# Distributed Joint Source-Channel Coding Using Quasi-Uniform Systematic Polar Codes
## 1 Introduction  
Polar codes, invented by [PERSON] [1] using a technique called channel polarization, are capable of achieving the symmetric capacity of any binary-input discrete memoryless channel (B-DMC) with low encoding and decoding complexity. Afterwards, the concept of source polarization was introduced in [2] as the complement of channel polarization. One immediate application of source polarization is the design of polar codes for source coding. Since the polarization phenomenon exists on both source and channel sides, it is of natural interest to integrate channel polarization and source polarization for joint source-channel coding (JSCC). From the perspective of information theory, this interest is motivated by the fact that the error exponent of JSCC is better than that of separate source-channel coding (SSCC) [3], which may indicate a performance loss of SSCC under the finite code-length. From a practical point of view, two main shortcomings of SSCC resulting in an unsatisfactory performance are summarized as follows:  
* With finite code-length, channel decoding error is unavoidable, such error may be disastrous for the source decoder.
* With finite code-length, in the output of source coding may exist residual redundancy, such redundancy is not exploited by the channel decoder to further improve the performance.  
Therefore, the JSCC scheme, which takes into consideration both source redundancy and channel error jointly, may have certain advantages over SSCC.  
In recent years, JSCC has attracted an increasing number of attention among researchers. The JSCC schemes based on turbo codes, LDPC codes, and polar codes can be found in [4; 5; 6; 7], and [8; 9; 10], respectively. In particular, the authors of [11] proposed a class of systematic polar codes (SPCs), called quasi-uniform SPCs (QSPCs), and showed that QSPCs are optimal for the problem of JSCC with side information and outperform classic SPCs [12].  
For the distributed source coding (DSC) problem, the Slepian-Wolf theorem [13] states that for two or more correlated sources, lossless compression rates of joint encoding can be achieved with separate encoding if a joint decoder is used at the receiver. This theorem has been known for a long time, but the practical DSC scheme was only recently proposed by [PERSON] and [PERSON] using syndromes [14]. Inspired by the syndromes approach, a large number of DSC schemes have sprung up based on powerful channel codes (CCs) such as turbo codes, LDPC codes, polar codes [15; 16; 17]. However, to obtain a good performance, these syndromes-based schemes usually need a considerably long code-length (\(10^{4}\)\(\sim\)\(10^{5}\)) and are sensitive to channel errors. Alternatively, the design of DSC scheme can be based on parity approach where a systematic code is used to encode the source and the source is recovered by parity bits. These kinds of schemes can be extended to distributed JSCC (DJSCC) [18; 19]. Our work falls into this category. Specifically, we focus on the design of parity-based DJSCC using polar/polar-like codes, where only parity bits are transmitted over the noisy channels. Each distributed source has one encoder to encode source message for both compression and error protection.  
The main contributions of this paper are summarized as follows. We propose a new class of polar-like codes, called punctured QSPCs, and show its asymptotic optimality. Then we propose a DJSCC scheme using two polar-like codes and show that the proposed scheme approaches achievable rates.  
The rest of this paper is organized as follows. The system model is given in Section 2. Section 3 introduces QSPCs and punctured QSPCs. In Section 4, we propose a DJSCC scheme based on two polar-like codes and analyze its performance limit. Section 5 presents the simulation results to verify the advantage of the proposed scheme and in Section 6 we conclude this paper.