Sequential Decoding of Tailbiting Convolutional

Sequential Decoding of Tailbiting Convolutional Codes for Hybrid ARQ on Wireless Channels Pal Orten Communication Systems Group Department of Signals and Systems, Chalmers University of Technology, phone: +46 31 772 1758, fax: +46 31 772 1748, email: [email protected] SE-412 96 Goteborg, Sweden.

Abstract | Sequential decoding of long constraint length convolutional codes is proposed for hybrid automatic repeat request packet data transmission on a Rayleigh fading channel. In this scheme computational complexity (the timeout condition) is applied for error detection. In order to improve the throughput, tailbiting codes are used. Furthermore, a method for estimating the initial state of the encoder is described, and for most packet lengths the tailbiting scheme with initial state/tail estimation yields signi cantly improved throughput compared to the case with a known tail. A throughput of 0.90 at a signal to noise ratio per coded bit of 15 dB shows this coding scheme to be competitive for wireless packet data services. Although interleaving must be performed over one packet only, high throughput is maintained also for correlated fading. Keywords | Sequential decoding, packet data, automatic repeat request (ARQ), tailbiting, systematic encoder, Fano algorithm, Rayleigh fading, wireless communication.

U

I. Introduction

P to now wireless communication systems have been designed primarily to support speech communication. The density of mobile phones increases rapidly and the users expect to have access also to other types of services. Third generation wireless systems should therefore provide a number of dierent services such as fax and image transmission, audio, Internet browsing and pure data transmission. These services demand a high degree of exibility from the system in general and thus also from the channel coding scheme [1]. Since data based applications are likely to dominate future wireless systems, packet switched networks will be designed. Packet based systems allow for the use of automatic-repeat-request (ARQ) schemes such that close to error free transmission can be achieved. Throughput is further improved when applying forward error correction (FEC) coding in addition to retransmission. Such hybrid ARQ schemes can be based on block or convolutional codes. Hybrid ARQ schemes using convolutional codes have been studied by a number of researchers for Gaussian and Rayleigh fading channels [2{6]. Convolutional codes are frequently proposed for use in wireless communication systems due to the eciency of the Viterbi algorithm which performs Maximum Likelihood (ML) sequence decoding. In addition the Viterbi algorithm can easily utilize soft decisions. There is however, one major drawback with convolutional codes applied to packet transmission: a number of tailbits equal to the memory of the encoder is needed to terminate the encoding to make dif-

ferent packets independent of each other. Obviously, these tailbits reduce the throughput, and to overcome this problem tailbiting convolutional codes have been proposed [7]. With tailbiting the last part of the data packet is used as the initial state of the encoder. In this way information bits are used to terminate the encoding process and ensure that the initial and starting states of the encoder are the same. With this scheme no redundant ush bits are added. The disadvantage of this scheme is that the initial state is not known and must be estimated by the decoder. Dierent such decoders are described in [7{9]. The conventional ARQ schemes described above require CRC (Cyclic Redundancy Check) bits in order to determine whether the packet contained errors or not. These check bits carry no information and they therefore reduce the system throughput. By using sequential decoding of long constraint length convolutional codes, the CRC bits are not needed. This is because the computational load, or decoding time, depends on the channel quality. Decoding time can thus be used to indicate erroneous packets (timeout condition) [10{14]. Due to the long constraint lengths applied with sequential decoding, a large number of tailbits are applied in conventional schemes to terminate the data packets and make them independent of each other. Since the number of tailbits is large, the reduction in throughput caused be the many tailbits is signi cant. By using long constraint length tailbiting convolutional codes and sequential decoding it should therefore be possible to eliminate the loss caused by both CRC- and tailbits. It has previously been shown that sequential decoding of long constraint length convolutional codes is a good candidate for channel coding in wireless data transmission [15, 16]. In this paper we therefore propose an algorithm for sequential decoding of long constraint length tailbiting convolutional codes, and show that the throughput can be signi cantly improved using such a coding scheme on a Rayleigh fading channel. The outline of this paper is as follows. Section II describes the basic concepts of sequential decoding and its application to packet data transmission. We then continue describing the principle of tailbiting, the tail (or initial state) estimation algorithm and the analysis in Section III. Section IV presents the results, and conclusions are given in Section V.

II. Sequential decoding for Packet Data

Sequential decoding is a sub-optimum way of decoding convolutional codes where only a fraction of the code tree is searched. This gives a substantially lower average computational load than what we would have with the Viterbi algorithm. However, for sequential decoding the number of computations is a random variable that depends on the channel conditions. When there is much noise on the channel, the decoder might therefore use much more time than on the average to decode a bit. The number of computations, C , needed to decode a bit with sequential decoding is given by the Pareto distribution [17, p. 361]

P (C  L) < AL? ;

(1)

by [21, p. 397]

Nt = 1 ?1P : R

(6)

It can be seen from the expression for the throughput (5) that if the constraint length is not very much smaller than the packet length, Lp, the fractional rate loss caused by the tailbits will have signi cant impact on the throughput. Since the constraint lengths used with sequential decoding are rather high, there is a potential for improving the throughput by eliminating the tail. III. Tailbiting with sequential decoding

When applying tailbiting together with sequential decodwhere A is a constant depending on the sequential decoding ing we have a problem somewhat similar to resynchronizaalgorithm used, and  is the Pareto exponent. The Pareto tion after buer over ow in continuous transmission. Such a resynchronization algorithm is described in [22]. There exponent is found from (see [17] or [18]) are however some important dierences. First, we have E (  ) 0 a limited number of bits or samples equal to the coded R=  ; (2) packet length to aid the state estimation. In addition, all bits in the packet should be correctly decoded. In continwhere R is the code rate and E0 () is the Gallager function, uous transmission a few erroneous bits in connection with given by (see e.g. [18{20]) an over ow may be tolerable, but can not be accepted in packet data transmission. ! 1+  q?1 QX ?1 X 1 With sequential decoding the constraint length is norP (k)P (rjk) +1 E0 () = ? log2 ; (3) mally very high (typically 30-50). Since trying all possible r=0 k=0 initial states is considered too complex for Viterbi decodwhere q is the number of input levels and Q is the num- ing, it would be completely impossible with sequential deber of output (quantization) levels, P (k) denotes a priori coding, because of the large number of states (equal to probabilities and P (rjk) denotes the probability of receiv- 2K ?1 ). To be able to resynchronize, it is common to aping r given that k was transmitted. The decoder observes ply systematic encoders together with sequential decoding. the total number of computations (or time) needed to de- We therefore propose an algorithm for decoding tailbiting code a data packet, and if it exceeds a certain value the codes produced by systematic encoders. data packet will be declared in error and a retransmission is requested. This is known as the timeout condition. For A. Tailbiting algorithm a packet with Lp information bits, the probability of reSince the initial state is given by the last information bits transmission PR will then be the probability that the total of the data packet, the coded data packet can be viewed number of calculations for the whole packet exceeds a cer- as being circular. This means that at the end of the data tain maximum, given by packet we can go on into the beginning of the packet as if the transmission were continuous. Furthermore, the decod? ; (4) ing may start anywhere in the packet, and proceed until we PR  Lp ACmax back at the starting point. As an estimate of the initial where Cmax is the maximum number of computations al- are state we use the systematic bits, and start the decoding lowed per information bit before asking for a retransmisi- process. 1 describes the principles of the algorithm. If son. If this value is too small, retransmissions are requested there areFig. errors in the initial state estimate, the metric will too often. On the other hand, if it is to large, time is wasted not grow as it would a correct initial state, and the on decoding a packet containing many errors. Both [10] decoding is terminated.with The then shifts one samand [12] argue that a reasonable value for the maximum ple further in the input buerdecoder and starts again. number of computations with ARQ is Cmax =Lp = 2. For a Ideally, when the systematic bits contain nodecoding errors the depacket transmission system, the parameter of most inter- coding process can continue around the \circular" packet. est is the system throughput. Assuming that each packet When coming back at the point where an acceptable iniis ended by a known tail, the throughput equals tial state was found, the algorithm proceeds into the packet again, as illustrated in Fig. 1. L R (5)  = N L + (Kp ? 1) ; The decoder must be able to decide whether the initial t p state estimate is correct, or whether the decoding should where R is the code rate, K is the constraint length of the be terminated and continued at a new starting position in code and Nt is the expected number of transmissions given the input buer. This can be done by observing the metric

Start initial state search

First bit in packet 200

Last bit in packet Start decoding

No noise, correct state Noise, correct state Noise and incorrect state

150

Sliding window searching for initial state

metric value

100

50

0

−50

End decoding

−100 0

50

100

150

#computations

Fig. 1. Description of the tailbiting algorithm. After the search Fig. 2. The metric change as function of the number of computations for the cases (1) no noise, correct initial encoder state, for a correct initial state has succeeded, the decoding may start. (2) Ec =N0 = 5 dB and correct initial encoder state, and (3) Decoding is performed around the circle and into the packet again Ec =N0 = 5 dB and wrong initial encoder state. to produce reliable estimates.

growth as function of the number of computations. Accumulated metric values for the cases of (I) no noise, (II) noise and correct initial state and (III) noise and incorrect initial state for an uncorrelated Rayleigh fading channel are plotted in Fig. 2. We see that normally an incorrect initial state will cause the metric to be negative and decrease, and the decoder will not be able to proceed into the code tree. By comparing the metric to thresholds at given depths in the decoding process, a decision can be taken on whether the initial state was found or not. We can now formulate the detailed steps of the algorithm:

of nding K ? 1 number of consecutive correct bits in a Lp -bit long packet is

PSE =

Lp X

i=K ?1

K ?1 (Lp ; i)
(1 ? p)i
pLp?i ;

(7)

where p is the probability of channel bit error and the factor K ?1 (Lp ; i) is the number of combinations such that there are K ? 1 consecutive correct bits and at the same time a total of i correct bits among the Lp bits. The factor k (n; l) can be calculated recursively as k X





k (n ? i; l ? i + 1) + nl ?? kk ; (8) 1. Place window on rst the K ?1 systematic soft decisions. i=1 2. Make hard decisions on the samples in the window, and use these as the initial state. where 0  k  l  n must be ful lled otherwise k (n; l) = 3. Start a decoding attempt. 0. Furthermore we have two criteria to stop the recursion: 4. If metric  < 1 , where 1 < 0, then terminate decod- k (i; i) = 1 and k (m; k) = m ? k + 1. Justi cation of this ing, and if the decoding depth d < Lp shift the window one recursive expression is given in the Appendix. The total k (n; l) =

position and go to 2. 5. If  > 2 and depth d  d1 then the initial state is declared found. 6. Decode until d = d0 , where d0  Lp + dD , dD is the decision depth. 7. Release the Lp decoded bits prior to d ? dD , and arrange them in proper order.

probability of retransmission will be PRT < PR + 1 ? PSE : (9) We have an inequality because the decoding might get started even with a few erroneous bits in the initial state, and initial state estimation failure and decoding failure (timeout), may coincide. Furthermore, in the expression for PSE , we have not included the case when the sliding B. Throughput with tailbiting window is placed on some of the last bits and some of the Obviously, the method described above requires that rst bits in the packet. Since the tailbits are eliminated there are K ? 1 consecutive error free bits found by hard the throughput is given by decision on the systematic part of the received sequence. This is true with rather high probability for moderate to (10) T = NR ; tT long data packets, but not for short packets and low SNRs. We are therefore likely to observe a loss for short pack- where NtT = 1=(1 ? PRT ) is the expected number of transets due to failed initial state estimation. The probability mission with tailbiting.

IV. Results

1 Some authors use  > 1, corresponding to nite mean value of the number of computations, as the criterion for when decoding is possible. Here we use  > 2 corresponding to nite variance of the number of computations. Looking at results from [12] this seems reasonable for analyzing hybrid ARQ systems

0.5 0.48 0.46 0.44 Throughput η

To investigate the performance of a system using the proposed algorithm, evaluation has been performed analytically and by simulations. In our communication system, information bits are encoded, then interleaved and sent to a BPSK modulator. The modulated signal is then transmitted on a Rayleigh fading channel, which is constant over the symbol period, and white Gaussian noise is added. We further assume that we have perfect channel phase estimation and synchronization in the receiver. Sequential decoding is performed using the Fano algorithm in combination with the proposed tailbiting scheme. For the simulations we use a systematic rate R = 1=2 encoder of constraint length K = 36 with generator polynomial 714461625313 (octal notation) having optimum distance pro le (ODP) [23]. We assume a retransmission to occur when the total number of computations exceed Cmax , and the following parameter settings for the initial state estimation were used: 1 = ?20, 2 = 30, d1 = 100 and dD = 200. Fig. 3 shows the throughput as function of the packet length, with signal to noise ratio Ec =N0 = 5 dB, which gives a Pareto exponent  = 2 [15, 16]. Ec is the energy per code bit and N0 is the single sided noise spectral density. As we can see analytical and simulated results match rather well. Tailbiting gives higher throughput than using a known tail for packet lengths above 300. The rapid degradation for packet lengths below 300 is due to initial state estimation failure. The algorithm requires that there must be K ? 1 consecutive error free channel symbols in order to start decoding. By reducing the constraint length slightly the probability of correct initial state estimation increases considerably, but this will increase the undetected error rate. However, the loss for the shorter packets is only critical for the lower SNRs. By increasing the Ec =N0 to that required for rate 2=3 and 3=4, see Table I, the initial state estimation has a low failure rate also for the shorter packets. This is illustrated in Fig. 4. Since the throughput is limited by the channel code rate, applying a higher code rate should result in a higher throughput. However, by increasing the code rate, we also increase the probability of retransmission due to the time out condition. To maintain the same probability of retransmission, the signal to noise ratio must be increased. A conservative estimate for when sequential decoding is feasible is obtained when the Pareto exponent is larger than 21 . The signal to noise ratio yielding  = 2 can be calculated by solving (2) for the Rayleigh fading channel and soft decisions. This was done in [15, 16], and some results are summarized in Table I for BPSK modulation. Using rate compatible codes [2], and an incremental redundancy ARQ scheme [14] we have a system that will automatically adjust the code rate to match the signal to noise ratio. Applying such a scheme together with the tailbiting sequential

0.42 0.4

0.38

Tailbiting, perfect state est. − analytical Tailbiting, perfect state est. − simulated Tailbiting, est. initial state − analytical Tailbiting, est. initial state − simulated Known tail (and initial state) − analytical Known tail (and initial state) − simulated

0.36 0.34 0.32 0.3 200

300

400

500 600 700 Packet length [bits]

800

900

1000

Fig. 3. Throughput as function of the packet length in information bits. Signal to noise ratio is Ec=N0 = 5dB giving channel bit error rate p  0:064, and the convolutional encoder used for simulations is a constraint length K = 36 systematic encoder with generator 714461625313 (octal notation).

decoding algorithm with the timeout condition we get the throughput plotted in Fig. 5. Here we have used packet length Lp = 200, Lp = 300 and Lp = 500 and a constraint length K = 36 code. The results are obtained analytically, and (10) is used as a very good approximation [14]. As we can see the gain in throughput by using tailbiting is increasingly signi cant as the packet length is reduced. However, for the lower signal-to-noise ratios the tailbiting scheme experiences some loss due to initial state estimation failure. For packet length Lp = 200 and Ec =N0  6 dB the tailbiting gives lower throughput than the conventional known tail scheme. TABLE I

Theoretical limits ( = 2) for sequential decoding on Rayleigh fading channel with soft decisions and BPSK modulation, for different code rates. The Ec =N0 values are given in dB.

R Ec =N0 R Ec =N0

1=4 -0.1 6=7 13.1

1=3 1.7 7=8 13.9

1=2 5.0 8=9 14.5

2=3 8.2 9=10 15.0

3=4 10.1 19=20 18.4

4=5 11.3 99=100 25.5

5=6 12.4 -

The results presented so far assume an uncorrelated channel (perfect interleaving). To obtain results for a correlated channel we have to use simulations and interleave over one coded packet. For the shorter packets we will then have smaller interleaving depths than for the longer packets, and we should therefore expect more loss as the packet size decreases. Simulation results showing the throughput as a function of the packet length for normalized doppler frequencies (normalized with the symbol period) equal to fd = 0:01 and fd = 0:1 are given in Fig. 6. Simulations are performed for the same half rate code used in Fig. 3 us-

0.5 0.48 0.46

0.74

0.44 Throughput η

0.72

0.42

0.7

Throughput η

0.68

0.4

0.38

0.66

Tailbiting, f = 0.01 d

0.36

0.64 0.62

Perfect est. R=3/4 Tailbiting R=3/4 Known tail R=3/4 Perfect est. R=2/3 Tailbiting R=2/3 Known tail R=2/3

0.6 0.58 0.56 0.54 200

300

400

500 600 700 Packet length [bits]

800

900

Known tail, fd = 0.01

0.34

Tailbiting, f = 0.1 d

Known tail, fd = 0.1

0.32 0.3 200

1000

Fig. 6.

300

400

500 600 700 Packet length [bits]

800

900

Throughput for dierent packet length simulated with 36 and normalized doppler frequency

Ec =N0 = 5 dB, K = fd = 0:01 and fd = 0:1.

Fig. 4. Analytical evaluation of throughput as function of the packet length for dierent signal-to-noise-ratios. At a higher signal tonoise-ratio a higher code rate can be used, and higher throughput is obtainable while the state estimation is more reliable. Results are shown for constraint length K = 36 with and without tailbit- ing a Jakes fading model [24]. As expected we have some ing for code rates 2=3 and 3=4. loss in throughput because of more timeouts. Since the

interleaving has to be done over one packet only, the loss is larger for the shorter packets. We also see some dierences for the two doppler frequencies. For long packets the reduction in throughput is dominated by decoding failure (timeouts). For the higher doppler frequency the symbols appear more random to the decoder than for the lower doppler frequency, where the symbols are more correlated. The higher doppler therefore gives a lower decoder failure rate and thus, a higher throughput. For short packets the success of the initial state estimation will dominate the throughput. For the lower doppler frequency a larger number of correlated symbols increases the probability that K consecutive error free bits can be found. For short packets and high doppler frequency the probability that K consecutive symbols are error free is quite low. This explains the crossings of the fd = 0:01 and fd = 0:1 curves in Fig. 6.

1

0.9

Throughput η

0.8

0.7

Tailbiting L =500 p

Tailbiting Lp=300

0.6

Tailbiting Lp=200

V. Conclusion

Known tail Lp=500 0.5

0.4 5

Known tail Lp=300 Known tail Lp=200 10

15 Ec/N0 [dB]

20

25

Fig. 5. Analytical evaluation of throughput as function of the signal to noise ratio. Results are shown for packet length equal to Lp = 200, Lp = 300 and Lp = 500, constraint length K = 36 with and without known initial state (tail) and uncorrelated fading assuming an incremental redundancy ARQ scheme.

It has been shown in this paper that sequential decoding of tailbiting convolutional codes will give a rather high throughput and is thus a good candidate coding scheme for packet data transmission in wireless communications. Futhermore, the presented results show that by applying tailbiting convolutional codes and the proposed decoding scheme, eliminating the need for known tailbits, the throughput is improved except for very short packets at low SNRs. The proposed ARQ scheme can be combined with incremental redundancy retransmission to get a system that adapts to the channel conditions yielding a throughput of 90 percent at signal-to-noise ratio per coded bit of 15 dB.

k (n; l) 0

1

k (n ? 1; l) 0 k (n ? 2; l ? 1)

[3] 1 0

[4] 1

k (n ? 3; l ? 2)

?n?k


l?k

[5]

Fig. 7. Illustration of the tree search giving the number of combi- [6] nations of k = 3 consecutive correct bits in a frame with n bits. The symbol 1 means a correct bit and 0 means an incorrect bit. As we can see the problem is sub-divided into smaller problems [7] of the same kind.

VI. Future Work

[8]

In order to construct an ecient incremental redundancy [9] ARQ scheme families of Rate Compatible Convolutional (RCC) codes should be found which are good for sequential decoding. Such codes may for instance be Rate Com- [10] patible Punctured Convolutional (RCPC) codes and nested convolutional codes searched with the criterion of optimum [11] distance pro le. Appendix

To understand the derivation of (8) consider Fig. 7. Here we have given an example were we want to nd k = 3 consecutive ones in a sequence ? of
n bits. Having found 3 consecutive ones, there are nl??33 possible combinations that end up with l ones totally. If the rst digit is zero there are n ? 1 digits left where you may nd k consecutive ones giving a total of l ones. This can then be expressed by k (n ? 1; l). If the rst digit was one and the next was zero we similarly have k (n ? 2; l ? 1) possible combinations. Since we have now already \used" one of the ones we will have l?1 in the expression. As we can see, this is a recursive problem as described in (8). It should be noted that this is generally an exponentially increasing problem. However, since the factors will be the same for many nodes it can be shown that it is actually growing as (n ? k)2 , making it possible to compute the combinations for practical values of the packet lengths and constraint lengths. Acknowledgments

[12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]

The author would like to thank Pal Frenger at the Communication Systems Group, Department of Signals and Systems, Chalmers University of Technology, for his partic- [23] ipation in the discussions leading to the derivation of (8). This work has been performed in the framework of the [24] project ACTS AC090 FRAMES, which is partly funded by the European Community. References [1] P. Frenger, P. Orten, T. Ottosson, and A. Svensson, \Rate compatible convolutional codes for multirate DS-CDMA systems," IEEE Transactions on Communications, In press. [2] J. Hagenauer, \Rate-compatible punctured convolutional codes

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