An amplify-and-forward relaying scheme based on network coding for Deep space communication

Abstract

Network coding, as a new technique to improve the throughput, is studied combined with multi-relay model in this paper to address the challenges of long distance and power limit in deep space communication. First, an amplify-and-forward relaying approach based on analog network coding (AFNC) is proposed in multi-relay network to improve the capacity for deep space communication system, where multiple relays are introduced to overcome the long distance link loss. The design of amplification coefficients is mathematically formulated as the optimization problem of maximizing SNR under sum-power constraint over relays. Then for a dual-hop relay network with a single source, the optimal amplification coefficients are derived when the multiple relays introduce non-coherent noise. Through theoretic analysis and simulation, it is shown that our approach can achieve the maximum transmission rate and perform better over single link transmission for deep space communication.

1. Introduction

Deep space communication plays a crucial role in particular for operations, efficient science-data downlink, and public outreach in case of human missions [1-2]. There are some specific challenges that would not encounter in the wireless communications on the earth, such as the long distance transmission resulting into great loss combined with the limited power on the space craft [3-6]. To solve the problems, lots of ways are considered in deep space communication, such as image compression and channel coding as well as application layer coding mechanism are simultaneously applied for image transmission over deep space channels[7,8]. At present, single link is always adopted to transmit information between the space craft and the earth, however, it is not an efficient way for the case of the long distance transmission and power limit [1]. Multi-relay structure or space network is a trend for future exploration demand [4,5].

In the future deep space exploration, the capacity of the communication system will have a huge demand [5]. Network coding is a new technique which allows intermedia nodes make a combination of its received messages and then send out, rather than just store and forward in the traditional communication network [9,10]. It is proved that there are advantages in throughput, load equalization, power deduction and so on [9-12]. Therefore, network coding is introduced to address the problems in deep space communication, which maybe bring with valuable things. Fountain coding is a application of network coding to overcome communication interruption [13]. A self-cooperative network coding is proposed for reliable transmission of the large file in deep space and low transmission delay, where sender makes use of time diversity technologies to offer cooperative information by itself [14].

Multi-relay structure is considered in this paper. On the surface, the increased terminals will introduce more overhead, more noise and lower reliability. In fact, it can utilize a small amount of space crafts to implement link segmentation to deduce the link loss at the same time. It will be a high benefit-cost ratio method when the related technique is introduced to guarantee the reliability and low noise. However, how to design the scheme for the relay nodes to cooperate with each other to maximize the transmission rate is a problem worth to study.

There are three main relaying schemes currently, including Amplify-and-Forward (AF), Decode-and-Forward (DF) and Compress-and-Forward (DF)[11,15]. The last two schemes have higher complexity while AF is only with linear complexity, which make it easy to carry out in deep space communication. Combined with analog network coding, an efficient amplify-and-forward (AFNC) scheme for deep space multi-relay network communication is proposed, where non-source nodes allow to combine its received messages according to the coding coefficients chosen random or fixed before transmission. Meanwhile, the sum power constraint over relays is set to lower the energy consumption inspired by the study of power allocation in [16]. A single source and dual-hop multi-relay network is considered in this paper. The amplification coefficient is designed to maximize the received signal-to-noise ratio (SNR) at sink node when the multiple relays introduce non-coherent noise and are subject to sum-power constraint. Then, the optimal amplification coefficient is derived in details for a dual-hop two-relay network with a single source. It is shown through theoretic analysis and simulation that this method is an efficient way to improve the channel capacity.

The organization of this paper is as follows. First, the characters of deep space communication and the principle of network coding are introduced as preliminaries in Section 2. Our model and the AFNC scheme are presented in Section 3. Then an illustrative example and the simulation are given in Section 4 and 5, respectively, which show that our approach have advantages in the transmission rate.

 

2. Definitions and Preliminaries

2.1 Deep Space Communication

In deep space communication, let PT be the transmit power at source and Pr be the received power at sink node, then we have

where GTx is the gain at the transmit antenna, GRx is the gain at the receiving antenna, d is the transmission distance, and λ is the wavelength of the signal. The coefficient indicates the power loss at the link.

Corresponds to the signal, the loss can be presented as

where

That is, the transmission between the space crafts can be formulated as

where x is the transmit signal, and z is the noise for deep space communication links which can be normalized as a N(0,1) Gaussian noise in general [17,18].

2.2 Network Coding

Network coding is a fusion of routing and coding. Its main idea is that the non-source nodes can make a linear or nonlinear combination of its received messages and then send out, where the intermedia nodes play a role of encoder or signal processer. It is proved that this technique can improve the throughput[9]. Analog network coding is a method used in physical network coding, which is mainly based on the relay amplify and forward mode[11,15]. Analog network coding can directly process the modulated signal, and its focus is how to recover a lossy signal rather than process the signal by mathematics.

Communication is always modeled as a directed graph G(V,E) , where V is the set of nodes, and E is the set of links. For a non-source node j ∈ V , its coding process is defined as

where In(j) denotes the set of incoming adjacent nodes of node j , and yi is the signal received at node j from node i , αi,j is the coding coefficients, xj is the transmit signal at node j after the coding process. At the sink node, the corresponding inverse transformation is processed to the received mixed messages to recover the source messages.

 

3. Network Model and AFNC Scheme

3.1 Network Model and Problem Formulation

Consider the deep space communication between the planets and the Earth in Fig. 1. Direct link (red link) is used at present, and it is very high cost to overcome the huge path loss and delay when the messages are transmitted directly between them. It is proved that relay network model could reduce the path loss and obtain better performance. On the other hand, a small amount of space crafts that have been sent into deep space could be used to build multiple paths to increase the channel capacity. A relay satellite divide the distance from Mars to Earth into parts, which reduce the total transmission power to some extent.

Fig. 1.Deep space relay network

A multi-relay two-hop model is considered in this paper, as in Fig. 2. The set of the relays is denoted as K = {1,2,...,K} , and the distance dij between node i and node j is labeled as in the figure. Half duplex and AF scheme is considered over relays. Combined with network coding, we propose AFNC relaying scheme, that is, non-source nodes can make a combination of its received signals, and the chosen rule of the coefficients is to maximize SNR to achieve the maximum transmission rate.

Fig. 2.Deep space relay network

Assume that each node k (non-source) simultaneously receives the signal from the source node S via different relay path, and there is no interference for all the output of channels. For this reason, we can ignore the time index, and the signal received at relay k can be expressed as

where is the channel response which represents the power loss in terms of distance, and zk is independent Gaussian noise with mean 0 and variance . For simple, the gaussian noise at all relays is denoted by z = [z1,z2,…,zK]T ∈ CK×1, and its correlation matrix is KR ≻ 0 ∈ CK×K with as the kth diagonal element. For a given antenna gain and communication setup, Ksh is considered as a constant in this paper. Hence, the received signal power at relay k is

Then, the relays amplify the received signal according to the coding coefficients, i.e., the transmission signal xk ∈ C at relay k can be expressed as

where αk ∈ C is the amplify coding coefficeint at the relay k. So we have the transmission signal power at relay k as

In order that the increased relays will not increase too much power consumption, we set a sum-power constraint over relays by the following formula

Compare with the transmission delay, the process time could be ignored. So we assume all relay nodes are instantaneous. Therefore, the received signal at the sink node is

where is the channel response from relay k to sink D, and hs,D = Ksh /dsD is the channel response directly from source node to sink node. For simple, we denote all hs,k as a K × 1 column vector hs = [hs,1,hs,2,…,hs,K]T, and denote all hk,D as a 1 × K row vector hD = [h1,D,h2,D,…,hK,D] . A coding coefficeint matrix is formed as A = diag(a1,a2,…,aK) ∈ CK×K. Therefore, the signal received at sink can be written in matrix form as

A coding sequence with length n and [2nR] codewords is used in our approach, the error probability of which approaches to 0 as the code length n → ∞. The codebook at source node is Gaussian codebook with distrubition N[0,Ps}, where Ps is transmit power at source. The relay satellites amplify and forward the received messages without delay, and the amplification coefficients αk are chosen such that the sum-power constraint (3-5) holds.

We will take the SNR as the measure of the quality. So the aim of the amplify coding coefficeint in our AFNC scheme is to maximize the SNR under the sum power constraint, which is formulated as follows

where α = [α1,α2,…,αK]T ∈ CK×1 is formed by the coding coefficients, and HD = diag(hD) ∈ CK×K. Obviously, it is a non-convex optimization problem. In general, it is very hard to solve it. However, in deep space communication, the number of the relays is always limited. So this optimal problem can be solved directly by Lagrange multiplier, or converted into convex optimization by the method of semi-definite relaxation proposed in [19].

3.2 Performance Analysis

We will give the upper bound of this network model. According to the cut-set bound, we have

We will calculate the upper bound of the two mutual information seperatively.

This upper bound can be taken as the capacity of a single input multiple output channel. Hence, we can take the relays as the multiple antenna at the receiver.

where hmax,D = max{h1,D,h2,D,…,hK,D} . That is, the above upper bound is obtained when the relay which has the mimimum path loss is adopted with maximum sum-power. This upper bound can be seen as the capacity of a multiple input single output channel. Then the relays can be taken as the multiple antenna at the source node.

To illustrate AFNC scheme, a dual-hop two-relay network is considered in this section for the communication from Mars to Earth, shown in Fig. 3, which can be modeled as in Fig. 4. S is the source (Mars), D is the receiver (Earth), and dl,l = 1,…,5 is the path length, k1 and k2 are relay satellites. There are three paths from S to D. Broadcast is adopted in our approach, that is, the source S broadcast the messages to k1, k2 and D, then relays k1 and k2 amplify and forward to D. Finally, the messages received at D is the combination of S, k1 and k2.

Fig. 3.A dual-hop two-relay deep space communication network from Mars to Earth

Fig. 4.A dual-hop two-relay network model

According to (3-1), the channel outputs of the relays are as follows:

where zS,k1, zS,k2 are the equivalent noise with gaussian distribution after amplifying.

Hence, the signal at the receiver is

where the noise is denoted by , and the terms including x represents the effective signal received at sink node. We assume the noise caused at intermedia nodes are non-coherent, so the total equivalent noise obey Gaussian distribution.

Next, the power received at sink node is

where E[|x|2] = Ps, E[|z|2] = δ2.

The equivalent signal is related to the amplification coefficients in our AFNC scheme while the total equivalent noise is only related to network parameters and some constant from expressions (4-3) and (4-4). Since the capacity of the equivalent channel is

where W is the bandwidth, N0 is spectral noise power density, and P is the power of effective signal. For the equivalent Gaussian channel, we concern how to choose the amplification coefficients to maximize the signal-to-noise ratio to achieve the maximum transmission rate, which is equivalent to maximize the power of signal received at sink node here. So the problem is formulated as maximizing the power of signal received at sink node with the sum power over relays limit. That is,

where Psum_max is the maximum sum-power, and

where δ12 = E[|zS,k1|2], δ22 = E[|zS,k2|2].

Lagrange multipliers is need to find the extreme value under the constraint. Let We define a function F(α) = E[|yD|2] + λg(α) as

Now we get the first-order partial derivatives to the variables αk1 and αk2 , respectively,

Then let them equal to 0, i.e.,

So we have

Substitute (4-12) and (4-13) into the sum-power constraint (4-6), we have

Then λ is derived from (4-14),

The two coefficients are obtained as follows by substituting λ into (4-12) and (4-13),

Finally, substitute (4-16) and (4-17) into (4-5), we have

So the transmission rate of our AFNC scheme in multiple relay network is

where δ2 = N0W . And this capacity is achieved when the coefficients are chosen according to (4-12) and (4-13).

 

5. Simulations

For intuition, we plot the transmission rate of our AFNC scheme in multiple relay network, and compare with that of the direct single link for deep space communication. We made the simulation based on Fig. 4. By data query, let d1=d4=3.844×105km, d2=d3=5.3×107km, d5=5.5×107km. The antenna gain is set to be 10, Psum_max = 100W, and spectral noise power density N0 = 0.01 . The Ka frequency band is utilized, and let λ=8mm [20]. The transmission rate of AFNC scheme is simulated as the transmit power Ps or the bandwidth W increase respectively, shown in Fig. 5 and Fig. 6.

Fig. 5.The transmission rate R increase with the transmit power PS

Fig. 6.The transmission rate R increase with the Bandwidth W

In Fig. 5, we let the bandwidth W=100MHz, and Ps is in [0,60]w. For fair, we add Psum_max with Ps as the transmit power at the source node for the direct link transmission. That is why the rate is nonzero when Ps is 0. We found that the rate of AFNC scheme has nonzero value after a certain power value because the noise is amplified rather than the signal when the transmit power is very small. In Fig. 6, we let Ps =10w, and W is in [50,500]MHz. Through obveration, we found that the result is consistant with theorectic analysis. No matter AFNC scheme or direct link transmission, the transmission rates increase with the bandwidth, and approach to the upper bound of the capacity in terms of SNR. On the other hand, bandwidth is rich for deep space communication. Hence, it is more important for the effect of transmit power on the transmission rate than that of the bandwidth. Furthermore, From both figures, with the same transmit power, the transmission rate R of AFNC scheme in relay network is significantly greater than that of the direct single link channel.

 

6. Conclusion

An AFNC scheme in multi-relay network is proposed to address the bottleneck of deep space communication. In our method, the non-source node could make a combination of its received messages with the coefficients satisfying the sum power constraint over relay space crafts. The optimal amplify coefficients are deduced for a dual-hop two-relay network, and the transmission rate is also given. It is shown that our AFNC scheme is advanced than the single direct link communication through theoretic analysis and simulation. The case of multi-source or multi-hop relay network will be left for feature work, together with the relation between the number of relays and the benefit-cost ratio.