STACK DECODING OF LINEAR BLOCK CODES FOR DISCRETE MEMORYLESS CHANNEL USING TREE DIAGRAM

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1 H PRASHANTHA KUMAR et. al.: STACK DECODING OF LINEAR BLOCK CODES FOR DISCRETE MEMORYLESS CHANNEL USING TREE DIAGRAM DOI:.297/ijct STACK DECODING OF LINEAR BLOCK CODES FOR DISCRETE MEMORYLESS CHANNEL USING TREE DIAGRAM H. Pashantha Kuma, U. Sipati 2, K. Rajesh Shetty 3 and B. Shankaananda 4,2,3 Depatment of Electonics and Communication Engineeing, National Institute of Technology Kanataka, India hpashanthakuma@gmail.com, 2 sipati_achaya@yahoo.co.in and 3 kshetty_nitte@yahoo.co.in 4 Vivekananda Institute of Technology, India bsnanda2@gmail.com Abstact The boundaies between block and convolutional codes have become diffused afte ecent advances in the undestanding of the tellis stuctue of block codes and the tail-biting stuctue of some convolutional codes. Theefoe, decoding algoithms taditionally poposed fo decoding convolutional codes have been applied fo decoding cetain classes of block codes. This pape pesents the decoding of block codes using tee stuctue. Many good block codes ae pesently known. Seveal of them have been used in applications anging fom deep space communication to eo contol in stoage systems. But the pimay difficulty with applying Vitebi o BCJR algoithms to decode of block codes is that, even though they ae optimum decoding methods, the pomised bit eo ates ae not achieved in pactice at data ates close to capacity. This is because the decoding effot is fixed and gows with block length, and thus only shot block length codes can be used. Theefoe, an impotant pactical question is whethe a suboptimal ealizable soft decision decoding method can be found fo block codes. A notewothy esult which povides a patial answe to this question is descibed in the following sections. This esult of nea optimum decoding will be used as motivation fo the investigation of diffeent soft decision decoding methods fo linea block codes which can lead to the development of efficient decoding algoithms. The code tee can be teated as an expanded vesion of the tellis, whee evey path is totally distinct fom evey othe path. We have deived the tee stuctue fo (8, 4) and (6, ) extended Hamming codes and have succeeded in implementing the soft decision stack algoithm to decode them. Fo the discete memoyless channel, gains in excess of.5db at a bit eo ate of -5 with espect to conventional had decision decoding ae demonstated fo these codes. Keywods: Extended Hamming Codes, Tee Diagam, Soft Decision Decoding, Discete Memoyless Channel, Fano Metic. INRODUCTION Eo contol coding (ECC) is commonly used to achieve eliable tansmission of infomation. Channel codes enable a decode to ecove fom eos poduced by noise in a communication channel. Codes ensue highe noise toleance at the eceive by adding edundancy into the use data to achieve bette sepaation of data sequences. ECC algoithms have constituted a significant enable in the telecommunications evolution, the intenet, digital ecoding and space exploation. The past decade has seen temendous gowth in availability and deployment of wieless sevices. This has been made possible by development of poweful signal pocessing algoithms to ensue efficient spectal usage/eo fee communication and development of hadwae platfoms on which these algoithms could be un. Thus developments in algoithm design and micoelectonics have gone hand in hand to ceate the infastuctue fo the infomation evolution that has tansfomed the way in which human beings live and wok []. Eo coecting codes can be divided into two classes accoding to the manne in which edundancy is added to the messages: block and convolutional. Block codes implement a one-to-one mapping of a set of k infomation symbols on to a set of n codewod symbols. We call this code as an (n, k) linea block code. The n-k symbols in a codewod ae a function of the infomation symbols, and povide edundancy that can be used fo eo coection and/o detection puposes. The minimum distance d min of a block code C is the smallest Hamming distance between any two codewods in the code. Both types of coding schemes have found pactical applications. Histoically convolutional codes have been pefeed, appaently because of the availability of the softdecision Vitebi decoding algoithm and the belief ove many yeas that block codes could not be efficiently decoded with soft-decisions. The main poblem is the fundamentally algebaic stuctue of block codes. Although this stuctue allows elegant algebaic decoding techniques to be applied when had decisions ae made, the eliance on finite field aithmetic fo decoding makes it difficult to exploit soft decisions. The beak though which enabled the possibility of using soft decision decoding (SDD) fo decoding block codes was povided by [2] who showed that any linea block codes can be epesented by a tellis, and that the Vitebi algoithm can theefoe be used fo soft decision decoding. Fo example, a (5, 4) paity check code is epesented by the paity check matix, H = [ ] Its syndome tellis is shown in Fig.. Fig.. Tellis epesentation of a (5, 4) paity check code It is inteesting to note that thee is no need to label the banches with the coded bits. A tansition between two states with the same level coesponds to coded bit. Linea block codes have tellises with a time-vaying numbe of states. This tellis is simple and has a egula stuctue. The minimum numbe of states can be quite lage, fo example, 2 64 fo the (28, 64) extended BCH code [3]. Although a cetain pemutation of the code achieves 2 43 states, which is still exceedingly lage fo pactical implementations of the Vitebi o 498

2 ISSN: (ONLINE) ICTACT JOURNAL ON COMMUNICATION TECHNOLOGY, MARCH 22, VOLUME: 3, ISSUE: Bahl-Cocke-Jelinek-Raviv (BCJR) algoithm [4]. In spite of exponential incease in computational complexity, the soft decision decoding using tellis diagam pefoms 2 to 3dB bette than had decision decoding (HDD) ove additive white Gaussian noise (AWGN) channel. This much amount of coding gain is vey significant. 3dB of coding gain can educe the equied bandwidth by 5% o incease data thoughput by a facto of 2 o incease ange by 4% o educe antenna size by 3% o educe tansmitte powe by a facto of 2. Theefoe collectively we can say that coding gain inceases the system pefomance o educes cost o both [5]. Since SDD inceases the eo coecting capability of the code by coecting moe numbe of soft eos and hencefoth inceases the coding gain compae to HDD. The potential of SDD ove HDD is illustated in Fig.2 fo the (5, 4) single paity check code. Fom the gaph we conclude that at bit eo ate (BER) of -5, SDD using Vitebi algoithm pefoms 2.3dB bette than HDD. pat of the tee. Whethe a paticula path is likely to be pat of the maximum likelihood path depends on the metic value associated with that path. The metic is a measue of the closeness of a path to the eceived sequence [7]. Evey linea block code can be epesented gaphically by means of a tee. Fig.3 epesents geneal tee epesentation fo an (n, k) systematic linea block code. Root S Fist level k th level n th level T e m i n a l n o d e s 2.3 db (n k ) paity symbols Fig.3. Tee epesentation of a binay (n, k) block code Fig.2. Pefomance fo the (5, 4) block code with HDD and SDD ove AWGN channel using Vitebi algoithm 2. TREE REPRESENTATION OF SYSTEMATIC LINEAR BLOCK CODES It is well known that the fixed amount of computation equied by the Vitebi algoithm is not always needed, paticulaly when the noise is light o signal to noise atio is high [6]. Fo example, assume that an (n, k) linea block code is tansmitted without eo ove a channel. The Vitebi algoithm will still pefom on the ode of 2 min{k, n-k} computations pe decoded infomation block, all of which is wasted effot in this case. In othe wods, it is often desiable to have a decoding pocedue whose effot is adaptable to the noise level. Sequential decoding using tee diagam is such a type of algoithm. Sequential decoding descibes any algoithm fo decoding channel codes which successively exploes the code tee by moving to new nodes fom an aleady exploed node. The pupose of tee seaching algoithms is to seach though the nodes of the code tee in efficient way, that is, without having to examine too many nodes, in an attempt to find the maximum likelihood path. Each node examined epesents a path though This tee has the following stuctues: ) Tee consists of n + levels. 2) Fo i k, thee ae 2 i nodes at the i th level of the tee. Thee is only one node s at the zeoth level of the tee called the initial node (o oot) of the tee, and thee ae 2 k nodes at the n th level of the tee, which ae called the teminal node of the tee. 3) Fo i k, thee ae two banches leaving evey node s i at level -i and connecting to two diffeent nodes at level (i+). One banch is labeled with an infomation symbol, and the othe banch is labeled with an infomation symbol. Fo k i n, thee is only one banch leaving evey node s i at level -i and connecting to one node at level (i+). This banch is labeled with a paity check symbol, eithe o. 4) The label sequence of path connecting the initial node s to a node s k at the k th level coesponds to an infomation sequence m of k bits. The label sequence of the path connecting the initial node s though a node s k at the k th level to a teminal node s n at the n th level is a codewod C. The label sequence of the tail connecting 499

3 H PRASHANTHA KUMAR et. al.: STACK DECODING OF LINEAR BLOCK CODES FOR DISCRETE MEMORYLESS CHANNEL USING TREE DIAGRAM node s k to node s n coesponds to the n-k paity check symbols of the codewod. The geneato matix G of an (8, 4) extended Hamming code with minimum Hamming distance d min =4 in systematic fom is given below. G The above geneato matix geneates all possible valid codewods in systematic fom. Fig.4 shows the tee epesentation of a binay extended Hamming code geneated by G. Fist Level S Fouth Level Eighth Level Fig.4. Tee epesentation of an (8, 4) Hamming code 3. DECODING WITH THE STACK ALGORITHM The tee epesentation of a linea block code can be used to facilitate stack decoding. In the Zigangiov - Jelinek (ZJ) o stack algoithm, an odeed list o stack of peviously examined paths of diffeent lengths is kept in stoage. Each stack enty contains a path along with its metic, the path with the lagest metic is placed on top, and the othes ae listed in ode of deceasing metic. Each decoding step consists of extending the top path in the stack by computing the Fano metics of its succeeding banches and then adding these to the metic of the top path to fom new paths, called the successos of the top path. The top path is then deleted fom the stack, its successos ae inseted and the stack is eaanged in ode of deceasing metic values. When the top path in the stack has highest Fano metic and also it is the end of tee, the algoithm teminates. Stack algoithm commonly uses a pobabilistic banch metic, namely, the Fano metic, which can be witten fo a continuous (o Gaussian) channel as [3] (2vl ) l Es M ( l vl ) log 2[ exp( 4 )] () whee, M( v ) is the banch metic fo the l th banch, E s is the enegy pe tansmitted bit and N is the one-sided noise powe density. Fo a discete memoyless channel with a unifomly distibuted souce and a cossove pobability p, the above Fano metic educes to M ( l vl ) log 2 p( l v ) log p( l ) R l 2 (2) Hee, R is the ate of the code in use, p( v ) is the channel tansition pobability of the eceived symbol given the tansmitted symbol v, p( ) is a channel output symbol pobability [8]. Fano s oiginal selection of this metic was based on a heuistic agument, and on occasion othe eseaches/designes have used othe metics. We assume a binay phase shift keying (BPSK) modulation, whee the bits c i {, } ae mapped to the tansmission bits x i {+, -} coesponding to the elation x ( ) c i i ; i [, n] (3) Afte tansmission ove the AWGN channel, we obtain the pobability distibution depicted in Fig.5. p(y ) p(y i x i = ±) p(y ) 2 3 N Fig.5. PDF fo eceived symbol y We assume that the y-axis in pevious figue is divided in to intevals of width y. In pactical systems, this value is often quantized. In ou decode analysis, the eceived signal is quantized to 3 bits, esulting in 2 3 diffeent quantization levels, using unifomly spaced quantization thesholds [9]. The block intepets 4 as the most confident decision that the codewod bit is a and intepets 4 as the most confident decision that the codewod bit is a. The values in between these epesent less confident decisions. Thus a binay input, continuous valued output has changed to 8-ay DMC. Below figue shows 8-level soft quantized DMC. y 5

4 ISSN: (ONLINE) ICTACT JOURNAL ON COMMUNICATION TECHNOLOGY, MARCH 22, VOLUME: 3, ISSUE: Table.(a). Fano metics fo binay input, 8-ay output DMC v The metics ae scaled by 9/.49 to obtain intege metics as shown below. v Table.(b). Scaled Fano metics Fig.6. Binay input 8-ay output DMC We now poceed to a desciption of the stack decoding algoithm by means of a set of ules fo updating the stack enties and fo backwad o fowad tanslations though the tee. Step : Load the stack with the oigin node in the tee, whose metic is taken to be zeo. Step 2: Compute the metics of the successos of the top path in the stack. Step 3: Delete the top path fom the stack. Step 4: Inset the new paths in the stack and eaange the stack in ode of deceasing metic values. Step 5: If the top path in the stack ends at teminal node in the tee with highest metic value, stop. Othewise, etun to Step 2. When the algoithm teminates, the top path with highest metic in the stack is taken as the decoded path. We pesent an example to illustate these ideas. Example : A binay (8, 4) extended Hamming code associated with the tee in Fig.4 is used to encode the infomation sequence x=(), esulting in the codewod v = (). This codewod is tansmitted ove the binay input, 8-ay output DMC with tansition pobabilities p( v) given by the enties [] shown in Table.. Table.. Tansition pobabilities p( v) fo binay input, 8-ay output DMC v Use this Fano metic and apply stack algoithm to obtain the tansmitted codewod. The esults ae shown in Table.2. Table.2. Decoding steps fo the stack algoithm Step (9) (-3) Step 4 (25) (9) Step 7 (3) (9) (-3) (-22) (-3) Step 2 (8) (-22) (-3) Step 5 (23) (9) (-3) (-22) (-3) Step 8 (9) (-) (-3) (-22) (-3) Step Step (-27) (-8) (-) (-) (-3) (-3) (-22) (-22) (-3) (-3) Step 3 (27) (-3) (-22) (-3) Step 6 (2) (9) (-3) (-22) (-3) contd., Step 9 (-9) (-) (-3) (-22) (-3) Step 2 (-9)Successful Decoding (-) (-3) (-22) (-3) Hee we can note that stack algoithm coected thee bit soft eos. We have checked the algoithm exhaustively fo many eo pattens of this kind and found that stack decode coected all of them. Hence, by employing this appoach, we wee able to coect many eo pattens of weight exceeding the eo coecting capability (had decision) of the code. Simulations have been pefomed by employing eight level soft quantization and known channel state infomation (CSI). Simulation esults (Fig.7 and Fig.8) quantify the bit eo ate (BER) fo HDD and stack decoding of (8, 4) and (6, ) Hamming codes. The sequence = is eceived. Eo bits (4 th, 5 th, and 6 th position) ae shown in dak. Using Eq.(2), the Fano metic is computed as follows. 5

5 H PRASHANTHA KUMAR et. al.: STACK DECODING OF LINEAR BLOCK CODES FOR DISCRETE MEMORYLESS CHANNEL USING TREE DIAGRAM 2 db Fig.7. Pefomance fo the (8, 4) extended Hamming code ove AWGN with HDD and Stack decoding 5 Fo an (8, 4) Hamming code, a BER of using HDD equies an SNR of 9.8dB while fo the soft decision stack algoithm, the same BER is achieved with 7.8dB. Hence, soft decision stack decoding pefoms 2dB bette than HDD..6 db in this pape can be used to decode any linea block code. Sequential decoding schemes have some dawbacks (such as vaiable decoding effot) that ae well known in the context of decoding of convolutional codes. Since block codes have a finite tee, the aveage numbe of computations and the decoding effot ae always bounded. Vey noisy eceived sequences typically equie a lage numbe of computations with a stack decode, sometimes moe than the fixed numbe of computations equied by the Vitebi algoithm; howeve, since vey noisy eceived sequences do not occu vey often, the aveage numbe of computations pefomed by a stack decode is nomally much less than fixed numbe pefomed by the Vitebi algoithm. It is well known that the (8, 4) single bit eo coecting extended Hamming code with had decision decoding coects only a single bit eo ove the span of the codewod, while soft decision stack decoding coects many thee bit pattens of soft eos. This in tun esults in a coding gain fo tansmission ove the AWGN channel when compaed to HDD. One of the main challenges in adoption and deployment of wieless netwoked sensing applications is ensuing eliable senso data collection and aggegation, while satisfying the low cost, low enegy opeating constaints typical of such applications. A wieless senso netwok is inheently vulneable to diffeent souces of uneliability due to tansient failues in cicuits and communication channels. The souces of uneliability can be classified into two categoies: () faults that change behavio pemanently, and (2) failues that lead to tansient deviations fom nomal behavio, temed as soft failues []. Hence it is necessay to povide a pope eo contol scheme to educe the bit eo ate in such applications. Asymmetic codes with low encoding complexity (encoding is usually pefomed at senso nodes which ae simple in constuction and have powe constaint), possessing modest eo coecting capability (because of low data ates and poximity between tansmitte and eceive) with modeately high decoding complexity (decoding is usually done at the base station which has geate esouces than the senso nodes) ae employed. The codes along with the decoding schemes poposed in this pape ae well suited to meet this equiement. REFERENCES Fig.8. Pefomance fo the (6, ) extended Hamming code ove AWGN with HDD and Stack decoding In a simila manne, it is obseved that the (6, ) Hamming code with HDD achieves a BER of -5 at an SNR of 8.6dB while the soft decision stack algoithm achieves the same BER at an SNR of 7dB. Thus, soft decision stack decoding pefoms.6db bette than HDD. 4. CONCLUSION A simple, efficient and nea optimal decoding scheme fo linea block codes using tee epesentation has been poposed in this pape. It is inteesting to notice that the technique poposed [] Daniel J. Costello and G. David Foney, Channel coding: The oad to channel capacity, Poceedings of IEEE, Vol. 95, No. 6, pp. 5-77, 27. [2] J.K. Wolf, Efficient maximum likelihood decoding of linea block codes using a tellis, IEEE Tansactions on Infomation Theoy, Vol. 24, No., pp. 76-8, 978. [3] Vladislav Sookine and Fank R. Kschischang, A Sequential Decode fo Linea Block Codes with a Vaiable Bias-Tem Metic, IEEE Tansactions on Infomation Theoy, Vol. 44, No., pp. 4-46, 998. [4] L.R. Bahl, J. Cocke, F. Jelinek and J. Raviv, Optimal decoding of linea codes fo minimizing symbol eo ate, IEEE Tansactions on Infomation Theoy, Vol. IT-2, pp , 974. [5] B. Thomson, Impoving bandwidth utilization with tubo poduct codes, available fo download at

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