# Adaptive Bit-Labeling Design for Probabilistic Shaping Based on Residual Source Redundancy
###### Abstract  
By using the residual source redundancy to achieve the shaping gain, a joint source-channel coded modulation (JSCCM) system has been proposed as a new solution for probabilistic amplitude shaping (PAS). However, the source and channel codes in the JSCCM system should be designed specifically for a given source probability to ensure optimal PAS performance, which is undesirable for systems with dynamically changing source probabilities. In this paper, we propose a new shaping scheme by optimizing the bit-labeling of the JSCCM system. Instead of the conventional fixed labeling, the proposed bit-labelings are adaptively designed according to the source probability and the source code. Since it is simple to switch between different labelings according to the source probability and the source code, the proposed design can be considered as a promising low complexity alternative to obtain the shaping gain for sources with different probabilities. Numerical results show that the proposed bit-labelings can significantly improve the bit-error rate (BER) performance of the JSCCM system.  
bit labeling; joint source-channel coded modulation; quadrature amplitude modulation (QAM); protograph LDPC codes +
Footnote †: journal: 2023, 25, 586. [[https://doi.org/10.3390/e25040586](https://doi.org/10.3390/e25040586)]([https://doi.org/10.3390/e25040586](https://doi.org/10.3390/e25040586))
For non-uniform sources, in [14], protograph LDPC codes are optimized under binary modulation with unequal power allocation. By using the source residual redundancy after source coding to achieve probabilistic shaping, a joint source-channel coded modulation (JSCCM) system was proposed [15]. In [15], the design of the source code parameter is targeted specifically at a given source probability, and thus the residual redundancy left after source coding can be properly used to shape the transmitted symbols. Accordingly, the source-channel code pairs optimized at one source probability may lead to extremely poor performance at another source probability due to the mismatch of the source code parameter and the source probability. However, this characteristic is undesirable for some systems with varying levels of data redundancy since the source-channel code pairs should be optimized at different source probabilities to ensure the PAS performance.  
In this paper, a low complexity method to obtain the shaping gain for sources with different probabilities is proposed for the JSCCM system. The main contributions of this paper are as follows:  
* By studying the effects of bit-labeling in JSCCM systems, it is found that good bit-labelings for different source codes or different source probabilities could be different.
* Based on the achievable system rate analysis, a new shaping scheme for the JSCCM system is proposed by optimizing the bit-labeling.
* In contrast to the fixed Gray labeling [16], the adaptive design of bit-labelings for the JSCCM system is proposed according to the source codes and the source probabilities. Since it is much simpler to switch between labelings than to optimize the source-channel code pairs for different source probabilities, it is attractive for systems with changing source statistics.  
The remainder of the paper is organized as follows. Section 2 presents the JSCCM system model. Section 3 proposes an adaptive design algorithm of the bit-labeling. Section 4 discusses the performance of the system using the adaptively designed labelings in comparison with the fixed labeling. Finally, Section 5 concludes the paper.