lentmaier, Kamil Sh. Zigangirov and Gerhard. Fettweis Vodafone Chair Mobile Commnications Systems, Technische Universität Dresden, Dresden, Germany s: {tavares, michael.lentmaier, Department of Electrical Engineering, Unviversity of Notre Dame, Notre Dame, IN 46556, USA Abstract In this paper, we present a new mltiple access scheme sing modlation which we call braided code division mltiple access (BCDMA). The principle behind this novel mltiple access techniqe is to combine in one single scheme a very efficient error correction code, interleaving for channel diversity exploitation and, simltaneosly, a mltiple access method for a large nmber of sers. BCDMA is based on braided codes, which are iteratively decodable codes defined on graphs and provide excellent performance with relatively low decoding complexity. There is a hge demand for higher data rates in next generation wireless systems. The typical mlti-ser scenario, where several sers access the same medim and the same resorces to perform commnication, reqires powerfl and efficient broadband transmission techniqes that enable the tilization of the spectrm resorces with very high efficiency. Orthogonal freqency division mltiplexing () is a transmission techniqe that elegantly addresses this problem and is crrently being considered as base technology for crrent and ftre wireless commnication systems. Moreover, error correcting codes are indispensable elements to improve the overall capacity of commnication systems. With the discovery of trbo codes [1], coding theorists and practitioners have learned how to approach Shannon s capacity with practical codes. Later, Gallager s low-density parity-check (LDC) codes [2] have been also recognized to be capacityapproaching [3]. In [4], the convoltional conterparts of Gallager s LDC block codes, i.e., LDC convoltional codes (LDC-CCs), were introdced. The LDC-CCs have the advantage that they are not limited to a fixed block length as in the case of the LDC block codes. Frthermore, the Tanner graphs nderlying the LDC-CCs have some interesting properties that enable the so-called pipeline decoding [4] and low-complexity encoding via shift-register operations. In [5] [9], the LDC-CCs were generalized to the case when the variable nodes and check nodes of the nderlying Tanner graph are sbstitted by some simple component codes (e.g., Hamming codes and convoltional codes). It was shown that these generalized LDC-CCs which were named braided codes (BCs) have excellent performance, good distance properties, and also that they can be continosly decoded with high speeds and low complexity sing pipeline decoders. In this paper, we present a novel mltiple access techniqe that is based on and BCs. We have denominated this new scheme braided code division mltiple access (BCDMA). 1 As we will discss throghot this paper, the BCDMA scheme reslts in a conceptal change in the organization of modems for mltiple access systems. This is de to the fact that error correction coding, interleaving for diversity exploitation, modlation and mltiple access which are generally processed by separate elements of a modem are now concentrated in one single entity that is derived from the typical two-dimensional array representation of the BCs. I. INTRODUCTION II. MULTILE CONVOLUTIONAL ERMUTORS Mltiple convoltional permtors (MCs) [11] are the fndamental elements in the constrctions presented in the following sections. We define an MC as a diagonal-type semi-infinite binary matrix of the form = ( p0,δ p 0,δ+1 p 0, p 1,δ+1 p 1, p 1, p t,t+δ p t,t+ 0 In, 0 δ and each row has exactly Γ 1 s and 0 s elsewhere. Moreover, each of the first colmns have at most Γ 1 s and all other colmns have exactly Γ 1 s. We also assme that there exist time instants t = t 1 and t = t 2 sch that p t1,t 1+δ = 1 and p t2,t 2+ = 1. In this context, Γ is called mltiplicity of the MC. The parameters δ and are called minimm and maximm permtor delays, respectively. The width of the permtor is defined as w = δ + 1. Frthermore, we say that an MC is periodic if and only if there exists a T sch that p t,t = p t+t,t +T, t, t Z +. The minimal vale of T for which this last eqation still holds is called period of the MC. An MC with period T can be constrcted from a T T mltiple block permtor sing the nwrapping procedre presented in [4]. III. BRAIDED CODES An ncomplicated way to nderstand the BCs is by taking a look at their encoders. Fig. 1 shows a general encoder strctre for BCs. In this pictre,, and are MCs with mltiplicities Γ 0, Γ 1 and Γ 2, and widths w 0, w 1 and w 2, respectively. Moreover, U t is a vector of information symbols that enter the systematic encoder at time instant t, and V t = U t, V t and V t are the vectors of coded symbols 1 Some basic principles of BCDMA were formlated in [10]. ).

2 U t U % t Fig. 1. V % t V % t Horizontal component code Vertical component code Encoder strctre for braided codes. that leave the encoder at time instant t. The feedback of paritybits of the horizontal encoder into the vertical one and vice versa, which can also be seen in Fig. 1, is characteristic of the BCs and reslts in very good error correction capabilities [5], [6]. In the remainder of this paper, we will focs on BCs that are constrcted sing binary rate R = 2/3 conventional convoltional codes as component codes. The BCs reslting from this constrction are called braided convoltional codes (BCCs) and they will have an overall rate R = 1/3. Moreover, we will assme symmetric MCs, i.e., Γ 0 = Γ 1 = Γ 2 = Γ and w 0 = w 1 = w 2 = w. In this sense, an implementation for a BCC encoder is exemplarily shown in Fig. 3, where the MCs, and with Γ = 5 and w = 10 correspond to the white cells of the array regions indicated by, and [ ] T, respectively. As we can note, this implementation is based on a two-dimensional memory array, which is processed in a sliding window fashion. The encoding of the information bits is then performed as follows. At the time instant t, the vector of information symbols U t = ( t,0,, t,γ 1 ) enters the BCC encoder and its elements are stored in their corresponding positions in. Then, the pairs of symbols ( t,j,ṽ t,j ), j = 0,, Γ 1, are read from the memory array and given as inpts to the horizontal component code that prodce as otpt the vector of symbols V t = (v t,0,,v t,γ 1 ) whose elements are stored in their corresponding positions in. A similar procedre is done for the vertical encoding. In this case, the pairs of symbols (ũ t,j,ṽ t,j ), j = 0,, Γ 1, are given as inpts to the vertical component code to prodce V t = (v t,0,,v t,γ 1 ) whose elements are stored in [ ] T. By doing so, in each time instant t the otpt of the BCC encoder is the vector of symbols = (V t,u t,v t ) = (v t,0,,,,, t,0, t,1, t,2, t,3, t,4,v t,0,, ), as it is shown by the dashed box in Fig. 3. v t,2,v t,3, A. BCCs and M-ary Modlation Schemes BCCs are essentially binary codes. However, we can se the array representation of the BCCs also to encode symbols of M-ary modlation schemes. In this case, each sed cell of the memory array will store q = log 2 (M) bits. In other words, the symbols t,j, ũ t,j, v t,j, ṽ t,j, v t,j and ṽ t,j will be binary vectors of length q. The encoding is then Inpt LLRs D 1 W τ D 1 D 2 O1 O1 D 2 D I Fig. 2. O2 O2 O 1 D I ipeline decoder for BCCs. Memory pipeline Hard decisions performed similar to the binary case, however, each parity symbol is obtained by encoding the q bits of each inpt symbol seqentially. For instance, in order to obtain the parity t,j,q 1 ) from the symbols t,j = symbols v t,j = (v t,j,0,, v ( t,j,0,, t,j,q 1 ) and ṽ t,j = (ṽ t,j,0,, ṽ t,j,q 1 ), the bits t,j,l and ṽ t,j,l enter the rate R = 2/3 component convoltional encoder to prodce the bit v t,j,l B. Decoders for BCCs for l = 0 to q 1. The BCCs can be decoded with a pipelined decoder architectre that enables continos, high-speed decoding. As it can be observed in Fig. 2, the main characteristic of this kind of decoder is the fact that all I decoding iterations can be performed in parallel by 2I concatenated identical processors, where i = 1,, I and e = 1, 2. Ths, after an initial decoding delay, the estimated vales for the code symbols are otpt by the pipeline decoder at each processing cycle in a continos manner. For the particlar case of BCCs, the 2I parallel processors are implementing an windowed version of the BCJR algorithm [12], [13]. In this case, each processor operates in a finite window of length W and calclates the a posteriori probabilities (A) of all code symbols (i.e., information and parity symbols). More specifically, the decoding of BCCs sing the pipeline decoder occrs as follows. At the t-th time instant, a vector of log-likelihood ratios (LLRs) R t corresponding to the vector of code symbols enters the memory pipeline as a priori information. Then, the read-logic I 1 transfers the necessary data from the memory pipeline to the horizontal component D (e) i code decoder D 1. On its trn, the processor D 1 performs the windowed BCJR decoding on the data and transfer the obtained extrinsic LLRs to the memory pipeline throgh the write-logic O 1. 2 After this, the processor D 1 starts processing new data. Observe in Fig. 2 that the crrent memory region sed by D 1 is separated from the vertical decoder D 1 by τ positions. This is to avoid that different decoders operate in overlapping sets of coded bits at the same time. When the extrinsic LLRs prodced by D 1 reaches the decoder D 1, vertical decoding is performed sing the extrinsic LLRs prodced by D 1 as a priori information. The new extrinsic LLRs obtained by D 1 are written into the memory pipeline and D 1 starts to process new data. This process repeats 2 I i and O i, with i {1, 2}, are determined by the MCs, and.

3 Vertical component code User 1 User 2 Dmmy cells t,4 t,3 t,2 t,1 Horizontal component code t,0 t,0 t,1 t,2 t,3 t,4 sbcarriers sbcarriers t 29 0 Transmitted code symbols at time t Fig. 3. Array representation of a braided convoltional code (BCC) and of a braided code division mltiple access (BCDMA) scheme for two sers. throgh all horizontal and vertical processors ntil the data reaches the vertical decoder D I. This last processor otpts the final LLRs and hard decision is done to determine the vales of the decoded bits. IV. BRAIDED CODE DIVISION MULTILE ACCESS In addition to the array representation of a BCC, the example shown in Fig. 3 can also be interpreted as the sbcarrier allocation pattern of a BCDMA system for two sers. In this case, we map each of the array cells at time t (i.e., cells inside the dashed box) to a sbcarrier of an system. For instance, the BCDMA scheme depicted in Fig. 3 wold need as basis for its realization an system with at least N = 30 sbcarriers. We assme that the white cells are associated with User 1 and gray cells are associated with User 2. The striped cells are dmmy cells, they do not contain any information. Moreover, in the Fig. 3, only the code symbols of User 1 are depicted in the array. If we nmber the sbcarriers from the left to right, the symbols transmitted by User 1 at time instant t, S 1 (t) = {v t,0,,,,, t,0, t,1, t,2, t,3, t,4,v t,0,, }, will be mapped to the sbcarriers v t,2,v t,3, C 1 (t) = {0, 5, 7, 8, 9, 10, 11, 12, 13, 18, 20, 21, 23, 26, 27}, respectively. Accordingly, the symbols of User 2 at time t, S 2 (t), will be mapped to the sbcarriers C 2 (t) = {1, 2, 3, 4, 6, 14, 15, 16, 17,19, 22, 24, 25, 28, 29}. We can observe that the MCs are the key components of or constrction. They are responsible for defining the BCCs, as well as, the sbcarrier allocation pattern of the sers. In Fig. 3, we denote, and [ ] T as being the sets containing the MCs (i) k, i {0, 1, 2}, k {0,, K 1}, of the K = 2 sers of the system. 3 The generalization for K 3 is straightforward. However, a fndamental condition that the sets (i), i {0, 1, 2}, mst flfill in order to define a BCDMA system is that their elements mst be orthogonal to 3 [ ] T in the set containing the inverse MCs [ k ]T, k {0,, K 1}.

4 Binary sorce BCDMA encoder modlator Binary sorce Interleaver Trbo encoder Π S/ M-QAM mapper Sbcarrier allocation modlator Sink Fig. 4. BCDMA decoder (a) demod. Rayleigh channel AWGN Sink Trbo decoder Inverse interleaver 1 Π /S (b) M-QAM demapper Sbcarrier deallocation demod. Systems nder stdy. (a) BCDMA system. (b) Bit-interleaved coded modlation (BICM) system sing trbo codes. Rayleigh channel AWGN each other. Considering the expression, this orthogonality condition can be formlated as p (i),k1 t,t p(i),k2 t,t = 0, t, t Z + and k 1 k 2, where p (i),k t,t is an element of the MC (i) k, i {0, 1, 2}, associated with the k-th ser of the system. In words, the condition imposed by means that the different sers of the BCDMA system do not share sbcarriers to transmit their symbols at the time instant t. Ths, they do not case interference to each other. A. Spectrm Management for BCDMA Systems In spite of having the sbcarrier allocation pattern determined by the error control coding scheme, the BCDMA system is still very flexible concerning the spectrm management for the several sers. Considering that the available spectrm is given by the nmber of sbcarriers N, some straightforward strategies for spectrm management are listed below: a) If N /3w < 2 the throghpt of the sers can increased or redced by increasing or redcing the mltiplicities Γ, respectively. Alternatively, different rows of the MCs can be overlapped into a single symbol to increase the throghpt, or the elements of the rows of the MCs can be spread across different symbols to decrease the throghpt. b) If N /3w 2 the above mentioned techniqes can also be applied. In addition, different rows of the MCs can be placed into orthogonal sb-bands of a single symbol to increase the throghpt. Observe that by applying the spectrm managements techniqes from above, the orthogonality condition in mst be redefined. B. Frther Remarks There are some points that are worth discssing when considering the constrctions presented in this section. As we can observe in Fig. 3, the sb-carriers belonging to a particlar ser are spread in freqency domain. This is a very important featre that enables the BCDMA system to exploit the freqency diversity of the channel. Another very important property that belongs the BCDMA system is the freqency hopping experienced by the sers in the sccession of the time instants, i.e., the sbcarriers sed by a particlar ser are different for the sccession of time instants. This last property implies that good and also bad parts of the spectrm are eqally (or almost eqally) distribted among the sers. In other words, it means that the BCDMA system is fair. Additionally, if BCDMA is spposed to be deployed in a celllar system with freqency rese, the freqency hopping is very important for averaging the inter-cell interference. V. SIMULATION RESULTS In this section, we evalate the performance of the BCDMA system depicted in Fig. 4(a) by means of compter simlations. As a reference system, we se the arrangement shown in Fig. 4(b), which employs the bit-interleaved coded modlation (BICM) ideas presented in [14] and analyzed in [15] to improve the code diversity of coded modlation on a Rayleigh channel. For the BCDMA system, the component convoltional codes have rate R = 2/3 and 4 states. Their generator matrices are given by G(D) = ( 1 0 1/(1 + D + D 2 ) 0 1 (1 + D 2 )/(1 + D + D 2 ) ). (3) On the other hand, the component codes of the trbo code of the BICM system have rate R = 1/2 and also 4 states. Their generator matrices are given by G(D) = ( 1 (1 + D 2 )/(1 + D + D 2 ) ). (4) Moreover, all MCs, interleavers and the sbcarrier allocation pattern of the BICM system have been randomly generated. For both systems, the nderlying system consists of N = 1024 sbcarriers and a cyclic prefix of length N C = 16, which is appended to the beginning of each transmitted symbol. Additionally, we se a Rayleigh flat fading channel in or simlations that is constant over the dration of one symbol bt varies independently from symbol to symbol. Moreover, in the demodlators of both systems from Fig. 4 a linear zero-forcing eqalizer with perfect channel knowledge is sed to mitigate the channel distortions. In or simlations, the BCDMA system was observed in six different configrations, which are listed below: 1. w = 50, Γ = 5, QSK modlation, block length L = bits. 2. w = 50, Γ = 5, 16-QAM modlation, block length L = bits. 3. w = 100, Γ = 10, QSK modlation, block length L = bits. 4. w = 100, Γ = 10, 16-QAM modlation, block length L = bits.

5 BER BCDMA w=50 QSK BCDMA w=50 16-QAM BCDMA w=100 QSK BCDMA w= QAM BCDMA w=150 QSK BCDMA w= QAM BICM N=21600 QSK BICM N= QAM BICM N=43200 QSK BICM N= QAM E b /N 0 [db] Fig. 5. BER performance of the BCDMA system 5. w = 150, Γ = 15, QSK modlation, block length L = bits. 6. w = 150, Γ = 15, 16-QAM modlation, block length L = bits. For the BCDMA configrations from above, the BCC encoders have been terminated with 2wΓ log 2 (M) zero bits, reslting in coding rates of R for QSK and R for 16-QAM. Moreover, the strategy b) of Section IV-A has been sed sch that N U = 90 sbcarriers are sed in each symbol to transport ser data. The BICM system has also been examined in different configrations. Here, interleaver lengths of N = and N = (reslting in terminated blocks of lengths L = and L = , respectively) have been combined with QSK and 16-QAM modlations. As in the case of the BCDMA system, the sbcarrier allocation modle of the BICM system assigns N U = 90 sbcarriers to ser data in each symbol. Fig. 5 shows the BER performance of the BCDMA system compared against the BICM system. In the simlations, a maximm of 100 decoding iterations were allowed. Moreover, at least 500 blocks were transmitted and at least 20 block errors have occrred in each simlation point. Astonishingly, the performance of the BCDMA system is abot 8 db better than the performance of the BICM system with similar parameters at a BER of This is qite an impressive reslt, since the component codes of both systems have almost the same trellis complexity and the interleavers of the trbo codes and of the BICM scheme are orders of magnitde bigger than the widths of the MCs sed in the BCCs. Frthermore, the steepness shown by the BER crves of the BCDMA system is an indication of the very good minimm distances of the nderlying BCCs. VI. CONCLUSIONS In this paper, we presented the general principles of a novel mltiple access scheme called braided code division mltiple access (BCDMA). Besides of providing a flexible and efficient medim access scheme for a large nmber of sers, the application of the BCDMA concept reslts in a conceptal change in the design of mltiple access modems. In this case, processing tasks that are traditionally considered in separate (i.e., error correction coding, interleaving for diversity exploitation, modlation and mltiple access) are now combined in one single entity. Moreover, the pipeline decoding of the nderlying braided convoltional codes (BCCs) enables very high speed VLSI implementations. Simlation reslts have also shown that the BCDMA system has a considerable gain in comparison with conventional systems based on bitinterleaved coded modlation. We consider BCDMA to be a promising research topic. Several points regarding this new techniqe remain open. For instance, the design of the MCs to simltaneosly maximize the error correction capabilities of the nderlying BCC, freqency diversity exploitation, fairness and interference averaging property mst be stdied. In addition, some other isses like decoding delay redction, VLSI implementation, rate compatibility, synchronization and channel estimation are also very interesting. ACKNOWLEDGMENT The athors are gratefl for the se of the high performance compting facilities of the ZIH at the TU Dresden. REFERENCES [1] C. Berro, A. Glaviex, and. Thitimajshima, Near Shannon limit errorcorrecting coding and decoding: trbo-codes, in roc. IEEE International Conference on Commnications, Geneva, Switzerland, May 1993, vol. 2, pp [2] R. Gallager, Low-Density arity-check Codes, MIT ress, Cambridge, MA, [3] D.J.C. MacKay, Good error-correcting codes based on very sparse matrices, IEEE Trans. Inform. Theory, vol. 45, no. 2, pp , Mar [4] A. Jiménez Feltström and K.Sh. 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