Opportunistic Multicasting Scheduling Using Erasure-Correction Coding over Wireless Channels

Multicast services over wireless communications have recently become more and more popular. Multicast gain has been explored in a worst-user (WU) approach by transmitting only one copy to all users in the multicast group, for example, [1]. In wireline networks, since user channels are fixed, the multicast throughput increases linearly with the multicast group size, . However, due to a possible large difference in the instantaneous channel gains of the various links from the base-station (BS) to users in a wireless fading environment, the BS may have to apply the lowest supportable rate (corresponding to the worst BS-user link), which results in very low bandwidth efficiency. Based on this fact, opportunistic scheduling for unicast transmission to explore the multiuser diversity by sending one copy to the user with the highest instantaneous channel gain has been extensively researched. Unfortunately, such a best-user (BU) approach does not make use of the multicast gain, which can yield low utility of resources, especially for a large multicast group size. While opportunistic scheduling for unicast transmission has been extensively researched, the idea of exploiting multiuser diversity into multicast has not been studied in an equivalent extent yet. In [2], a threshold- multicast scheduling scheme at MAC layer is proposed in which in each time-slot the BS sends multicast packet if there are at least users that have sufficiently good channel to receive the packet. Another opportunistic multicast approach is proposed in [3–6], in which, in each time-slot, one copy is sent to only users with the best channel quality of the multicast group. The transmission rate is selected as the supportable rate of the worst user in these best users. In this way, in each transmission, only user can reliably receive the packet while the other ( ) users with insufficient channel gains cannot. To cover the whole multicast group, the use of retransmission has been discussed and proposed in [4–6]; however, those schemes are either inefficient or too complex to implement. In [7–9], proportional fair schemes have been studied aiming to maximize throughput while maintaining the fairness between multicast users and multicast group. These studies assume perfect knowledge of the channel responses of all users in the multicast group at the BS.

In this paper, we propose an opportunistic multicast scheduling scheme that can jointly explore multicast gain, multiuser diversity, and time/frequency diversity in a wireless fading environment. In the proposed scheme, each packet is sent only once to all users in the multicast group at a transmission rate determined by a selected channel gain threshold and an erasure-correction coding is used to deal with possible erasures when the instantaneous signal-to-noise ratio (SNR) of a BS-user link happens to be inadequate. Reed-Solomon erasure-correction code is applied to a block of transmitted packets such that erased packets can be recovered as long as the number of erased packets in a block does not exceed the erasure correction capability, that is, . As each packet can be transmitted in a time or a frequency slot, erasure-correction coding to a block of transmitted packets effectively explores the time/frequency diversity in a wireless fading environment. The selection of channel gain threshold and erasure correction code parameters are jointly optimized for best multicast throughput. Furthermore, to study the role of channel knowledge, the proposed scheme is considered in two cases: (i) with full channel gain knowledge and (ii) with only partial knowledge of fading type and average SNR. An analytical framework has been developed to evaluate the multicast throughput of the proposed erasure-correction coding opportunistic multicast scheduling (ECOM) scheme as well as the BU and WU approaches. We prove that the effective multicast throughput (i.e., the multicast rate that each user can receive) of WU and BU asymptotically converges to zero as the group size increases while that of our proposed scheme is bounded from zero depending on the SNR. Numerical results illustrate that for small multicast group size, full channel gain knowledge can offer better multicast throughput than partial channel knowledge; however, for large group size, the difference in multicast rates of these two cases is just negligible. Besides, performance evaluation shows that with the ability of combining both gains, the proposed scheme outperforms both BU and WU for a wide range of SNRs.

Furthermore we consider extending ECOM for applications to Orthogonal Frequency Division Multiplexing (OFDM) systems. In particular, we explore frequency diversity in a frequency-selective fading environment by sending coded packets over subcarriers. However, since there is a correlation in channel gains among the subcarriers, deep fade on one subcarrier may result in insufficient instantaneous SNR on neighbouring subcarriers. Hence, we have investigated the effects of correlation in subcarrier channel gains on the achievable multicast throughput of the proposed scheme. Numerical results indicate that by exploring frequency diversity, we can significantly reduce the delay with negligible degradation in multicast throughput.

The rest of this paper is organized as follows. In Section 2 the proposed ECOM schemes are described and the analytical framework on multicast throughput of the proposed schemes and BU and WU is provided. Then performance evaluation and comparisons are discussed to illustrate the trade-off between multicast gain and multiuser diversity and the significance of full and partial channel knowledge. In Section 3, ECOM scheme is extended for applications to OFDM systems. The effects of correlation in subcarrier channel responses in a frequency-selective fading environment on the multicast throughput and the trade-off between throughput and delay are discussed. Finally, Section 4 provides concluding remarks.

In this section, we consider to extend the application of the proposed ECOM scheme to broadband OFDM systems in a frequency-selective fading environment by exploiting frequency diversity. OFDM divides the entire transmission bandwidth into many subchannels, each with sufficiently narrow bandwidth such that the corresponding subchannel response can be regarded as being frequency-flat. To apply ECOM scheme to OFDM systems, one approach could be transmitting coded packets on one selected subcarrier in many independent time-slots as in Section 2. Since packets are coded in blocks and sent in time, each user needs to receive the entire block before decoding, and hence, this introduces a delay. One modification to reduce this delay is to send many coded packets simultaneously on many subcarriers, that is, exploring frequency diversity.

Nevertheless, in multicarrier OFDM systems, fading in neighbor subcarriers can be correlated so that deep fade in one subcarrier can also result in deep fade in other nearby subcarriers. As a consequence, for large fading correlation, the multicast rate can be greatly reduced as packets sent on these subcarriers are likely to be erased. At this point, two questions arise: how this correlation factor affects the multicast throughput and how we can take advantage of frequency diversity with as little as possible degradation in the achievable multicast rate. These two questions will be addressed in this section. As discussed in Section 2, for large group size, the throughput performance of ECOMP is as good as ECOMF. Furthermore, ECOMP requires only the knowledge of average SNR and fading type of BS-user links. Therefore, when applied to OFDM the ECOMP scheme is more suitable than ECOMF as it can significantly reduce feedback signaling. Hence, in this section we focus only on ECOMP scheme. First, using only frequency diversity to study the effect of correlation on multicast rate, all the coded packets are sent in only one time slot, each on one subcarrier. ECOMP is then expanded to make use of both time diversity and frequency diversity to investigate the trade-off between throughput and delay.

3.1. OFDM System Model and Correlation Factor among the OFDM Subcarriers

Consider a wireless downlink OFDM system with subcarriers over a bandwidth ,where is the subcarrier bandwidth, as shown in Figure 8. Cyclic prefix is not shown in the diagram for simplicity, as it does not affect the following study. In a given time slot , is a block of complex signals in frequency domain where is the transmitted signal on frequency . is assumed to have zero mean and power . These transmitted signals are transformed into samples in time domain through the discrete inverse Fourier transform with sampling time . These time samples are serially transmitted through a multipath fading channel and are contaminated by additive white Gaussian noise with noise power. The received signal can be expressed as

(38)

Let be the number of resolvable paths in the multipath fading channel, with being the tap gain of the th resolvable path. Similar to Section 2, 's are assumed to remain unchanged in a given time-slot and vary independently from one time-slot to another. The impulse response of the multipath channel in time domain can be written as

(39)

In our study, 's are assumed to follow power delay profile in COST 259 [12], that is, with .

From (38), the received signal in frequency domain (after FFT block in Figure 8) can be given by

(40)

where is the fast Fourier transform (FFT) of and denotes the channel response in frequency domain, which is the FFT of (39), that is,

(41)

indicates the fade level of the signal on subcarrier and hence determines the channel quality at that frequency. Since with , . Since 's remain unchanged in a given time-slot and vary independently from one time-slot to another, 's also remain unchanged in a given time-slot and vary independently from one time-slot to another.

From (41), the correlation in channel responses of subcarriers and can be calculated as follows:

(42)

and ,

(43)

As shown in (43), this correlation factor depends on the distance between two subcarriers and the number of resolvable paths, . Figure 9 further illustrates this correlation factor for an OFDM system of 256 subcarriers over a multipath Rayleigh fading channel with 2, 4, 8, and 256 tap gains. It can be seen in Figure 9 that this correlation is large when the frequency separation between and is small; hence if one subcarrier is in deep fade, the other is also likely to be in deep fade which may result in packet loss on both subcarriers if we use these two for transmission. On the other hand, when this correlation factor is small, deep fade on one subcarrier is less likely to affect the other subcarrier. Moreover, it is observed from Figure 9 that, for a given level of correlation, when the number of resolvable paths increases, the frequency separation decreases. For example, when the number of resolvable paths increases from 2 to 4, the minimum frequency separation to achieve a correlation of 0.3 decreases from 105 to 52 subcarriers, which is approximately two times. The same observation applies when increases from 4 to 8. However, when is larger than 8, this observation is no longer valid as shown in Figure 9. Hence, multipath fading channel introduces frequency diversity that can be used, especially for from 2 to 8.

3.2. ECOMP for OFDM

Using the system model as described in Section 2.1 to support multicast scenario for users, we first derive the relationship between average SNR and the instantaneous SNR on each subcarrier and then describe the operation of ECOMP in the case of OFDM.

Applying (41), the instantaneous channel gain for user on subcarrier is given by

(44)

Let be the transmitted signal on subcarrier at time-slot with normalized power of 1, the average SNR on subcarrier of user at time-slot can be expressed as

(45)

and the instantaneous SNR on subcarrier of user at time-slot is given by

(46)

where .

Following the discussions in the previous section, similar to (4) and (5), the effective multicast rates for WU and BU in OFDM system can be given by

(47)

3.2.1. ECOMP for OFDM Using Only Frequency Diversity

Expanding ECOMP to OFDM system, the transmission rate to send multicast packet on each subcarrier can be selected based on the channel gain threshold as follows:

(48)

To recover the erased packets for users with inadequate instantaneous SNR, ECOMP makes use of RS-code with the same encoding scheme as described in Section 2. For simplicity, without loss of generality, we assume that the number of RS coded packets is equal to the number of subcarriers, that is, , and hence, the whole RS-coded packet block is sent in one time-slot with each coded packet transmitted on one subcarrier as illustrated in Figure 10 for time-slot .

The optimization problem for multicast rate when applying ECOMP to OFDM then becomes

(49)

where is the probability that a given user can receive at least nonerased packets on all the subcarriers. At the first look, this probability is similar to that in the previous section; however, it is noted that, in the scenario of OFDM, channel gains in subcarriers are correlated and a simple expression using binomial distribution as in the previous section is not applicable. For this, we use simulation to study the throughput performance of ECOMP for OFDM. In particular, we investigate the effects of the number of resolvable paths on the effective multicast throughput and compare the performance of the proposed scheme with that of BU and WU. In our simulations, multicasting is done on a multicast group of users. For simplicity, an OFDM system with subcarriers and RS is considered.

1. (1)

   Effects of the Number of Resolvable Paths on Multicast Throughput

    

In Figure 11, the effect of different fading conditions on the achievable normalized throughput over multipath fading channels is examined at the same average SNR of 20 dB with the number of resolvable paths , and 11. The results in Figure 11(a) show that the effective multicast throughput depends on the number of resolvable paths. When increases, the correlation among the subcarriers reduces, that is, less chance that packets are erased at the same time, and hence the achievable multicast rate increases as increases. This is not the case for unicast where the ergodic capacity is independent of the number of resolvable paths as shown in [13–15]. However, as shown in Figure 11(b), this gain comes with a cost: at the same code rate, as the number of resolvable paths increases, the throughput curve becomes more sensitive to the channel gain threshold .

1. (2)

   Performance Comparison

    

Figure 12 compares the effective multicast rates of the WU, BU, and ECOMP schemes in a 5-tap multipath fading environment for a wide range of SNR. Similar to the performance comparison in Section 2, BU is better than WU in the low SNR region while WU outperforms BU after the crossover point of around 5 dB. Combining both multicast gain and multiuser diversity, the throughput performance of ECOMP is superior to both BU and WU in the considered SNR range and asymptotically converges to WU at high SNR. However, it is noted that the improvement in throughput of ECOMP on OFDM system is smaller than that in the case of single carrier. For instance, at 5 dB, as compared to BU and WU, ECOMP offers an improvement in effective multicast rate of 10 times in the case of multicarrier (Figure 12) and 18 times in the case of single carrier (Figure 6). This can be explained by the fact that, due to the high correlation in frequency responses, channel gains of adjacent subcarriers are similar, and consequently, changes in the SNR threshold to adjust multicast gain and multiuser diversity give less effect on the multicast throughput than in the case of independent time slots. In other words, correlation in channel responses will decrease the benefits of combining multicast gain and multiuser diversity.

3.2.2. ECOMP for OFDM Using Both Time and Frequency Diversity

When applying ECOMP to OFDM by sending all coded packets on all subcarriers, it can be seen that we gain times reduction in the delay as each RS block can be sent in only one time-slot. However, as the channel gains of OFDM subcarriers are correlated, if one subcarrier of a given user is in deep fade (i.e., is very low), it is likely that the subcarriers close to it are also in deep fade and the packets that are sent on these subcarriers will likely be erased. To compensate for the erased packets ECOMP has to select a lower transmission rate and lower RS code rate to gain more erasure correction capability and hence this reduces the multicast rate. To enhance this throughput performance, it is necessary to keep the correlation among the subcarriers in use as low as possible by increasing their frequency separation. As shown in Figure 9, this required frequency separation depends on the number of resolvable paths. In other words, by transmitting coded packets on subcarriers far from each other we can achieve lower correlation for higher multicast throughput. However, fewer packets can be transmitted on one time-slot and as a consequence more time-slots are needed for transmitting each RS block, introducing more delay.

Based on the above discussions, we consider the extended erasure-correction coding-based opportunistic multicast (EECOM) scheme, in which the BS encodes data packets using RS erasure code to form a block of -coded packets in the same way as in Section 2, but, instead of sending all -coded packets on all subcarriers in one time-slot (as in Figure 11), these -coded packets are sent over time-slots on only a subset of OFDM subcarriers equally spaced with a frequency separation of . For the same assumption of , . Figure 13 illustrates the transmission mechanism of EECOMP for .

The optimization problem for EECOM can be modified from (49) as in the following:

(50)

where is the probability that a given user can receive at least nonerased packets over time-slots. For the same reason as in the last part, the probability does not have a closed-form mathematical expression and the throughput of EECOM is investigated by simulations. Similar to Section 3.2, our simulation results are based on a group of 100 users in an OFDM system with subcarriers.

Figure 14 depicts the performance of EECOMP for different numbers of time-slots. In this graph, the horizontal axis represents the number of time-slot in scale. It is observed that as the number of time-slot increases, the multicast throughput monotonically increases, which illustrates the trade-off between throughput and delay. When , EECOMP exploits only frequency diversity by sending all coded packets of one RS block in one time-slot on all subcarriers. The effective multicast throughput for EECOMP is the same as in Figure 11(a). When , EECOMP exploits only time diversity by sending only one coded packet in each time-slot on one subcarrier and the multicast rate in this case is the same as in Figure 2(b), that is, around 3.34 b/s/Hz which is about 25% higher than that in the case of . Moreover, when is large enough, the effective throughput for EECOMP is approximately equal to that of . For example, when is larger than 32 for 8-tap channels, 64 for 4-tap channels, or 128 for 2-tap channels, the achievable multicast rates for EECOMP are roughly the same as the case of , which correspond to a correlation factor of 0.3 or less (by calculating the frequency separations in these cases and referring to Figure 9 for the correlation values). This indicates that there is a significant delay reduction with virtually no penalty in multicast rate at these points and is reduced when the number of resolvable paths increases; for example, is reduced by 2, 4, 8 times when , respectively. However, when is very large, as shown in Figure 9, the correlation between two subcarriers of a given frequency separation is not much changed with . Hence, for larger , the delay cannot be reduced further.

In addition, it is observed that for points in Figure 14 with the same multicast rate, they yield the same correlation level shown in Figure 9. For instance, at for and for , the multicast rate is about 3.3 b/s/Hz/user (Figure 14). At these points, the subcarrier separations are 32 and 64 subcarriers when and , respectively, for the same correlation factor of about 0.7 (Figure 9). The same observation applies for other points with approximately the same effective multicast throughput in Figure 14.

We have proposed and studied an erasure-correction coding-based opportunistic multicast scheduling scheme aiming at exploiting multicast gain, multiuser diversity, and time/frequency diversity to enhance the throughput performance over wireless fading channels. In the proposed scheme, the BS sends each packet only once at a transmission rate determined by a channel gain threshold and using erasure correction capability of RS( ) to recover erased packets due to insufficient instantaneous SNR on BS-user links. RS coding scheme is applied to a block of packets and coded packets are sent in time or frequency slots to effectively explore time/frequency diversity. The channel gain threshold and erasure code rate are jointly optimized for best multicast throughput.

On frequency-flat fading channels, the selection of channel gain threshold is considered in two cases of full channel knowledge and partial knowledge of average SNR and fading type of wireless channel. An analytical framework has been developed to analyze the effective multicast throughput of BU, WU, and of the proposed scheme. In this framework, we prove that while the effective multicast rates of both BU and WU asymptotically converge to zero as the multicast group size increases, this effective multicast rate of the proposed scheme is bounded from zero and depends on the average SNR. We further prove that, for the proposed ECOM scheme, the benefit of full channel knowledge is only pronounced at small multicast group sizes. As the group size increases, partial knowledge of channel response is sufficient in providing approximately the same throughput performance but significantly reducing resources (bandwidth, power) for feedback signalling.

In addition, numerical results illustrate that multiuser diversity is most pronounced at low SNR region since the difference in supportable rates of various users is large while multicast gain is superior at high SNR region where the difference in channel gain is compressed by the log-function that results in small difference in supportable rates among the users. The throughput comparison illustrates that with the ability of combining multicast gain and multiuser diversity, the proposed scheme outperforms both BU and WU for a wide range of SNR.

Furthermore, in this paper, we have extended ECOM for applications to OFDM system aiming at exploiting both time and frequency diversity in a frequency-selective fading environment. The effects of frequency correlation on multicast rate are investigated and our study shows that by exploiting both time and frequency diversity, we can significantly reduce transmission delay with negligible degradation in multicast throughput.