A soft decision decoding of product BCH and Reed-Müller codes for error control and peak-factor reduction in OFDM

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1 A soft decision decoding of product BCH and Reed-Müller codes for error control and pea-factor reduction in OFDM Yves LOUET *, Annic LE GLAUNEC ** and Pierre LERAY ** * PhD Student and ** Professors, Departent of Electronics SUPELEC, High Engineering Institute - Capus of Rennes Avenue de la Boulaie - BP 8, 355 CESSON-SEVIGNE - FRANCE Abstract - Orthogonal Frequency Division Multiplexing (OFDM) is a very attractive odulation schee for data transission in ultipath channels. But the high Pea-to-MEan Power Ratio (PMEPR) of the odulated signal is a ajor drawbac and aes the use of soe circuits very tricy : for instance, power aplifiers at the eission side could have nonlinear characteristics and a high PMEPR could exhibits interodulation noise and distortions. A very recent theory based on Reed-Müller codes controls the PMEPR and exploits the correction capacity of the code to correct error propagation : the PMEPR is reduced to a level of 3 db and the distance of the code is the quarter of the codeword (in this specific application of Reed-Müller and OFDM). But the correction capacity is liited and have to be balanced with the code rate. This paper first presents a product coding schee based on BCH and Reed-Müller codes to iprove error correction and to eep the PMEPR propriety of the odulated signal. Next, we present a soft decision decoding based on Chase s algorith associated to Reed-Müller codes. Then, we analyse the BER perforances of both hard and soft decoding associated to the product coding schee odelling AWGN and nonlinear TWTA power aplifiers. With this soft decision decoding, we obtain a db coding gain copared to hard decoding when using product codes and about 5 db coding gain copared to an uncoded syste, at BER=0-4. Balanced with the global code rate, BCH(55,3,7) with Reed-Müller codes have higher coding gain copared to other product codes. Nonlinear distortions degrade BER perforances and the coding gain is still about db at BER=0-4. The applications concerns high data rate and broadband transissions using OFDM where power efficiency is a ajor challenge. Key-Words - OFDM, pea-factor, coding, Reed-Müller codes, soft decoding, nonlinear aplifiers. Bacground. The OFDM odulation The advantages of the OFDM odulation are now well nown. Based on the concept of transitting the data in parallel QAM odulated sub-carriers using frequency division ultiplexing, the OFDM odulation requires no adaptation to instantaneous channel responses and is easy to ipleent thans to the progresses in the electronics field these recent years []-[4]. The DAB (Digital Audio Broadcasting) and the DVB-T (Digital Video Broadcasting by terrestrial networs) technologies are based on OFDM. But the ajor drawbac of OFDM is the high pea-factor of the odulated signal (defined as the instantaneous power to the ean power ratio, designed as the PMEPR). This could exhibits distortions and interodulation noise when the signal is passing through nonlinear circuits lie power aplifiers. Let N be the nuber of carriers, C, ={0,...,N-}, the coplex inforation sybols vector and T S the OFDM sybol length. The coplex envelope of the odulated signal is : S ( t ) = N = 0 C and the PMEPR of S(t) is : e t iπ TS ()

2 PMEPR 0 t TS [ S ] = () T ax S S TS 0 S We can show, that without processing, PMEPR S(t) =0Log(N) (db). The PMEPR characterises the signal s fluctuations and the idea is to reduce the dynaic of S(t), i.e. its PMEPR.. The Reed-Müller coding schee A novel idea based on Reed-Müller codes [4] reduces the PMEPR of a signal on the one hand and exploits on the other hand the correction capacity of the code to correct errors : the PMEPR is reduced to a level of 3 db, theoretically apart fro N. This coding is based on the following theore which is the lin between the OFDM odulation and the Reed-Müller codes : let RM(,) be a first order Reed-Müller code of paraeter and G its generator atrix of rows x i. For a h -PSK odulation, the ey theore is as follow [5] : h T D = xπ ( ) xπ ( + ) + G u (3) = is a 3 db PMEPR vector for any perutation π of the sybols and for any inforation vector u {0,,..., h }. D=(d 0,,d,,d - ) is called a Golay sequence. Then, using the vector h (iπ d / ) C = e and the relation (), S(t) is a 3 db PMEPR signal. The code rate τ depends on and then on N= (the nuber of carriers) according to :! Log + h( + ) τ = h (4) and the Haing distance d H of the code is : N d H = = = (5) 4 4 Nevertheless, fro (4), the code rate decreases highly with N (or with ) and Fig. shows its evolution for a BPSK odulation (for higher constellation sizes, the code rate is still ore wea) : dt Fig. : code rate for a BPSK odulation According to (5), the code has correction capacity if N 6. This iplies a code rate τ 0.5 in BPSK and τ 0.4 in QPSK. These paraeters (N=6, QPSK) have been chosen for the coputer siulations. Fig. presents the OFDM bloc diagra used. RANDOM DATA BER ANALYSIS CODING DECODING MODULATE (IFFT) DEMODULATE (FFT) NON LINEAR AMPLIFIER Fig. : the OFDM syste diagra Properties of product codes AWGN. Introduction The principle of product codes is to cobine two codes C (n,, d ) and C (n,, d ). Placing the inforation bits in an array, the coding is as follow : the rows are coded with C and the n coluns are coded with C. The result code C = C C has paraeters : - n = n n - = - d = d d - R = R R where R is the code rate of C and R i the code rate of C i. The idea is to build ore powerful codes with large distances.. Application to OFDM The idea is to associate BCH and Reed-Müller codes for both error correction and pea-factor reduction. In our application, Reed-Müller code (RM) paraeters are n = 3, = 3 and d = 4. For equal BCH error

3 capacities, we have coputed product codes perforances for : P = BCH(3,6,7) RM, P = BCH(63,45,7) RM, P 3 = BCH(7,06,7) RM, P 4 = BCH(55,3,7) RM, Results are shown in Fig. 3. The BER is first estiated for hard Reed-Müller decoding in an AWGN channel. The nuber of carriers of the OFDM process is equal to 6, for a QPSK odulation on each carrier. According to the relation (), the odulated sybol C is : iπ C = e d h for a Golay sequence D=(d 0,,d - ). Then, the apping of the OFDM-Reed-Müller coding schee does not respect a Gray coding. Moreover, the all codewords aes the use of the MAP (Maxiu A- posteriori Probability) soft decision too coplex. That s why a soft decision decoding using Chase's algorith has been developed. This decoding selects the least reliable sybols according to a reliability function and replaces the by their nearest neighbours in the constellation. Then, it decodes the new sequences. Fig. 3 : BER for Reed-Müller hard decoding (with AWGN) o : P = BCH(3,6,7) RM, : P = BCH(63,45,7) RM, : P 3 = BCH(7,06,7) RM, : P 4 = BCH(55,3,7) RM,.3 Coents Fro Fig.3, for the sae error correction capacity (here d=7), we can first see that the ore the BCH code rate is close to, the ore its associated product code with Reed-Müller has a bad BER. If we have a loo to P, P, P 3 and P 4 code rates, they are respectively equal to 0., 0.9, 0.34 and So, the code rate has to be balanced with the BER of the product code. To iprove the BER of these product codes, we have set a soft decision decoding for the Reed-Müller codes. 3 A soft decision decoding for Reed- Müller codes 3. Introduction 3. The reliability function Let S be the received sybol, we first ae a hard decision to S and decide C as the eitted sybol []. For each sybol C l C we calculate a reliability function : where Λ l Pr( C / S ), l = Log Pr( C l / S ) Pr( x / x ) = e σ π ( x x ) σ (7) ( 6 ) In the constellation, we define S(x,y) and C l (x l,y l ). With independent AWGN(0,σ ) in X and Y and according to Bayes 's theore, we can show that : Λ l, l Pr( x, y / x, y ) d = Log = Pr( x, y / xl, yl ) l σ d where d l and d are respectively the euclidean distance between S and C l and between S and C. For each sybol, we calculate 3 associated reliable functions and affect it the iniu of the three values. Then, in the all sequences, we select the sybols which have the iniu affected values (which correspond to the least reliable sybols) and change the with their nearest neighbours in the constellation. For the coputer siulations, we have chosen N=6 (=4) and a QPSK odulation on each carrier (h=). So as to perfor the decoding, we have to decide how any L least reliable sybols we choose to apply Chase's theore. According to the ethod [7], we (8)

4 have taen L=4. This iplies a increasing coplexity of 4 + decoding copared to hard decoding. 3.3 Siulation results of product codes with a soft decision decoding associated to Reed- Müler codes Results are shown in Fig.4. The soft decoding is applied to the Reed-Müller codes. The channel is AWGN and the BER is estiated after the BCH decoding. signal S in (t), the input/output relationship of the aplifier can be expressed as : S out = f ( S in i[ ϕ in + g ( Sin )] ) e (9) where S out (t) is the output of the coplex signal. f(.) and g(.) describe AM/AM and AM/PM distortions. In this paper, we have focused on the TWTA (Traveling- Wave Tube Aplifier) odel where : A β 0 A f ( A) = ; g( A) = (0) + α A + βa A is the instantaneous envelope aplitude of the input signal. f has axiu value for A ax = /α [6]. Fig. 4 : BER for BCH and Reed-Müller product codes with hard (P and P ) and soft (P 3 and P 4 ) decoding : P = BCH(55,3,7) RM, : P = BCH(3,6,7) RM, : P 3 =BCH(55,3,7) RM, : P 4 = BCH(3,6,7) RM, We can see that the coding gains between hard and soft decoding is ore iportant for BCH(55,3,7) RM than for BCH(3,6,7) RM. They are respectively equal to db and.6 db at BER=0-4. But even if P 3 BER perforances are less significant copared to P 4, these results have to be balanced with the global code rate of the product code : for P 4, the code rate is equal to 0.09 and equal to for P 3. So the use of a soft decoding associated to BCH(55,3,7) RM product code sees to be a good coproise. 4 Nonlinear aplifier odels We have used nonlinear characteristics to focus on the PMEPR benefits of the Reed-Müller codes towards power efficiency. Power aplifiers have nonlinear characteristics in aplitude and phase (nown respectively as AM/AM and AM/PM transfer distortions). Assuing a coplex baseband representation of the odulated signals and an input (a) (b) Fig. 5 : AM/AM (a) and AM/PM (b) TWTA transfer curves For a N length uncoded sequence (with no particular propriety) whose coponents are c i, i {0,,N-}, the pea aplitude of the OFDM associated envelope signal is equal to Nax( c i ). In this paper, we have set the condition A ax =Nax( c i ). The advantage of using a Reed-Müller code to generate Golay sequences is to have a pea aplitude value after the IFFT of only N ax( c i ) instead of Nax( c i ). Fig.5 shows the AM/AM and AM/PM distortions used in this paper. We have taen [6]:

5 α = ; β 0 = π/ ; β = Siulation results for AM/AM and AM/PM distortions in an AWGN channel Nuerical values for β 0 and β iposes a axiu angle constellation rotation of π/6 what explains the BER degradations for AM/PM distortions. The sae coents are valuable for AM/AM distortions. The results are on Fig.6 (a) and (b) where we have used a BCH(3,6,7) for all the product codes siulations. Fig 6 (a) : AM/AM + AWGN distortions Fig. 6 (b) : AM/PM + AWGN distortions : BCH(3,6,7) RM, hard decoding : BCH(3,6,7) RM, soft decoding iproved the BER perforances and obtain a good coproise : a global code rate of for a coding gain of about db between hard and soft decoding at BER=0-4. But the nuber of carriers is only 6 what is quite wea for an OFDM application in severe channels with ultipath or fading. We are carrying out our researches on this point. References : [] Accepted paper : Y. Louët and A. Le Glaunec, Pea-Factor reduction in OFDM by Reed- Müller channel coding : a new soft decision decoding algorith, Proceedings of MELECON 000, May 000, Cyprus. [] B. Le Floch, M. Alard and C. Berrou, Coded Orthogonal Frequency Division Multiplex, Proceedings of the IEEE, Vol. 83, No. 6, pp , 995. [3] J.A.C. Bingha, Multicarrier odulation for data transission : An idea who has coe IEEE Coun. Mag., Vol. 8, pp.5-4, 990. [4] J. Leonard and JR. Ciini, Analysis and siulation of a Digital Mobile Channel Using Orthogonal Frequency Division Multiplexing, IEEE Transactions on Counications, Vol. 33, No. 7, 985. [5] J.A. Davis and J.Jedwab, Pea-to-ean power control in OFDM, Golay Copleentary Sequences, and Reed-Muller Codes, IEEE Trans. On Inforation Theory, Vol. 45, No. 7, 999. [6] E. Costa, M. Midrio and S. Populin, Ipact of Aplifier Nonlinearities on OFDM Transission Syste Perforance, IEEE Counications Letters, Vol. 3, No., pp 37-39, 999. [7] D. Chase, "A class of algoriths for decoding bloc codes with channel easureent inforation", IEEE Trans. on Inforation Theory, Vol. 8, No., Conclusion This paper ais to present a channel coding schee to correct errors and to reduce the pea-factor of the OFDM odulated signal. We have developed a soft decision decoding based on Chase s algorith and obtain a db coding gain at BER=0-4 between hard and soft decoding. By using BCH(55,3,7) and Reed-Müller codes as a product schee, we have