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Considering the fact that sensors are energy-limited and the wireless channel conditions in wireless sensor networks, there is an urgent need for a low-complexity coding method with high compression ratio and noise-resisted features. This paper reviews the progress made in distributed joint source-channel coding which can address this issue. The main existing deployments, from the theory to practice, of distributed joint source-channel coding over the independent channels, the multiple access channels and the broadcast channels are introduced, respectively. To this end, we also present a practical scheme for compressing multiple correlated sources over the independent channels. The simulation results demonstrate the desired efficiency.

In recent years, the growing concern for our environment and society has led to applications ranging from detecting chemical leaks to monitoring underground parking. Wireless Sensor Networks (WSNs) have received significant attention to fulfill these requirements. WSNs consist of a large number of cheap sensors, e.g. intelligent sensor nodes, micro-cameras, which are densely allocated and self-organized to efficiently and reliably perform complex tasks in inaccessible situations.

While having a significant impact throughout society with their rich application space, WSNs also bring challenges to information and network technologies. The constrained energy of the sensors leads to limited processing capabilities and transmitting power. Furthermore, the impacts of noise, fading and multi-user channels [

Addressing the above issues, Distributed Joint Source-Channel Coding (DJSCC) has attracted considerable research as an accepted efficient technique. DJSCC derives from the Distributed Source Coding (DSC), which is an important source coding development. DSC allows the separate encoding of correlated sources to be as efficient as joint encoding in traditional Shannon's theory, which is stated by the Slepian-Wolf theorem [

This paper summarizes the developments of DJSCC over various channels in WSNs. In [

This paper is meant to generalize a framework for the development of DJSCC in WSNs, and is by no means exhaustive. The organization is as follows. A brief review of DSC is presented in Section 2, especially the noise immunity of the DSC. The previous theoretical results and implementing methods of the DJSCC for the asymmetric case, the independent channels case, the MAC case and the BC case are presented in Section 3. Our proposed scheme is provided as an example showing the efficiency of DJSCC for the independent channels case in Section 4. Section 5 concludes the paper.

Slepian-Wolf theorem [_{1} and _{2}:
_{i}_{i}, i

Reference [_{1} and the other source _{2} (called the side information) can be regarded as the input and the output of the virtual channel respectively. Thus, the DSC can be processed as the channel coding problem. Based on this idea, the first practical DSC scheme called DISCUS [

The binning/syndrome approach and the parity approach are major methods for practical DSC due to the notion in [

In the asymmetric case, one of the sources _{2}_{1}_{1}_{2}

In the asymmetric case, the separation theorem is valid. [_{1}_{1} ≥ _{1} | _{2}) the encoding rate of _{1}

An extension to lossy DJSCC in this case is given in [

Reference [

The syndrome-based approach can still be used in DJSCC by treating the syndromes as a special type of parity bits. Reference [

Instead of compressing only one source, in the independent channels case, all the sources are independently encoded and transmitted over their respective noisy channels, as shown in

In the independent channels case, the separation theorem still holds [_{1}_{2}_{i}_{i}, i_{1}_{2}_{i}_{i}

Typical designs [

Most of the works on DJSCC over MAC concentrate on the necessary and sufficient conditions for sending the correlated sources with arbitrarily small probability of error. The practical algorithms are rare to the best of our knowledge.

Reference [_{1},_{2}) ∼ _{1},_{2}), the sources transmit their codewords _{i}, i_{1} × _{2}, _{1},_{2})). The achievable region sending _{1}_{2}_{1},_{2},_{1},_{2},_{1},_{2})_{1}∣_{1}) _{2}∣_{2}) _{1},_{2}). The MAC region and the Slepian-Wolf region are the special cases of the region in (6). Generalizing the results to sources with a common part _{1}) = _{2}), corresponding achievable rate region is listed in Theorem 1 of [

Although an instructive counter example is presented in [_{1},_{2},_{1},_{2},_{1},_{2},_{1}∣_{1}) _{2}∣_{2}) _{1},_{2}). Note that the correlation among the sources condenses the region. Recently, [

Using the similar strategies in [

Here one important point should be added, that there is a special case of MAC, discrete memoryless (dm) Asymmetric MAC (AMAC), over which the separation theorem can be applied [

The problem of transmitting correlated sources over BC has been investigated as early as in 1987 [_{V}_{1},…,_{VK}_{|}_{U}_{1},…,_{K}_{i}, i

In [_{U}

Then the necessary and sufficient condition for joint source-channel coding over BC is presented in Theorem 6 in [

Using ideas from [

A brief summary of this section is given here. The first two cases, the asymmetric case and the independent channels case, are comparatively simple, and have been adequately studied. However, in the MAC case and the BC case, a number of problems remain to be solved; for example, the optimal rate regions of the general DJSCC settings and corresponding code design methods are all undetermined. If the interference of the adjacent channels can be considered as a kind of noise in WSNs, the MAC case degrades to the independent channels case. The research on the independent channels case will play a fundamental role in the study on DJSCC. Thus, we propose a DJSCC scheme for multiple correlated sources over the independent channels in the following section.

A practical DJSCC scheme for multiple correlated sources is proposed in the scenario of the independent channels case. Compared with the practical designs listed in subsection 3.2, the proposed scheme stresses two points. Firstly, the DJSCC for three correlated sources is investigated. Intuitively, this leads to more efficient redundancy utilization than considering pairs. In contrast, the approaches in [

In the proposed scheme, an efficient coding strategy with a single LDPC code is designed to cope with both source and channel coding. And the theoretical limits of DJSCC for multiple sources are derived. The simulation results demonstrate the desired efficiency of the proposed DJSCC scheme.

The proposed scheme is inspired from the parallel channel model introduced in [_{1}, X_{2}_{3}_{j}_{i}_{i}_{ij}, ∀_{i}_{i}_{i}_{i}_{i}_{i}k_{i}_{i}_{i}_{i}_{i},b_{i}_{i}_{i}

Since _{i}_{i}_{i}

At the joint decoder, the received different parts of the information bits from different encoders will re-produce an integrated codeword for decoding, and they act as side information for each other. Therefore, the initial LLRs for different fractions in message-passing algorithm for decoding [

Suppose that the actual noisy channel for the source _{i}_{1}_{2} and _{3} fractions are replaced by the side information which are received from _{2}_{3}_{2}_{3}

However, there is a more complex situation in determining the correlation models for some fractions of _{2}_{3}_{2}_{3} fraction, the decoded version X̂_{1} of _{1}_{3}_{1} and the received version of _{3}_{2}

The remainder of this subsection will give the constraint condition of _{i}_{ij}^{2}), ^{2}^{2}

_{1}, X_{2}, X_{3}

_{2} as an example, the overall channel capacity of _{2} is:

Since
_{2}∣have the lower bound:

This implies the formula (12) by substituting (10) into the (17). Similarly, other results of this theorem can be achieved.

In order to compare with the previous theoretical results in subsection 3.2, we consider the case of two correlated sources _{1}_{2}_{1} = _{2} = 1 - _{3} = 0, _{1} = 1-_{2} = _{3} = 0, where 0≤a≤1. The total encoding rate can be obtained as:

Neglecting the noisy influence on the side information bits, we can get C(p,σ^{2})=C(p)=1–H(p). Introducing it into formula (18), the same representation of formula (5) is achieved. Thus, the independent channels theory in subsection 3.2 is a special case of our theorem. This is a novel and instructive extension of the previous independent channels theory. Considering the noise interference on the side information bits, as the scheme we proposed, is more reasonable in practical applications.

In order to evaluate the performance of the proposed DJSCC scheme, we simulate it for the two sources and three sources cases. The length of original source sequence is fixed to ^{8}. The LDPC code we used is defined by the degree distribution (19) from [

It is assumed that the cross-over probability between the two sources ^{c}_{1} + _{2} = 1/^{c}_{b}_{0}

We also simulate two separate designs with the same total encoding rate ^{s}, R^{c}

In our paradigm, it is assumed that _{ij}^{c}_{i}_{i}_{i}_{1}_{2}_{3}_{1}_{2}_{3}_{1}_{2}_{3}_{1}_{2}_{3}_{b}/N_{0}_{1}, X_{2}_{3}_{1}_{3}

In this paper, a survey of the theories and implementation approaches of DJSCC for the asymmetric case, the independent channels case, the multiple access channel case and the broadcast channel case is presented. We have proposed an efficient framework of DJSCC for the independent channels case. Our results extend the existed theoretical limits of DJSCC to a more practical scenario by considering the noisy side information and multiple correlated sources. The simulate results verify the limit-approaching performance of the proposed scheme.

It can be found that the problem of finding the necessary and sufficient conditions of the most general setting in the MAC case and the BC case remains open to this day. Moreover, corresponding efficient coding strategies also have a tremendous space for further development. One of the main issues for practical deployment of DJSCC is the correlation model. In practical WSNs, it is usually hard to determine the exact joint distributed function; moreover, the correlation may be time-varying. Thus, the adaptive DJSCC scheme is a meaningful and challenging problem. Another future direction is introducing the multiterminal source coding to DJSCC. Since the multiple access channels and the broadcast channels are all the multi-user channels, it is obvious that the distributed compression of multiple correlated sources should be considered. However, the existing designs for DJSCC mainly constrained on processing two sources. The proposed scheme we presented here represents a first step towards this goal.

This work is partially supported by National Science Foundation of China under grant 60702049 and 60502036, National Science & Technology Pillar Program (2008bah37b04), and the 111 Project (No. B08004). The authors would like to thank all of the reviewers for their detailed comments that have certainly improved the quality of our paper.

The system model of DJSCC for the asymmetric case.

The system model of DJSCC for the independent channels case.

The DJSCC rate region for reliable transmission in the independent channels case: the intersection between the capacity region and the Slepian-Wolf rate region.

The system model of DJSCC for the MAC case.

The system model of DJSCC for the BC case.

Structures of the outputs of the encoders: Only the gray squares are transmitted for each encoder.

The parallel channel model of the source _{2}_{3}_{1}

BER of the _{1}

BER of the _{2}

BER of the sources _{1}, X_{2}_{3}