EVALUATION OF THE PERFORMANCE OF BCH CORRECTING CODES ON by vve15535

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									Proceedings of the 11th WSEAS International Conference on COMMUNICATIONS, Agios Nikolaos, Crete Island, Greece, July 26-28, 2007    340




           PERFORMANCE EVALUATION OF BCH CORRECTING CODES ON
                A FADING CHANNEL USING OFDM MODULATION

                                 A.Seddiki1, A.Djebbari1, J.M.Rouvaen2, A. Taleb-Ahmed3
                       1
                           Laboratoire de Télécommunication et Traitement Numérique du Signal,
                                        Université de Sidi-Bel-Abbès, 22000, Algérie
                                 2
                                   Laboratoire de Radio Détection et Traitement de Signaux,
                                    Université de Valenciennes, UMR/CNRS 8520, France
                             3
                               LAMIH, Université de Valenciennes, UMR/CNRS 8530, France
                                                         E-mail : seddiki_ali@msn.com


         Abstract: In this paper, we evaluate the                           good state, where the probability of error is small,
         performance       of     BCH       (Bose-Chaudhuri-                or in bad state, where the probability of error is
         Hocquenghem) correcting codes when used to                         larger. The dynamic of the channel are modeled as
         protect data over a land mobile channel using                      a first-order Markov chain, where in [6,7] showed
         OFDM        (Orthogonal      Frequency      Division               its accuracy for a Rayleigh fading channel and in
         Multiplexing) modulation. To deal with memory                      [8] presented a way to match the parameters of GE
         channels, the Gilbert-Elliott (GE) model was                       model to the land mobile channel. However, all
         considered to simulate a Rayleigh fading channel                   studies dealing with GE channel was considered in
         and BCH codes to analyse the error process.                        the case of single carrier modulation and multiple
         Relating GE parameters to the physical quantities                  carriers modulation was not considered [4,5,8,9]. In
         determining the fading statistics, we simulated the                this work, we consider the performance of different
         effect of introducing OFDM parameters with                         binary BCH codes [2,3] using OFDM system
         respect to the parameters of the channel error                     [13,15,16]. We use an interleaved GE channel to
         probability function (e.g., mobile speed, modulation               evaluate performance for land mobile channel using
         type, delay constraint, and parameters of error                    probability of error as a function of channel
         correcting codes). Simulation results using OFDM                   parameters, interleaver parameters, error correcting
         modulation rather than single BPSK modulation                      code, and type of modulation. We focus in our
         shows, for different BCH codes, significant                        model on 3 parameters. First to cover a large range
         performance.                                                       of mobile communication, we considered the
                                                                            impact of different values of Doppler frequencies
         Keywords: OFDM, Block coding, Rayleigh fading,                     (which reflects the mobile speed) for the choice of a
         Gilbert-Elliott channel, Interleaving, land mobile                 critical threshold SNR in which the channel is in
         channel.                                                           the good state. Second we analysed correlation
                                                                            between block codes length and interleaving depth
         1. INTRODUCTION                                                    to show how to keep an acceptable delay constraint
                                                                            in a real situation. The third parameters was the
            Studies of the performance of error correcting                  number of carriers used in the OFDM modulator to
         codes are most often concerned with situations                     see how performance is improved with respect to
         where the channel is assumed to be memoryless                      mobile speed, interleaver depth, and block codes
         allowing to simplified theoretical analysis [1]. In                length.
         situation, where memory is accounted for,                             The remainder of the paper is organized as
         analytical and result studies are few and obtained                 follows. In section 2, an overview of GE model
         via simulation [4,5]. The received signal in a                     with consideration of interleaving effect is
         mobile digital system is known to display Rayleigh                 described. Section 3, the outstanding of matching
         statistics. This Rayleigh fading is characterized in               GE model to land mobile channel is considered.
         the digital domain by having burst errors. To deal                 Section 4, gives a brief introduction to OFDM
         with such a complicated channel model, it is                       modulation. Simulation results are indicated in
         possible to use a less complex one that reflects the               section 5 showing the effect of different parameters.
         essential properties of the complicated one. For a                 Finally, conclusions are drawn in section 6.
         channel with memory, the Gilbert-Elliott (GE)
         channel is one of the simplest models [4,6,9]. GE                  2. INTERLEAVED GE CHANNEL
         model provides a useful discrete model where its
         parameters can be readily related to the statistics of               The GE channel is a first-order, discrete-time,
         the fade. In this model for a slowly varying                       stationary, Markov chain with two states, one good
         channel, the channel is assumed to either be in a                  and one bad, denoted G and B. the probability that
Proceedings of the 11th WSEAS International Conference on COMMUNICATIONS, Agios Nikolaos, Crete Island, Greece, July 26-28, 2007                          341



         the channel state changes form G to B and form B                            better the interleaver can be expected to work, and
         to G are denoted by b and g, respectively (Fig.1).                          if m is infinite, the performance would be the same
                                                                                     as for a memoryless channel (m         ∞, we obtain
                                                                                           ∞                 ∞
                                                     b                               b’= P (B), and g’= P (G)).
               1-b                                                       1-g
                                                                                       If the GE channel is observed at n consecutive
                              G                                      B               instants of time, the probability that the channel is
                                                                                     in the bad state d times, 0 ≤ d ≤ n, is given by
                                                     g
                                                                                                Pn(d) =
                          Fig.1 The Gilbert-Elliott channel model
                                                                                                P∞(G)( Pn(d\GG)+ Pn(d\GB))
                                                                                                  P




          The probability that the channel is in the good and                                   + P∞(B)( Pn(d\BG)+ Pn(d\BB)),                 1≤d<n
         the bad state at the kth instant of time are denoted                                   P∞(G)(1-b)n-1 ,
                                                                                                  P                                           d=0
         by Pk(G) and Pk(B), respectively, with matrix
                P                     P
                                                                                                P∞(B)(1-g)n-1 ,
                                                                                                  P                                           d=n
         notation Pk=[ Pk(G), Pk(B)].
                          P       P




          The probability of being in state G at time k, given                                                                          (6)
         that the channel is in state B at time 0, will be                             Here Pn(d\GG) is the conditional probability of
         denoted Pk(G\B).
                      P
                                                                                     being d times in the bad state, conditioned on being
          Let T denote the transition matrix for the channel,                        in the good state both the first and the last instants
                                                                                     of times, and the other conditional probabilities are
                                         ⎡1 − b               b ⎤              (1)   defined accordingly in appendix.
                                      T =⎢
                                         ⎣g                  1- g⎥
                                                                 ⎦                   If a t-error correcting block code of length n is used
         So that                                                                     where the interleaving depth is m, the probability of
                                              k +1       k                           a codeword error is,
                                          P      =   PT                        (2)
                                                                                              n        ⎡ d ⎛⎛ d ⎞
         From (1) and (2), we can see that how fast the                               Pcw = ∑ Pn ( d ) ⎢ ∑ ⎜ ⎜ ⎟ Pe ( B ) i (1 − Pe ( B )) d − i
                                                                                                              ⎜⎜ ⎟
         channel is changing from one state to the other                                    d =0       ⎢ i=0 ⎝ ⎝ i ⎠
                                                                                                       ⎣
         depends on b and g. For the channels that we are                                                        n−d
                                                                                                                                ⎛n − d ⎞
         interested in, the channel is slowly changing                                                  ⋅ ∑ ⎜                   ⎜ j ⎟  ⎟
                                                                                                          j = max( 0 ,t +1− i ) ⎝      ⎠
         compared to the symbol rate, and hence b+g<< 1.
          The stationary distribution is denoted P∞= [ P∞(G),                                                                                      ⎞⎤
                                                                                                               ⋅ Pe (G ) j (1 − Pe (G )) n − d − j ⎟ ⎥
                                                                                                                                                   ⎟
                                                                     P     P




         P∞(B)] and is found to be
           P



                                                                                                                                                   ⎠⎦
                                                                                                                                                    (7)
                                      P∞=[ g/(b+g), b/(b+g)]
                                          P                                    (3)
                                                                                      We assume here that d symbols are received when
         Finally, the probabilities of error for the good and                        GE channel is in bad state, and n-d symbols
         bad states are denoted by Pe(G), Pe(B), respectively.                       received when GE channel is in good state.
           Because of the large degradation of the                                    Indexes i and j denote the number of symbols in
         performance caused by the memory of the channel,                            error when the GE channel is in the state B and G
         a way to improve the performance is to use an                               respectively [13].
         interleaver in order to make code symbols less                               In (7), the parameters b and g in Pn(d) are replaced
         independent [9,13].                                                         by using equation (3),(4), and (5). In the following
           The interleaver used is a block with m rows and n                         section, we describe the way to calculate the GE
         columns (for block coding, n is equal to block                              parameters in function of the statistic characteristics
         length to avoid ‘wrap-around’ effect), where the                            of Rayleigh fading channel [9].
         bits that are to be transmitted are fed in row-wise
         and fed out column-wise. Then, the corresponding                            3. MATCHING GE CHANNEL TO LAND
         transition probabilities b’ and g’ for an interleaved                          MOBILE CHANNEL
         GE channel if observed m moments of time later
         [13], are                                                                     There are several works [1,8,13] which give a
                                                                                     constructive way to match the GE channel model to
                     b’= P∞(B)(1-(1-b-g)m)                                     (4)   a flat Rayleigh fading channel by choosing different
                                                                                     matching parameters(level for signal to noise
                     g’= P∞(G)(1-(1-b-g)m)                                     (5)   ration, level-crossing rate, and arbitrary thresholds).
                                                                                       However, the way that threshold is chosen affect
         From (4) and (5), the effect of the interleaving                            more the accuracy of the model, and for this reason
         depth can be clearly seen. The larger value of m, the
Proceedings of the 11th WSEAS International Conference on COMMUNICATIONS, Agios Nikolaos, Crete Island, Greece, July 26-28, 2007                                                            342



         the parameters of the GE model are carefully                                          Where Pe(λ) is the symbol error probability for a
         treated.                                                                              given value of λ , which depends on the modulation
           Assuming that the channel fades slowly with                                         scheme used. We shall concentrate on using OFDM
         respect to a bit interval the parameters of the model                                 modulation technique.
         can be related to various physical quantities. The
         Rayleigh fading results in an exponentially                                           4. OFDM MODULATION
         distributed multiplicative distortion of the signal.
                                                                                                Orthogonal frequency Division Multiplexing
         Hence, the probability density function of the SNR,                                   (OFDM) is a very attractive modulation scheme for
         λ, is given by [1,7,9]                                                                data transmission in multipath fading. OFDM can
                                                                                               effectively randomise burst errors caused by
                                               1            −λ                                 Rayleigh fading, which comes from interleaving
                               f (λ ) =             e            λ0
                                                                      ,   λ≥0
                                               λ                                               due to the parallelisation. The FFT-based OFDM
                                                                                         (8)   system is represented in figure 2.

         Where λ0 is the average SNR.                                                           Serial data
                                                                                                              Serial to
                                                                                                              Parallel              Signal      IFFT     P/
                                                                                                                                                              Guard
                                                                                                                                                              interval    D/A    Up
                                                                                                Input         converter             mapper               S    insertio    LPF    converte
          The channel is said to be in the good state while                                                                                                   n                  r


         the SNR is above a threshold λT and once the SNR
         falls below λT the channel goes into the bad state.                                                                                                                       GE
                                                                                                                                                                                 Channel

         The stationary probabilities of finding the GE
         channel in respective states with respect to λT are,                                   Serial data
                                                                                                              Parallel to
                                                                                                              Serial                Signal      FFT      S/
                                                                                                                                                              Guard
                                                                                                                                                              interval    LPF    Down
                                                                                                Output        converter             demapp               P    removal     A/D    converte
                                                                                                                                    er                                           r

                                                    ∞

                              P ∞ (G ) =            ∫ f (λ ) dλ = e
                                                                          −ρ 2
                                                                                                                            Fig.2: FFT-based OFDM system
                                                    λ
                                                    T

                                                                                        (9)    Because of dividing an entire channel bandwidth
                                                                                               into many narrow subbands [14,16,17], the
         Where ρ2=-λT/λ0. And                                                                  frequency response over each individual subband is
                                                                                               relatively flat, and the distribution of data over
                                               λT                                              many carriers means that the selective fading
                            P∞ (B) =           ∫ f (λ ) dλ = 1 − e
                                                                                 −ρ 2
                                                                                               causes some bits to be in errors. The
                                                0
                                                                                               implementation of an error correcting code make
                                                          (10)                                 possible to avoid errors by using a forward error
          Using the level crossing rate and the SNR density                                    correction. Let N be the number of carriers, Ci,
         function, the transition probabilities can be found as                                i={0,.., N-1}, the complex information symbols
         follows [8]                                                                           vector, and T the OFDM symbol length. The
                                           ρ f d T 2π                                          transmitted signal over a symbol duration T is
                                g=                                                  (11)
                                               e −ρ − 1
                                                        2
                                                                                               [14,15,18],
                                b = ρ f d T 2π                                      (12)
                                                                                                                  ⎛ N−1                        ⎞
                                                                                                    S( c,t ) = Re ⎜ ∑ Ci exp( j2π ( f0 + if )t ⎟                         0 ≤ t ≤T
         Where T is the symbol interval (specified in terms                                                       ⎝ i=0                        ⎠
         of symbol rate Rs=1/T), and f d = vf c c is the                                                                                 (15)
                                                                                                 The codeword Ci consists of N symbols chosen
         Doppler frequency (maximum Doppler speed), with                                       from an M-ary modulation method. All of the
          v the vehicle speed, f c the carrier frequency and c                                 codewords form the set Ci. For MPSK,
         the light speed.
           The error probabilities in respective states in the                                                                      2π
                                                                                                                                j      ( ai )
         GE channel are taken to be the conditional error                                                       Ci = e              M
                                                                                                                                                       ai ∈ Z M                 (16)
         probabilities of Rayleigh fading channel,
         conditioned on being in the respective state,
                                                                                                 The duration of an OFDM symbol T is N times
                                        ∞
                                                                                               the duration of the symbols Ci plus the duration of
                                        ∫
                          1
               Pe (G ) = ∞       f (λ )Pe (λ )dλ                                    (13)       the cyclic prefix or guard band. The complex
                        P (G ) λ
                                           T                                                   envelope of the transmitted signal, sampled at 1/T
                                       λT                                                      is,
                                       ∫ f (λ )P (λ )dλ
                               1
               Pe ( B ) =                                                               (14)                    ~            N −1
                            P ∞ ( B)                                                                            S ( c ,n ) = ∑Ci exp( j 2πni / N )
                                                            e
                                       0
                                                                                                                                                                                (17)
                                                                                                                                         i =0
Proceedings of the 11th WSEAS International Conference on COMMUNICATIONS, Agios Nikolaos, Crete Island, Greece, July 26-28, 2007                                                   343



                                                                                                 Figures 4 and 5 shows the BER plots vs. threshold
         5. SIMULATION RESULTS                                                                   SNR λT for BCH code (7,4,1) for two values of
                                                                                                 average SNR λ0, 10dB and 20dB, with different
             In order to cover a wide range of mobile                                            Doppler frequencies. From the figures, it is clear
         communication environments, and give different                                          that an increase in f d T (which reflects the mobile
         model of fading channel with different degree of
                                                                                                 speed) improves the performance of the BCH code
         correlation, we consider the product f d T as an                                        and for large values of f d T , the errors tend to be
         independent parameter performed with the values                                         more random (independent) as the transition
          f d T = 0.001, 0.01, 0.05, and 0.1.                                                    probabilities b and g increase leading to good
         The OFDM scheme based on BPSK type with                                                 performance since BCH code is capable to correct
         parameters fixed for ifftsize=2048 (size of inverse                                     such random errors.
         Fast Fourier Transform), guard time=128, guard
                                                                                                            -2
         period type using half cyclic extension of the                                                    10
         symbol, and number of carriers N=512 and
         N=1024.
                                                                                                            -3
            Having seen that the GE model can be used to, in                                               10
         an accurate way, to estimate the code error
         probability for block coded transmission over the
         land mobile channel using a single carrier BPSK                                                    -4
                                                                                                           10
         modulation [13], we will now evaluate the effect

                                                                                                    BER
         performance of using multiple carriers BPSK
         modulation (OFDM) and the effect of the choice of                                                  -5
                                                                                                           10
         error correcting code over an interleaved GE
         channel. Figure 3 presents the model scheme used.
                                                                                                            -6
                                                                                                           10            Fdt=0.01
                                                                                                                         Fdt=0.05
            Random         Channel                             OFDM
                                                               IFFT                                                      Fdt=0.1
            Data           Coding              Interleaver
                           BCH codes                           Modulate                                                  Fdt=0.001
                                                                                                            -7
                                                                                                           10
                                                                                                                 0   2           4   6       8       10     12     14    16   18
                                                                           Gilbert-Elliot                                                Threshold snr (dB)
                                                                             Channel

                                                                                                    Fig.5: BER for (7,4,1)BCH code vs. λT , λ0=20dB, N=512
            BER            Channel             De-             OFDM
            Analysis       Decoding            -interleaver    FFT
                           BCH codes                           Modulate                                     0
                                                                                                           10



                               Fig.3: Simulation system model                                               -1
                                                                                                           10



           We consider the case with perfect interleaving                                                   -2

         (memoryless channel: m ∞, b’= P∞(B), g’= P∞(G)).
                                                                                                           10




                                                                                                            -3
                                                                                                           10
                   -2
                                                                                                     BER




                  10
                                                                                                            -4
                                                                                                           10



                   -3
                  10                                                                                        -5
                                                                                                           10



                                                                                                            -6
                                                                                                           10
                   -4
                  10                                                                                                 (23,13) BECC
                                                                                                                     (14,6) BECC
                                                                                                                     (23,12) Golay
            BER




                                                                                                            -7
                                                                                                           10
                                                                                                                 0           5           10                   15    20        25
                                                                                                                                              Mean snr (dB)
                   -5
                  10
                                                                                                 Fig.6: BER for three codes with BPSK,                    f d T =0.01, λT =10dB
                   -6
                  10           Fdt=0.01                                                            In comparison with figure 6 (with burst error
                               Fdt=0.05                                                          correcting codes BECC and Golay codes [8]), the
                               Fdt=0.1
                               Fdt=0.001
                                                                                                 use of an OFDM-BPSK system gives a significant
                   -7
                  10                                                                             BER improvement than a single BPSK modulation.
                       0   2         4     6       8       10     12      14       16       18
                                               Threshold snr (dB)
                                                                                                 One can observe from the figures 4 and 5, is that
                                                                                                 the BER is less sensible to small variation of λT
            Fig.4: BER for (7,4,1)BCH code vs. λT , λ0=10dB, N=512
                                                                                                 (which is a critical parameter for GE channel),
                                                                                                 hence the choice of the exact value of this
Proceedings of the 11th WSEAS International Conference on COMMUNICATIONS, Agios Nikolaos, Crete Island, Greece, July 26-28, 2007                                             344



         parameter is not critic for performance. Figure 7                               Code               Original code                 Minimal    Code rate
         shows the impact of the variation of the average                                length                                           distance
         SNR λ0, where we use a (15,7,2)BCH code with                                    7                  (7,4,1)                       3          57%
         approximately the same rate and error capability 2.
                                                                                         15                 (15,7,2)                      5          46%

                                                                                         31                 (31,16,3)                     7          51%
                   -2                                                                    63                 (63,30,6)                     13         47%
                  10
                                                                                         127                (127,64,10)                   21         50%
                   -3                                                                    255                (255,123,19)                  39         48%
                  10


                   -4                                                                               -2
                  10                                                                               10
            BER




                   -5                                                                               -3
                  10                                                                               10


                   -6
                  10                                                                                -4
                                                                                                   10


                   -7           ean
                               M snr=10dB
                  10
                                                                                             BER
                                                                                                    -5
                               M snr=15dB
                                ean                                                                10
                                ean
                               M snr=20dB
                                ean
                               M snr=25dB
                   -8
                  10                                                                                -6
                       0   2       4        6       8       10     12   14   16     18             10
                                                Threshold snr (dB)

          Fig.7: BER for (15,7,2)BCH code vs. λT ,
                                                                                                                BCH(255,123,19)
                                                               f d T =0.01, N=512                   -7
                                                                                                   10           BCH(127,64,10)
                                                                                                                BCH(63,30,6)
                                                                                                                BCH(31,16,3)
           In this figure, the BER decrease with the increase                                                   BCH(15,7,2)


         of the level of SNR λ0 , the channel tends to stay
                                                                                                                BCH(7,4,1)
                                                                                                    -8
                                                                                                   10
         more in the good state than the bad state(b>>g), the                                           0   2       4             6   8     10 12 14     16   18   20   22
                                                                                                                                            ean
                                                                                                                                           M snr (dB)
         burst errors length is small in this case and the
         binary BCH code correcting capability is efficacy.                               Fig.8: BER for BCH code vs. λ0 , λT=10dB, f d T =0.003,
         We can see more performance using ofdm that the                                                   N=512, D=20ms
         case of figure 6.
           We now compare the performance of different                                              -2
                                                                                                   10
         BCH codes for the situation where the interleaving
         is not perfect due to the delay constraint.
                                                                                                    -3
                                                                                                   10
                                Delay = 2nϕτ                                 (16)

           Where ϕ is the interleaving depth, τ is                                                  -4
                                                                                                   10
         information rate, and n is the block code length. We
         use here:
              • Maximum delay due to the interleaver =
                                                                                             BER




                                                                                                    -5
                                                                                                   10
                   20ms;
              • Information rate τ = 9.6kbit/s;
                                                                                                    -6
              • Normalized Doppler frequency f d T equals                                          10
                   0.003 (corresponding to a vehicle speed of                                                    C (2 5 2 ,1 )
                                                                                                                B H 5 ,1 3 9
                   20 m/h);                                                                         -7           C (1 7 4 0
                                                                                                                B H 2 ,6 ,1 )
                                                                                                   10
              • Number of carriers N =512 and N =1024.                                                          C (6 ,3 ,6
                                                                                                                BH 40)
                                                                                                                C (3 ,1 ,3
                                                                                                                BH 16)
              • The block codes length with code rate range                                                      C (1 ,7 )
                                                                                                                B H 5 ,2
                                                                                                                 C (7 ,1
                                                                                                                B H ,4 )
                   of 50% used in our simulation:                                                   -8
                                                                                                   10
                                                                                                        0   2       4             6   8     10 12 14    16    18   20   22
                                                                                                                                           ean
                                                                                                                                          M snr (dB)

                                                                                          Fig.9: BER for BCH code vs. λ0 , λT=10dB, f d T =0.003,
                                                                                                            N=1024, D=20ms
Proceedings of the 11th WSEAS International Conference on COMMUNICATIONS, Agios Nikolaos, Crete Island, Greece, July 26-28, 2007                                              345



           Figures 8 and 9 show that the outstanding                                                                 REFERENCES
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         APPENDIX                                                                                                    Effect of Imperfect Interleaving for the Gilbert-
                                                                                                                     Elliot Channel”, IEEE Trans. Comm., vol. 47, no. 5,
            The conditional probabilities are defined as [9],                                                        pp.681-688, May 1999.
                         min( d +1,n − d )
                                             ⎛ n − d − 1⎞ ⎛ d − 1 ⎞                                                  [13] W. Y. Zou and Y. Wu, “COFDM: An
         Pn ( d GG ) =         ∑
                               u =2
                                             ⎜
                                             ⎜          ⎟⎜
                                                        ⎟⎜
                                             ⎝ u −1 ⎠ ⎝u − 2⎠
                                                                  ⎟.(1 − b )n − d −u b u −1 (1 − g )d −u +1 g u −1
                                                                  ⎟                                                  overview”, IEEE Trans. Broadc., vol. 41, no. 1, pp.
                                                                                                                     1-8, March 1995.
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                         min( d ,n − d )
                                           ⎛ n − d − 1⎞ ⎛ d − 1 ⎞                                                    reduction in OFDM by Reed6Muller channel
         Pn ( d GB ) =       ∑u =1
                                           ⎜ u − 1 ⎟ ⎜ u − 1 ⎟.(1 − b )
                                           ⎜
                                           ⎝
                                                      ⎟⎜
                                                      ⎠⎝
                                                                ⎟
                                                                ⎠
                                                                               b (1 − g )d −u g u −1
                                                                       n − d −u u
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                                           ⎛ n − d − 1⎞ ⎛ d − 1⎞
         Pn ( d BG ) =        ∑
                              u =1
                                           ⎜ u − 1 ⎟ ⎜ u − 1⎟.(1 − b)
                                           ⎜
                                           ⎝
                                                      ⎟⎜
                                                      ⎠⎝
                                                               ⎟
                                                               ⎠
                                                                            b (1 − g )d −u g u
                                                                      n−d −u u −1
                                                                                                                     Computation and Reduction of the Peak-to-Average
                                                                                                                     Power Ratio in Multicarrier Communications”,
                                                                                                                     IEEE Trans. Comms., vol. 48, No 1, pp. 37-44, Jan.
                         min( d ,n−d +1 )
                                                                                                                     2000
                                             ⎛ n − d − 1⎞ ⎛ d − 1⎞
         Pn ( d BB ) =         ∑
                               u =2
                                             ⎜ u − 2 ⎟ ⎜ u − 1⎟.(1 − b)
                                             ⎜
                                             ⎝
                                                        ⎟⎜
                                                        ⎠⎝
                                                                 ⎟
                                                                 ⎠
                                                                                 b (1 − g )d −u g u −1
                                                                        n−d −u +1 u −1
                                                                                                                     [16] Y. Li, J. Moon, “Increasing data rates through
                                                                                                                     iterative coding and antenna diversity in OFDM-
                                                                                                                     based      wireless      communication”,        IEEE
                                                                                                                     Globecom’01, San Antonio, TX, 2001
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         réception dans un système OFDM”, Suplec
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         interleaved space-time coding for OFDM in block
         fading channel”, IEEE VTC’04, Milan, Italy, May
         2004.

								
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