A Coparison of Convolutional and Turbo Coding Schees For Broadband FWA Systes Ioannis A. Chatzigeorgiou, Miguel R.. Rodrigues, Ian J. Wassell and Rolando Carrasco Abstract The block fading characteristics of fixed wireless access (FWA) channels do not allow the exploitation of tie diversity through the use of powerful codes cobined with interleaving. We deonstrate that turbo codes are not necessarily the optial solution in block fading channels. Convolutional codes, carefully selected so as to present the sae decoding coplexity as turbo codes, achieve siilar perforance when used in systes without antenna diversity. When antenna diversity is exploited, turbo codes outperfor convolutional codes only for a large nuber of antennas. Index Ters FWA, turbo codes, convolutional codes, decoding coplexity, OFM. B I. INTROUCTION roadband fixed wireless access (FWA) systes can provide high data rate counications where traditional landlines are either unavailable or too costly to be installed []. Such systes ay operate over links where the line-ofsight (OS) coponent is very sall or even nonexistent, serving residential or sall office/hoe office (SOHO) subscribers. Recently, the IEEE 80.6a standard has included orthogonal frequency-division ultiplexing (OFM) as one of the available transission techniques to cobat ultipath delay spread, which is experienced in broadband FWA systes []. Moreover, turbo codes are also included as one of a nuber of possible channel codes for these systes. Turbo codes have proved to be powerful in both the additive white Gaussian noise (AWGN) channel [3] as well as the perfectly interleaved Rayleigh fading channel [4]. However the broadband FWA channel suffers fro slow fading and consequently a large nuber of consecutive sybols in the sae block experience identical fading. The block fading experienced in such a channel prevents the exploitation of tie diversity in order to iprove perforance through the use of powerful coding cobined with interleaving owing to delay considerations. Hoshyar et al. [5] showed that turbo and convolutional codes, cobined with OFM, have nearly the sae bit error rate (BER) perforance in block fading channels, when antenna diversity is not exploited. in et al. [6] showed that, when antenna diversity is exploited, turbo codes outperfor convolutional codes only at high signal-tonoise ratios in Rayleigh slow-fading channels. The otivation for this paper is to copare turbo-coded with convolutionally coded OFM systes in realistic broadband FWA scenarios. Furtherore, in contrast to [5] and [6] coplexity considerations are taken into account, i.e., the decoders are carefully configured so that the coparisons are perfored with equal decoder coplexity. In particular, we consider broadband FWA systes both with and without antenna diversity. Although turbo codes have already been proposed for ultiple antenna systes [7], a thorough perforance coparison between turbo coding and convolutional coding in such cases has not been perfored. II. SYSTEM MOE In this paper, we copare the perforance of turbo coding to that of convolutional coding in broadband FWA systes, both with and without antenna diversity. Six interi BFWA channel odels have been adopted by IEEE 80.6a [8]. In this paper we consider the SUI3 odel, which corresponds to average suburban conditions. The SUI3 odel has three fading taps at delays of 0, 0.5 and µs, with relative powers of 0 db, -5 db and -0 db. The oppler spread is 0.4 Hz and the RMS delay spread is 0.64 µs, when oni-directional antennas are used. When ultiple transit and ultiple receive antennas are eployed, the envelope correlation coefficient is 0.4. A. Single-input, single-output (SISO) coded OFM The SISO coded OFM odel, which eploys one transit and one receive antennas, is shown in Fig.. I. A. Chatzigeorgiou (tel: 44-3-76707, fax: 44-3-767009, e- ail: ic3@ca.ac.uk), M. R.. Rodrigues (e-ail: rdr3@ca.ac.uk) and I. J. Wassell (e-ail: ijw4@ca.ac.uk) are with the aboratory for Counication Engineering, University of Cabridge, Cabridge CB3 0F, United Kingdo. R. Carrasco is with the Counications and Signal ing Group, School of Electrical, Electronic and Coputer Engineering, University of Newcastle, Newcastle upon Tyne NE 7RU, United Kingdo (e-ail: r.carrasco@ncl.ac.uk). This work is sponsored by EPSRC under Grant GR/S46437/0. Fig.. SISO odel of a coded OFM syste
The source bits are input to a channel encoder. In this paper we consider both convolutional coding and turbo coding. For the case of convolutional coding, the encoder uses a recursive systeatic (RSC) code with code rate /. In the case of turbo coding, the encoder is a parallel concatenation of two recursive systeatic convolutional encoders, as described in [3]. The source bits feed the first constituent encoder, while the second encoder is fed by an interleaved version of the original data. The interleaver is assued to be rando and has a size of bits. The output of the turbo encoder consists of the systeatic bits of the first RSC encoder, the parity bits of the first RSC encoder and the parity bits of the second RSC encoder. In order to increase the code rate, puncturing of the parity bits can be applied. Puncturing patterns specify which bits are allowed to pass and which are rejected. For exaple, if patterns 0 and 0 are applied to the first and second encoders respectively, all bits located in even positions within the parity strea of the first RSC encoder and all bits located in odd positions within the parity strea of the second RSC encoder will be rejected, achieving an overall code rate of /. The coded bits are shuffled by a block interleaver, which is inserted in order to reduce the effects of the block-fading channel. After channel encoding and block interleaving, the binary signal is apped onto odulation sybols. The strea of odulation sybols is converted to N parallel streas. At each signaling interval, a block of N parallel sybols S={S,S,,S N } is odulated using an inverse fast Fourier transfor (IFFT). The IFFT operation is denoted by an NxN atrix Q -. The output of the IFFT at each signaling interval is a block of N channel sybols s={s,s,,s N }, where s is given by: s = Q S () Before transission of s over the channel, a cyclic prefix is appended to prevent inter-sybol interference (ISI) and interchannel interference (ICI). At the receiver, the cyclic prefix of each OFM block is reoved, at each signaling interval. The relationship between the block of receive sybols r={r, r,, r N }, after the cyclic prefix reoval, and the block of transit sybols s, before the cyclic prefix insertion, can be expressed as: r = Hs n () where n is a length N, white Gaussian noise signal block and H is a NxN circulant atrix that contains the channel ipulse response. It is assued that the receiver has perfect knowledge of the channel. Circulant atrices can be diagonalized by the Fourier transforation atrix. Therefore, if Q and Q - correspond to the FFT and IFFT operations, atrix H can be rewritten as: H = Q ΛQ (3) where Λ is a NxN diagonal atrix whose eleents λ() =,,N, are the eigenvalues of H and correspond to the discrete Fourier transfor (FT) of the channel ipulse response. The eleents λ() are also the channel gains experienced by the OFM sub-carriers. The FFT operation following the serial-to-parallel conversion, provides the received odulation sybols R={R,R,,R N } fro the received block of sybols r, since R=Qr. Substituting for r gives: ( n) = Q[ ( Q ΛQ) s n] = ΛS N R = Qr = Q Hs since S=Qs and N=Qn. At each signaling interval, the soft deapper uses the received block of sybols R to calculate the a-posteriori loglikelihood ratios (Rs) of the received bits, which are required by the channel decoder. We use the notation b(l,) to express the l-th bit conveyed by the -th OFM subcarrier, where l depends on the odulation schee used and =,,N. The a-posteriori R of b(l,), given vector R was received, is equal to: ( b(,) R) { b( l,) = R } b( l,) = 0 R { } (4) Prob l = ln (5) Prob can be expressed as the su of two ters, naely an a- priori R A and an extrinsic R E : ( b( l,) R) ( b( l,) ) b( l, ) A E ( R) = (6) Since no a-priori knowledge is assued in our odel, A is equal to zero and so = E. Taking into account the odulation schee used, can be further siplified. ue to space liitations, only the R expressions for QPSK odulation are given but siilar expressions for higher odulation levels can also be easily derived. In QPSK two bits are grouped to for a sybol. The a-posteriori Rs of b(l=,) and b(l=,), given vector R was received, is given by: j λ ( ) ( b( l =,) R) = 4 SNR λ( ) Re { e R } j λ ( ) ( b( l =,) R) = 4 SNR λ( ) I { e R } After the R values are de-interleaved, they are used by the channel decoder to produce estiates of the transitted bits. The convolutionally coded OFM syste uses the Viterbi algorith (VA) to decode the received bits. The decoder of the turbo-coded OFM syste consists of two soft-input soft-output decoders. The first decoder uses a-priori inforation to produce soft estiates of the transitted bits by processing the Rs of the received systeatic bits and the Rs of the received parity bits of the first RSC encoder. Extrinsic inforation is extracted fro the soft estiates and acts as a-priori inforation for the second decoder. Siilarly, the second decoder processes the Rs of the received interleaved systeatic bits as well as the Rs of the received parity bits of the second RSC encoder, to produce better estiates of the transitted bits as well as extrinsic inforation, which will be used as a-priori inforation by the first decoder at the next iteration. The trellis-based decoding algoriths considered in this paper are the optial Maxiu (7)
3 A-Posteriori algorith in the log doain (log-map), also known as BCJR algorith [9], the soft output Viterbi algorith (SOVA) [0] and the ax-log-map algorith []. B. Multiple-input, ultiple-output (MIMO) coded OFM The MIMO coded OFM odel, which eploys N T transit antennas and N R receive antennas, is shown in Fig.. intervals. The relationship between the blocks of receive sybols r and r and the blocks of transit sybols s and s can be expressed as: r = H s H s n r = H which is equivalent to: ( s ) H ( s) n' (8) r H ( ) ( ) = r' H H s' n' H s n (9) where H j is an N N circulant atrix that contains the channel ipulse response fro transit antenna j to the receive antenna, and n and n are noise vectors during two consecutive signaling intervals. Based on (3) and (4), we can derive the relation between the receive odulation sybols R and R and the transit odulation sybols S and S : R Λ S ( ) ( ) = R' Λ Λ S' N' Λ N (0) Fig.. MIMO odel of a coded OFM syste After channel encoding and block interleaving, the binary signal is apped onto odulation sybols. For an arbitrary nuber of transit antennas, Tarokh et al. [] generalized the space-tie coding schee introduced by Alaouti [3]. For siplicity, we have ipleented Alaouti-type spacetie block coding (STBC) with N T = transit and N R receive antennas. Assue that S and S are two consecutive blocks of odulation sybols each of length N, conveyed during two consecutive signaling intervals, which are input to the STBC encoder. Each of the j outputs of the STBC encoder, where j= or in the case of Aalouti-type STBC, is linked with a transit antenna. uring the first signaling interval, S j= =S and S j= =S will be the blocks of sybols at streas j= and j= respectively, since N T =. uring the second tie interval S j= =-(S ) and S j= =(S) will be the blocks of sybols at streas j= and j= respectively, where S represents the conjugate of S. OFM odulation is perfored on the blocks of sybols S j= and S j= during the first signaling interval and S j= and S j= during the second signaling interval. Based on (), blocks s j= =s and s j= =s are generated and transitted siultaneously during the first signaling interval, whereas s j= =-(s ) and s j= =(s) are generated and transitted during the second signaling interval. Before transission over the channel, a cyclic prefix is appended to each block to prevent ISI and ICI. At the receiver, we assue that N R = antenna is deployed in order to siplify the analysis. Expressions can be easily extended to cover the case for N R >. The cyclic prefix of each OFM block is reoved, at each signaling interval. Assue that r and r are two consecutive blocks of receive sybols of length N each, received during two consecutive signaling where Λ j is an N N diagonal atrix whose eleents λ j (), =,,N, correspond to the FT of the channel ipulse response fro transit antenna j to the receive antenna and N and N correspond to the FT of the noise vectors n and n. As in the SISO odel, the soft deapper calculates the a- posteriori Rs of the received bits. Since R and R are blocks of odulation sybols received in two consecutive tie intervals, we use the notation b(l,) to express the l-th bit conveyed by the -th OFM sub-carrier during the first tie interval and b (l,) to express the l-th bit conveyed by the -th OFM sub-carrier during the second tie interval. Based on (5) and (6), the a-posteriori R expressions of b(l,) and b (l,), given vectors R and R, for QPSK odulation are given by: ( b(, ) R' ) = 4 SNR Re{ ( λ ( ) ) R λ ( ) ( b(, ) R' ) = 4 SNR I{ ( λ ( ) ) R λ ( ) ( b' (, ) R' ) = 4 SNR Re{ ( λ ( ) ) R λ ( ) ( b' (, ) R' ) = 4 SNR I{ ( λ ( ) ) R λ ( ) () () Finally, the R values are de-interleaved and used by the channel decoder to produce estiates of the transitted bits. III. COMPEXITY OF THE ECOING AGORITHMS The decoding algoriths considered for turbo decoding are log-map, ax-log-map and SOVA. The Viterbi algorith is used for convolutional decoding. A coplexity analysis was presented in [4] but in order to siplify the coparison it was assued that logical and atheatical operations have siilar coplexity. A ore thorough investigation was perfored in [5], where each operation is quantified as a nuber of equivalent additions. In our analysis, the
4 coplexity expressions were re-derived since the coplexity estiations for SOVA and for log-map in [5] were rather pessiistic. More specifically, it is assued in [5] that SOVA operates in the trace-back ode [6] and that the table look-up in the log-map algorith is equivalent to 6 additions. In contrast, in our analysis SOVA operates in the less coplex register exchange ode, as described in [0], at the expense of additional storage requireents. A look-up operation in log- MAP is considered to be equivalent to 3 coparisons, resulting in a coplexity of 3 equivalent additions, since only 8 values need to be stored in a look-up table [4]. In Table I the nuber of equivalent additions for the various operations is shown. Table II-IV list the coplexity of the decoding algoriths, expressed in ters of the nuber of equivalent additions, for a code rate of /, as a function of the encoder eory order M. The additional coplexity of the branch etric calculations due to a-priori inforation exploited by the turbo decoder has also been taken into account. survivor path. In practice, the actual coplexity of SOVA is lower than the predicted coplexity for sall eory orders. As the eory order increases, the exponential ter in the coplexity expression doinates and the expression converges to the actual coplexity, as shown in Fig. 3. For this reason, SOVA is not considered when the eory order of the turbo decoder is low and a coparison with the MAPbased algoriths would not be fair. TABE I: NUMBER OF EQUIVAENT AITIONS PER OPERATION Operations Nuber of Equivalent Additions Addition, Subtraction Multiplication, ivision Coparison Maxiu, Miniu ook-up Table 3 TABE II: COMPEXITY OF VITERBI AGORITHM Nuber of Equivalent Additions 6 M Path Metric Calculations 4 M Hard ecision 3 Overall Coplexity 0 M 3 TABE III: COMPEXITY OF SOVA Fig. 3. Coplexity of the iterative decoding algoriths noralized by the coplexity of the conventional Viterbi algorith Based on our calculations, which are presented graphically in Fig. 4, the coplexity of a turbo decoder with eory order M=, which applies log-map with 7 iterations, is coparable to that of a siilar turbo decoder that applies ax-log-map with iterations or a convolutional decoder with a eory order of M=8 that applies the conventional Viterbi algorith. Nuber of Equivalent Additions M Path Metric Calculations 5 M Hard ecision 3 Trace-back Procedure 4 (5M) Overall Coplexity 7 M 4 (5M) 4 TABE IV: COMPEXITY OF OG-MAP AGORITHM Nuber of Equivalent Additions M Path Metrics (Forward proc.) 9 M Path Metrics (Backward proc.) 9 M Soft ecision 8 M 3 Overall Coplexity 48 M 3 TABE V: COMPEXITY OF MAX-OG-MAP AGORITHM Nuber of Equivalent Additions M Path Metrics (Forward proc.) 4 M Path Metrics (Backward proc.) 4 M Soft ecision 8 M 3 Overall Coplexity 8 M 3 The coplexity of SOVA corresponds to the worst-case scenario, according to which the truncation depth of the traceback procedure is always 5M and all the decoded bits across each diverging path differ fro the corresponding bits of the Fig. 4. Coplexity coparison between turbo decoding and convolutional decoding IV. SIMUATION RESUTS In our siulations, the turbo encoder consists of a rando interleaver with size =000 bits and two identical terinated RSC codes with rate /, octal generator polynoial (,5/7) and eory order M=. The parity bits of the constituent RSC codes are punctured alternately so as to achieve an overall rate of /. The convolutional encoder uses an
5 RSC(,753/56) code with rate / and eory order M=8. An OFM/6-QAM syste with N=56 sub-carriers is used. In Fig. 5 the perforance of the turbo-coded OFM syste is copared to that of the convolutionally coded OFM syste. For coparison purposes the perforance of these systes in the AWGN channel has been also included. We note that even though turbo codes outperfor convolutional codes in the AWGN channel, the perforance of both codes is equivalent in the SISO SUI3 channel (i.e., antenna diversity is not exploited). In the MIMO SUI3 channel (i.e., antenna diversity is exploited) turbo codes perfor better than convolutional codes as the nuber of antennas increases. This is due to the fact that the underlying channel approaches a non-fading AWGN channel. However a very large nuber of antennas is required for turbo codes to achieve a noticeable coding gain over convolutional codes. Furtherore, the perforance of convolutional codes is better than turbo codes using the ax-log-map algorith with iterations in SISO as well as low diversity MIMO systes (Fig. 6). Fig. 5. BER coparison between turbo (applying the log-map algorith) and convolutional codes Fig. 6. BER coparison between turbo (applying the ax log-map algorith) and convolutional codes V. CONCUSIONS AN FUTURE WORK In this paper, we copare the perforance of turbo codes to that of convolutional codes for broadband FWA scenarios. In particular, the various decoders are carefully configured so as to present siilar coplexity. It has been shown that convolutional codes perfor siilarly to turbo codes for BERs up to 0-4 in SISO systes, while turbo codes eventually outperfor convolutional codes in MIMO systes using a large nuber of antennas. Consequently, turbo codes do not offer an advantage to broadband FWA systes, which use a liited nuber of antennas. These results are of practical interest for the deployent and design of high perforance and broadband FWA systes. Future work will involve coparisons to convolutional codes specifically designed for block fading channels [7], which achieve axiu code diversity without necessarily exhibiting axiu free Haing distance. REFERENCES [] H. Bolcskei et al, Fixed Broadband Wireless Access: State of the Art, Chalenges and Future irections, IEEE Co. Mag., pp.00-08, Jan. 00. [] IEEE 80.6a Standard for ocal and Metropolitan Area Networks, Part 6: Air Interface for BFWA Systes-Aendent : Media Access Control Modifications and Additional Physical ayer Specifications for - GHz, Jan. 003 [3] C. Berrou, A. Glavieux, Near Optiu Error Correcting Coding and ecoding: Turbo-Codes, IEEE Trans. on Co., vol. 44. no. 0, pp. 6-7, Oct. 996. [4] J. P. Woodard,. Hanzo, Coparative Study of Turbo ecoding Techniques: An Overview, IEEE Trans. on Vehicular Tech., vol. 49, no. 6, pp. 08-3, Nov. 000. [5] R. Hoshyar, S.H. Jaali, A.R.S. Bahai, Turbo Coding Perforance in OFM Packet Transission, Proc. VTC 000, vol., pp.805-80, May 000. [6]. in,. J. Ciini, C.I. Chuang, Coparison of Convolutional and Turbo Codes for OFM with Antenna iversity in High-Bit-Rate Wireless Applications, IEEE Co. etters, vol.4, no.9, pp.77-79, Sept. 000. [7] A. Stefanov, T. uan, Turbo-Coded Modulation for Systes with Transit and Receive Antenna iversity over Block Fading Channels: Syste Model, ecoding Approaches, and Practical Considerations, IEEE Journal on Selected Areas in Counications, vol.9, no. 5, pp.958-968, May 00. [8] V. Erceg et al., Channel Models for Fixed Wireless Applications, IEEE 80.6a cont. IEEE 80.6.3c-0/9r4, June 003. [9]. R. Bahl, J. Cocke, F. Jelinek and J. Raviv, Optial ecoding of inear Codes for Miniising Sybol Error Rate, IEEE Trans. Inforation Theory, pp. 94-87, Mar. 974. [0] J. Hagenauer, P. Hoeher, A Viterbi Algorith with Soft-ecision Outputs and its Applications, Proc. Globeco 89, vol. 3, pp. 680-686, Noveber 989. [] W. Koch, A. Baier, Optiu and Sub-optiu etection of Coded ata distributed by tie-varying Inter-Sybol Interference, Proc. IEEE Globeco 90, pp.679-684, ec. 990. [] V. Tarokh, H. Jafarkhani, A. Calderbank, Space-Tie Block Codes fro orthogonal designs, IEEE Trans. on Inf. Theory, vol. 45, pp. 456-467, May 999. [3] S. M Alaouti, A Siple Transit iversity Technique for Wireless Counications, IEEE Journal on Selected Areas in Counications, vol. 6, Oct. 998, pp. 45-458. [4] P. Robertson, E. Villebrun, P. Hoeher, A Coparison of Optial and Sub-optial MAP ecoding Algoriths operating in the og oain, Proc. ICC 995, vol., pp.009-03, June 995. [5] P. H.-Y. Wu, On the coplexity of turbo decoding algoriths, Proc. VTC 00, vol., pp.439 443, May 00. [6] J. Hagenauer, E. Offer,. Papke, Iterative ecoding of Binary Block and Convolutional Codes, IEEE Trans. on Inf. Theory, vol. 4, no., pp. 49-445, March 996. [7] R. Knopp, P. A. Hublet, On Coding for Block Fading Channels, IEEE Trans. on Inf. Theory, vol.46, no., pp.89-05, Jan. 000.