by using the latest signal processor. Let us assume that another factor of can be achieved by HW implementation. We then have ms buffering delay. The total delay with a 0x0 interleaver is given in Table III. TABLE III TOTAL DELAY WITH 0 X 0 INTERLEAVER AND SPEED FACTOR OF 0 ( ms BUFFERING DELAY) Bit rate kbit/s kbit/s kbit/s Mbit/s Delay 0. ms.9 ms. ms. ms The whole frame need not be received in order to realize that a retransmission might be necessary, and so this may reduce the delay for a packet transmission system. In a system employing sequential decoding the quality can be traded for lower delay since overflow is related to buffer size. It is then possible to have a system that can adapt to the user requirements by having dynamic buffer sizes. CONCLUSION Theoretical performance limits for sequential decoding on fading channels have been calculated for hard and soft decisions, for different modulation methods and coding rates. Simulation results show a good match with the theoretical bounds. For a rate / code with coherent BPSK, sequential decoding can be done at E b /N 0 equal to 8 db. Simulation results also show the degradation when using non ideal interleaving. For a block interleaver of size 0 x 0 the loss is less than db compared to perfect interleaving. All results are without utilizing diversity and some gain will be achieved with diversity. Future mobile communication system must provide a range of different data type services, and the results from this work show that sequential decoding is a good candidate coding scheme for the next generation system. FUTURE WORK Future work on sequential decoding on fading channels might include non perfect channel phase estimation and diversity combining. Performance loss with shorter codes more applicable to packet oriented transmission, and some kind of tailbiting algorithm, could also be investigated. ACKNOWLEDGMENTS This work has been partly financed by Nera Telecommunications, Norway, and partly by the project ACTS AC090 Frames which is founded by the European community. REFERENCES [] P. Frenger, P. Orten, T. Ottosson, and A. Svensson, Rate matching in multichannel systems using RCPCcodes, in Proc. IEEE Vehicular Technology Conference, Phoenix, Arizona, USA, 99, pp -. The authors would like to acknowledge the contribution of their colleagues from Siemens AG, Roke Manor Research Limited, Ericsson Radio Systems AB, Nokia Corporation, Technical University of Delft, University of Oulu, France Telecom CNET, Centre Suisse d Electronique et de Microtechnique SA, ETHZ, University of Kaiserslautern, The Royal Institue of Technology, Instituto Superior Tecnico, Integracion y Sistemas de Medida, S.A. [] C. Berrou, A. Glavieux and P. Thitimajshima, Near Shannon limit error control-correting coding and decoding: Turbo- codes (), Proceedings IEEE Int. Conf. on Communications, Geneva, Switzerland, 99, pp. 0-00. [] P. Jung, J. Plechinger, M. Dötsch, and F. Berens, Rate Compatible punctured Turbo-codes for future mobile radio applications, To be submitted. to IEEE Transactions on Vehicular Technology. [] A. J. Viterbi and J. K Omura, Principles of Digital Communication and Coding, McGraw-Hill, 99. [] G. C. Clark Jr. and J. B. Cain, Error Correction Coding for Digital Communications, Plenum Press, 98. [] S. Lin and D. J Costello Jr., Error Control Coding: Fundamentals and Applications, Prentice Hall, 98. [] R. G. Gallager, Information Theory and Reliable Communication, John Wiley and Sons, 98. [8] J. B Andersson and S. Mohan, Sequential Coding Algorithms: A Survey and Cost Analysis, IEEE Transactions on Communications, Vol. COM-, No, pp 9-, February 98. [9] S. G. Wilson, Digital Modulation and Coding, Prentice Hall, 99. [0] J. G. Proakis, Digital Communications, McGraw-Hill, 99. [] P. Orten, Simulation of the Fano Algorithm for DSP Implementation on Inmarsat B High Speed Data, Technical Report No. MET 9 00, ABB Corporate Research, Norway, 99, [] R. Johannesson, Robustly Optimal Rate One-Half Binary Convolutional Codes, IEEE Transactions on Information Theory, Vol. IT-, No., pp -8, July 9. [] V. K. Bhargava, D. Haccoun and P. P Nuspl, Digital Communications by Satellite, John Wiley and Sons, 98. [] J. A. Catipovic and A. B. Baggeroer, Performance of Sequential Decoding of Convolutional Codes Over Fully Fading Ocean Acoustic Channels, IEEE Journal of Oceanic Engineering, Vol., No, pp -, January 990. [] K. Muhammed and K. Ben, On the performance of sequential and viterbi decoders for high-rate punctured convolutional codes, IEEE Transactions on Communications, Vol, No, pp 8-9, November 99. [] N. Shacham, Performance of ARQ with sequential decoding over one-hop and tow-hop radio links, IEEE Transactions on Communications, Vol. com-, pp-80, October 98. [] A. Drukarev and D. J. Costello, Hybrid ARQ error control using sequential decoding, IEEE Transactions on Information Theory, Vol. IT-9, pp. -, July 98. [8] S. Kallel and D. Haccoun, Sequential decoding with ARQ and code combining: A robust hybrid FEC/ARQ system, IEEE Transactions on Communications, Vol., No., pp -80, July 988.
Overflow rate 0 0 0 0 0 0 0 8 9 0 Fig.. Overflow rate as function of E b /N 0 for coherent BPSK with different interleaving lengths and soft decisions, f d = 0.0. BER 0 0 0 8 Fig.. Simulated Bit error rate for coherent BPSK on Rayleigh fading channel with sequential decoding of systematic convolutional code with constraint length, perfect interleaving is assumed, 0, 0 and 0 decoding errors respectively for the three simulation points. BER requirements for data services in future mobile communication systems. Very few alternative methods can operate satisfactorily at the same low E b /N 0 values. The results presented are without utilizing diversity, and the expected diversity gain makes sequential decoding even more promising []. If the frame size (number of bits between each flushing) is too small the BER performance will degrade since most errors arise from end effects in the Fano decoding algorithm. This effect can be reduced by optimizing the step size. In a circuit switched application where the frames can be long this will not be a problem. Fast enough hardware for sufficiently low overflow probability should be no problem since even software solutions exist for medium data rates ( kbit/ s) []. ARQ Systems Perfect interleaving Interleaving 0x0 Interleaving 0x0 Interleaving 0x0 0 9 8 9 0 The decoder will know when there is an overflow and will then take appropriate actions. Resynchronization of the decoder in case of an overflow can be quite cumbersome. When an overflow occurs one can alternatively ask for a retransmission of the current frame. In case of an overflow it is often so that there would have been undetected errors as well due to noisy channel conditions. With this scheme we have a reliable ARQ system [][]. Code combining ARQ schemes have been proposed in [8]. These retransmission scenarios require the use of limited length frames with tail bits. We need a number of tail bits equal to the code memory, and since the constraint length of a code used with sequential decoding is high, the frames must be rather long to ensure that the fractional rate loss is not too high. There will be a trade off to be done between fractional rate loss and throughput. Overflows due to deep fades can be detected soon and retransmission can be asked for at an early stage of the decoding process. Multiple Rate and Variable Rate One can support multiple rates by using codes of different rates R = k/n. This does not affect the complexity of a sequential decoder the same way that it does for Viterbi decoding. One could also use puncturing of the code to obtain different rates with the same encoder and decoder. It should be mentioned that puncturing has a more detrimental effect on sequential decoding than on Viterbi decoding []. This is the case for both overflow rate and bit error rate. Also, when puncturing a code it is not necessarily possible to obtain codes that are good enough to come close to the theoretical bounds. The metric should be changed when changing code rate since the Fano metric is code-rate dependent. Recent results have shown that it is advantageous to match the code to the channel data rate in a multichannel system like multicode CDMA []. By puncturing such that the code is rate compatible it should also be possible to have variable rates. One can then use coding with incremental redundancy in packet switched networks which is expected to be important in a future UMTS system. What data rates can be supported with this coding scheme depends on what doppler frequencies can be expected. It is reasonable to believe that one will have lower doppler requirements for the higher data rates. Delay For data transmission one can afford some delay in most applications. The delay with sequential decoding is dominated by the need for buffering and the interleaving delay. For a fixed size interleaver, the interleaving delay will be low for high data rates. When the data rate is low, small input buffers are required, but the interleaving delay will be large (same fixed interleaver size used). One would like to keep the overflow rate constant. The overflow rate is proportional to ( sb) ρ, s being the speed factor of the decoder and B is the buffer size. In order to keep the overflow rate constant the buffer size must be inversely proportional to the speed factor. Since the speed factor is inversely proportional to the data rate, the buffer size is proportional to the data rate. The buffering delay is then constant. To get an idea about the interleaving delay consider the application simulated in this paper. The speed factor of two can be increased to at least 8
8 0 0 0 Overflow rate 0 0 Fig.. The as function of E b /N 0 for BPSK on Rayleigh fading channel with soft decisions. achieved by soft decision compared to hard decision is given in the table. The gain is larger for the higher code rates. TABLE II THEORETICAL LIMIT FOR SEQUENTIAL DECODING ON RAYLEIGH FADING CHANNEL WITH SOFT DECISION AND BPSK MODULATION. Code rate R E b /N 0.9 db. db 8.0 db 0.0 db. db Gain over hard decision SIMULATION MODEL AND RESULTS Simulation model description rate R=/ rate R=/ rate R=/ rate R=/ rate R=/ 0 0 0. db. db.0 db 8.9 db. db Simulations are done in order to see what we loose with a non ideal decoder, and to obtain results for non-perfect interleaving. The source bits go through a systematic convolutional encoder with constraint length and code rate R = /. The generator polynomial in octal notation is, which is an optimum distance profile systematic code []. The coded bits are interleaved and then input to a transmitter filter and a BPSK modulator. The modulated signal is transmitted on a Rayleigh fading channel with additive white gaussian noise. At the receiver the signal is coherently demodulated. Perfect phase estimates are assumed. The convolutional code is sequentially decoded using the Fano algorithm (see e.g. [] for a description of the Fano algorithm). In order to simulate computational complexity and overflow, a cost function is applied based on a DSP implementation of the decoder []. This decoder has speed factor of approximately and a data buffer corresponding to 0 channel symbols at kbit/s. Results on computational complexity (overflow) The theoretical calculations done in the previous section assume a memoryless channel, and in order to verify those 0 0 8 0 8 Fig.. The overflow rate as a function of average E b /N 0 for coherent BPSK with perfect interleaving, obtained by simulations. results by simulations, perfect interleaving is assumed. Fig. shows the simulated overflow rate for hard and soft decisions. We see that the E b /N 0 point where the overflow rate gets low corresponds rather well with theory. Above this limit one can achieve as low overflow probability as wanted by using a faster HW for implementing the decoder. The simulated decoder is a rather slow decoder since the speed factor is. Hardware implementations will have much higher speed factors. Results for some finite interleaving lengths are given in Fig.. The normalized doppler frequency, f d = F d T b, is 0.0, F d being the doppler frequency and T b being the bit duration. For these simulations a block interleaver was used. As we can see, by using an interleaver depth of 0, we loose less than db compared to perfect interleaving. Some gain will also be achieved by utilizing diversity. Results from [] show that for a non coherent system with coding rate R = / and soft decisions, the gain by utilizing optimal diversity is approximately db. Almost the whole gain is achieved by just two paths diversity. Bit error rate results BER results are not that interesting to investigate since they can easily be improved, with a very modest complexity increase, by increasing the constraint length. Some results from simulations are however, provided in Fig. to show what can be expected. In the simulations, we had 0, 0 and 0 decoding errors at E b /N 0 equal to 8, 9 and 0 db respectively. Overflowed frames are not included. It should also be noted that the step size in the Fano algorithm is not optimized to yield minimum error probability since the overflow rate was a more critical parameter for this slow decoder. APPLICABILITY IN FUTURE MOBILE SYSTEMS Error Rate Soft decision Hard decision We see from the results presented in the previous sections that sequential decoding of convolutional codes can fulfil the
and the i th moment is From this it follows that if the ρ, then the mean value of the number of computations will not be bounded. A equal to unity corresponds to R = R 0, where R 0 is the computational cut off rate. We also see that if ρ then the variance of the number of computations is not bounded. One therefore wishes to operate the sequential decoder at a signal to noise ratio (SNR) where the is above. The as a function of E b /N 0 can be found by evaluating Eq. (). The Pareto exponent for hard and soft decisions on an additive white gaussian noise (AWGN) channel can be found in many references (e.g. []). For a Rayleigh fading channel and hard decisions it can be found by inserting the bit error rate for the given modulation and detection method into the expression for the Gallager function, and then solving for the signal to noise ratio that satisfies Eq. (). Fig. and Fig. shows the as a function of E b /N 0 for BPSK, DPSK and coherent and non coherent FSK with different coding rates. The E b /N 0 value is per information bit. The bit error rates for Rayleigh fading channels and different modulation/detection methods can be found in [0]. As we can see from Fig. and Fig., the varies for different modulation and detection methods and for different coding rates. Table I gives the theoretically lowest E b /N 0 ratio where a sequential decoder can operate on a Rayleigh fading channel with hard decisions. It should be mentioned that the calculations done here assume a memoryless channel, and we therefore need perfect interleaving to obtain these results. TABLE I THEORETICAL LIMIT FOR SEQUENTIAL DECODING ON RAYLEIGH E b /N 0 E C i [ ] FADING CHANNEL WITH HARD DECISION Coherent BPSK DPSK Theoretical Limits with Soft Decision ρa = lim ( L i ρ ). () L i ρ Coherent Binary FSK Noncoherent FSK R=/ 9. db.8 db. db.8 db R=/ 0. db.9 db. db.9 db R=/.0 db. db.0 db 0. db R=/ 8.9 db.9 db.9 db.0 db R=/. db. db. db 8. db When doing sequential decoding soft decisions reduces the computational complexity. For AWGN channels the soft decision gain is between and db depending on coding rate (see []). For a Rayleigh fading channel we normally achieve much more coding gain by using soft decisions than what we achieve for AWGN channels. It is therefore interesting to see how much can be gained for the computational complexity of a sequential decoder with soft decision on a Rayleigh fading channel. Assuming BPSK modulation on a Rayleigh fading channel with AWGN, the probability density of received level r, given that symbol was transmitted, is equal to f R ( r ) = f X ( x) * f N ( n), () 8 BPSK DPSK rate R=/ rate R=/ rate R=/ rate R=/ rate R=/ 0 0 0 0 0 0 0 0 Fig.. The as function of E b /N 0 for BPSK and binary DPSK for different code rates and hard decisions. 8 Coherent Non coherent rate R=/ rate R=/ rate R=/ rate R=/ rate R=/ 0 0 0 0 0 0 0 0 80 Fig.. The as a function of E b /N 0 for coherent and non coherent FSK for different code rates and hard decision. where f X ( x) and f N ( n) are the density functions of the Rayleigh and Gaussian distributions respectively, and * denotes convolution. Due to symmetry we have f R ( r 0) = f R ( r ). (8) The expression for the Gallager function will now be [] - + ρ E 0 ( ρ) = log P( k) f R ( r k) r k = 0 + ρ dr, (9) where the integral is over the continuous output space. This expression can be evaluated numerically to give the theoretical performance with soft decisions. Fig. shows the Pareto exponent for different code rates for BPSK modulation. The theoretical limits are summarized in Table II. Also the gain
Sequential Decoding in Future Mobile Communications Pål Orten and Arne Svensson Dept. of Information Theory, Chalmers University of Technology, S- 9 Göteborg, Sweden phone: + 8, fax: + 8, email: pal.orten@it.chalmers.se ABSTRACT Performance for sequential decoding on Rayleigh fading channels has been obtained theoretically and by simulations. Results show that sequential decoding of long constraint length convolutional codes is a good candidate coding scheme for data oriented services in future mobile communication systems. The applicability of such a coding and decoding scheme for future mobile systems is discussed. INTRODUCTION Future wireless systems will have to provide a variety of services with many different data rates up to around Mbit/s. The system must provide quality that varies with user requirements and applications. For some applications like video transmission it might also be advantageous to have time varying data rates. For the realization of such a complex system, advanced coding schemes will have to be applied. Research is being done in order to come up with coding schemes that are flexible- and powerful enough to fulfil the requirements of the various services in a future mobile communication system [][]. Impressing coding gains obtained with Turbo coding [] makes it a promising candidate, subject to much study for future mobile applications []. This paper studies the possibility of using sequential decoding algorithms for data services in a future mobile system. Sequential decoding algorithms were developed before the optimum Viterbi algorithm was found. After that sequential decoding has been used to decode long constraint length codes that are too complex for Viterbi decoding. Sequential decoding is described in most coding theory textbooks [], [], [], []. A number of different sequential decoding algorithms and a comparison of them is presented in [8]. We will derive performance limits for sequential decoding for different modulation and detection methods for communication on the Rayleigh fading channel. The theoretical calculations are supported by simulations and used to indicate the effect of non ideal interleaving. Then the applicability of sequential decoding in UMTS (Universal Mobile Telecommunication System) is discussed. PERFORMANCE LIMITS ON RAYLEIGH FADING CHANNEL Calculating the theoretical limits The decoding complexity of a sequential decoder is linearly dependent on the constraint length of the code (not exponentially as with Viterbi decoding). A sequential decoder is therefore capable of decoding very long constraint length codes, and very low bit error rates are therefore possible. The main problem with a sequential decoder is that the number of computations required for decoding one bit is a random variable that depends on the channel. Long buffers are therefore needed to cope with the varying delay in the decoder. Since the number of computations (and then decoder delay) is a random variable, and the buffers are finite, buffer overflows are unavoidable. The total bit error rate (BER) is given by P b = P e + P of, () where P e is the undetected bit error rate and P of is the error contribution from overflows. Since the complexity is linearly dependent on the constraint length one can normally choose the constraint length long enough so that P of dominates the total error rate. The system must therefore be designed such that the overflow probability is low enough to fulfil the system requirements. There are several alternative actions to be taken in case of overflow: () One may try to resynchronize the decoder and continue decoding, () the frame could be erased (more on this later), or () one could output the uncoded bits in case of a systematic encoder. If the channel is too noisy it is obvious that the decoder will have severe problems. In the following we calculate the signal-to-noise ratios for which sequential decoding is possible on a Rayleigh fading channel. These calculations will be supported by some simulation results. It has been shown, (see e.g.[]) that the number of computations C required to decode one information bit for a sequential decoding algorithm is given by the so called Pareto distribution P( C L) < AL ρ, () where A is some constant and ρ is the. The value of the is found from (see [] or []) R E 0 ( ρ) =, () ρ R being the code rate and E 0 ( ρ ) is the Gallager function, given by (see e.g. [], [9] or []) Q q - + ρ E 0 ( ρ) = log P( k)p( r k) r = 0 k = 0 + ρ, () where q is the number of input levels and Q is the number of output levels, P( k) denotes a priori probabilities and P( r t) denotes the probability of receiving r given that k was transmitted. Since the distribution of computations is Pareto it is easily shown that the density function will equal f C ( L) = ρal ( ρ + ), ()