Joint Power Allocation for Multicast Systems with Physical-Layer Network Coding

Network coding (NC) was first proposed for wireline systems to improve the network's achievable rate about ten years ago [1]. It was then shown in [2, 3] that by coding the data stream, rather than separated symbols, the NC can provide a higher achievable rate than the traditional Shannon information theory does. In order to apply the network coding in wireless communication, a lot of investigations have been conducted to improve the network's achievable rate by making wireless channel coefficients similar to the wireline channel ones [4‐8]. In [4‐6], the idea of network coding is applied in the physical-layer of wireless networks, denoted by physical-layer network coding (PLNC), in order to enlarge the achievable rate. The PLNC is to jointly process multiple symbols without decoding them considering that the aim of the transmission is to tell the symbols to the destination rather than the relay [7‐9]. Here, it is the data stream, instead of any separate symbol, that is tackled by NC. As such, NC improves the achievable rate from the data stream perspective.

For the physical-layer network coding, there are basically two types of NC, namely, the amplify-and-forward (AF) NC [10] and the decode-and-forward (DF) NC [11]. For the AF-NC, the mixed messages received by the relay are first amplified properly without requiring decoding, and then forwarded to the destinations. For the DF-NC, the relay first decodes these mixed messages to obtain all of the individual messages, and then mixes the separated messages in order to forward them to the destinations. In [12], the AF-NC and DF-NC are compared by both theoretical analysis and simulations, drawing a conclusion that the AF-NC is superior to DF-NC in both transmission rate and link error probability. It is interesting to note that similar results are reported in [13].

The power allocation issue in physical-layer network coding system attracts a lot of attention since a superior performance could be realized by allocating the power properly. In [14], a power allocation scheme has been proposed to maximize the capacity of systems with only one single relay. In [15], an optimization framework has been formulated to allocate the powers for the DF-NC without consideration of AF-NC. A similar power allocation scheme is also proposed for the XOR-based NC [16]. However, the focus of the power allocation schemes is only on the single-cast instead of the multi-cast transmission model. In fact, the multi-cast system is widely used in wireless communications, where the physical-layer network coding can be applied since the nature of broadcast is available for PLNC in the multi-cast model.

As mentioned earlier in this paper, the core of the network coding is to properly combine or mix multiple messages without decoding them separately. Although the performances of network coding are very attractive from the information theory perspective, implementation of these performances incurs very high complexity [17]. In order to decrease the complexity of network coding in practical wireless communication systems, the lattice is introduced to realize the network coding [18].

The lattice is intrinsically a codebook similar to the traditional constellation modulation. There are basically five aspects to judge whether a lattice is good or not, including the packing feature, the covering feature, the mean squared error due to quantization, the error probability, and the closed property. Namely, a good lattice should be as compact as possible, and should be large enough to cover all of the elements; moreover, the quantization error and the error probability of the transmission in AWGN channels should be as small as possible; furthermore, any linear combination of two elements in the lattice should belong to this lattice [19, 20].

To the authors' best knowledge, most of the existing investigations focus on the multiple-access relay channels or unicast model rather than the multi-cast relay channel. There is only one published paper [21] that studies the power allocation problem for the multi-cast model with physical-layer network coding [21], where the system model is composed of two sources, N relays and two destinations. With the novel system model, an isolated precoder and a distributed precoder are designed to achieve the full relay diversity gain. However, power allocation is performed only for the sources while equal power is allocated for all relays. Considering the fact that the network's achievable rate is jointly determined by all of the sources and relays, the performance of the whole system could be further improved if source and relay powers are jointly optimized. In this paper, the joint power allocation issue for both sources and relays will be investigated in order to maximize the network's achievable rate. The difficulty lies in that the joint optimization problem is not convex with respect to (w.r.t.) the whole parameters, leading to a nonconvex optimization problem. The goal of this paper is to develop an iterative algorithm to tackle the nonconvex optimization problem for joint power allocation in the physical-layer network coding. The rest of the paper is organized as follows. Section 2 presents the multi-cast system model and the joint power allocation-based mathematical model. Section 3 first establishes an optimization problem for the joint power allocation based on the criterion of achievable rate maximization. Then, it is disclosed that the proposed problem is not convex w.r.t. the whole parameters. By neglecting the noise power of the second-hop channel under the medium to-high-SNR regimes, and replacing the original cost function with its bound, a convex problem w.r.t. the whole parameter set is obtained such that the convex optimization method can be used to solve the joint power allocation problem. Note that the high-SNR approximation is very accurate due to the very small power of the noise in the second-hop channel. As an alternative, we also design an iterative algorithm to solve the nonconvex problem by using the convex features of the original cost function w.r.t. part of the parameters. In Section 4, the lattice-based network coding that uses the joint power allocation schemes is studied. Section 5 presents the simulation results, with comparison to the existing power allocation schemes, in terms of achievable rate and cumulative distribution function. Section 6 concludes the paper.

Notations

denotes the absolute value; means the summation of a vector; means a transpose; is the Frobenius norm; denotes the complex Gaussian distribution with the zero mean and the unit covariance.

Consider a multi-cast system consisting of two sources, K distributed relays and two destinations, where the physical-layer network coding (PLNC) is employed. Suppose that both sources want to transmit symbols to both destinations, while there is no direct link between S1 (or S2) and d2 (or d1) due to the pathloss and the large-scale fading as discussed in [21]. The system model is shown in Figure 1, where all of the relays are half-duplex and use the amplify-and-forward relaying protocol. Each source transmits a frame consisting of K symbols, which utilize K time slots and two subcarriers from both sources to both destinations. On each subcarrier, one symbol from S1 and another from S2 are transmitted to the relay on the first subcarrier; in the same time slot, both symbols are mixed by the same relay and then retransmitted to both destinations on the second subcarrier. Note that all of the relays are working in the round-robin way proposed in [21]; namely, only the k th relay works in the k th time slot.

The symbols of S1 and those of S2 are, respectively, denoted as and . In the k th time slot, the symbol of S1, , and that of S2, , are transmitted to the k th relay and both destinations on the first subcarrier. Thus, the symbol received by d1, d2 and the k th relay can be, respectively, given by

where and are the direct link from S1 to d1 and that from S2 to d2 in the k th time slot; and are the k th symbol's power coefficients for S1 and S2, respectively; and denote the first-hop channel coefficients from S1 and S2, respectively, to the k th relay in the k th time slot; and are the additive white Gaussian noises (AWGNs) with the distribution at d1 and d2 in the k th time slot on the first subcarrier; is the AWGN with the distribution . Note that there is no direct link between S1 (S2) and d2 (d1) due to the path loss and large-scale fading as discussed in [21]. On the second subcarrier in the k th time slot, both sources remain silent [21], yet the k th relay amplifies the mixed symbols as

and then retransmits to both d1 and d2. Note that is the power allocation coefficient of the retransmitted symbol for the k th relay. The amplifying power gain in (4) from the k th relay can be written as

The symbol received at d1 from the k th relay in the k th time slot can be described as

Similarly, that at d2 can be given by

where and are the channel coefficients for the k th relay to d1 and d2 on the second subcarrier, respectively. All of the channel coefficients are distributed with . and are the AWGNs at d1 and d2 on the second subcarrier with the distribution of .

By far, as shown in (1) and (2), both d1 and d2 have received two symbols during two subcarriers on the k th time slot. Since the direct link between S1 and d1 (or S2 and d2) is quite good, d1 and d2 can successfully detect the symbol of S1 and that of S2, respectively. By using the detected and as well as the interference subtraction method, the received symbols after the interference cancellation in (6) and (7) can be expressed as

Note that this subtraction is called as analogue network coding(ANC) [21, 22]. In (8) and (9), the power coefficients and are optimized in [21] with a fixed value of the power coefficient . In the subsequent sections, we will jointly optimize all of the power coefficients, , , and to maximize the achievable rate of the whole network. Assume that full channel state information (CSI) of the whole networks is available for every node, which is considered to be feasible in the slow-varying scenario.

In this section, the optimization problem for the joint power allocation in the multi-cast system with the physical-layer network coding is first established using the criterion of achievable rate maximization. As the problem is nonconvex, that is, difficult to solve, it is then modified to obtain a convex power allocation problem by using the high-SNR approximation. As an alternative, an iterative algorithm whose convergence is guaranteed is also developed to optimize the power coefficients of all sources and relays.

Since the lattice plays a very important role in the performance of network coding for the wireless communication, its construction method is of great importance to achieve the potential performance of network coding while having quite low complexity. The famous algorithm of Construction A has widely been employed to design the lattice [18]. Upon completing the design of lattice offline, the messages are modulated to this lattice online and then transmitted to all relays. With the mixed lattice codes at each relay, the relay amplifies the mixed lattice codes and then retransmits them to the destinations. Note that the mixed lattice codes received at each relay are still in the lattice due to the closed property of the lattice. At the destination, each message is demodulated by using the lattices. The detailed flow chart is presented in Figure 2.

It has been shown that the power allocation problem is very important for the lattice-based network coding [25, 26]. In [25], the feasibility of different powers for two sources has been studied, where two sources want to exchange messages with the help of only one relay. It is also disclosed that the system performances would be enhanced if the sources use different transmit powers. However, there is no optimization of the power allocation for the sources. In [26], the power allocation is optimized for the underlying system in [25]; namely, the powers for the two sources are optimized by using an upper bound of the achievable rate. Unfortunately, the power for relay is fixed without any optimization. Thus, the joint power allocation schemes proposed in this paper can be used in the lattice-based network coding, where the power for the source lattice and that for the relay lattice are jointly optimized.

In this section, simulation results are demonstrated to show the performances of the two proposed joint power allocation schemes in terms of the achievable rate and cumulative distributed function (CDF). In the simulations, all of the channel entries are i.i.d. complex Gaussian with zero mean and unit covariance, where all of the channel fading coefficients are independent from each other. The number of sources and that of destinations are set to 2, and that of the relays is set as in all figures except for in Figure 3. The existing method proposed by [21] is also simulated in comparison with the proposed schemes. All of the curves are realized by the Monte Carlo numerical simulations, where the number of Monte Carlo simulations is set to 5000.

Figure 3 presents the achievable rate as a function of the source power with the fixed relay power  dB and . From this figure, we can see that the proposed power allocation scheme using the high-SNR approximation achieves a better achievable rate than the existing one [21] in the multi-cast system with the lattice-based physical-layer. Moreover, the gain increases with the growing value of SNR.

Figure 4 presents the achievable rate versus the subtotal power of all relays realized by the proposed two schemes and the existing method. Note that the total power available for the whole system can be expressed as , where and are, respectively, the subtotal power of the sources and that of the relays. With the same total power for the whole system, the existing power allocation scheme [21] fixes the powers for all relays, and only allocates powers for sources. However, our proposed schemes jointly optimize the source and relay powers with the same total power available for the whole system. It is shown that the proposed schemes outperform the existing one with a large gain, which increases with the growing value of SNR. On the other hand, the existing scheme only optimizes the source power without consideration of the relay power, leading to a loss of achievable rate with the increase of the total power. Figure 5 gives similar curves as a function of with fixed  dB. Obviously, the proposed two schemes outperform the existing scheme, especially in the high-SNR regime.

Figure 6 presents the cumulative distribution function for two SNR regimes, where Case I means  dB and Case II denotes  dB. It is seen from Figure 6 that the proposed power allocation scheme-based on the derived bound function is always better than the existing one at any value of CDF. For example, at 5 b/s/Hz in Case I, the error rate drops from 20% to 10%. At 10 b/s/Hz in Case II, the error rate drops from 30% to 20%.

In this paper, we have studied the joint power allocation problem for multi-cast systems with physical-layer network coding-based on the maximization of the achievable rate. To deal with the nonconvex optimization problem, a high-SNR approximation is employed to modify the original cost function in order to obtain a convex minimization problem, where the approximation is shown to be asymptotically optimal at the high-SNR regime. As an alternative, an iterative algorithm has been developed by utilizing the convexity property of the cost function w.r.t. a part of the whole power coefficients. Considering the low complexity of the physical-layer network coding in the multi-cast system, the lattice-based network coding that uses the proposed joint power allocation schemes has been suggested. The simulation results have shown the effectiveness of the proposed schemes.