VIEWS: 0 PAGES: 7 CATEGORY: Education POSTED ON: 5/8/2010 Public Domain
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 performance of the more powerful code is degraded when the threshold SNR λT is greater than the [1] L. Ahlin, “Coding Methods for the mobile radio average SNR λ0, and using a more powerful code channel”, Nordic Seminar on Digital Land Mobile implies that the block length is increased, therefore Communication, Feb.1985, Espoo, Finland. the interleaved depth has to be reduced in order to [2] R. E. Blahut, “Theory and Practise of Error keep an acceptable constraint. As a result, the Control Codes”, Addison-Wesly, Menlo Park, interleaver is much better for a BCH code with California, 1983. smaller block length. One can observe also from the [3] S. Lin, D. J. Costello, Jr., “Error Control figures is that the increase of the number of carriers Coding: Fundamentals and Applications”, Prentice N lead to more performance for block code with hall, NJ, 1983. small length when λT≥λo (burst errors occur when [4] E. N. Gilbert, “”Capacity of a burst noise the channel is in the bad state), and once the SNR channel”, Bell Syst. Tech. J., vol.39, pp. 1253-1266, λo is above the threshold λT, the channel is in the Sept. 1960. good state and the efficacy of block code with large [5] E. O. Elliot, “estimates of error rates for codes length come evident. on burst-noise channels”, Bell Syst. Tech. J., vol.42, pp. 1977-1997, Sept. 1963. [6] H. S. Wang, N. Moayeri, “Finite-state Markov 6. CONCLUSION channel—A useful model for radio communication channel”, IEEE Trans. Veh.Technol., vol. 44, pp. In this paper, we simulated the performance of 163-171, feb. 1995. block code transmission over the land mobile [7] R. Krishnamurthi, “An Analytical Study of channel ( represented by the GE channel model) Block Codes in a Portable Digital Cellular System”, using OFDM based on BPSK modulation type and Ph.D. Thesis, SMU, 1990. considered the impact of several parameters on the [8] G. Sharma, A. Dholakia, and A. A. Hassan, performance of BCH codes. The increase in “Simulation of error trapping decoders on a fading f d T (which reflects the mobile speed) improves the channel”, in Proc. 1996 IEEE Vehicular technology Conf., Atlanta, GA, Apr. 28-May 1, 1996, pp. 1361- performance of the BCH code and for large values 1365. of f d T , the errors tend to be more independent as [9] J.R Yee and E. J. Weldon, Jr., “Evaluating of the transition probabilities b and g increase leading the performance of error correcting codes on a to efficacy correction of random errors. Gilbert channel”, IEEE Trans.Comm., vol. 43, no. We analysed the effect of the threshold λT on the 8, pp. 2316-2323, Aug. 1995. GE channel and we saw that improving is obtained [10] B. Wong, C. Leung, “On Computing when λ0 is large than the threshold λT. Using undetected error probabilities on the Gilbert interleaving depth according to the block code channel”, IEEE Trans. Comm., vol. 43, no. 11 length with respect to delay constraint, it was seen pp.2657-2661, Nov.1995. that BCH codes with small length give more [811] J.G. Proakis, Digital Communications, 2nd performance. ed., McGraw Hill, 1989. [12] L. Wilhelmson, Laurence B. Milstein, “On the 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. [14] Y. Louet, A. Le Glaunec, “Peak-factor 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 coding: a new soft decision decoding algorithm”, Proceedings of MELECON 2000, May 2000, Cyprus. min( d ,n−d ) [15] V. Tarokh and H. Jafarkhani “On the ⎛ 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 Proceedings of the 11th WSEAS International Conference on COMMUNICATIONS, Agios Nikolaos, Crete Island, Greece, July 26-28, 2007 346 [17] S. Tertois, A. Le Glaunec, “Symétries du problème de correction des non linéarités à la réception dans un système OFDM”, Suplec international technical report, 2002. [18] Y. Li, J. Moon, “Performance analysis of bit- interleaved space-time coding for OFDM in block fading channel”, IEEE VTC’04, Milan, Italy, May 2004.