A simple low rate turbo-like code design for spread spectrum systems

Transcription

1 5 A imple low rate turbo-like code deign for pread pectrum ytem Durai Thirupathi and Keith M. Chugg Communication Science Intitute Dept. of Electrical Engineering Univerity of Southern California, Lo Angele Abtract In thi paper, we introduce a imple method to contruct very low rate recurive ytematic convolutional code from a tandard rate- convolutional code. Thee coden turn are ued in parallel concatenation to obtain very low rate turbo like code. The reulting code are rate compatible and require low complexity decoder. Simulation reult how that an additional coding gain of.9 db in additive white Gauian noie (AWGN) channel and 2 db in Rayleigh fading channel i poible compared to rate-/3 turbo code. The low rate coding cheme increae the capacity by more than 25% when applied to multiple acce environment uch a Code Diviion Multiple Acce (CDMA). I. INTRODUCTION It i a well known fact that, for a given input block ize, in an additive white Gauian noie (AWGN) channel, reduction in the code rate beyond a certain threhold ha diminihing return []. Furthermore, if low rate convolutional code are contructed by pecifying the generator polynomial for each coded bit, then a ubtantial encoding complexity incurred. In addition, for a convolutional code of a given contraint length, it may be difficult to achieve additional coding gain [6]. A coniderable amount of reearch ha been done in thi area in the pat few decade [7, 9, ]. Such code find their application in ytem where bandwidth expanion ha no additional penalty (e.g. pread pectrum ytem) or in ytem which are power limited with abundance of bandwidth to utilize (e.g. ultra wide band ytem). After the dicovery of turbo code [2], very low rate turbolike code uch a uper-orthogonal turbo code (SOTC) [4], turbo adamard code (TC) [3], etc. have been introduced. Both thee code ue a binary bi-orthogonal block code characterized by where i the input information ize, i the output code word length and i the minimum ditance of the block code. SOTC i a parallel concatenation of two uper-orthogonal recurive convolutional code. In the cae of TC, information bit are fed into a recurive ingle parity check encoder, the output of which i mapped onto a bi-orthogonal ignal et. Therefore, in order for a one-to-one mapping it i enough to have a 2-tate finite tate ma- Thi work wa upported in part by Army Reearch Office and National Science Foundation under the grant DAAD and ANI , repectively. chine. The diadvantage of the SOTC i that the decoding complexity increae dramatically with decreae in code rate. On the other hand, TC, which i contructed uing weaker contituent code ha lower rate of convergence which might not be a deired property for practical application. So, in deigning very low rate turbo-like code for practical application, one of the critical iue to deign trong enough contituent convolutional code with decoding complexity of the overall code being almot independent of the code rate. In thi paper, we introduce a technique to contruct uch a cla of low rate convolutional code. Thee code are then ued to build very low rate turbo code. Since thee code are contructed from rate- parent code, the complexity of the reulting encoder and decoder i very low. The ret of the paper i organized a follow. Section II briefly review the cla of uper-orthogonal convolutional code. Section III decribe the algorithm that i ued in thi work to contruct low rate code. Application to pread pectrum ytem conidered in Section IV. Numerical reult and concluion are given in Section V and VI, repectively. II. BRIEF REVIEW ON SUPER-ORTOGONAL CODES Block code are characterized by the parameter "!$#&%' where i the length of the codeword, the input alphabet ize and!(#&% the minimum ditance between any two codeword. The ratio ' ) i the code rate. Orthogonal binary block code are characterized by ( ) and the correponding code rate i '. ere, the ize of the code (i.e. the number of codewordn the code) i alo. In other word, for a given tuple, the encoding proce election of one of the poible codeword. A bi-orthogonal code i a block code that include an orthogonal code and it correponding complementary et. The length of thi code remain the ame but the input alphabet ize increaed by one, reducing the code rate to *+, -. In the ret of thi paper, we will refer to thee codeword a ignal. One way of generating orthogonal (hence bi-orthogonal) ignal et i by uing adamard matrice. A imple method to generate a adamard matrix of ize./ i through recurion i.e., a adamard matrix of ize./ can be generated recurively from a adamard matrix of ize.2 denoted by, where = []. The recurion i given by ()

2 6 [ 6 [ Data Orthogonal convolutional code K bit hift regiter (2 k,k) Orthogonal code Bi-orthogonal convolutional code uper-orthogonal convolutional code + 2 k Fig.. Encoder tructure for Orthogonal, Bi-orthogonal and Super-orthogonal code where 6 repreent the complement of. The adamard matrix of ize 9.: and it correponding complementary matrix form the required bi-orthogonal ignal et. Orthogonal convolutional code (OCC) of rate ) - are deigned uch that for any given tate, the output ignal on all the outgoing and incoming branche are pairwie orthogonal. Since thee ignal are elected from an orthogonal block code of ize, the input alphabet ize hould be. A block diagram of thi encoder i given in Figure. A the rate of the convolutional code i, only one data bit will be fed to the finite tate machine (FSM) for each length output. Therefore, the memory of the finite tate machine hould be large enough (;< to be precie) to acce all the ignal. In the trelli tructure of the bi-orthogonal convolutional code (BOCC), the ignal on the branche that merge into a tate are orthogonal wherea thoe that originate from a tate are antipodal or vice vera. Since the code ize i => for rate, the memory of the finite tate machine hould be at leat to acce all thee ignal. Figure 2 how a ection of trelli tranition of thee code. The dotted line correpond to an input?@? and the olid line correpond to an input?&a?. Fig. 2. A ection of trelli tranition of Orthogonal code, Bi-orthogonal code, (c) Super-orthogonal code Super-orthogonal convolutional code (SOCC) are a clever engineering contruction of binary trelli coden uch a way that the ignal on the trelli tranition from and to any given tate are pairwie antipodal (a hown in Figure 2 (c)). In order to accomplih thi kind of trelli tructure, the number of tate (c) hould be at leat B=>. The total number of poible ignal (which i twice that of the code rate) that can be ued in a trelli tep i ame a that of the total number of tate. So, when the code rate i decreaed, the code ize increae and hence the total number of tatencreae. Thi, in turn increae the complexity of the decoder. ence, for thi cla of code to be conidered for very low rate application (rate CD - E ), it i imperative that thi dependence be removed without diturbing the trelli tructure. III. CODE CONSTRUCTION In order to contruct very low rate code, we tart with a good, F -tate, rate- recurive ytematic convolutional code and apply a combination of repetition and bit flipping. By good code, we mean that the ignal emerging from and converging to any given tate differ in maximum poible bit location. Our algorithm can be decribed a follow: G Divide the tate pace into equal and even ubet (IJ#KML NPO&OQO&M 3 ) uch that the cardinality of R2I # 3 F. G For each I #, aign a one-to-one mapping with a combination of repetition and flipping of the correponding output code ymbol. G Depending on the memberhip of the current tate SUT@V A'P "UOQO&O&,FW;2 3, apply the mapping to the output. The mapping that i ued can be decribed recurively. The following example illutrate the contruction algorithm. Let the number of tate F be UE. Let thi pace be divided into four ( 3Y ) ubet. Let the required rate Z be /8. Let [ denote the two coded output bit of the parent code and the correponding current tate be S T \V 3 ANU PO&O&OQU ]. Apply the following oneto-one mapping to obtain the required rate: If ST_^8I, then the coded output i [[[P[. If S T ^8I`, then the coded output i [ [[ [. If S T ^8Ia, then the coded output i [[ [. If S T ^8Icb, then the coded output i [ [P[. I Standard PCCC R = /2 N State R = /2 N State Mapper: State/Rate Dependent Mapper: State/Rate Dependent Fig. 3. Parallel concatenation of the propoed low-rate encoder tructure The chematic of the reulting encoder i given in Figure 3 where it i hown applied to each of the two contituent code of the turbo code. Figure 4 how the change in the trelli tructure by the application of our algorithm to a four tate rate-/2 parent code in order to obtain a rate-/4 code. In thi cae, the tate pace i divided into two ubet of two tate each. I correpond to tate d)sues f and I ` correpond to tate d S ` S a f. Let be the coded output of the parent rate-/2 code at a given

3 m m Ž m time intant. Then, if the current tate belong to I, the new coded output i whereaf the current tate belong to I `, the output i 6. Since thi contruction methodology i baed on rate-/2 code with arbitrary number of tate, the following obervation can be made. G code rate (turbo code) uch a /7,/8,/5,/6,/3,/32, etc. can be obtained directly while other rate can be achieved by puncturing. G code rate independent of the number of tate which tranlate to the code complexity being a deign parameter [5]. Z 2 Z Fig. 4. Change in the trelli tructure with application of our algorithm parent code reulting lower rate code In addition, note that for any given tate in the trelli, ignal on all the incoming and outgoing branche are pairwie antipodal. In other word, for rate that are reciprocal of multiple of 4, code contructed by thi method will alway be uperorthogonal [] in nature. Figure 5 how the trelli tructure of the traditional uper-orthogonal code for rate ) along with that of the imple uper-orthogonal code that reult from our contruction procedure. Notice that the traditional SOCC ue all the g poible ignal while the imple code ue only ignal. In other word, the propoed code ue only a ubet of the complete bi-orthogonal ignal et which in turn lead to reduction in code complexity. Fig. 5. Trelli diagram of imple SOCC traditional SOCC Let the code rate be ). Let B be the matrix repreentation of the complete bi-orthogonal ignal et. Then B i of ize -hib=>mj(.k. Let l repreent a ubet of the bi-orthogonal ignal et uch that the ize of l i %.n. For large enough, we aume ocp. That i, for low encoding rate, our encoder elect ignal from a relatively maller ubet of the complete bi-orthogonal ignal et. By contruction, a rqt, with fixed, everal column of l can be made to be ame. So, it may be poible to puncture one or more of thee column to obtain any arbitrary rate with minimal coding lo. The decoder i a tandard turbo decoder [3] (with log domain um-product algorithm). The decoding algorithm can be implemented a min*(.) operation which i defined a minu vwmxn 3 ; ln y z{ry 3 min }vw,xnc; ln M $~y- z BO (2) IV. APPLICATION TO SPREAD SPECTRUM SYSTEMS In thi ection, the application of the propoed low rate turbolike code to pread pectrum ytem, epecially to code diviion multiple acce (CDMA), i conidered. The chematic of the code-pread ytem uing low rate code and conventionally coded (rate-/3 turbo code) and pread ytem i given in Figure 6. Three different channel, namely AWGN channel, block fading channel and multi-uer channel with AWGN are conidered. b b Low rate code R = /n igher rate code R = /n 2 d Scrambling d Spreading Fig. 6. Low rate coded and crambled ytem Conventionally coded and pread ytem A imple emi-analytic expreion to etimate the degradation in ignal to noie ratio due to the preence of multiple uer in a low rate coded ytem can be derived. The matched filter output that correpond to a coded ymbol on a trelli tranition can be written a x 3 ƒ # ƒ T,ˆ T Š ˆ # T (3) under perfect power control and ynchronization. ere, i zero mean Gauian random variable with variance FŒP, i the total number of uer and ƒ i the coded ymbol energy. Under the aumption that the multiple accenterference can be modelled a Gauian, it i traight forward to how that the additional information bit-to-noie ratio needed to achieve the ingle uer performance can be approximated by ƒ{ ) š F*ŒN { M ; œž š (4) Ÿ;2

4 Ž Ž where š i the bit-to-noie ratio needed for ingle uer performance for a pecific bit error rate (uually UA ). Equation (4) can be generalized to ƒ{ F ŒN { M ; œ{ (5) ;: where repreent additional preading involved. V. NUMERICAL RESULTS The algorithm i applied to 6-tate rate- ) recurive ytematic convolutional parent code ued in [2]. The feed forward and feedback polynomial of thi code are - and in octal, repectively. Two uch code are ued to build the turbo code. The interleaver ize i 24 bit. The tate pace of each contituent code i divided into four ubet (m = 4) of four tate each. The mapping technique explained in the example in Section II i applied. Reult hown are after 5 iterationn AWGN channel and iterationn fading channel. 2 Bit Error Rate 2 (c) E /N b o Fig. 8. Performance of variou PCCC cheme with rate /32 contituent code. Code contructed uing the new algorithm Traditional SOTC (c) Turbo-adamard code No. of ignal ued Conventional SOTC Code rate = / Code rate = / Code rate = / TABLE I COMPARISON OF NUMBER OF ADDITIONS REQUIRED TO COMPUTE TE BRANC METRIC IN TRADITIONAL SOTC AND SIMPLE SOTC BER Rate = /3 Rate = / E b /N o Fig. 7. Performance of Rate-/3 parent (turbo) code v Rate-/63 code in AWGN channel Figure 3 how the coding gain achievable uing the propoed technique to reduce the code rate from ) ) to rate )E in AWGN channel. Note that an additional coding gain of.9 db i poible. The performance degradation of thi code with repect to the correponding traditional SOTC i evaluated and the reult are hown in Figure 8. The degradation i le than. db at BER of UA. For completene, we have alo included the performance of the bet known low rate turbo-like code (TC, code rate 7/37). The reult for TC i obtained after performing 5 iteration. Note that there i a degradation of only.2 db. owever, the propoed code tructure reult in about 75% complexity reduction at the decoder compared to the traditional SOTC. In addition, the computational complexity of the propoed cheme per iteration i much leer than that of TC. Furthermore, the convergence rate of TC i approximately three time lower than that of the propoed cheme. Thi an undeired property for hardware implementation. ere, we compare the complexity of the propoed cheme with that of the traditional SOTC in detail. The complexity i calculated in term of total number of operation needed to decode one information bit. Since the code rate and number of iteration are ame for both the code tructure, the number of tate the only parameter that decide the complexity for a given interleaver ize. Since the traditional SOTC for a rate ) )E require a total of g tate compared to the new cheme which require taten total, a complexity reduction of about 75% i achieved. To be precie, in the preence of AWGN, both hard and oft decoding of convolutional code that ue bi-orthogonal ignal et involve the calculation of correlation between the received ignal with the bi-orthogonal ignal et. The complexity of the propoed cheme and the traditional SOTC cheme i compared baed on the total number of operation required to calculate thi correlation ince it ha direct influence on the total number of operation required to decode an information bit. One advantage of uing adamard matrix (of ize *.9 ) to generate the bi-orthogonal ignal et i that it ha a nice property that the correlation can be calculated in 8./ operation. The total number of ad-. 3 ditionnvolved (neglecting negation) in order to calculate the branch metric i given in Table for both the traditional SOTC and the propoed code. For a given code rate Z, the table give the number of computation needed by the propoed coding The complexity analyi of TC i not conidered in thi work.

5 cheme if ignal are ued from the complete bi-orthogonal ignal et. The lat column correpond to uing all the ignal in the ignal et i.e., the computational complexity of the traditional SOTC. Note that a tremendou amount of complexity reduction i achievable with the new coding cheme. Bit Error Rate 2 Rate /63 coding+ crambling Rate /3 coding+ preading E /N b o Fig. 9. Performance of Rate-/3 parent code v Rate-/63 code in flat Rayleigh fading channel. Simulation reult for Rayleigh fading channel are hown in Figure 4. In thi cae, an additional channel interleaver i alo included. The fading amplitude i kept contant over a block of 63 coded ymbol and independent among block. Simulation reult how that an additional coding gain of 2 db can be achieved with the application of low rate coding. Note from Low rate Coding Analyi Coding+Spreading Analyi Low rate Coding Simulation Coding+Spreading Simulation Gauian. Due to complexity, imulation were carried out only for a maximum of 8 uer. VI. CONCLUDING REMARKS We have propoed a imple method to generate a cla of very low rate turbo code from rate- contituent code that can be ued for combined coding and preading in pread pectrum multiple acce ytem. Apart from being rate compatible for variou rate, thee code allow for low complexity encoder and decoder implementation. REFERENCES [] S. Dolinar, D. Divalar, and F. Pollara, Code Performance a a function of Block Size, TMO Progre Report [2] C. Berrou, A. Glavieux, and P. Thitimajhima, Near Shannon limit errorcorrecting coding and decoding: turbo-code, in Proc. ICC 93, Geneva, Switzerland, May 993, pp [3] L. Ping, W. K. Leung and K. Y. Wu Low rate Turbo-adamard Code, in Proc. ISIT 2. [4] P. Komulainen and K. Pehkonen, Performance evaluation of uperorthogonal turbo coden AWGN and flat Rayleigh fading channel, IEEE JSAC, vol.6, no.2,pp.96-25, Feb [5] D. Thirupathi and K. M. Chugg, A imple contruction of low rate convolutional code with application to low rate turbo-like code deign, accepted for publication in Globecom 22. [6] P. D. Papadimititiou and C. N. Georghiade, On Aymptotically optimum rate /n convolutional code for a given contraint length, IEEE Comm. Letter, Vol. 5, NO., pp , Jan. 2. [7] A. J. Viterbi, Very low rate code for maximum theoretical performance of pread pectrum multiple-acce channel, IEEE J. Select. Area Commn., vol. 8, pp , May 99. [8] K. M. Chugg, A. Anataopoulo, and. Chen, Iterative Detection: Adaptivity, Complexity Reduction, and Application, Kluwer Academic Publiher, MA, 2. [9] J. P. Chaib and. Leib, Very low rate Trelli/Reed-Muller (TRM) Code, IEEE Tran. on Comm., vol. 47, No., pp , Oct [] A. Viterbi, CDMA: Principle of Spread Spectrum Communication, Addion-Weley, Reading, USA, 995. [] P. D. Shaft, Low rate convolutional code applicationn pread pectrum communication, IEEE Tran. on Comm., vol-com 25, No. 8, pp , Aug Additional E b /N o required Additional number of uer Fig.. Additional šªm«k c (in db) required to maintain ingle uer performance in multiple acce channel Figure 5 that in a heavily loaded ytem, with ingle uer detector, a ignificant reduction in ƒ BF Œ required to maintain ingle uer performance i poible by uing the low rate coded cheme 2 intead of conventionally coded (rate /3 turbo code) and pread cheme. Multiple accenterference i modelled a In thi cae, uer are eparated by crambling equence