FOR wireless applications on fading channels, channel

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1 160 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 16, NO. 2, FEBRUARY 1998 Design and Analysis of Turbo Codes on Rayleigh Fading Channels Eric K. Hall and Stephen G. Wilson, Member, IEEE Abstract The performance and design of turbo codes using coherent BPSK signaling on the Rayleigh fading channel is considered. In low signal-to-noise regions, performance analysis uses simulations of typical turbo coding systems. For higher signalto-noise regions beyond simulation capabilities, an average upper bound is used in which the average is over all possible interleaving schemes. Fully interleaved and exponentially correlated Rayleigh channels are explored. Furthermore, the design issues relevant to turbo codes are examined for the correlated fading channel. Turbo interleaver design criteria are developed and architectural modifications are proposed for improved performance. Index Terms Codes, concatenated coding, fading channels, interleaved coding, Rayleigh channels. I. INTRODUCTION FOR wireless applications on fading channels, channel coding is an important tool for improving communications reliability. Turbo codes, introduced in [1], have been shown to perform near the capacity limit on the additive white Gaussian noise (AWGN) channel. As a powerful coding technique, turbo codes offer great promise for improving the reliability of communications over wireless channels where fading is problematic. To date, only limited attention has been given to the performance of turbo codes on fading channels [2], [3]. In this work, we explore both the performance and design of turbo codes for fully interleaved channels and correlated Rayleigh slow-fading channels. The organization of the paper includes a brief overview of turbo codes followed by a discussion of the channel model. We then proceed to discuss the turbo code average upper bounding technique along with an examination of two-codeword probability bounds on correlated and independent Rayleigh fading channels. Simulation results are then presented for typical turbo schemes, followed by the results and conclusions from our simulations and the average bound. We conclude with an examination of the design of turbo codes on correlated fading channels. II. SYSTEM MODEL Turbo codes, introduced in [1], are, in essence, parallel concatenated convolutional codes (PCCC). The turbo en- Manuscript received October 1996; revised April 25, This work was sponsored by the National Science Foundation under Grant NCR and NASA/LeRC under Contract NAG This paper was presented in part at CISS 96, Princeton, NJ, March 1996 and the 6th Mini-Conference on Communications in conjunction with IEEE GLOBECOM 96, London, U.K., November 1996 The authors are with the Department of Electrical Engineering, University of Virginia, Charlottesville, VA USA. Publisher Item Identifier S (98) coder is constructed from simple recursive systematic convolutional (RSC) encoders connected in parallel and separated by interleavers. The decoder uses an iterative, suboptimal, soft-decoding rule where each constituent RSC is decoded separately. The constituent decoders then participate in sharing of bit-likelihood information in an iterative fashion. The constituent decoders traditionally use the BCJR algorithm [4], which is a MAP symbol decoding algorithm for block and convolutional codes. While the global turbo decoder is not ML, it has been shown to perform within 0.7 db of the Shannon limit on the AWGN channel for bit-error rates (BER s) of 10 and message lengths of [1]. In this paper, we consider coherent BPSK signaling over a nondispersive Rayleigh slow-fading channel. With appropriate sampling, the discrete representation of this channel is where is an integer symbol index, is a BPSK symbol amplitude ( ), and is an i.i.d. AWGN component with zero mean and power spectral density. The fading amplitude is modeled with a Rayleigh pdf, for. With sufficient channel interleaving (fully interleaved), the s are independent. Without sufficient channel interleaving, we adopted an exponentially correlated channel model as in [5] and [6]. In this model, the continuoustime, autocorrelation function is given by where is the Doppler bandwidth and is the lag parameter. For Rayleigh channels, the turbo decoder must be modified to incorporate the appropriate channel statistics. In the MAP algorithm, this corresponds to formulating the transition metric ( s). For a fully interleaved channel and known fading amplitudes (side information, SI), the transition metric from [7] is given as The probability is conditionally Gaussian,. For the fully interleaved channel without side information (NSI), the transition metric is given as (1) (2) /98$ IEEE

2 HALL AND WILSON: TURBO CODES ON RAYLEIGH FADING CHANNELS 161 To formulate and, we use the law of total probability to write with Hamming weight produces a codeword with Hamming weight. Substituting into (6), the average upper bound for word and bit error can be expressed as (3) In [8], it is noted that the evaluation of the integral in (3) has no known closed form. To avoid this problem, it is proposed in [8] to assume that is Gaussian in the region of most probable. From this assumption, (3) is approximated as (4) (5) The term is not a function of the conditioning ( ), therefore making its computation unnecessary in the MAP algorithm [9]. It should also be noted that for the Rayleigh channel with average energy of 1,. III. PERFORMANCE BOUNDING The ability to evaluate turbo codes in regions of high signal-to-noise requires lengthy simulations or an analytic bounding technique. In [10] and [11], an average upper bound is developed for turbo codes. It is shown that this bound is very useful in determining the error floor as well as understanding the impact of constituent encoder choice and block size on performance for the AWGN channel. Here, we apply this bound to the Rayleigh fading channel. A. Derivation of the Average Upper Bound Consider the traditional union upper bound for the ML decoding of an block code. Without loss of generality, we assume that the all-zeros codeword was sent, and we write the upper bound on the probability of word error as and In (8) and (9), is an expectation with respect to the distribution. This average upper bound is attractive because relatively simple schemes exist for computing from the state transition matrix of the RSC [10], [11]. With, the performance of turbo codes can be studied on various statistical channels by formulating the two-codeword probability for the channel of interest and using (8) or (9). B. for Fully Interleaved Channels with SI For the average upper bound, exact two-codeword probabilities or tight upper bounds are required. On the fully interleaved channel with SI, the exact probability of incorrectly decoding a codeword into a codeword which differs from in bit positions indexed by is (8) (9) (10) Here, is the tail integral of a standard Gaussian density with zero mean and unit variance defined as Here, is the number of codewords with Hamming weight and is the probability of incorrectly decoding to a codeword with weight. For a turbo code with a fixed interleaver, the construction of requires an exhaustive search. Due to complexity issues involved in this search, [10] and [11] propose an average upper bound constructed by averaging over all possible interleavers. The result of this averaging can be thought of as the traditional union upper bound, but with an average weight distribution. As in [11], the average weight distribution can be written as (6) To compute the average word error probability, we must average over the channel gains. The result is a multidimensional integral given as (11) If the fading amplitudes are independent, the indexes of the differing bit positions are of no importance only the weight of the incorrect codeword matters. Therefore, we can formulate the two-codeword probability in terms of only the Hamming distance of the codewords as (7) where is the number of input words with Hamming weight and is the probability that an input word (12)

3 162 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 16, NO. 2, FEBRUARY 1998 and For the average upper bound, we need an exact representation of or a tight upper bound. The exact evaluation of (12) is very difficult. To solve this problem, we examine four options. The first option is to simplify (12) to a form that can be evaluated via numerical integration. The other three options avoid the problem of numerical integration by seeking closed form upper bounds for. Option 1 (Exact): In [12], it is noted that can be expressed in the alternative form given by Substituting into (13) Option 3 (Bound): Another option for upper bounding on the fully interleaved fading channel with SI is to upper bound the function in the integrand of (12). For this approach, we have two possibilities. First, consider the -function bound With (19), we bound the function of (12) as (19) (20) Substituting this bound into (12), we observe a product of integrals, each having closed-form solutions, simplifying the bound to (21) (14) Since all the fades are independent, the -dimensional integral for reduces to a product of integrals over each. Furthermore, these integrals have closed-form solutions allowing us to write Option 4 (Bound): As an alternative to (19), consider the -function bound From this bound, the (22) function of (12) can be bounded as (15) (23) While (15) has no known closed-form solution, it can be evaluated via numerical integration over the single variable. For the average upper bound, we may wish to avoid numerical integration for every value of and so we next examine three bounding options. Option 2 (Bound): To avoid numerical integration in (15), consider upper bounding the integrand as Substituting this bound into (12) and evaluating the integrals, we upper bound as (24) (16) This bound will be tight for large values of and will allow us to upper bound with the closed-form expression In Fig. 1, the four options for the evaluation of the twocodeword probability are illustrated for. The slope of the curves in this figure is determined by the diversity order which, for independent fading, corresponds to the codeword distance. For example, Fig. 1 shows a reduction in of two orders of magnitude from db. In the semilog plot, the slope of the curve is where (17) (18) slope Comparing the bound options to the exact curve, in regions of low SNR, Option 4 is the tighter of the three bounds. However, in this region, note the weakness of Option 2. In regions of high SNR, the roles are somewhat reversed, with Option 2 being almost exact, while Options 3 and 4 differ from the exact by a scale factor.

4 HALL AND WILSON: TURBO CODES ON RAYLEIGH FADING CHANNELS 163 Fig. 1. Comparison of P2(d) expressions for the fully interleaved Rayleigh channel with SI and d =5. C. for Fully Interleaved Channels with NSI For the fully interleaved channel with NSI, we use the bound developed by Hagenauer in [8]. This bound is based on the simplified decoding metric (5), and is given by assumption, the two-codeword probability can be bounded by an expression which is a function of and given as (27) where and. (25) D. for Correlated Channels with SI For the exponentially correlated fading channel with SI, we will appeal to bounds developed in [5] and [6] for.in [6], it is noted that tight bounds on require knowledge of the positions of differing symbols in the codewords and. This bound is given as (26) In this expression, is the Hamming distance between the two codewords, and indicates the time spacing between differing code symbols with duration. For example, (100101) compared to the all-zeros codeword has and and. Note that there will be s. For the union bound, we can loosen the bound in (26) as in [6] and [5] by making the pessimistic assumption that all differing symbols are adjacent (, ). With this IV. RESULTS A. Simulation Results For the low signal-to-noise region, analytic evaluation of turbo codes has proven very difficult. Therefore, we examine performance based on simulations. Simulations will consider rate-1/3 turbo schemes using the 16-state RSC with generator (21/37). This scheme is often denoted (1, 21/37, 21/37) due to its construction via parallel concatenation. The 16-state RSC has 6, one less than the optimum free distance code which has 7. However, this is the encoder used in original paper of turbo codes [1]. For the fully interleaved channel, we have plotted simulation data for rate-1/3 turbo schemes with different block sizes in Figs. 2 and 3. Here, we are considering input frames of length 420, 5000, and bits. In each plot, a capacity limit is shown for reference (see the Appendix for a derivation of these limits). In all simulations, the turbo decoder uses the BJCR algorithm with modifications found in [7]. The results for 420 are shown after eight iterations, while both 5000 and are shown after 15 iterations. It should be noted the simulations without side information were done using the optimum metric rather than the simplified metric shown previously [13]. In all cases, a single turbo interleaver is used with the interleaver fixed for all simulated frames. For 420, a helical interleaver is used which has been shown to be effective on the AWGN channel [14].

5 164 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 16, NO. 2, FEBRUARY 1998 Fig. 2. Simulations on the fully interleaved Rayleigh channel with side information (SI). Fig. 3. Simulations on the fully interleaved Rayleigh channel without side information (NSI). For block sizes greater that 1000, it has been observed that randomly generated interleavers generally perform better than deterministic interleaver designs [15]. Therefore, for 5000 and , the fixed interleaver is generated randomly and used without optimization. With SI, it can be observed that for , the performance of is within 0.7 db of the capacity limit on the Rayleigh channel. However, even for 420, the performance is much better than uncoded BPSK which achieves BER at 44 db. Therefore, these rate-1/3 turbo codes are capable of coding gains exceeding 40 db. Without channel side information, the performance degrades approximately 0.8 db, consistent with the corresponding capacity limits (see the Appendix). Furthermore, the performance for remains within 0.7 db of the capacity limit for BER. In Figs. 4 and 5, the performance of the same turbo schemes with 420 and 5000 is shown for various fading bandwidths. It can be observed that performance deteriorates rapidly as decreases (fading process slows). It should also be noted that for large blocks, the penalty for decreased fading

6 HALL AND WILSON: TURBO CODES ON RAYLEIGH FADING CHANNELS 165 Fig. 4. Simulations on the exponentially correlated Rayleigh channel with k = 420 and SI. Fig. 5. Simulations on the exponentially correlated Rayleigh channel with k = 5000 and SI. bandwidth is less severe. For a BER and 0.01, the 420 scheme suffers roughly a 4-dB penalty versus fully interleaved performance, while 5000 is only penalized 2 db. Similar results can be observed for 0.001, implying that large blocks contribute greater diversity to the system. B. Bound Results We now evaluate the average upper bounding technique for various channel models and various turbo schemes (Figs. 6 12). It should be noted that the tightness of these bounds to actual simulation data from a specific interleaving scheme is questionable for several reasons. These include the union bound, the averaging over all interleavers, and the bounding of. This fact is illustrated in Fig. 6. It should also be noted that the true performance of turbo codes does not diverge at low SNR, as indicated by the bounds. In fact, the change in slope of the BER bound curves around 10 is not an effect of the channel, but rather an artifact of the union bound attributable to overcounting [11].

7 166 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 16, NO. 2, FEBRUARY 1998 Fig. 6. Simulation and average bound for (1, 21/37, 21/37), K = 420 (SI and NSI). Fig. 7. Average upper bound results on the fully interleaved Rayleigh channel with SI using different two-codeword probabilities and the (1, 5/7, 5/7) turbo scheme with K = 10. In Fig. 7, a plot for small block length illustrates the effects of the different bounds outlined previously. Motivated by this figure, the bound results in the remainder of this work will use the bound given in (24). This bound offers the best compromise between performance in regions of low SNR while not requiring numerical integration. In Figs. 8 and 10, the effects of increased block length can be observed. In [10], it is noted that the interleaver gain for increased block length is proportional to for all RSC s. Therefore, for an increase in block length from 100 to 1000, the performance increase is 1/10, which can be observed in the figures. The slopes of BER bounds will eventually converge to the minimum distance of the ensemble of codes. Therefore, the effects of using encoders with better free distance can be seen in Figs This effect is most evident is regions of high SNR. In regions of low SNR where our bounds are less useful, the actual weight spectrum becomes more important in influencing performance.

8 HALL AND WILSON: TURBO CODES ON RAYLEIGH FADING CHANNELS 167 Fig. 8. Average bounds for the fully interleaved Rayleigh channel with K = 100 and SI. Fig. 9. Average bounds for the fully interleaved Rayleigh channel with 16-state RSC s and SI. TABLE I ENERGY DEGRADATION DUE TO CORRELATED FADING AT HIGH SNR FOR K = 100 For the exponentially correlated channel, Fig. 12 shows the average bound results for 100 and the (1, 21/37, 21/37) turbo code. As in the simulations, performance degrades as the correlation of the channel increases ( decreases). In fact, this degradation can be predicted if we return to the twocodeword probability. Approximating (27) by dropping the last two terms, the term can be viewed as an energy degradation factor multiplying. Therefore, if we assume that the minimum distance dominates, as is the case at high SNR, we can approximate the difference relative to the fully interleaved channel. These differences are shown in Table I as well as Fig. 12. While these bounds give insights into achievable performance and how to choose constituent encoders, they tell little regarding the performance of the best interleaving scheme for

9 168 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 16, NO. 2, FEBRUARY 1998 Fig. 10. Average bounds for the fully interleaved Rayleigh channel with K = 100 and NSI. Fig. 11. Average bounds for the fully interleaved Rayleigh channel with 16-state RSC s and NSI. a given block size and constituent encoder. In can be said that the best scheme performs better than the ensemble bound but performance could be much better. This is due to the variation in the achievable minimum distances within the class of interleaving schemes for a given block size and constituent encoder. For example, over the class of all interleavers, the (1, 21/37, 21/37) turbo scheme with 420 has a worst case interleaver which yields a minimum distance of ten. However, it is known that this scheme using a helical interleaver can achieve a minimum Hamming distance of 22 [16]. Therefore, in regions of high SNR, actual performance will be much better than the average bound due to the dramatic differences in the diversity orders. Again, at low SNR, the multipliers on low-weight events contribute to make the situation less clear. V. DESIGN FOR CORRELATED FADING To examine the design of turbo codes on correlated channels, we will consider the union upper bound on block-error probabilities for a specific turbo code rather than the ensemble average considered above. The union bound for the probability

10 HALL AND WILSON: TURBO CODES ON RAYLEIGH FADING CHANNELS 169 Fig. 12. Average bounds for the exponentially correlated Rayleigh channel with (1, 21/37, 21/37), K = 100 and SI. of bit error of an turbo code is given as (28) where is the information weight of and is the two-codeword probability. A bound for this probability is given in (26). As in [6], we can loosen and simplify this bound to The term is referred to as the phrase length product and can be used as a design parameter [6]. A. Interleaver Design Issues Using (30) and (31), we can make some statements regarding the design of turbo codes in areas of high signal-to-noise. In this region, error events will be dominated by codewords having minimum Hamming distance. Therefore, we can rewrite (30) as (29) If we substitute (29) into (28), we have (30) where the approximation arises from for small and represents an effective information weight multiplier for the class of error events of distance defined as (31) (32) From this approximation and the definition of from above, we develop the following design objectives for the turbo interleaving scheme: 1) maximize, referred to as maximizing the diversity order 2) among the class of interleaving schemes achieving the maximum, minimize. The first criterion is identical for turbo code design on AWGN channels, and involves creating interleavers that prevent short merges in both constituent trellises. However, the traditional turbo scheme using only one interleaver might have more than one interleaver that achieves the maximum. Therefore, our second criterion states that, within the class of interleavers achieving maximum, choose the one which minimizes. From (31), we observe that this statement reduces to maximizing the phrase length product for sequences achieving. In fact, this product is maximized when the weight is evenly spread throughout the entire codeword.

11 170 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 16, NO. 2, FEBRUARY 1998 Fig. 13. Fig. 14. Turbo code fragments. Turbo codeword after traditional serialization. B. Turbo Modifications In its classical representation, a PCCC turbo scheme having two constituent encoders is generally drawn with only one interleaver preceding the second constituent encoder. In this scheme, codewords are formed by the serialization of the systematic and parity streams. In this scheme, transmitter latency is reduced since there is no additional buffering stage following encoding. Receiver latency is also reduced since the first constituent decoding can be done as the codeword is received. Despite these benefits in terms of latency, the traditional serialization procedure will suffer problems on correlated fading channels due to clustering of weight within codewords. This clustering occurs since the event involves short merges in both constituent trellises. Figs. 13 and 14 illustrate this clustering effect. With a single turbo interleaver, this clustering of weight is always going to be present for sequences that produce low Hamming weight. However, the clustering can be reduced by adding an additional interleaver either before the first encoder or before the serialization of the systematic sequence (Fig. 15). The extra interleaver should improve performance by increasing the phrase length product for low output weight sequences, as well as reducing the correlation between adjacent systematic symbols for the constituent decoders. The systematic interleaver should operate over the entire input frame, with the design being arbitrary provided that it is sufficiently different from the turbo interleaver. A possible candidate is the row/column interleaver which reads data into a matrix by rows while reading the data out by columns. For situations with relaxed constraints on system latency, another possible interleaving scheme is to append a block interleaver following the serialization procedure as in Fig. 16. Block interleavers have been shown to perform well in mitigating fading effects by reducing the effective correlation time of the channel. The ability of this scheme to reduce channel correlation is directly related to the size of the block interleaver or the interleaver depth. Unfortunately, the additional latency for the scheme is proportional to the size of the interleaver. If received codewords are buffered at the receiver before decoding, the addition of a block interleaver designed to scramble over codewords only increases latency at the transmitter. For the design of these interleavers, the row/column interleaver again is a strong candidate due to the fact that it provides uniform spacing between formerly adjacent symbols. Figs. 17 and 18 show simulation results illustrating the benefits of the alternative interleaving schemes for exponentially correlated channels with SI. In Fig. 17, a (1, 21/37, 21/37) turbo code is used where Here, a helical interleaver is used and For the block interleaving scheme, a row/column interleaver with dimensions is used. For the systematic interleaving scheme, a row/column interleaver is used. The size of the systematic interleaver is matched to the input block length ( 420) while the block interleaver is matched to the codeword length ( ). In the simulation, for a BER of 10, the block and systematic interleaving schemes offer coding gains of 2 and 3 db, respectively. While these gains do not match the coded performance on the fully interleaved channel, they have been obtained with relatively little effort (i.e., latency and extra hardware). If the goal was the performance of the fully interleaved channel, the size of the block interleaver could be increased. In Fig. 18, a turbo design with large block and stronger degree of channel correlation ( 0.001) is considered. Here, the turbo interleaver uses the same random design used in the simulations. The block interleaver is a row/column design while the systematic interleaver is a row/column design. Despite the strong channel correlation, the additional interleaving yields coding gains of 5 and 7 db, respectively. The increase in the coding gains over the previous example is largely related to the size of the systematic and block interleavers. Again, by increasing the size of the block interleaver, the fully interleaved performance can be approached. In both previous examples, it can be noted that interleaver design is more critical as decreases. VI. CONCLUSIONS In this paper, we have shown via simulations that turbo codes are capable of performance very near the capacity limit on fully interleaved fading channels. Furthermore, we have shown simulation results and bounds to indicate the performance of turbo systems under conditions of both independent and correlated fading. The bound results give indications of achievable performance as well as the effects of block length and constituent encoder choice. We concluded by proposing interleaving options as well as structural modifications to improve turbo code performance on correlated fading channels. APPENDIX With coherent BPSK signaling, the discrete fading channel model is given as, where is the channel output and is an input BPSK signal with energy constraint.

12 HALL AND WILSON: TURBO CODES ON RAYLEIGH FADING CHANNELS 171 Fig. 15. Systematic interleaving for fading channels. Fig. 16. Block interleaving for fading channels. Fig. 17. Simulation results for turbo modifications on the exponentially correlated Rayleigh channel with K = 420, BT s = 0.01, and SI. The variable is an AWGN component, and is the channel gain with Rayleigh distribution and is independent of. Channel capacity is defined as the maximum over the input distribution of the mutual information between the channel output and input. For the fading channel, if the fading amplitude is known, the mutual information is conditioned on this knowledge. For this case, we write the capacity expression as (33) (34)

13 172 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 16, NO. 2, FEBRUARY 1998 Fig. 18. Simulation results for turbo modifications on the exponentially correlated Rayleigh channel with K = 5000, BT s = 0.001, and SI. and simplifies to (35) The steps above are based on the independence of and, conditional probability rules, and the law of total probability. is the mutual information between and conditioned on knowledge of the channel gain. is the expectation over the distribution. Based on the independence of and, this distribution can be written as For symmetric channels with a finite input alphabet (i.e., BPSK), the maximization in the capacity definition is achieved by an equiprobable input distribution. From the determination of and, the capacity expression is (38) In this expression, is the Rayleigh pdf with average power of 1 and is the Gaussian pdf with a mean of and variance. The term is defined as (39) (40) If channel side information is not available, the channel capacity is written (41) (42) (43) (36) The channel gain emerges in this expression as we examine, which can be written as By symmetry, is Substituting back into for and using the simplifications (37) (44)

14 HALL AND WILSON: TURBO CODES ON RAYLEIGH FADING CHANNELS 173 Fig. 19. Channel capacity on the fully interleaved Rayleigh channel with coherent BPSK signaling. TABLE II CAPACITY LIMITS FOR COHERENT BPSK ON THE INDEPENDENT RAYLEIGH FADING CHANNEL solution to the following equation [17]: For BPSK signaling with the assumption of equiprobable inputs, and, the true boundary is found from the solution to where (45) Equations (38) and (44) can be computed using numerical integration and the results are plotted in Fig. 19. Notice that the lack of SI costs about 1 db in for codes of rate 1/4 to 1/2. From the noisy channel coding theorem, a code exists that will give arbitrarily small error performance provided the rate of the code is less than the capacity of the channel. By equating the code rate and channel capacity, we can determine the smallest such that arbitrarily small error performance is achievable. Through appropriate puncturing of the parity sequences, turbo codes of any rate are attainable. However, turbo code rates of 1/2 and 1/3 are most commonly found in the literature. Using (38) and (44), the smallest are shown in Table II for these code rates based on the following relation: From the converse to the channel coding theorem, the probability of error can be lower bounded in regions where the code rate is greater than the channel capacity from the (46) Unfortunately, for fading channels with BPSK signaling, (46) has no closed-form solution due to the complexity of the capacity expressions and. Therefore, we simply show a vertical line to indicate the minimum required for zero error performance in Figs. 2 and 3. REFERENCES [1] C. Berrou, A. Glavieux, and P. Thitimajshima, Near Shannon limit error-correcting coding and decoding: Turbo codes, in Proc. IEEE Int. Conf. Commun., 1993, pp [2] S. L. Goff, A. Glavieux, and C. Berrou, Turbo-codes and high spectral efficiency modulation, in Proc. IEEE Int. Conf. Commun., 1994, pp [3] P. Jung, Novel low complexity decoder for turbo codes, Electron. Lett., pp , Jan [4] L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, Optimal decoding of linear codes for minimizing symbol error rate, IEEE Trans. Inform. Theory, pp , Mar [5] F. Gagnon and D. Haccoun, Bounds of the error performance of coding for nonindependent Rician-fading channels, IEEE Trans. Commun., vol. 40, pp , Feb [6] S. Shamai and G. Kaplan, Achievable performance over the correlated Rician channel, IEEE Trans. Commun., vol. 42, pp , Nov [7] P. Robertson, Illuminating the structure of code and decoder of parallel concatenated recursive systematic (turbo) codes, in Proc. IEEE GLOBECOM Conf., 1994, pp [8] J. Hagenauer, Viterbi decoding of convolutional codes for fading- and burst-channels, in Proc. Int. Zurich Seminar, [9] A. S. Barbulescu and S. S. Pietrobon, A simplification of the modified Bahl decoding algorithm for systematic convolutional codes, in Proc.

15 174 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 16, NO. 2, FEBRUARY 1998 Int. Symp. Inform. Theory Appl., Sydney, NSW, Australia, Nov. 1994, pp [10] S. Benedetto and G. Montorsi, Unveiling turbo codes: Some results on parallel concatenated coding schemes, IEEE Trans. Inform. Theory, vol. 42, pp , Mar [11] D. Divsalar, S. Dolinar, R. J. McEliece, and F. Pollara, Transfer function bounds on the performance of turbo codes, TDA Progr. Rep , JPL, Cal Tech, Aug [12] J. Craig, A new, simple and exact result for calculating probability for two-dimensional signal constellations, in Proc. IEEE MILCOM, 1991, p [13] E. K. Hall, Performance and design of turbo codes on Rayleigh fading channels, Master s thesis, Univ. Virginia, Charlottsville, [14] A. S. Barbulescu and S. S. Pietrobon, Terminating the trellis of turbocodes in the same state, Electron. Lett., vol. 31, pp , Jan [15] S. Dolinar and D. Divsalar, Weight distributions for turbo codes using random and nonrandom permutations, TDA Progr. Rep , JPL, Cal Tech, Aug [16] W. J. Blackert, E. K. Hall, and S. G. Wilson, An upper bound on turbo code free distance, in Proc. IEEE Int. Conf. Commun., Dallas, TX, June 1996, pp [17] S. G. Wilson, Digital Modulation and Coding. Englewood Cliffs, NJ: Prentice-Hall, Stephen G. Wilson (S 65 M 68) received the B.S.E.E. degree from Iowa State University, Ames, the M.S.E.E. degree from the University of Michigan, Ann Arbor, and the Ph.D. degree in electrical engineering from the University of Washington, Seattle. He is currently Professor of Electrical Engineering at the University of Virginia, Charlottesville. His research interests are in applications of information theory and coding to modern communication systems, specifically data compression of still and moving imagery for digital transmission, and digital modulation and coding techniques for satellite channels, wireless networks, spread spectrum technology, and transmission on time-dispersive channels. Prior to joining the University of Virginia faculty, he was a Staff Engineer for the Boeing Company, Seattle, WA, engaged in system studies for deep-space communication, satellite air-traffic-control systems, and military spreadspectrum modem development. He also acts as consultant to industrial organizations in the area of communication system design and analysis and digital signal processing, and is the author of the graduate-level text Digital Modulation and Coding. Dr. Wilson is presently Area Editor for Coding Theory and Applications of the IEEE TRANSACTIONS ON COMMUNICATIONS. Eric K. Hall received the B.S.E.E. degree from Duke University, Durham, NC, in 1994 and the M.S.E.E. degree from the University of Virginia, Charlottesville, in He spent the summer of 1996 working for Lockheed Martin Tactical Communications Systems, Salt Lake City, UT. Currently, he is working toward the Ph.D. degree in electrical engineering at the University of Virginia. His research deals with practical and theoretical aspects of error-control coding. Mr. Hall is a member of Eta Kappa Nu and the IEEE Communications and Information Theory Societies.