Low Complexity Soft-Input Soft-Output Hamming Decoder

Transcription

1 Low Complexty Soft-Input Soft-Output Hammng Der Benjamn Müller, Martn Holters, Udo Zölzer Helmut Schmdt Unversty Unversty of the Federal Armed Forces Department of Sgnal Processng and Communcatons Holstenhofweg 85, Hamburg, Germany Abstract We nvestgate a low complexty Soft-Input Soft-Output (SISO) Hammng Der. The Decodng s based on error patterns whch belong to the same syndrome. It s shown that t s suffcent to nvestgate error patterns wth one and two errors to gan up to 1.35 db compared to hard decson decodng. The proposed decodng algorthm has a lnearly rsng complexty, O(N c), wth the word length N c. The further consderaton of error patterns wth three errors whch belong to the determned syndrome gan further 0.2 db and mproves the qualty of the soft-output due to the ncreased number of comparsons wth vald words. However, ths also ncreases the complexty of the decodng process to O(N 2 c ). We present smulaton results for soft decodng of Hammng s up to a word length of 63 bt. Furthermore, we present results for turbo decodng wth the 63, 57-Hammng as a component. Index Terms Syndrome based soft decodng, Hammng Code, low complexty, soft-output, turbo decodng I. INTRODUCTION The decodng algorthms of Hammng-s were nvestgated n several papers. The exhaustve maxmum lkelhood decodng of a (63, 57)-Hammng would requre the comparson of 2 57 vald words. Chase reduced the complexty by checkng a fxed number of the error patterns wth a slght performance degradaton [1]. A further reducton of consdered error patterns was obtaned n [2]. Based on ths approach we show that the group of error patterns can be reduced to sngle and double errors to gan up to 1.35 db compared to hard decson decodng (HDD). In contrasts to the proposed decodng technque there are Trells-based decodng algorthms [3] [4] whch have a quadratc complexty O(N 2 c ) for Hammng and Reed-Muller s. A further dfferent approach was consdered n [5], whch s based on systematc bt-flppng. In [6] MAP decodng s performed based on Hadamard transforms, whch leads to a complexty of O(N c ldn c ). One of the here proposed decodng algorthms can acheve a lnear complexty wth a performance degradaton of 0.2 db compared to maxmum lkelhood decodng. For a (63, 57)-Hammng, only 32 vald words have to be compared n terms of ther soft nformaton whch s equvalent to a lnear complexty O(N c ), wth the word length N c. In addton, the proposed soft-output decodng makes the algorthm sutable for further applcatons lke turbo decodng. II. ENCODING AND TRANSMISSION The encodng of the message bts a can be performed by a modulo 2 vector matrx multplcaton of a and the generator matrx G c a G (1) The expresson s equvalent wth c = (a G) mod 2. The generator matrx of systematc 7, 4-Hammng s gven by G = (2) } {{ } I } {{ } P c s modulated, so that a logcal zero s equvalent to a +1 and a logcal one s equvalent to a 1, x {+1, 1}. The modulated sgnal x s dstorted by the addtve whte Gaussan nose (AWGN) w and results n the receve sgnal y, y = x + w. (3) III. HARD DECISION DECODING For hard decson decodng (HDD) t s requred to derve the bt sequence c from the dstorted sgnal y. The syndrome z can be calculated as follows z c H T (c + e) H T, (4) where H s the party check matrx and e s the error pattern belongng to the syndrome. In ths manner, every syndrome leads to exactly one sngle error pattern and the decodng can be performed based on a syndrome table. If the error pattern s 0, the syndrome s also 0 whch means that the receved word s a vald word and no decodng s requred. Error patterns wth 2 (duets) or 3 errors (trplets) whch belong to the same syndrome are not taken nto account for the decodng and the dstorted word c s corrected to

2 ĉ c + e. (5) In fact, every double error s ded to a vald but wrong word. Ths explans the poor performance of HDD for Hammng s, whch are llustrated n Fg. 2, 3, 4 and 5 for the dfferent word lengths. IV. SYNDROME BASED SOFT DECISION DECODING For the syndrome based soft decson decodng t s requred to calculate the log-lkelhood ratos (LLR or L-values) from the receved sgnal y, whch fnally leads to L (x y) = ln ( exp L (x y) = ln exp P (x = +1 y) P (x = 1 y), (6) E b (y 1) 2) ( E b (y + 1) 2) = 4E b y, (7) assumng that a logcal zero and one have the same probablty [7]. Let us assume that the syndrome of a dstorted bt sequence of a 7, 4-Hammng s z = (0 0 1). The possble error patterns are collected n matrx E wth ts elements e j, E = , (8) where the second to fourth row bears the duets and the ffth to eghth row bears the trplets. The error patterns for all syndromes are determned n advance and stored n a lst. The sze of the lst rses quadratcally for double errors and cubcally for trple errors (see Tab. I). Every row of E s multpled by the absolute value of log-lkelhood ratos of the receved sgnal L (x y). Afterwards, the resultng row vector s added up. The vector wth the lowest sum of L-values suggests the error pattern wth the hghest probablty of a correct decodng. For the 7, 4-Hammng N e1 + N e2 = 4 error patterns have to be multpled by L (x y) to cover sngle and double errors. If t s requred to take trplets nto account, N e3 = 4 more error patterns have to be multpled by L (x y). In order to estmate the complexty, the number of duets belongng to one syndrome s gven by N e2 = 1 N c! N c (N c 2)! 2! = N c 1, (9) 2 wth N c beng the length of the word. The number of trplets can be calculated smlarly by N e3 = 1 ( N c! N c (N c 3)! 3! N ) e2 = N c 1 Nc (10) Eq. 9 and 10 show that the complexty rses lnearly for the duets and quadratcally for the trplets. Table I shows the number of duets N e2 and trplets N e3 belongng to one syndrome. TABLE I NUM OF DUETS N e2 AND TRIPLETS N e3 BELONGING TO ONE SYNDROME, SIZE OF ERROR PATTERN LISTS FOR ALL SYNDROMES N c N e2 Lst sze N e3 Lst sze V. SOFT-OUTPUT DECODING In general, soft-output decodng provdes output values for teratve or turbo decodng. In order to generate soft-outputs, the followng algorthm s proposed. The probablty values of a word are gven by P ( c j = c j y j ) = exp ( L (x j y j ) ) 1 + exp ( L (x j y j ) ). (11) In the next step, the probablty values are multpled columnwse for the gven error pattern of every row. P = j { P ( cj = c j y j ) f e,j = 0 1 P ( c j = c j y j ) f e,j = 1 (12) Now P s normalzed, so that the sum of the normalzed probabltes P over all rows s equal to 1, P = 1. The normalzed probabltes P are gven by P P = P. (13) P can be nterpreted as the probablty of correct decodng for the gven error pattern of row. In a last step, the probablty that x j = +1, for a gven receved word y, s calculated by the sum of P over all rows, f e,j = c j, where c j s defned as the logcal receved bt sequence. ˆ ndcates the estmaton of the new probabltes after the soft decodng. ˆP (x j = +1 y) = e,j= c j P (14) Due to the normalzaton, so that P = 1, the probablty of ˆP (x j = 1 y) can be calculated by ˆP (x j = 1 y) = 1 ˆP (x j = +1 y). (15) In order to exchange the nformaton for turbo decodng t s requred to calculate L-values from the derved probabltes.

3 VI. TURBO DECODING Wth the ablty of soft-output decodng, we can utlze the soft-outputs for turbo decodng. In order to do so, the systematc data bts have to be end twce. The frst encodng can be performed as shown n Eq. 1. The second encodng requres an addtonal nterleaver pror to the encodng. We only nvestgate an nterleaver of the length of the systematc bts of one word. The nterleaved data bts are bt reverse to the unnterleaved data bts. Ths short nterleaver enables smlar delays as for the orgnal length. Fnally the unnterleaved and the party bts of the frst enr and the second enr are multplexed to the word whch s transmtted. Fg. 1 shows the schematc structure of a turbo der. The receved word y s separated nto systematc bts and the party bts of the frst enr and the party bts of the second enr. Based on the soft-nput the enrs calculate soft-outputs as descrbed n secton V. The enr output, subtracted by ther soft-nput, result n the extrnsc nformaton of the decodng process. Ths extrnsc nformaton s drected to the other enr and s added to the soft nformaton of the receved systematc bts. TABLE II SIMULATION RESULTS FOR REQUIRED E b / IN db FOR A BIT ERROR RATE OF AND THE RESULTING CODING GAIN (UNCODED 8.37 db FOR = ) N c K c HDD Gan Duets Gan Trplets Gan (7,4) HDD (7,4) duets (7,4) trplets unon bound y Demux Party 1 Party 2 Int. Fg. 1. Systematc Int. Der 1 Der Schematcs of turbo decodng VII. SIMULATION RESULTS Dent. Ded systematc Fg. 2. Bt error rate for dfferent types of decodng for a 7, 4-Hammng The extenson of the word length, up to 15 bt, results n a further performance gan. It s also apparent that the dfference between duet decodng and trplet decodng rses. The codng gan amounts to 0.95 db for the HDD and 2.23 db for the duet decodng. Further 0.1 db can be ganed by trplet decodng. (15,11) HDD (15,11) duets (15,11) trplets unon bound For the smulaton results Hammng s of a word length for 7 tll 63 bt were nvestgated. Fg. 2, 3, 4 and 5 llustrate the performance of the dfferent decodng strateges for a certan word length. Tab. II summarzes the results for all non-teratve s. Fg. 2 shows the bt error rate of the 7, 4-Hammng for dfferent types of decodng. It s shown that the decodng performance of the duet and trplet decodng s qute smlar and very close to the unon bound whch s an upper bound for the bt error probablty after maxmum lkelhood decodng. For the evaluaton we focus on a =. The codng gan amounts to 0.31 db for the HDD and 1.66 db for the duet decodng. Fg. 3. Bt error rate for dfferent types of decodng for a 15, 11-Hammng The 32, 26-Hammng obtaned the best results for the non-teratve s, for duets as well as for trplets. Fg. 4 shows that the codng gan amounts to 2.35 db for the duet decodng and 2.49 db for the trplet decodng.

4 (31,26) HDD (31,26) duets (31,26) trplets unon bound process. In addton, the assumpton of a normalzaton for P = 1 can lead to naccurate soft-outputs. Fg. 4. Bt error rate for dfferent types of decodng for a 31, 26-Hammng Qute a smlar pcture can be drawn for the 63, 57-Hammng whch has the hghest rate (R c = 0.905) of the consdered Hammng s (see Fg. 5). The codng gan s lower than for the 32, 26-Hammng (2.26 db gan for the duet and 2.45 db gan for the trplets), but the bt error curve falls more sharply. It s also shown that the dfference between duet and trplet decodng s the hghest wth 0.19 db. Fg. 5. (63,57) HDD (63,57) duets (63,57) trplets unon bound [8] Bt error rate for dfferent types of decodng for a 63, 57-Hammng Fg. 6 shows the smulaton results of the bt error rate of turbo decodng wth the 63, 57-Hammng as a component. The soft-outputs for the turbo decodng were calculated as shown n secton V for double and trple errors. The turbo der consderng sngle error, duets and trplets performs 0.36 db better than the non-teratve trplet der. Turbo decodng only consderng sngle and double errors leads to no further gan compared to non-teratve duet or trplet decodng. In fact, the codng gan decreases by 0.5 db compared to non-teratve duet decodng. Ths can be explaned wth the small numbers of comparson wth other vald words whch lead to an naccurate soft-output after the decodng (63,57) duets (63,57) trplets (69,57) Turbo duets (69,57) Turbo trplets Fg. 6. Comparson of non-teratve decodng and turbo decodng for duet and trplet decodng VIII. CONCLUSION We have proposed a soft-nput Hammng der whch ether consders error patterns wth up to 2 or up to 3 errors belongng to the determned syndrome. The complexty of duet decodng rses lnearly wth the word length and quadratcally for trplet decodng. The duet decodng can gan up to 1.35 db compared to hard decson decodng (= ). Consderng error patterns up to 3 errors gans further 0.2 db, where the gan for trplet decodng ncreases wth the word length. Furthermore, we proposed a soft-output der based on the soft-nput Hammng der. The soft-outputs where utlzed for turbo decodng. It was shown that duet decodng s unsutable for turbo decodng, due to the poor qualty of the soft-outputs. The turbo decodng based on trplet decodng (69, 57-Hammng ) showed a performance gan of 0.36 db compared to the trplet decodng of the 63, 57-Hammng. These results were obtaned wth the smallest possble nterleaver of one data word length. REFERENCES [1] D. Chase, Class of algorthms for decodng block s wth channel measurement nformaton, Informaton Theory, IEEE Transactons on, vol. 18, no. 1, pp , Jan [2] J. Snyders, Reduced lsts of error patterns for maxmum lkelhood soft decodng, Informaton Theory, IEEE Transactons on, vol. 37, no. 4, pp , July [3] J. Wolf, Effcent maxmum lkelhood decodng of lnear block s usng a trells, Informaton Theory, IEEE Transactons on, vol. 24, no. 1, pp , Jan [4] T. Kasam, T. Takata, T. Fujwara, and S. Ln, On the optmum bt orders wth respect to the state complexty of trells dagrams for bnary lnear s, Informaton Theory, IEEE Transactons on, vol. 39, no. 1, pp , Jan [5] Smon Hrst and Bahram Honary, A smple soft-nput/soft-output der for hammng s, n Proceedngs of the 8th IMA Internatonal Conference on Cryptography and Codng, London, UK, 2001, pp , Sprnger-Verlag.

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