A study of turbo codes for multlevel modulatons n Gaussan and moble channels Lamne Sylla and Paul Forter (sylla, forter)@gel.ulaval.ca Department of Electrcal and Computer Engneerng Laval Unversty, Ste-Foy, Quebec Canada G1K 7P4 (418) 656-3555, (418) 656-3159 fax ABSTRACT - In ths paper, we present smulaton results for a system combnng turbo codes and multlevel modulaton. Ths scheme s shown to be an excellent soluton when t comes to transmsson over bandlmted channels. Its performance n moble rado channels can be mproved usng OFDM. 1. Introducton Snce the frst publcatons regardng turbo codes, varous studes have been carred out. Most of these studes llustrated performances or dealt wth the optmzaton of the structure of turbo codes usng bnary modulaton. The requrement for hgher nformaton transmsson rates on bandlmted channels have led some researchers to consder turbo codes wth spectrally effcent modulatons. In [2], an applcaton of turbo codes to trells coded modulaton (TCM) called parallel concatenated trells coded modulaton s presented. At the encoder ths scheme uses two recursve systematc convolutonal coders (RSC) of rate R = b ( b 1), b beng even, and two nterleavers for a throughput of b bts/sec/hz 2 2b 2 wth -QAM. The decodng s performed teratvely by the symbol MAP decoder followed by a bt relablty calculaton module. In [3], a turbo encoder employng two Ungerboeck codes n ther recursve form as component codes has been used n combnaton wth TCM. Ths turbo TCM (TTCM) uses an nterleaver operatng over symbols of 2 bts for an 8 PSK modulaton. The varables passed between symbol MAP decoders are vectors of four loglkelhood ratos (LLRs), one for each possble nformaton group value. These two models have n common a major modfcaton on the classcal turbo encoder to ft the TCM model at the expense of an ncrease n complexty at the decoder. Ths paper presents some results obtaned wth turbo codes and multlevel modulatons n Gaussan and moble rado channels by applyng the so called pragmatc approach [1]. Secton 2 deals wth the encoder structure and the sgnal mappng scheme. Secton 3 presents the channels models used n the smulatons and the Secton 4 presents the decodng operaton. The last two sectons are devoted to the smulatons results and some concludng remarks. 2. Codng and modulaton schemes The encoder s made up of two RSCs wth constrant length k = 3 and polynomal generators (5,7). The RSCs are separated by an unform block nterleaver. For short frame transmsson applcatons, the nterleaver was set to a sze of 192 bts whch s compatble wth the IS-95 CDMA cellular standard. Another block sze of 1022 bts has been used. Ths sze, after addng 2 bts for termnatng the RSCs, match a COFDM symbol wth bnary modulaton. The output of the 1/3 turbo encoder s punctured to obtan a 1/2 codng rate. The system presented here s dfferent from the TCM approach n the sense that t uses a codng rate R = 1/2 for a 16 pont constellaton. The output sequence of the encoder (u 1,c 1,u 2,c 2,...) s modulated by two 4-ASK n quadrature resultng n a 16-QAM system where the I and Q components are ndependently coded (usng Gray code). The ASK symbols {A k, B k } are then transmtted through the channel. 3. Channel models Generally, for purpose of smulaton, two approaches are used to model a moble rado channel. The frst one deals wth the multpath phenomenon and the second one deals, n addton to multpath, wth frequency selectvty [7]. A Raylegh flat fadng channel has been used to smulate the multpath phenomenon. In ths case, the receved symbol, (x k, y k ), s gven by x k = α k A k y k = β k B k (1) where (α k, β k ) are two ndependent Raylegh dstrbuted random varables and ( k, q k ) are the addtve whte Gaussan nose components wth varance σ 2. The frequency selectvty has been ntroduced by means of ntersymbol nterference (ISI). In an ar-to-ground k q k 0-7803-4314-X/98/$10.00 1998 IEEE
communcaton, the channel s characterzed by a drect lne of sght (LOS) and a multpath component wth a relatve delay [5]. Ths delayed path creates some ISI on the receved sgnal (2) wth n representng the relatve delay of the dffuse component. Ths model fts the Rcan fadng channel. For applcatons n mcrowave LOS rado channels, a smlar model has been consdered, the Rummler channel whch s a two path model [6] (3) where a s the overall attenuaton, and the term wthn parenthess represents the nterference between two rays havng a relatve delay τ 0 and producng a mnmum ampltude at the notch frequency f 0. b s the relatve ampltude of the multpath ray. The outputs of the channel are coherently demodulated and the soft nputs of the decoder are then (X k, Y k ). 4. Decoder structure x k = A k α k n A k n y k = B k β k n B k n The decodng of ths multlevel modulaton system s much smplfed by use of the pragmatc approach [1]. The turbo decoder, mplemented wth the SOVA [4] algorthm, and optmzed for bnary modulaton, can then be used. Wth ths approach, the decoder must be preceded by a module that computes the LLR of each bt contaned n an ASK symbol. The LLR can be expressed as follows: Λ( u ) k, (4) wth u k,1 = u k and u k,2 = c k and (a 1,, a 0, ) are the realzatons of symbol A k condtonal to u k, = 1 and u k, = 0 respectvely. A good approxmaton of ths expresson s gven by k q k ( )τ 0 Hf () a 1 b e j2π f f 0 = ( ) = σ 2 ------ 2 2 1 --------- 2σ 2 ( X a ) 2 exp k 0, = 1 log----------------------------------------------------------------------------- 2 1 --------- 2σ 2 ( X a ) 2 exp k 1, = 1 (5) Although ths approxmaton can lead to good performance, the expresson of Λ(u k ) n (5) s an overestmaton of ts real value. A better approxmaton can be obtaned by defnng the bt-mappng n one dmenson of the constellaton wth unt energy as follows (6) where 2d s the dstance between two ponts. Ths equaton can be decomposed nto two terms. Λ(u k ) can now be approxmated by the frst term on the second part of ths equaton. Fgure 1 show three curves of Λ(u k ) versus X k. The sold lne represents the true expresson n (4), the * lne represents the frst approxmaton n (5) and the o lne the last approxmaton. The approxmaton n (7) s closer to ts real value and as t wll be shown later, leads to a better performance. 5. Smulatons results Λ( c ) = X 2 k k Λ( u k ) = X k A = 2d ( 1 2u ) k k, 1 d ( 1 2u k, 2 ) sgn ( 1 2u k, 1 ) Λ( u k, 1 ) = 2dX k σ 2 --------- 1 2σ 2 4d 2 2dX k cosh (7) ------ log-------------------------------------------------------------------- 2 --------- 1 2σ 2 4d 2 2dX k cosh The smulatons have been carred out usng the Sgnal Processng WorkSystem (SPW) package. The BER was computed after 3 teratons of the decoder. Fgure 2 shows for a Gaussan channel 3 BER curves. These curves demonstrate that turbo codes combned wth 16-QAM (TC-QAM), even wth short frame sze and non random block nterleaver, gve a better performance than conventonal TCM wth the same throughput. The LLRs were computed as n (5). The codng gan over the TCM can be ncreased by usng the second approxmaton n (7). In fact, by usng ths equaton to compute the LLR, the performance of the TC-QAM can be mproved as s shown n Fgure 3.
For the slow Raylegh flat fadng channel, the results wth QPSK modulaton appear on Fgure 4. A BER of 10-5 at a SNR around 7.5 db s acheved by the TC. On the channels wth ISI, we ntended to compare the turbo COFDM (T-COFDM) wth the TC-QAM. On the Rce model, the multpath component was delayed by an amount of 10 µsec. relatve to the LOS path and the carrer to multpath rato C/M was set to 10 db. A frame sze of 1022 bts was used. The T-COFDM has a codng gan of 1db over the TC-QAM at a BER of 10-4 (Fgure 5). LLR - Eq. 4 * Eq. 5 o Eq. 7 On the Rummler channel the relatve delay between the two paths was set to 16 µsec. and the attenuaton factor, b, set to 0.2. In ths case also, the T-COFDM outperforms the TC-QAM; the BER curve s shown n Fgure 6. 6. Concluson In ths paper we presented a study of turbo codes combned wth multlevel modulatons. The system descrbed here could be an alternatve to the conventonal TCM and s less complex than the PCTCM and TTCM. In frequency selectve channels, a combnaton of TC wth OFDM results n a power and spectrally effcent system. Fgure 1: Λ(u k ) versus X k. 7. References 1. S. Goff, A. Glaveux, and C. Berrou, Turbo-codes and hgh spectral effcency modulaton, Proc., IEEE Int. Conf. on Commun., pp. 645-9, May 1994. 2. S. Benedetto, D. Dvsalar, G. Montors, and F. Pollara, Parallel concatenated trells coded modulaton, Proc., IEEE Int. Conf. on Commun., (May), pp. 974-8, 1996. 3. P. Robertson and T. Worz, Coded modulaton scheme employng turbo codes, Electroncs Letters, vol. 31, pp.1546-7, Aug. 31st 1995. 4. J. Hagenauer and L. Papke, Decodng turbo codes wth the soft-output Vterb algorthm (SOVA), Proc., IEEE Int. Symp. on Inform. Theory, p. 164, June 1994. 5. J. Proaks, Dgtal Communcatons, thrd edton. New York: McGraw-Hll, Inc.,1995. 6. W. D. Rummler et al., Multpath fadng channels models for mcrowave dgtal rado, IEEE Communcaton Magazne, vol. 24, Nov. 1996. 7. W. C. Lee, Moble Communcaton Engneerng, McGraw-Hll, 1982. Fgure 2: Comparson of TC-QAM wth TCM.
Fgure 3: Comparson of the results obtaned by computng the LLR usng dfferent approxmatons. Fgure 5: BER curves of T-COFDM and TC-QAM on a Rcan channel wth ISI. Fgure 4: BER curves of bnary TC and uncoded QPSK on a Raylegh channel. Fgure 6: BER curves of T-COFDM and TC-QAM on a Rummler channel.
A study of turbo codes for multlevel modulatons n Gaussan and moble channels Lamne Sylla and Paul Forter (sylla, forter)@gel.ulaval.ca Department of Electrcal and Computer Engneerng Laval Unversty, Ste-Foy, Quebec Canada G1K 7P4 (418) 656-3555, (418) 656-3159 fax ABSTRACT - In ths paper, we present smulaton results for a system combnng turbo codes and multlevel modulaton. Ths scheme s shown to be an excellent soluton when t comes to transmsson over bandlmted channels. Its performance n moble rado channels can be mproved usng OFDM. 1. Introducton Snce the frst publcatons regardng turbo codes, varous studes have been carred out. Most of these studes llustrated performances or dealt wth the optmzaton of the structure of turbo codes usng bnary modulaton. The requrement for hgher nformaton transmsson rates on bandlmted channels have led some researchers to consder turbo codes wth spectrally effcent modulatons. In [2], an applcaton of turbo codes to trells coded modulaton (TCM) called parallel concatenated trells coded modulaton s presented. At the encoder ths scheme uses two recursve systematc convolutonal coders (RSC) of rate R = b ( b 1), b beng even, and two nterleavers for a throughput of b bts/sec/hz 2 2b 2 wth -QAM. The decodng s performed teratvely by the symbol MAP decoder followed by a bt relablty calculaton module. In [3], a turbo encoder employng two Ungerboeck codes n ther recursve form as component codes has been used n combnaton wth TCM. Ths turbo TCM (TTCM) uses an nterleaver operatng over symbols of 2 bts for an 8 PSK modulaton. The varables passed between symbol MAP decoders are vectors of four loglkelhood ratos (LLRs), one for each possble nformaton group value. These two models have n common a major modfcaton on the classcal turbo encoder to ft the TCM model at the expense of an ncrease n complexty at the decoder. Ths paper presents some results obtaned wth turbo codes and multlevel modulatons n Gaussan and moble rado channels by applyng the so called pragmatc approach [1]. Secton 2 deals wth the encoder structure and the sgnal mappng scheme. Secton 3 presents the channels models used n the smulatons and the Secton 4 presents the decodng operaton. The last two sectons are devoted to the smulatons results and some concludng remarks. 2. Codng and modulaton schemes The encoder s made up of two RSCs wth constrant length k = 3 and polynomal generators (5,7). The RSCs are separated by an unform block nterleaver. For short frame transmsson applcatons, the nterleaver was set to a sze of 192 bts whch s compatble wth the IS-95 CDMA cellular standard. Another block sze of 1022 bts has been used. Ths sze, after addng 2 bts for termnatng the RSCs, match a COFDM symbol wth bnary modulaton. The output of the 1/3 turbo encoder s punctured to obtan a 1/2 codng rate. The system presented here s dfferent from the TCM approach n the sense that t uses a codng rate R = 1/2 for a 16 pont constellaton. The output sequence of the encoder (u 1,c 1,u 2,c 2,...) s modulated by two 4-ASK n quadrature resultng n a 16-QAM system where the I and Q components are ndependently coded (usng Gray code). The ASK symbols {A k, B k } are then transmtted through the channel. 3. Channel models Generally, for purpose of smulaton, two approaches are used to model a moble rado channel. The frst one deals wth the multpath phenomenon and the second one deals, n addton to multpath, wth frequency selectvty [7]. A Raylegh flat fadng channel has been used to smulate the multpath phenomenon. In ths case, the receved symbol, (x k, y k ), s gven by x k = α k A k y k = β k B k (1) where (α k, β k ) are two ndependent Raylegh dstrbuted random varables and ( k, q k ) are the addtve whte Gaussan nose components wth varance σ 2. The frequency selectvty has been ntroduced by means of ntersymbol nterference (ISI). In an ar-to-ground k q k 0-7803-4314-X/98/$10.00 1998 IEEE
communcaton, the channel s characterzed by a drect lne of sght (LOS) and a multpath component wth a relatve delay [5]. Ths delayed path creates some ISI on the receved sgnal (2) wth n representng the relatve delay of the dffuse component. Ths model fts the Rcan fadng channel. For applcatons n mcrowave LOS rado channels, a smlar model has been consdered, the Rummler channel whch s a two path model [6] (3) where a s the overall attenuaton, and the term wthn parenthess represents the nterference between two rays havng a relatve delay τ 0 and producng a mnmum ampltude at the notch frequency f 0. b s the relatve ampltude of the multpath ray. The outputs of the channel are coherently demodulated and the soft nputs of the decoder are then (X k, Y k ). 4. Decoder structure x k = A k α k n A k n y k = B k β k n B k n The decodng of ths multlevel modulaton system s much smplfed by use of the pragmatc approach [1]. The turbo decoder, mplemented wth the SOVA [4] algorthm, and optmzed for bnary modulaton, can then be used. Wth ths approach, the decoder must be preceded by a module that computes the LLR of each bt contaned n an ASK symbol. The LLR can be expressed as follows: Λ( u ) k, (4) wth u k,1 = u k and u k,2 = c k and (a 1,, a 0, ) are the realzatons of symbol A k condtonal to u k, = 1 and u k, = 0 respectvely. A good approxmaton of ths expresson s gven by k q k ( )τ 0 Hf () a 1 b e j2π f f 0 = ( ) = σ 2 ------ 2 2 1 --------- 2σ 2 ( X a ) 2 exp k 0, = 1 log----------------------------------------------------------------------------- 2 1 --------- 2σ 2 ( X a ) 2 exp k 1, = 1 (5) Although ths approxmaton can lead to good performance, the expresson of Λ(u k ) n (5) s an overestmaton of ts real value. A better approxmaton can be obtaned by defnng the bt-mappng n one dmenson of the constellaton wth unt energy as follows (6) where 2d s the dstance between two ponts. Ths equaton can be decomposed nto two terms. Λ(u k ) can now be approxmated by the frst term on the second part of ths equaton. Fgure 1 show three curves of Λ(u k ) versus X k. The sold lne represents the true expresson n (4), the * lne represents the frst approxmaton n (5) and the o lne the last approxmaton. The approxmaton n (7) s closer to ts real value and as t wll be shown later, leads to a better performance. 5. Smulatons results Λ( c ) = X 2 k k Λ( u k ) = X k A = 2d ( 1 2u ) k k, 1 d ( 1 2u k, 2 ) sgn ( 1 2u k, 1 ) Λ( u k, 1 ) = 2dX k σ 2 --------- 1 2σ 2 4d 2 2dX k cosh (7) ------ log-------------------------------------------------------------------- 2 --------- 1 2σ 2 4d 2 2dX k cosh The smulatons have been carred out usng the Sgnal Processng WorkSystem (SPW) package. The BER was computed after 3 teratons of the decoder. Fgure 2 shows for a Gaussan channel 3 BER curves. These curves demonstrate that turbo codes combned wth 16-QAM (TC-QAM), even wth short frame sze and non random block nterleaver, gve a better performance than conventonal TCM wth the same throughput. The LLRs were computed as n (5). The codng gan over the TCM can be ncreased by usng the second approxmaton n (7). In fact, by usng ths equaton to compute the LLR, the performance of the TC-QAM can be mproved as s shown n Fgure 3.
For the slow Raylegh flat fadng channel, the results wth QPSK modulaton appear on Fgure 4. A BER of 10-5 at a SNR around 7.5 db s acheved by the TC. On the channels wth ISI, we ntended to compare the turbo COFDM (T-COFDM) wth the TC-QAM. On the Rce model, the multpath component was delayed by an amount of 10 µsec. relatve to the LOS path and the carrer to multpath rato C/M was set to 10 db. A frame sze of 1022 bts was used. The T-COFDM has a codng gan of 1db over the TC-QAM at a BER of 10-4 (Fgure 5). LLR - Eq. 4 * Eq. 5 o Eq. 7 On the Rummler channel the relatve delay between the two paths was set to 16 µsec. and the attenuaton factor, b, set to 0.2. In ths case also, the T-COFDM outperforms the TC-QAM; the BER curve s shown n Fgure 6. 6. Concluson In ths paper we presented a study of turbo codes combned wth multlevel modulatons. The system descrbed here could be an alternatve to the conventonal TCM and s less complex than the PCTCM and TTCM. In frequency selectve channels, a combnaton of TC wth OFDM results n a power and spectrally effcent system. Fgure 1: Λ(u k ) versus X k. 7. References 1. S. Goff, A. Glaveux, and C. Berrou, Turbo-codes and hgh spectral effcency modulaton, Proc., IEEE Int. Conf. on Commun., pp. 645-9, May 1994. 2. S. Benedetto, D. Dvsalar, G. Montors, and F. Pollara, Parallel concatenated trells coded modulaton, Proc., IEEE Int. Conf. on Commun., (May), pp. 974-8, 1996. 3. P. Robertson and T. Worz, Coded modulaton scheme employng turbo codes, Electroncs Letters, vol. 31, pp.1546-7, Aug. 31st 1995. 4. J. Hagenauer and L. Papke, Decodng turbo codes wth the soft-output Vterb algorthm (SOVA), Proc., IEEE Int. Symp. on Inform. Theory, p. 164, June 1994. 5. J. Proaks, Dgtal Communcatons, thrd edton. New York: McGraw-Hll, Inc.,1995. 6. W. D. Rummler et al., Multpath fadng channels models for mcrowave dgtal rado, IEEE Communcaton Magazne, vol. 24, Nov. 1996. 7. W. C. Lee, Moble Communcaton Engneerng, McGraw-Hll, 1982. Fgure 2: Comparson of TC-QAM wth TCM.
Fgure 3: Comparson of the results obtaned by computng the LLR usng dfferent approxmatons. Fgure 5: BER curves of T-COFDM and TC-QAM on a Rcan channel wth ISI. Fgure 4: BER curves of bnary TC and uncoded QPSK on a Raylegh channel. Fgure 6: BER curves of T-COFDM and TC-QAM on a Rummler channel.