Evaluating the Performance of Raptor Codes for DVB-H by using the Gilbert-Elliott Channel

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1 Evaluating the Performance of Raptor Codes for DVB-H by using the Gilbert-Elliott Channel Leif R. Wilhelmsson Ericsson Research Nya Vattentornet, SE Lund, Sweden Abstract The Gilbert-Elliott channel is used to evaluate the performance of Raptor codes for file download in the standard Digital Video Broadcasting for Handhelds (DVB-H). This allows for an analytical evaluation without assuming the log-normal fading affecting the different time-slices to be uncorrelated. The approach is also a complement to performing evaluation using prototypes in laboratory, which during the standardization work was found to be rather cumbersome when the channel was slowly varying. The choice of performance criterion is discussed, and some of the results that at first might seem counter intuitive are found to be a consequence of the used criterion. Finally, the coding gain obtained by using Raptor codes in addition to MPE- FEC is evaluated for various speeds of the receiver, showing a substantial gain, although noticeable reduced when the receiver is moving slowly. I. INTRODUCTION In wireless communications, the transmitted signal is typically subject to shadowing in addition to the fast fading. The shadowing effect on the transmitted signal is very different than that of the fast fading. Particularly, as the correlation distance for the fast fading is on the order of 10 cm, the correlation distance for the shadowing is typically in excess of 10 meters. Virtually all systems for wireless communications employ some kind of forward error control (FEC) coding. For the FEC code to be efficient, the different code symbols should ideally experience uncorrelated fading. In case of fast fading, this can often be achieved by means of interleaving. In order for interleaving to be effective, the interleaving depth should be large enough to essentially average out the fading statistics, which for typical speeds is feasible if, say, an interleaver corresponding to a few ms is used. Interleaving is, however, typically not a feasible way to deal with shadowing, since the size of the interleaver for typical data rates would be enormous, and the imposed delay in many cases would not be acceptable. For services with no real-time requirements, it is in principle possible to reduce the required fading margin by using powerful coding that spans a sufficiently long time. For the standard Digital Video Broadcasting for Handhelds (DVB-H) [1], [2], this powerful coding can be achieved by introducing an extra layer of coding on the application layer. A concept denoted time-slicing is introduced in DVB-H. This essentially means that data belonging to different files or different programs are sent in different time-slots. The amount of data in a timeslice, including overhead for e.g. error correction, is limited to 2 Mbits, [1], [2]. With a typical transmission rate of, say, 10 Mbits/s this means that the duration of a time-slice is on the order of 0 ms. Typical streaming applications, like television, might be expected to require some 0 kbits/s. This means that a time-slice must be received every 5 seconds or so. In case of file download, the file does not fit within one time-slice, one can choose the interval between the used time-slices somewhat arbitrary. During the DVB-H standardization work with an application layer-forward error correcting code (DVB-H AL-FEC), it was found that in case large files are to be delivered a significant coding gain can be obtained by using Raptor codes [3]. This is due to the combination of Raptor codes and time-slicing which means that the transmission of a complete file might take considerably longer than the correlation time for the shadowing. Raptor codes are erasure codes that works on the application layer. The amazing thing with Raptor codes is that they are able to perform close to an ideal code at a complexity which allows for very large codewords, see [4]. Due to this it is possible to use codewords that span over as many timeslices that are needed to deliver the file, thus allowing for a considerable power saving gain. The power saving gain possible to achieve was in [3] estimated by assuming that the shadowing affecting different time-slices were independent. This makes sense if the distance between the different time-slices is sufficiently long relative to the speed of the receiver. To assess the case when the channel was only slowly varying, and also qualitatively see the effect of different Doppler frequency, the following approach was taken in the standardization work [5]. First error traces were generated in a laboratory environment, the traces were on Transport Stream (TS) packet level. These traces were then used to determine the performance of a Raptor code by means of simulation. While performing these simulations it was observed that when the channel is slowly varying, the traces must be very long in order to give accurate results. In fact, in [5], for the case of low Doppler frequency (1 Hz and 10 Hz) no error traces were available at the relevant SNR values and also in case of high Doppler frequency (80 Hz), the error traces were in some cases too short to give statistically reliable results. In this work we try to close the gap between either having to use a very simple model, that is known to be valid only /07/$ IEEE

2 for moderate and large speeds of the receiver, and having to generate data in the laboratory, which even in this case might be rather cumbersome in case the receiver is moving at slow speed. The approach to achieve this is by using the well-known Gilbert-Elliott (GE) channel model [6], [7] and choose the parameters properly. By doing this we are able to estimate the gain obtained by using Raptor codes also when the receiver is moving slowly. Moreover, the model gives a good indication of when the assumption of independent shadowing used in [3] is valid. The results can in principle also be used to choose a suitable interval for the time-slices in case a certain speed of the receiver is assumed. The remaining part of the paper is organized as follows. In Section 2 the GE channel is described and some useful statistical properties are derived. Section 3 shows how the parameters of the GE channel can be chosen to reflect some key properties of a commonly used model for the shadowing fading. Numerical results are provided in Section 4 for some different environments, speeds etc. for realistic DVB-H parameters. Finally, Section 5 concludes the paper. II. SOME USEFUL STATISTICAL PROPERTIES OF THE GE CHANNEL The GE model dates back to the early s [6], [7]. In this model for a slowly varying channel, the channel is assumed to either be in a good state the error rate is small or in a bad state the error rate is large. The GE model is depicted in Figure 1. Here the good and the bad state are denoted G and B, respectively, as the probabilities that the channel changes from the good state to the bad state and vice versa are denoted b and g, respectively. The probabilities that the channel is in the good and the bad state at the mth instant of time are denoted P m (G) and P m (B), respectively. Henceforth, boldfaced typesetting is used for vectors and matrices, and we let P m =[P m (G),P m (B)] denote the probability distribution at time instant m. The conditional probability that the channel is in state B at one instant of time and in state Gminstants of time later is denoted P m (G B), and the other three conditional probabilities are denoted accordingly. The dynamics of the GE channel is conveniently described by considering the transition matrix, T, of the channel, given by [ ] 1 b b T =. (1) g 1 g Using standard methods, the eigenvalues of T are found to be λ 1 =1and λ 2 =1 b g, with corresponding eigenvectors S 1 =[1, 1] T and S 2 =[b, g] T, T denotes transpose. Consequently, we can write S = [ 1 b 1 g T = SΛS 1, (2) ] [ 1 0, Λ = 0 (1 b g) ]. (3) 1 b G Fig. 1. b g B The Gilbert-Elliott channel. 1 g In considering the problem of finding the probability distribution at time m given the probability distribution at time 0, we immediately have P m = P 0 T m = P 0 SΛ m S 1. (4) As m, the impact of the initial state will diminish, and we therefore let P = [P (G),P (B)] denote the stationary distribution. This is easily shown to be P =[g/(b + g),b/(b + g)]. (5) As an example, if we assume that the channel at time t =0 is in the good state, then the probabilities that the channel is in the good and bad state at time m, respectively, are found to be P m =[P (G)(1 λ m 2 )+λ m 2,P (B)(1 λ m 2 ) λ m 2 ]. (6) Similarly, if P 0 =[0, 1], corresponding to that the channel at time t =0is in the bad state, we find P m =[P (G)(1 λ m 2 ),P (B)(1 λ m 2 )]. (7) Hence, we see that the influence of the initial state is an exponentially decreasing function, and the rate of the decay is given by λ 2. To proceed, let us define a random variable s k as follows { 1 if the channel is in state G s k = (8) 1 if the channel is in state B It then makes sense to talk about the average of the GE channel as well as the correlation of the channel. We immediately have E[s k ]=P (G) P (B) =2P (G) 1. (9) Moreover, the covariance function of the GE channel is found to be r(m) =E[s k s k+m ] (E[s k ]) 2 =4P (G)(1 P (G))λ m 2, (10) and, consequently, the correlation function equals ρ(m) = r(m) r(0) = λm 2. (11) When evaluating the performance of a code when used for the GE channel, the probability distribution for being in the bad state d times out of n is of interest. In [8] the following theorem was proved: Theorem 1: If the Gilbert-Elliott channel is observed at n consecutive instants of time, the probability that the channel is in the bad state d times, 0 d n, is given by

3 P (G)(1 b) n 1 d =0; P P n (d) = (G)(P n (d GG)+P n (d GB)) +P (B)(P n (d BG)+P n (d BB)) 1 d<n; P (B)(1 g) n 1 d = n, P n (d GG) = P n (d GB) = P n (d BG) = P n (d BB) = min(d+1,n d) i=2 ( n d 1 i 1 )( ) d 1 i 2 (1 b) n d i b i 1 (1 g) d i+1 g i 1, min(d,n d) i 1 i 1 i=1 (1 b) n d i b i (1 g) d i g i 1, min(d,n d) i 1 i 1 i=1 (1 b) n d i b i 1 (1 g) d i g i, min(d,n d+1) i 2 i 1 i=2 (1 b) n d i+1 b i 1 (1 g) d i g i 1. Here P n (d GG) is the conditional probability of being d times in the bad state, conditioned on being in the good state both the first and the last instants of time, and the other conditional probabilities are defined accordingly. III. MATCHING THE GE CHANNEL MODEL TO LOG-NORMAL FADING When shadowing is taken into account, this is typically done by introducing a fading margin. In essence, if the required SNR (in db) for the system is denoted, (C/N) T, then the fading margin is how much larger the average SNR (in db), C/N, must be to obtain the required link performance. The shadowing is often taken to be log-normally distributed, and the channel quality is then considered too poor in case the log-normal fading causes the power to be reduced by more than C/N - (C/N) T. Let state B of the GE channel correspond to the situation that the channel quality is too poor and let σ denote the standard deviation of the log-normal fading. The probability that the GE channel is in state B is then given by ( ) P C/N (C/N) T (B) =Q, (12) σ Q(x) = 1 e t2 /2 dt. (13) 2π In [9], the correlation of the fading process has been found to be accurately modeled as x ρ(d) =e d/d0, (14) d 0 is referred to as the correlation distance. To see how this is matched to the GE model, let T denote the time step TABLE I GE PARAMETERS FOR SOME DIFFERENT T.SEE THE EXAMPLE FOR FURTHER DETAILS. T b g ρ(1) = λ 2 d/d 0 2s s s s s s s s that is used in the discrete time GE model, and let v denote the speed of the receiver (measured in m/s). This means that the receiver moves Tv during an instant of time of the GE channel. Comparing (11) and (14), it therefore follows that λ 2 = e Tv/d0. (15) Using (12) and (15), b and g are easily determined as is illustrated in the following example: Example: Consider at channel C/N = 18dB,σ = 5.5 db, and d 0 = m. Moreover, suppose that the receiver is moving at km/h (11.1 m/s). and (C/N) T =10dB.Using (12), we find b/(b+g) =0.07. Furthermore, taking T =5s, we have λ 2 = e Tv/d0 = = 1 b g. Consequently we have b = and g = Clearly, the ratio b/(b+g) will not depend on the dynamics of the channel. It is also intuitively clear that when T is increased the GE channel will become closer to being memoryless. In Table I, the obtained parameters are shown for some different T. Here the time step in the GE channel is increased so that the correlation function (11) will decrease because λ 2 in decreased. If instead the time step would have been kept constant, the correlation function (11) would have decreased to the same value, but because m had increased. For illustration, λ 2 and the distance normalized with the correlation distance are also given. IV. NUMERICAL RESULTS To evaluate the impact of correlated shadowing, and also to estimate the coding gain obtained by Raptor codes, the chosen parameters were taken similar as those used in the standardization work [10], the performance of Raptor codes was evaluated using error traces. For basic properties of DVB-H, see e.g. [1], [2]. In this work, the following assumptions are used when evaluating the performance: The amount of user data in a time-slice is 1.5 Mbits. This corresponds to a code rate of 3/4 of the MPE-FEC and time slice of maximum size. The duration of a time-slice is on the order of 0.2 s., and the log-normal fading remains constant during this time. The Raptor code requires 2% coding overhead to be successful. The probability of erasure is 1 in the bad state and 0 in the good state.

4 80 Acquisition prob. = 0.90 Acquisition prob. = 0.95 Acquisition prob. = Acquisition prob. = 0.90 Acquisition prob. = 0.95 Acquisition prob. = 0.99 Acquisition time [Time slice bursts] Acquisition time [Time slice bursts] C/N [db] C/N [db] Fig. 2. Acquisition time versus C/N in case of a 4MB file for three different requirements on the probability of acquisition. v =km/h,d 0 =m. Fig. 3. Acquisition time versus C/N in case of a 4MB file for three different requirements on the probability of acquisition. v =90km/h,d 0 =m. The threshold the GE channel is assumed to change state, C T is 15.5 db. According to [11], this corresponds reasonably well (with in 1 db) with the case that the modulation is 16-QAM and the code rate of the convolution code is 2/3. σ =5.5dB. Unless stated otherwise, the distance between the timeslices is 5 s. Since the number of time-slices required to deliver a file will be a stochastic variable, it is commonplace to discuss the performance in terms of acquisition probability. In [10], the average number of time-slots was determined for an acquisition probability of 95 %. In Figure 2, the acquisition time for three different acquisition probabilities are depicted as a function of the SNR on the channel. The speed of the receiver is relatively small ( km/h) and the correlation distance is m. Although C T =15.5 db, the average SNR on the channel must be considerably higher in order for the acquisition time to being close to the minimal. The corresponding figure for the situation that the receiver is moving at higher speed (90 km/h) is shown in Figure 3. Since the channel in this case will be less correlated between two time-slices, and correlation typically has a negative impact on the coded performance, one would expect the acquisition time to be reduced. As can be seen this is also the case for C/N up to, say, 25 db. However, for large C/N values the opposite will sometimes be the case. To explain this, which seems counter intuitive, consider the situation the probability of beginning in the good state is close to one and the channel is slowly changing. This means that there is a rather high probability that the first timeslice which carries data for the file experiences a good channel, and that the channel remains in the good state throughout the delivery of the entire file. Now, suppose the figure of merit to determine the performance is the probability of achieving acquisition, and this is, say, taken to be 95%. If P (G)(1 b) (n 1) > 0.95, (16) then n time slices will suffice. However, P (G), is independent of how fast the channel is changing, and if sufficiently close to one, it is beneficial that b is small, i.e., that the channel is slowly changing. It should be emphasized that this effect is not due to the GE-channel model, although it is simple to illustrate what happens. This effect can be expected also for a real-world channel. An interesting trade-off is of course what a suitable time between the time-slices is, in order to deliver the file as quickly as possible but using as few time-slices as possible. Clearly, one would use the smallest number of time-slices (or nearly so) by ensuring that the channel was uncorrelated between the bursts. However, this would, especially in case of large files, imply a rather large delay. In Figure 4 this trade-off is considered by plotting the acquisition time as a function of the time duration between the time-slices. Referring to Figure 4, it is seen that in case C/N = 18 db, there is no reduction in the number of required time-slices by increasing the time between time-slices above 5s. It can also be seen that in case of high C/N, the time-slices can be sent closer in time without needing additional time-slices, as in case of small C/N, there is a noticeable gain by increasing the time between the time-slices. The explanation to this behavior is as follows. If C/N, is increased then also g will be increased and the duration the channel is in the bad state will on average be shorter. Thus, less time between the time-slices is required to ensure that the channel will not remain in the bad state for two consecutive time-slices. One major reason for using Raptor codes is that the required fading margin can be reduced when coding can be done over several time-slices. In [3] the gain of using a Raptor code was estimated to be about 4 db in case a3mbfilewastobe delivered with a probability of 90 %. In case without Raptor coding, the rate of the MPE-FEC was 3/4, resulting in that 16 time-slices were needed. Since the coding is within each time-slice, the delivery will be successful only if all time-slices can be correctly decoded, i.e., the channel is sufficiently good for all 16 time-slices. When Raptor codes were considered, the

5 1 110 SNR = 14 db SNR = 18 db SNR = 22 db MPE FEC MPE + Raptor Acquisition time (Time slice bursts) Required SNR [db] Time between bursts [s] Speed [km/h] Fig. 4. Acquisition time versus the time between the bursts in case of a 4MB file for three different C/N values. File size = 4 MB, v =km/h, d 0 = m. Fig. 5. Comparison of the required SNR to achieve 90% acquisition probability for MPE-FEC and Raptor codes as a function of the speed of the receiver. File size = 3MB, d 0 =m. See text for further details. code rate of the MPE-FEC was increased to 6/7, and the rate of the ALG-FEC was taken to 14/16 so that the overall code rate was kept the same. In this case the file is successfully delivered as long as at least 14 time-slices are correctly decoded. To compensate for the increased code rate of the MPE-FEC, it was assumed that the required C/N for successful decoding of the MPE-FEC had to be increased by 1.5 db. In Figure 5, the same parameters are used when comparing the two coding strategies, but without assuming the log-normal fading to be uncorrelated. The required C/N is taken to be 14.5 db in case no Raptor code is used, corresponding to the performance of a typical receiver in case 16 QAM and rate 1/2 for the convolutional code, [11]. In case the Raptor code is used, the required C/N istakentobe16db.referring to Figure 5, the performance with and without Raptor codes for the above parameters is depicted as a function of the speed of the receiver, we define the coding gain obtained by using a Raptor code as how much the C/N can be reduced. As can be seen, when the receiver is moving at about km/h or more the obtained coding gain is close to 4 db which agrees with what was reported in [3]. However, for lower speeds, the coding gain becomes smaller for two reasons. First, the performance for the Raptor code is degraded due to that the channel is no longer memoryless. Second, in case no Raptor code is used all 16 time-slices must be correctly received for the file to be correct. For this case a correlated channel is in fact advantageous, since the probability that the channel remains in the good state is increased, compare (16). V. CONCLUSION The performance of Raptor codes, when used for file delivery in DVB-H, was considered. By using the Gilbert- Elliott channel to model the effect of log-normal shadowing, the performance was analytically studied without invoking the assumption that different time-slices were affected by uncorrelated shadowing. The trade-off between delivering a file using as few time-slices as possible and delivering a file as quickly as possible was treated by considering the number of required time-slices as a function of the time duration between the time-slices. It was found that if the timeslices are transmitted so that the receiver has moved about one correlation distance, there is very little to be gained by increasing the distance between the time-slices further. Finally, the result on coding gain for Raptor codes reported in [3] was generalized to the case also the impact of correlated shadowing was considered. ACKNOWLEDGMENT The author would like to thank the reviewer with detailed insight within the area of AL-FEC and the standardization of AL-FEC for DVB-H for valuable comments. REFERENCES [1] ETSI EN 2 4 Digital Video Broadcasting (DVB); Transmission System for Handheld Terminals (DVB-H), European Telecommunication Standard, Nov. 04 Available at [2] G. Faria, J. A. Henriksson, E. Stare, and P. Talmola, DVB-H: Digital broadcast services to handheld devices, Proc. IEEE vol. 94, no. 1, pp , Jan 06. [3] E. Stare, A. Lindberg, and M. Watson, Simplified model to assess the gain by using application layer generic forward error correction (ALG- FEC) for file delivery, Doc. TM-CBMS 19. [4] A. Shokrollahi, Raptor Codes, IEEE Trans. on Inform. Theory, pp , No. 6, June 06. [5] DVB TM-CBMS1341, Application layer forwards error correction simulations for small file download, June 05. [6] E. N. Gilbert, Capacity of a burst-noise channel, Bell Syst. Tech. J. vol. 39, pp , Sep. 19. [7] E. O. Elliott, Estimates of error rates for codes on burst-noise channels, BSTJ vol. 42, pp , Sep [8] L. Wilhelmsson and L. B. Milstein, On the effect of imperfect interleaving for the Gilbert-Elliott channel, IEEE Trans. on Commun., pp , May [9] M. Gudmundssson, Correlation model for shadow fading in mobile systems, IEE Electronic Letters, pp , Nov [10] DVB TM-CBMS1361, Proposal for simulations for evaluations of application layer FEC for file delivery, June 05. [11] ETSI EN Digital Video Broadcasting (DVB); Transmission System for Handheld Terminals (DVB-H); Validation Task Force Report, May 05 Available at