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4-ary network coding for two nodes in cooperative wireless networks: exact outage probability and coverage expansion
- Jin-Taek Seong^{1} and
- Heung-No Lee^{1}Email author
https://doi.org/10.1186/1687-1499-2012-366
© Seong and Lee; licensee Springer. 2012
Received: 30 December 2011
Accepted: 21 November 2012
Published: 22 December 2012
Abstract
The fact that sensor nodes are powered by limited-capacity batteries makes power efficiency; one of the most critical issues in wireless sensor networks (WSNs). Advanced communication techniques combined with network coding and cooperative schemes have attracted considerable attention as ways to improve power efficiency in wireless transmission as well as to achieve high throughput and spectral efficiency in WSNs. In this study, we consider cooperative wireless networks with two nodes and one base station, and investigate the effect of using non-binary network coding on the enhancement in power efficiency. First, we derive the exact and general outage probability in our network coding scheme. We show that full diversity order can be obtained using a non-binary network code with GF(4) in the considered network. We use this result to study the extent to which the coverage area of a wireless source node can be expanded by network coding without increasing transmit power. Our results indicate that the benefit in terms of coverage expansion is substantial. The results included in this study show the influence of optimal power allocation on power efficiency. The optimum ratio of power allocation varies according to the wireless channel environments and the field size of network codes.
Keywords
- Wireless sensor network
- Outage probability
- Non-binary network coding
- Cooperative network
- Power efficiency
- Coverage expansion
- Power allocation
1. Introduction
In wireless sensor networks (WSNs), sensor nodes operate on the limited energy source of onboard batteries, making power efficiency a key issue because replacement or recharging of batteries is difficult. The very high-energy expenditure of WSNs makes long-range message transmission undesirable. Consuently, there are several ways to improve power efficiency, such as optimal transmit power allocation [1–5].
Channel fading is one of the underlying causes of performance degradation in wireless networks. One naïve approach to combating fading is to increase the transmit power. A more advanced method is to use diversity techniques, which can be employed without increasing the transmit power. To date, many diversity techniques have been developed and employed in time, frequency, and space domains. Cooperative networking is a modern approach that aims at increasing spatial diversity via user cooperation. Each user participates in collaboration and shares the benefit of using a virtual antenna array in transmitting information to a receiver that is available through another user’s antenna [6]. Ahlswede et al. [7] proved that network coding achieves optimality in terms of the flow rate for a single-source multicast scenario. This would be impossible to achieve by simply routing or by replicating the data. Many studies have since been conducted to verify that network coding provides advantages over existing cooperative network schemes [8–13].
Analyses of outage probability in cooperative networks are presented in [6, 14–19]. Chen et al. [15] showed that binary network coding (BNC), based on the arithmetic of a Galois Field of size 2, i.e., GF(2), provides improved diversity gains and bandwidth efficiencies in wireless networks in which each user employs a simple decode-and-forward (DF) scheme that assumes a perfect inter-user channel. In practice, there exist channel errors between users, as discussed in [16], where the authors proposed an adaptive DF scheme with BNC. It was recently shown in [17] that BNC is not optimal for achieving full diversity in a system of multiple users and relays. However, it has also been shown that full diversity order can be achieved using non-binary network coding (NBNC) with GF(q) for q > 2 [17–19].
The remainder of this article is organized as follows. In Section 2, we describe cooperative schemes, channel model, and outage probability. In Section 3, the exact outage probability in cooperative networks is derived and analyzed for different network coding schemes. Power efficiency techniques based on outage probabilities are described in Section 4. Finally, we conclude this study in Section 5.
2. System description
2.1. Cooperative schemes
Cooperative transmission schemes can be divided into two categories based on the method employed to process messages at intermediate nodes: the amplify-and-forward (AF) scheme and the DF scheme, both are widely used relay protocols [6]. In the AF scheme, an intermediate node receives a noisy signal of the source’s message, amplifies it in non-regenerative mode, and forwards it to a destination. In the DF scheme, a relay node decodes the source’s message, re-encodes it, and forwards it to the destination. We focus on the second of these cooperative transmission protocols, i.e., the DF scheme.
We consider a cooperative scheme for wireless networks as shown in Figure 1. There are two source nodes, nodes 1 (N1) and 2 (N2), and two phases, the broadcasting and the relay phases, in the cooperative scheme. In the broadcasting phase, source nodes N1 and N2 transmit messages, S_{1} and S_{2}, respectively. In the relay phase, when both nodes successfully recover the transmitted messages, the messages are re-encoded and then forwarded to the BS. When a node is unable to successfully perform decoding, it repeats its message in the relay phase. When receiving repeated messages, BS as a destination performs maximum ratio combining (MRC) of these messages, and recovers the transmitted messages. In this study, we assume that the transmission rate is selected to be sufficiently lower than the capacity of each channel so that near perfect decoding of messages can be accomplished with the use of a channel code. Thus, for all wireless channels, the received messages are either completely corrupted, and therefore not available at the receiving end, or considered error-free.
At the BS, the set of all possible received messages is {S_{1},S_{2,}Z_{1,}Z_{2}}, where the subscript denotes the index of the source node. The first two messages are received in the first phase, and the latter two are linearly combined and sent from the sources in the relay phase. The alphabet of the combined message, Z_{1} and Z_{2}, is selected to be a finite field. The two finite fields considered in this study are GF(2) and GF(4).
where H_{2} is the network coding matrix with its elements drawn from GF(2) and S is the source message vector. The arithmetic should follow that of GF(2). Most existing network coding schemes are based on BNC.
where H_{4} is the network coding matrix composed of elements from GF(4) and the arithmetic operations are those of GF(4).
The core idea of cooperative communication systems is to alleviate the negative effects of communication channels, such as fading and noise, and to increase the probability of successful message reception via cooperation. With a closer look at the rows of H_{4}, we note that any two rows of H_{4} are linearly independent, while those of H_{2} may not be. This means that as long as any two messages out of the four, {S_{1}S_{2}Z_{1}Z_{2}}, are received correctly, NBNC-4 can correctly decode the correct transmit messages S_{1} and S_{2}. This is not possible with the BNC scheme. For example, the last two rows of H_{2} are dependent on each other. Thus, with the reception of only Z_{1} and Z_{2}, the BNC scheme cannot decode the messages S_{1} and S_{2} accurately. For a network of two-user cooperation, this desirable behavior can be attained by increasing the field size to 4. This behavior was first observed in [17]. In this article, our focus again is to show how this favorable behavior can lead to power efficiency in terms of coverage expansion, and to study how the transmit power should be allocated differently between the two sensors given a fixed power budget.
2.2. Channel model
where k ∈ {1, 2} denotes the transmission phase (broadcasting or relay phase), and i, i ∈ {1, 2}, denotes the transmitted node (N1 or N2). Let j denote the received node for j ∈ {1, 2, d}, where d denotes BS. The transmitted and received signals are given as x_{ i,j,k } and y_{ i,j,k } with i ≠ j. P_{ i } denotes the transmit power at the i th node. The channel gain is represented by h_{ i,j,k }, which consists of the fading term p_{ i,j,k } and the path loss coefficient q_{ i,j,k }, i.e., h_{ i,j,k } = p_{ i,j,k }q_{ i,j,k }. Here, we assume that the fading term p_{ i.j,k } is random and the path loss coefficient q_{ i,j,k } depends on the distance between nodes i and j. Noise n_{ i,j,k } is AWGN with a normal distribution $\mathcal{N}\left(0,{N}_{0}\right)$ having a zero mean and power spectral density N_{0}. The path loss coefficient is modeled as q_{i,j,k} = (d_{0}/d_{i,j})^{α/2}, where 2 < α < 6 is the path loss exponent, d_{ i,j } is the distance between nodes i and j, and d_{0} is the reference distance. In this study, we use d_{0} = 1 and α = 3, and |h_{i,j,k}| is assumed to be Rayleigh distributed such that the channel energy of power |h_{i,j,k}|^{2} is exponentially distributed. We assume that the fading term p_{ i,j,k } is a complex-valued, independent and identically distributed Gaussian in each dimension with a zero mean and 1/2 variance. The average power of h_{ i,j,k } is then represented by the average power of q_{ i,j,k }, which depends on the distance between the transmitter and the receiver. All channel gains are assumed to be reciprocal, i.e., h_{ i,j,k } = h_{ j,i,k }. The instantaneous signal-to-noise ratio (SNR) of each channel is denoted as γ_{i,j,k} := |h_{i,j,k}|^{2}P_{ i }/N_{0}, where P_{ i }/N_{0} is the transmit SNR at the source node i.
2.3. Outage probability
where Γ_{i,j} = σ_{i,j}^{2}P_{ i }/N_{0} is the average SNR at the receiver j, σ_{i,j}^{2} is the variance of the channel gain h_{ i,j,k } which depends only on the distance such that σ_{i,j}^{2} = σ_{i,j,1}^{2} = σ_{i,j,2}^{2}. The outage probability P_{ out }(γ_{i,j,k}, R) is a function of the average SNR and the transmission rate.
3. Outage probability for 4-ary network coding
In this section, we aim to derive the outage probability that allows us to investigate the effects of different outage events, transmit power allocations, channel gains, and field sizes (GF(2) versus GF(4)) in network coding, on power efficiency. This analysis is somewhat different from that given in a recent article [17] that studied outage probabilities under a number of approximations: (i) they did not consider all possible outage scenarios (for a full consideration see [14]), (ii) all channel outages are treated with the same transmit powers, the same average channel gains, and thus the same average channel SNRs. Our analysis is exact and generalized, with consideration of different transmit powers, rates, and average channel gains. This generalized analysis framework enables us to conduct not only a diversity order analysis, but also a complete outage probability analysis as a function of SNR. These results help us investigate the coverage area expansion and the OPA problems. Our outage probability analysis shows that the diversity order achievable with NBNC-4 is three, instead of two, as obtained in [15]. It should be noted that full diversity order is obtainable in the considered network channel.
3.1. Outage events in the cooperative network
Transmitting messages for two nodes according to the four scenarios
Case | N1 | N2 | ||
---|---|---|---|---|
Broadcasting | Relay | Broadcasting | Relay | |
1 | S _{1} | Z _{1} | S _{2} | Z _{2} |
2 | S _{1} | Z _{1} | S _{2} | S _{2} |
3 | S _{1} | S _{1} | S _{2} | Z _{2} |
4 | S _{1} | S _{1} | S _{2} | S _{2} |
3.2. Outage probability for 4-ary network coding
In the following, we focus on the derivation of outage probability for the NBNC-4 scheme. First, network coding in the relay phase is performed. Message transmission consists of two phases as described in the previous section. We analyze the outage event based on MRC. In this study, we assume that the instantaneous SNRs for the broadcasting and relay phases are mutually independent.
We define the transmission rate for each node as R_{1} and R_{2}, respectively. We consider the outage probability for N1, which is identical to that for N2 as a result of symmetry. NBNC-4 in the relay phase follows the network coding method specified in (2).
where ${A}_{1}:=\frac{2{r}_{1}^{2}{r}_{2}{N}_{0}^{3}}{{\sigma}_{1,d}^{4}{\sigma}_{2,d}^{2}}+\frac{{r}_{1}^{2}{r}_{2}{N}_{0}^{3}}{2{\sigma}_{1,2}^{2}{\sigma}_{1,d}^{2}{\sigma}_{2,d}^{2}}+\frac{{r}_{1}^{2}{r}_{2}{N}_{0}^{3}}{2{\sigma}_{1,2}^{2}{\sigma}_{2,1}^{2}{\sigma}_{1,d}^{2}},{A}_{2}:=\frac{{r}_{1}{r}_{2}^{2}{N}_{0}^{3}}{{\sigma}_{1,d}^{2}{\sigma}_{2,d}^{4}}+\frac{{r}_{1}{r}_{2}^{2}{N}_{0}^{3}}{{\sigma}_{2,1}^{2}{\sigma}_{1,d}^{2}{\sigma}_{2,d}^{2}},\text{and}\phantom{\rule{2pt}{0ex}}{A}_{3}:=\frac{{r}_{1}^{2}{r}_{2}{N}_{0}^{3}}{{\sigma}_{1,2}^{2}{\sigma}_{1,d}^{4}}.$
The outage probability analysis for BNC is performed similar to the analysis performed for the NBNC-4 scheme, with the result that the outage probabilities for BNC are identical to those for NBNC-4, except for the first case, i.e., P_{out,binary}^{2} = P_{out,4 − ary}^{2}, P_{out,binary}^{3} = P_{out,4 − ary}^{3}, and P_{out,binary}^{4} = P_{out,4 − ary}^{4}. The reason for this is that the outage events, in each of Cases 2, 3, and 4, for the BNC scheme, are identical to those of NBNC-4. The only difference comes from Case 1.
where ${B}_{1}:=\frac{{r}_{1}{r}_{2}{N}_{0}^{2}}{{\sigma}_{1,d}^{2}{\sigma}_{2,d}^{2}},{B}_{2}:=\frac{{r}_{1}^{2}{r}_{2}}{2{\sigma}_{1,2}^{2}{\sigma}_{1,d}^{2}{\sigma}_{2,d}^{2}}+\frac{{r}_{1}^{2}{r}_{2}}{2{\sigma}_{1,2}^{2}{\sigma}_{2,1}^{2}{\sigma}_{1,d}^{2}},{B}_{3}:=\frac{{r}_{1}{r}_{2}^{2}{N}_{0}^{3}}{{\sigma}_{2,1}^{2}{\sigma}_{1,d}^{2}{\sigma}_{2,d}^{2}},\text{and}\phantom{\rule{0.5em}{0ex}}{B}_{4}:=\frac{{r}_{1}^{2}{r}_{2}{N}_{0}^{3}}{{\sigma}_{1,2}^{2}{\sigma}_{1,d}^{4}}.$
3.3. Outage probability comparison of different transmission schemes
In order to investigate the influence of different channel gains, we assume that the transmit powers of the two nodes are equal, i.e., P_{1} = P_{2}, and we use the same transmission rates R_{1} = R_{2} = 1 b/s/Hz. As shown in Figure 3, we evaluate the effect of variances of the channel gains. We can observe that the NBNC-4 scheme achieves a diversity order of three, in contrast to a diversity order of two for both the BNC and the non-cooperative schemes. In Figure 3a, we set all variances of the channel gains as σ_{1.d}^{2} = 1, σ_{1.2}^{2} = 2, σ_{2.d}^{2} = 125. This means that the link quality between N2 and BS is better than the other. Since the variance of the channel gain depends on the distance, the case of Figure 3a reflects the channel environment where N2 is close to BS. In Figure 3b, σ_{1.d}^{2} = 1 and σ_{1.2}^{2} = σ_{2.d}^{2} = 8, which means the link quality from N2 to BS is higher than that from N1 to BS, with equal power allocation (EPA). This setting has a geometrical meaning such that N2 is located in the middle of N1 and BS. In Figure 3c, we consider the case where N2 is located closer to N1, by setting σ_{1.d}^{2} = 1, σ_{1.2}^{2} = 125, and σ_{2.d}^{2} = 2 with EPA. Note that the diversity orders for the three different schemes still hold. The diversity order for the non-cooperative scheme is still two, owing to the time diversity obtained by using MRC at BS.
4. Power efficiency enhancement schemes
In this section, we consider two approaches for enhancing power efficiency. One is to increase the field size in network coding and assess its effect on power efficiency. The other is to allocate a given level of transmit power to the two source nodes. In this study, power efficiency is expressed in terms of both outage probability and coverage expansion.
4.1. Coverage expansion
4.1.1. Location of source nodes
Based on this 2D setting, the outage probability from the source N1 to BS can readily be analyzed by substituting the variances in the relevant outage expressions given in Section 3.2.
4.1.2. Evaluation of coverage area expansion
The contour of outage probabilities evaluated at 10^{–4} for the source N1 is plotted in Figure 4, where the blue and red lines indicate the results of using the BNC and NBNC-4 schemes, respectively. Figure 4 shows that the position of N1 is expanded by the 4-ary network code. We assumed that the transmit power of both nodes is P_{1}/N_{0} = P_{2}/N_{0} = 20 dB and R_{1} = R_{2} = 1 b/s/Hz. Suppose that the source N1 is located at (2, 0). Then, the 4-ary network code achieves an outage probability of 10^{–4} or less, whereas the binary code does not. The contour of the outage probability at 10^{–4} for N1 has been extended with the use of NBNC-4, as compared to the use of BNC.
4.2. OPA
The other power efficiency technique investigated in this study is transmit power allocation. We investigate this problem for the two network codes, using the outage analysis framework developed in Section 3.2.
4.2.1. Formulation of OPA
Hasna and Alouini [2] attempted to minimize outage probability under a total transmit power constraint. Based on a symbol-error-rate analysis with M-PSK and M-QAM modulations, power allocation schemes for DF protocols are presented in [4, 20], where the authors considered MRC receivers. A power allocation problem for Nakagami fading channels is considered in [21]. We assume that each node knows all the channel state information by using an appropriate channel feedback scheme. We investigate the outage performance of optimal transmit power allocation subject to a total power constraint. In other words, the OPA solution is obtained based on minimization of the outage probability given under a total power constraint.
We investigate the effect of variances of channel gains on the optimum ratio of power allocation, while the outage probability is minimized.
4.2.2. Discussion for various link qualities
In this section, we discuss optimal transmit power allocation for various channel environments. We consider the position of nodes as follows: source node (N1) is located at coordinate (1, 0), BS is at (0, 0), the relay node (N2) is free to move around in the 2D space. We investigate the effect of the position of the relay node N2 on optimal transmit power allocation. In addition, we aim to investigate the effect of the size of finite fields, used in the underlying network coding scheme, on the results of optimum power allocation.
A noteworthy observation in Figure 8 is that there are two different approaches for obtaining the optimum ratio. One is the analytical approach of solving the optimization problems (29) and (30), which are based on the approximated outage probabilities (20) and (23). Another observation is the results obtained from exhaustive numerical evaluations of the exact outage probabilities (19) and (22) as a function of ρ for both the BNC and NBNC-4 schemes. Note that the results from the two approaches are almost identical. This validates the optimization problem set up in (25).
Now returning to our discussion of the optimum ratio ρ, Figure 8 shows that the optimum ratio ρ is, approximately, less than 0.8 and larger than or equal to 0.5 for the two network coding schemes. In more exact terms, when the relay N2 moves closer to BS, i.e., x → 0, the transmit power P_{1} rises to 0.78P_{t} (ρ = 0.78), while the transmit power P_{2} for the relay N2 goes to 0.22P_{t}. 78% of the total transmit power should be allocated at the source N1 for optimum results. The technical reason for this result is found from close investigation of (20) and (23) approximated, such that the channel variance σ_{2,d}^{2} becomes much larger than the other fixed parameters, and the approximated outage probabilities are dominated mainly by the two terms ${\scriptscriptstyle \frac{{A}_{1}}{{P}_{1}^{2}{P}_{2}}}$ and ${\scriptscriptstyle \frac{{A}_{3}}{{P}_{1}^{3}}}$. Note that P_{1} is taken to the second and third powers in these terms, while P_{2} is at its first power. Therefore, it is easy to see that more power should be allocated to P_{1} than to P_{2} in order to obtain a smaller outage probability. The result that more transmit power should be allocated to the source N1 rather than to the relay N2 as x → 0 is reasonable, since the role of the relay becomes decreasingly critical as it moves away from the source and becomes closer to BS.
On the other hand, we consider the other case in which relay N2 is moved closer to source N1. In the BNC case, we note that the optimal ratio approaches 1/2, i.e., the transmit powers P_{1} and P_{2} approach P_{ t }/2. In the NBNC-4 case, however, a very interesting behavior is observed. More transmit power P_{1} should be used at the source rather than at the relay to achieve the minimum outage probability. This phenomenon is more interesting with NBNC-4 in the case where relay N2 is closer to source N1.
The optimum ratio increases as the size of the finite field used in network coding is increased from 2 to 4. We can observe from Figure 8 that the optimum ratio of power allocation for the NBNC-4 scheme is generally much greater than that of the BNC at any position of x. In other words, more transmit power should be used at source N1 to obtain smaller outage probability. This is because the combined messages Z_{1} and Z_{2} are maximally used in NBNC-4. Recall the two different network coding matrices, H_{2} for BNC and H_{4} for NBNC-4, defined in (1) and (2), respectively. The rank of any (2 × 2) submatrix, i.e., any two rows of H_{4}, is always 2, while that of H_{2} is not always 2 (some may be 1). The crucial difference between the two network coding schemes can be seen in Case 1 in Section 3.2. This is the outage event considered in (8). With the NBNC-4 scheme, it is possible for only BS to recover the original messages S_{1} and S_{2} with the availability of only Z_{1} and Z_{2}. This is not possible with the BNC scheme. Figures 7 and 8 show this in detail. In other words, they show how the crucial difference in Case 1 affects the result of OPA, as well as the corresponding outage probability results.
5. Conclusions
In WSNs, sensor nodes operate from finite capacity energy sources, i.e., onboard batteries; thus, designing a system with high power efficiency is a key issue. In this study, the power efficiency is investigated as the size of finite fields for the linear network coding is increased from 2 to 4, and as the allocation of transmit power, i.e., the power used at the source node versus the power at the relay node, is varied. To evaluate the benefits of these techniques, we derived the outage probability expressions for the considered network coding schemes. We then analyzed the diversity order for the network coding schemes, one with GF(2) and the other with GF(4). Our results indicate that the diversity order using GF(4) is three, but that the diversity order using the binary network code is only two. We studied the effects of increased field size on the expansion of the network coverage area. Coverage area expansion by only changing the field size in network coding, without increasing the transmit powers, is a creditable and interesting research result of this study. Our result indicates that the power efficiency benefit of GF(4) as compared to that of GF(2) is substantial and, it manifests not only in increased diversity order but also in noteworthy coverage area expansion.
In future work, it will be meaningful to verify that the proposed NBNC scheme can be extended to a larger-scale network, where more sensor nodes are involved in cooperative transmission.
Declarations
Acknowledgments
This study was supported by the National Research Foundation (NRF) of Korea grant funded by the Korean government (MEST) (Do-Yak Research Program, No. 2012–0005656, Haek-Sim Research Program, No. 2012–047744).
Authors’ Affiliations
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