Achieving Low Outage Probability with Networ Coding in Wireless Multicarrier Multicast Systems Juan Liu, Wei Chen, Member, IEEE, Zhigang Cao, Senior Member, IEEE, Ying Jun (Angela) Zhang, Senior Member, IEEE, Huaiyu Dai, Senior Member, IEEE State ey Laboratory on Microwave and Digital Communications Tsinghua National Laboratory for Information Science and Technology (TNList) Department of Electronic Engineering, Tsinghua University, Beijing 84, China Department of Information Engineering, The Chinese University of Hong ong Department of Electrical and Computer Engineering, NC State University, Raleigh, NC 7695 Email: {eeliujuan, wchen, czg-dee}@thu.edu.cn, yjzhang@ie.cuh.edu.h, huaiyu_dai@ncsu.edu. Abstract In wireless cellular systems, it is an important and challenging tas to reliably multicast to numerous users that require the same contents at one transmission. In this paper, we propose a networ coding based multicast scheme for wireless cellular OFDM systems. The base station encodes source pacets with linear networ coding and multicasts the coded pacets to the target users over multicarriers. Thus, the users can correctly recover the source message as long as they successfully receive a certain number of pacets. The reliability of coded wireless multicast is characterized by the user outage probability, which can be greatly reduced by efficiently exploiting frequency diversity gain via networ coding. We show that the BS shall adjust the data transmission rate per carrier to strie a good balance between the reliability at each subcarrier and the redundancy among coded pacets. It is also found that full diversity gain and no diversity gain should be exploited in the high and low SNR regimes, respectively. Simulation results also reveal that networ coding generally provides great advantage for reliable wireless multicasting to a large number of users. Index Terms Wireless multicast, networ coding, OFDM, outage probability. I. INTRODUCTION Many emerging applications such as mobile TV and video conferencing aim to deliver the same large volume of data to multiple mobile users []. Such applications are particularly well suited for multicast transmissions, which broadcast the same content to a group of users at a single transmission. Wireless multicast/broadcast technologies have found wide application in wireless digital broadcast/multicast systems, such as Digital Video Broadcasting-Terrestrial/Handheld (DVB-T/H). Multimedia broadcast/multicast services have also been specified in next-generation wireless cellular systems, such as Long Term Evolution-Advanced (LTE-A) and WiMAX (Worldwide Interoperability for Microwave Access) systems [], which adopts Orthogonal Frequency Division Multiplexing (OFDM) as the air interface to combat multi-path fading effect. In wireless environments, it is a great challenge for the Base Station (BS) to reliably multicast to all of the target This wor was supported by the National Basic Research Program of China under Grant CB6, and NSFC founded ey project under Grant No. 688 and No.69. mobile users with diverse channel conditions. To ensure the successful reception for each user, the BS should multicast at a data rate adapted to the worst channel condition among the users within a multicast group. This, however, may lead to a capacity limitation problem [], especially when the size of the multicast group becomes large [4]. And the popular Automatic Repeat request (ARQ) protocol does not wor well in wireless multicast scenarios, where retransmissions of pacets that some users did not get are redundant for the others that already successfully receive them. Recently, rateless coding or networ coding has been applied to improve the reliability of wireless multicast networs [5] [9]. And considerable research interests have been aroused to combine networ coding and ARQ to offer reliable wireless multicasting services []. It seems that the unreliability and broadcast nature of wireless channels mae coded wireless multicast as a natural choice. In contrast to the previous wor, we consider to apply a networ coding based multicast scheme to reliably multicast real-time multimedia traffics to multiple users with no feedbac information in wireless cellular OFDM systems. The BS shall adopt linear networ coding to encode the pacets of a message, and then multicast the coded pacets to the target users on multiple subcarriers. A user will decode the original message, if he correctly receives at least a certain number of coded pacets. By selecting an optimal data rate per carrier, the networ coding based multicarrier multicast scheme can efficiently exploit frequency diversity gain to greatly reduce the outage probabilities for all the target users. It is found that the BS should multicast the source pacets to the target users without networ coding and thus utilize no diversity gain, when the users have relatively low SNRs. In contrast, when the users have relatively high SNRs, the BS should multicast the source pacets with repetition coding and hence tae full advantage of the frequency diversity gain. And the proposed multicast scheme can obtain a much higher throughput gain over the non-coding multicarrier multicast scheme in different scenarios, as shown by simulation results. The remainder of this paper is organized as follows. Section II introduces the system model and presents the networ coding based wireless multicarrier multicast scheme. Section
III analyzes the performance of the proposed scheme and figures out the optimal multicast parameters. Sections IV and V present simulation results and conclusions, respectively. II. PROBLEM STATEMENT We consider a wireless OFDM system, where the BS assigns S subcarriers to a multimedia traffic and transmits its data to target mobile users. Let h,s denote the channel coefficient for user on subcarrier s. Suppose that the time resource is slotted and the channels follow flat Rayleigh bloc fading. That is, the channel coefficient h,s stays constant during each timeslot while varying independently and identically in different timeslots. Also, we assume that the h,s varies independently across different subcarriers. Thus, h,s can be characterized by a complex Gaussian random variable with zero-mean and variance σ. Liewise, the Additive White Gaussian Noise (AWGN) at the receiver of each user can also be characterized by a complex Gaussian random variable with zero-mean and varianceσ. Suppose that the total transmission power is equal to P watt and the BS transmits at each subcarrier with an equal power P = P S. The BS transmits a pacet on a subcarrier s at a data rate r bit/s/hz. Let γ = P denotes the transmitted SNR at each σ subcarrier. The user can correctly decode the pacet if the instantaneous channel capacity C,s = log( + h,s γ) is greater than or equal to the data rate r. When C,s < r, an outage event occurs and the outage probability is given by p,s = Pr{C,s < r} = exp( r ). Hence, the outage γ σ probability p,s is an increasing function of the data rate r and can be denoted by p,s = f (r) for all s. Due to wireless channel fading, each user may fail to decode the pacets on some subcarriers and thus cannot recover the original message. And due to independent and different channel conditions, different users could miss different pacets. Hence, it is not a trivial tas to successfully deliver the same contents to numerous mobile users at one transmission. In this paper, we study a networ coding based multicast scheme to reliably serve multiple users requiring the same program over multi-carriers with no feedbacs. As shown in Fig. (a), the BS de-multiplexes the data bits of a message into n parallel original source pacets. Then, it encodes these pacets into S coded ones with linear networ coding and multicasts a coded pacet to users over a subcarrier. The flow from the source (or the BS) to each user is thus equal to n. From the max-flow min-cut theorem [5], each user can decode the original message only if it correctly receives at least n coded pacets. Suppose that the multimedia program shall be transmitted at a total target data rate R bit/s/hz. When the BS multicasts a coded pacet at a data rate r per carrier, the users should successfully decode at least n = R/r pacets to recover the original message. To explain it more clearly, we then give a simple example. As depicted in Fig. (b), the BS multicasts pacets a and b and their exclusive-or a b on subcarriers,, and, respectively. Three users try to receive the pacets. User and user can correctly decode the original pacets a and b since they both correctly receive at least two pacets. While user Figure. R source De-multiplex Networ coding Subcarrier Subcarrier Subcarriern SubcarrierS (a) Illustration of the transmitter structure subcarrier user (b) A simple example: = S = decoded The networ coding based wireless multicarrier multicast scheme. who successfully receives only one pacet cannot recover the original message. In the next section, we will try to find the optimal transmission parameters n and r to maximize the average user throughput. III. PERFORMANCE ANALYSIS A. The Average User Throughput In this paper, we aim at achieving a very low outage probability for all the users by applying the proposed networ coding based multicast scheme. That is, with the help of networ coding, all of the users are supposed to be well served simultaneously at one transmission. When the proposed multicast scheme is applied, user fails to recover the message if he correctly receives less than n coded pacets from S subcarriers. Accordingly, the outage probability of user, denoted by p, can be computed as ( ) S p = ( f (r i (f (r S i, () i where n { R/r,,S}. The user outage probability p depends on the data rate, r, and the threshold number of pacets, n. Hence, it can be expressed as g (r,n), a function of r and n. Thus, each user can correctly decode the original message with the probability g (r,n), and obtain the average throughput R( g (r,n. Therefore, the average user throughput can be given by ( ) T = R( g (r,n = R g (r,n). ()
It can be greatly improved by reducing the average outage probability, given by p out = g (r,n). () Then, we will carry out a detailed discussion on finding the optimal multicast parameters to minimize the average outage probability. B. Outage Minimization As stated above, we should minimize the average outage probability p out subject to a total target data rate R so as to improve the average user throughput. Hence, the optimization problem can be formulated as min r,n s.t. p out (r,n) = g (r,n) { r n R, n S = {,,,S}. The first constraint indicates that each user should receive the coded pacets at a total target data rate R to recover the original message. The second constraint presents the feasible value of the parameter n. The optimal solution to Problem (4) is denoted by (r, n ) and the minimum average outage probability is denoted byp out. This problem is a mixed Integer Programming (IP) one and generally it is not a trivial tas to derive its optimal solution. In the following, we endeavor to find the optimal multicast parameters step by step. ) Optimal data rate per carrier r : At first, we show that the parameter r taes the form of R n, since a higher data rate leads to a reduced user outage probability. Lemma. For a given n, the average outage probability p out (r,n) is a decreasing function of the data rate r, i.e., p out (r,n) p out ( R n,n), for all r [R n, ). Proof: From Eq. (), the outage probability g (r,n) for each user can be represented in terms of the regularized incomplete beta function, as follows g (r,n) = ( S ) i ( f (r i (f (r S i = (S n+) ( ) S f (r). t S n ( t) dt Since f (r) = exp( r ) is an increasing function of γ σ r, there is f (r) f ( R n ) for any r R n. Hence, we have g (r,n) g ( R n,n) and g (r,n) (4) ( ) ( R g n,n, r R ), (5) n which completes the proof. From this lemma, the average outage probability p out can be reduced by multicasting at a data rate r = R n on each subcarrier. That is, the optimal data rate per carrier must be equal to r = R n. ) Optimal threshold number of pacetsn : By substituting r = R n into (4), we have the following equivalent problem min n S p out (n) = g ( R,n). (6) n Let n denote the solution to Problem (6). Notice that (6) is an IP problem that generally is NP hard. Fortunately, the optimal parameter n can be obtained by applying one-dimensional search methods. To give more insight, we will discuss the optimal parameters in different scenarios as follows. High-SNR regime This accounts for the case, where the target users are close to the BS and have sufficiently high received SNRs. Theorem. For finite σ, the optimal solution to (6) is n =, when the average SNR is sufficiently high, i.e., γ. Proof: When the average SNR γ gets sufficiently high, there exists f ( R n ) and f γσ ( R n ). From Eq. (), we have g ( R n For n S, there exists,n) ( S i ) ( γσ γσ ) S i ( ) ( ) S n+. S g ( R n,n) g (R,) = g. That is, each user achieves the minimum outage probability g, when n is set to be one in the high-snr regime. Hence, we have p out = g ( R n,n) g. (7) So there is n =, when the average SNR is high enough. From this theorem, one can see that the BS should apply repetition coding to multicast a pacet and its copies to the target users over S subcarriers in the high-snr regime. By substituting r = R and n = into () and (), we have [ ( )] S ( g = exp [ ( T = R R γ σ R γ ) S R σ σ S ], ) S γ S, when the target users have sufficiently high received SNRs. Since the outage probability g decays with the average SNR as γ S, each user can achieve the frequency diversity gain in the highest order S. Homogeneous case In this case, all the users are assumed to be located together and roughly have almost the equal path loss. Without loss of generality, we assume that the users experience i.i.d. flat Rayleigh fading across all subcarriers, i.e., σ = σ. Hence there is g (r,n) = g(r,n). Then, it is important to find an optimal n to minimize p out (n) = g( R n,n). Since the function g( R n,n) is a unimodal function of n, we can carry out a modified Golden-section search procedure to find the optimal (8)
parameter n within O(logS) steps []. The computational complexity is very low even for a very large S value. As presented in Theorem, there is n = when the average SNR γ is high enough. In contrast, when the average SNR is very low, there is n = S, as demonstrated by simulation results later. Hence, the BS should select optimal parameters according to the users location in the homogeneous case. Heterogeneous case In this case, the target users are located in different places and have different SNRs. Intuitively, the BS should multicast at a data rate adapted to the furthest users to guarantee the quality of the received signal for all the users. However, this scheme does not wor well in practical scenarios, as shown by simulation results. Instead, the proposed scheme with the optimal parameters r and n should be applied. The minimum user outage g * 4 6 8 S=, R= S=4, R=4 S=6, R=6 Simulation results High SNR approximation 4 5 Average SNR (db) (a) The minimum user outage g vs. SNR IV. SIMULATION RESULTS In this section, we present numerical and simulation results to demonstrate the performance of the proposed scheme. Consider a wireless cellular OFDM system, where users attempt to receive the same contents over S subcarriers. Without loss of generality, the bandwidth of a subcarrier is set to be Hz. We will demonstrate simulation results in two cases: ) homogeneous case, where all the users are assumed to experience i.i.d. flat Rayleigh fading across all subcarriers; and ) heterogeneous case, where all the users are uniformly distributed in the cell with its radius equal to m and with the path loss exponent equal to.5. A. Homogeneous Case We first demonstrate the performance of the proposed scheme in the homogenous case in Fig.. In this experiment, it is assumed that σ = σ = and = 8. Specifically, Fig. (a) and Fig. (b) plot the minimum outage probability g and the optimal threshold number of pacets n versus the average SNR, respectively. As shown in these figures, the numerical results demonstrated by line curves are validated by simulation results plotted by the diamond mars. It is shown in Fig. (a) that the minimum user outage probability g dramatically degrade with the increase of the average SNR. This is because that each user can receive coded pacets with a higher probability when the average SNR increases. Accordingly, the optimal threshold n decreases with the increase of the average SNR. There is n = S, when the average SNR is relatively very small. And the optimal threshold n decreases gradually till it reaches one, i.e., n =, when the average SNR becomes relatively high enough. In this case, the user outage probability g decays exponentially with the average SNR as γ S. From the above, one can see that there is no need of applying linear networ coding in wireless multicast scenarios in the low SNR regimes. Instead, the traditional multicarrier multicast scheme should be applied. Specifically, each message generated from the multimedia traffic shall be divided into S smaller pacets, each being multicast to all the target users on a subcarrier. And with the increase of the average SNR, more and more frequency diversity should be exploited. Figure. The optimal threshold n * 6 5 4 S=, R= S=4, R=4 S=6, R=6 Simulation results 4 5 Average SNR (db) (b) The optimal threshold n vs. SNR The outage performance in the homogeneous case. In the high-snr regime, each message is paced in a pacet. And the BS shall apply repetition coding to transmit the pacet and its S copies on the subcarriers. In this way, the BS chooses to exploit the full frequency diversity gain to improve the system outage performance. B. Heterogeneous Case In Fig., we demonstrate the average throughput that each individual user can obtain by applying the proposed networ coding based multicast scheme in the heterogeneous case. In this experiment, we assume that the target users are uniformly distributed in a cell. Let d denote the distance of user from the BS. Set S = = 56 and R = 64 bit/s/hz. For comparison, we simulate four multicast schemes as follows: ) the proposed multicast scheme with the optimal parameters r = and n = n ; ) the proposed multicast scheme with the transmission parameters r = (d max ) and n = n (d max ), where d max is the users maximum distance from the BS and n (d max ) denotes the optimal threshold when the variance σ is equal to dmax;.5 ) the proposed multicast scheme with the transmission parameters r = (d min ) and n = n (d min ), where
Figure. The average user throughput The average user throughput 7 6 ( *,n * ) 5 ( * (d max ),n * (d max 4 ( * (d ),n * (d min min Traditional multicarrer multicast 5 5 The distance from the BS (m) (a) The average SNR is db. 7 6 ( *,n * ) 5 ( * (d max ),n * (d max 4 ( * (d min ),n * (d min Traditional multicarrer multicast 5 5 The distance from the BS (m) (b) The average SNR is 5dB. The outage performance in the homogeneous case. transmitting at the parameters adapted to the furthest users may be wrong, since all the users receive nearly zero throughput, as shown in Fig.. In contrast, the networ coding based multicast scheme which adapts its transmission parameters to the closest users can guarantee the service quality of the users near the BS, however, cannot well serve the relatively further users. It is also observed that the optimal networ coding based multicast scheme significantly outperforms the noncoding one for any transmitted SNR. This owes to the fact that by exploiting frequency diversity gain via coding, each user can recover the original message if he can correctly decode any n coded pacets. Hence, there is no need of correctly receiving all the pacets transmitted from S subcarriers. V. CONCLUSIONS In this paper, we investigate the problem of applying networ coding to provide reliable multicast in wireless OFDM systems. With networ coding, all the target users can decode the original message if they correctly receive the coded pacets at a total target data rate. The average outage probability can be minimized by optimizing the multicast parameters. It is shown that networ coding based wireless multicast can reliably serve the target users at one transmission. We also show that in the low-snr regime, no redundancy should be brought to the pacets and the traditional multicarrier multicast scheme should be applied. However, more frequency diversity gain should be exploited with the increase of the average SNR. In the high-snr regime, the BS should multicast the original message with repetition coding over different subcarriers, thus maing full use of the frequency diversity gain. By applying coded wireless multicast, a significant performance gain can be obtained in various scenarios. d min is the users minimum distance from the BS and n (d min ) denotes the optimal threshold when the variance σ is equal to d.5 min ; 4) the traditional non-coding multicarrier multicast scheme, where the BS multicasts to all the users at the smallest data rate r = R S on each subcarrier. Without coding, each user should decode all of the S pacets to recover the original message. Fig. (a) and Fig. (b) plot the average user throughput for the average transmitted SNR equal to db and 5dB, respectively. One can see that the closer users from the BS can achieve a higher average throughput due to less path losses no matter what ind of multicarrier multicast scheme is applied. Among the four schemes, the networ coding based multicast scheme with the optimal parameters can get the highest average user throughput for different SNRs. This is because that with the help of networ coding, a good balance between reliability and efficiency can be achieved for all the users, when the optimal parameters are chosen. As shown in Fig. (b), all the users can nearly have the highest average throughput, when the optimal networ coding based multicast scheme is applied for a relatively higher transmitted SNR. It seems lie this multicast scheme can beat the large-scale channel fading in some cases. However, the intuitive thin of REFERENCES [] U. Varshney, Multicast over wireless networs, Communications of the ACM, vol. 45, pp. 7, Dec.. [] J. She, F. Hou, P. H. Ho, and L. L. Xie, IPTV over WiMAX: ey success factors, challenges, and solutions, IEEE Commun. Mag., vol. 45, pp. 87 9, Aug. 7. [] H. Seo, S. wac, and B. G. 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