Compressive cooperation for gaussian half-duplex relay channel
© Liu et al.; licensee Springer. 2012
Received: 1 March 2012
Accepted: 11 July 2012
Published: 23 July 2012
Motivated by the compressive sensing (CS) theory and its close relationship with low-density parity-check code, we propose compressive transmission which utilizes CS as the channel code and directly transmits multi-level CS random projections through amplitude modulation. This article focuses on the compressive cooperation strategies in a relay channel. Four decode-and-forward (DF) strategies, namely receiver diversity, code diversity, successive decoding and concatenated decoding, are analyzed and their achievable rates in a three-terminal half-duplex Gaussian relay channel are quantified. The comparison among the four schemes is made through both numerical calculation and simulation experiments. In addition, we compare compressive cooperation with a separate source channel coding scheme for transmitting sparse sources. Results show that compressive cooperation has great potential in both transmission efficiency and its adaptation capability to channel variations.
Compressive sensing (CS) [1, 2] is an emerging theory concerning the acquisition and recovery of sparse signals from a small number of random linear projections. Recently, it is observed that CS is closely related to the well-known channel code called low-density parity-check (LDPC) codes [3, 4]. In particular, when the measurement matrix in CS is chosen to be the parity-check matrix of an LDPC code, the CS reconstruction algorithm proposed by Baron et al.  is almost identical to Luby’s LDPC decoding algorithm . It is the similarity between CS and LDPC codes that inspires us to propose and study compressive transmission, which utilizes CS as the channel code and directly transmits multi-level CS random projections through amplitude modulation.
Since CS has both source compression and channel protection capabilities, it can be looked on as a joint source-channel code. When the data being transmitted is sparse or compressible, a conventional scheme will first use source coding to compress the data, and then adopt channel coding to protect the compressed data over the lossy channel. Compressive transmission brings some unique advantages over such a conventional scheme. First, since CS uses random projections to generate measurements regardless of the compressible patterns, it reduces complexity at the sender side. This will benefit thin signal acquisition devices, such as single-pixel camera  and sensor nodes. Second, it improves robustness. It is well-known that compressed data are very sensitive to bit errors. In the conventional scheme, when the channel code is not strong enough to protect data in a suddenly deteriorated channel, the entire coding block or even the entire data sequence may become undecodable. In contrast, CS random projections directly operate over source bits, and sporadic bit errors will not affect the overall data quality.
In this article, we will focus on studying the cooperative strategies for compressive transmission in a relay channel, or compressive cooperation. We consider a three-terminal Gaussian relay channel consisting of the source, the relay and the destination. Such a relay channel is first introduced by van der Meulen  in 1971, and has triggered intense interests since then. However, most previous researches on cooperative strategies are based on binary channel codes, such as LDPC codes [9–11] and turbo codes [12–14]. This article presents the first work that applies CS as the joint source-channel code in a relay channel. In particular, we present four decode-and-forward (DF) strategies, three of which resemble those for binary channel codes and the fourth one is peculiar to CS because it takes the arithmetic property of CS into account. We theoretically analyze the four strategies and quantify their achievable rates in a half-duplex Gaussian relay channel. Numerical studies and simulations show that all strategies except the receiver diversity strategy have high transmission efficiency and small implementation gap while code diversity scheme has the most stable performance. We further compare compressive cooperation with a separate source channel coding scheme, and results show that using CS as a joint source-channel code has great potential in a relay channel.
The rest of the article is organized as follows. Section 2 describes the channel model. Section 3 overviews compressive cooperation and analyzes the information-rate bound in a half-duplex Gaussian relay channel. Section 4 studies four DF schemes and their respective achievable rates. Section 5 reports results of numerical studies and simulations. Section 6 concludes the article.
where z r and are Gaussian noises perceived at R and D.
where is the perceived Gaussian noise at D. Finally, the destination D decodes original message from received signals during BC and MAC modes.
Compressive transmission overview
Compressive transmission in a relay channel
where is a power scaling parameter to match with sender’s power constraint.
where B is also a Rademacher matrix, and w contains m2measurements.
Under these power constraints, the corresponding scaling parameters αs 1, αs 2 and α r can be derived, where the average power of symbol A1u, A2u and B u are determined by the row weight of corresponding sampling matrix and sparsity probability of u.
where H(u) is the entropy of u and m1 and m2 determines the cost time slots for the BC mode and MAC mode, respectively. If the base of the logarithm in entropy computation is 2, the rate R is expressed in bits per channel use. It should be noted that the rate R in Equation (11) is related with the symbol energy , and E r . For the compressive transmission along a link channel, when the corresponding transmission power is larger, higher quality of measurements could be derived and the number of needed measurements for source recovery could be smaller. Therefore higher rate could be achieved from large allocated transmission energy.
In such compressive transmission system, the encoding complexity is rather low because the calculation of measurements at the source node only involves the sums and differences of a small subset of the source vector. The complexity of the belief-propagation based decoding algorithm is O(TMLQ log(Q)) , where L is the average row weight, Q is the dimension of transmitted message in belief propagation process, T is the iteration number and M is the number of received measurements.
where I(X;Y) represents the mutual information conveyed by a channel with input X and output Y. The supremum is taken over t(0≤t≤1) and all the joint distributions p(x1x2w) up to the alphabet constraints on X1X2and W.
In order to approach the capacity, a parameter which is not explicitly shown in (12) also needs to be optimized. It is the correlation between x2and W, i.e. the codeword sent by the source and the relay in the MAC mode. The source and the relay send identical messages at one extreme (r=1), while they send entirely different messages at the other extreme (r=0). In the design of cooperative LDPC code, it is observed that the optimal achievable rate can be well approximated by the better case of r=0 and r=1. We make the same simplification and only consider cases r=0 and r=1. In these two extreme cases, several terms in (12) can be simplified:
The mutual information terms in (12) are determined by both channel signal-to-noise ratio (SNR) and the input alphabet. The input alphabet is determined by the modulation scheme in conventional transmission and jointly determined by the binary source and the measurement matrix in compressive transmission. It is impossible to compute a general information rate curve for compressive cooperation.
The achievable rate of the compressive transmission is a function of channel SNR when the properties of source message and measurement matrix are given, and two functions are defined for ease of presentation. When all measurements come from the same channel with SNR P, the achievable rate is denoted by R(P). When measurements are received from different channels, the achievable rate is denoted by R((γ1,P1),…,(γ k ,P k )), where k is the number of channel realizations. γ i and P i (1≤i≤k) are the time proportion and SNR of the i th channel realization.
Compressive cooperative strategies
In this section, we specify four DF strategies for compressive cooperation, namely receiver diversity, code diversity, successive decoding and concatenated decoding. The first three strategies resemble those for binary channel codes, and the last one does not have binary counterpart in conventional relay communication, because it combines the arithmetic property of CS with the signal superposition process of MAC mode.
Both r=0 and r=1 are considered. When r=1, the binary message u is treated as a whole. The transmitted signals x 1 , x 2 and w are the CS measurements of u obtained with matrix A1, A2 and B, respectively. When r=0, message u is looked on as the concatenation of two parts, or . The source transmits the measurements of u 1 in BC mode, while it transmits measurements of u 2 in MAC mode. The relay decodes u 1 and then transmits the measurements of u 1 in MAC mode. Therefore, the two transmitted signals w and x 2 in MAC mode are the CS measurements of different parts of u.
The purpose of cooperative strategy design is to choose appropriate matrices A1, A2and B such that the original message can be recovered at the destination with the minimum number of channel uses. Since we fix to adopt Rademacher sampling matrices in our system, the choice of A1, A2 and B is decided by the number of the rows of these matrices, which also implies the selection of time proportion t of BC mode. The number of needed measurements for successful CS reconstruction depends on their reliability, which should be ensured by appropriate energy allocation at the source and relay during the BC and MAC mode.
DF strategies for r= 1
In this scheme, the source in BC mode and the relay in MAC mode transmit the same set of CS measurements (A1=A2=B), such that m1=m2 and t=0.5. At the destination, the two noisy versions of the same measurement, received in BC and MAC mode, are combined into one through maximal ratio combining (MRC) . The SNR of the combined signal is the sum of SNRs of the received signals from independent Gaussian channels. As the SNR of is , and the SNR of is P2 as defined in (13), the SNR of the combined signal is .
where the supremum is taken over all time and power allocations that satisfy the constraint (4). The term tR(P sr ) expresses the constraint that the relay should be able to fully decode the source message at the end of BC mode.
where z1 and z2 are unknown realizations corresponding to and . The numbers of measurements in y1and y2 are m1 and m2 respectively. Be noted that above equation is derived by dividing respective power scaling parameters from received signals. The SNRs of y1and y2remain and P2.
DF strategies for r= 0
Intuitively, when channel condition is good, the source can transmit new information to the destination during MAC mode. The destination receives the measurements of message u1 in BC mode, and receives superposition of measurements of u1and u2 in MAC mode. We propose two different decoding strategies and corresponding matrices design for r=0.
Successive decoding is commonly used in conventional relay network. The destination first decodes message u1from the signals received in both BC and MAC mode. The information about message u2 is treated as noise at this stage. After u1 is decoded, the destination removes the information about u1 from the signals received in MAC mode, and then decode u2.
By definition (5), the SNR of is .
Although successive decoding is the capacity achieving decoding algorithm in Gaussian relay channel, it may not be optimal in compressive cooperation. This is due to the fact that the achievable rate of compressive transmission R(P) has a very different form from Shannon capacity . The intuition behind concatenate decoding is that higher efficiency may be achieved by jointly decoding u1 and u2 rather than treating u2 as noise when recovering u1.
In all the four achievable rate expressions, the supremum is taken over all possible time proportion t and transmission powers that satisfy the energy constraint (4).
Numerical study and simulations
This approximation is reasonable because otherwise a source needs to do per measurement energy allocation to achieve the optimal performance.
Evaluating compressive cooperation strategies
In the formulation of the proposed four DF schemes, the supremum is taken over all possible time proportion and transmission powers that satisfy (4). The analytical solution to the optimization problem is hard to find since R(P) is unknown. Therefore, we first obtain R(P) for compressive transmission through simulations, and then compute the achievable rates of the four DF strategies through numerical integration. Baron et al.  have reported that there is an optimal row weight Lopt≈2/p beyond which any performance gain is marginal. We slightly adjust L to 15 and use eight −1’s and seven 1’s. For simplicity, we use the amplitude modulation of only one carrier wave. The performance for quadrature amplitude modulation (QAM) can be easily deduced from our reported results.
Using this property, it can be derived that the rate of receiver diversity is no greater than that of code diversity.
The comparison between the code diversity scheme for r=1 and the two r=0 schemes draws a consistent conclusion as in conventional relay channels. First of all, the performance difference between r=0 schemes and r=1 schemes is not significant. Second, r=0 schemes show advantage when channel SNR is high, but r=1 schemes perform better when SNR is low. Our numerical results show that the achievable rate of r=0 schemes is higher than r=1 schemes when SNR is higher than 13 dB. Although the two r=0 schemes exhibit similar performance, concatenate decoding appears to be better than successive decoding when channel SNR is higher than 13 dB.
In Figure 4, simulation results of three DF schemes are compared with the highest numerical rate computed when r is either 0 or 1. It can be seen that the implementation gap is within 1.4 dB for all three schemes. During simulation, we observe that code diversity has very stable performance at both high and low SNRs. The performance of the two r=0 schemes has a slightly larger variation. In addition, when channel SNR is lower than 12 dB, both r=0 schemes degrade to two-hop transmission, i.e. . Considering the fact that r=0 schemes do not significantly improve channel rate at high SNR, and code diversity is easier to implement, it is a wise choice to stick to code diversity scheme in practical systems.
Actually, when wireless channel state information is unknown for the source node, the channel code based on CS measurements can be generated limitlessly and transmitted until some predefined recovery quality is achieved at the receiver. The reason is that more redundancy will be achieved through increasing the number of CS measurements, which can help overcome the channel noise as shown in . This rateless property makes the compressive cooperation communication system much easier to adapt to channel variation compared with traditional LDPC codes.
In the end of this part, we analyze and compare the computational complexity of the four DF strategies.
Comparing compressive transmission with a separate source channel coding scheme
We can find that compressive transmission does not incur any rate loss to the reference schemes when the channel SNR is below 15 dB. After 15 dB, the rate achieved by compressive transmission starts to saturate. We have performed other experiments which show that the saturation SNR is determined by the sparsity of the Rademacher measurement matrix. If a denser measurement matrix is used, compressive transmission will cover a larger dynamic range. Even with the current setting, compressive transmission shows a better channel adaptation capability than the reference schemes.
This article proposes compressive transmission which utilizing CS random projections as the joint source-channel code. We describe and analyze four DF cooperative strategies for compressive transmission in a three-terminal half-duplex Gaussian relay network. Both numerical studies and simulation experiments are carried out to evaluate these strategies’ achievable rates. We have compared compressive cooperation with a conventional separate source channel coding scheme. Results show that the proposed compressive cooperation has great potential in wireless relay channel because it not only has high transmission efficiency, but adapts well with channel variations.
The authors would like to thank the anonymous reviewers, whose valuable comments helped to greatly improve the quality of the article.
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