## 1 Introduction  
The rapid development of modern information and communication technologies, such as the Internet of Things, wireless communication, and cloud [1, 2, 3, 4, 5], makes remote healthcare, or tele-healthcare, simpler than before, as shown in Figure 1. In recent years, wireless sensor networks (WSNs) have matured to be used to improve quality of life, which is regarded as one of the key research fields in the healthcare application industry and has attracted more and more attention [6, 7, 8, 9, 10, 11, 12, 13]. It is well known that, sensor nodes in WSNs generally adopt small embedded devices such as some wearable devices, and the battery power carried by a single node is limited. In the complex medical information environment, the application of WSNs is usually oriented to some large-scale monitoring fields, which is necessary to collect massive amounts of monitoring data. In order to realize the effective data acquisition of WSNs, it is very important to reduce the energy consumption of nodes. Multiple Phase Shift keying (MPSK) is provided in IEEE 802.15.4c standard [14, 15]. Detection of coded MPSK signal has important research and application value by improving transmission efficiency, reducing the energy consumption of node transmission and improving the life cycle of healthcare network. In the literature, much progress has been achieved regarding the detection of coded MPSK signals. The work in [16] proposes a simplified soft decision metric for MPSK without the noise variance knowledge, and no performance loss is observed. In [17], the iterative decoding of LDPC coded MPSK is considered over time correlated Rayleigh fading channels. However, this research is developed under a coherent channel, which is clearly not suitable for our purposes. Under a phase noncoherent channel, the pilot-aided LLR for LDPC coded MPSK orthogonal frequency division multiplexing transmission is developed in [18]. However, it cannot be directly tailored for WSNs, wherein much attention is paid to low complexity, low cost, low power consumption. It is notable that much focus on the detection of uncoded binary phase shift keying (BPSK) signals in IEEE 802.15.4 WSNs has been witnessed in recent years [19; 20; 21; 22; 23]. Multiple-symbol detection (MSD) scheme for offset quadrature phase shift keying (O-QPSK) receivers is also considered in [24; 25; 26; 27]. However, all of these are developed without channel coding. In this work, we propose simple bit LLR calculation schemes for coded MPSK signals. Unlike the traditional noncoherent detection scheme which was equipped with an LLR extraction scheme with high complexity to achieve the best possible reliability, we pay all of our attention towards the simple design to balance complexity and reliability. We summarize our main contributions as follows. * As a benchmark, the optimal bit LLR under a phase noncoherent channel is first given. For this scheme, both zero-order Bessel function and perfect instantaneous fading amplitude is involved. Thus, we propose a simple calculation configuration without CSI, which greatly simplifies the optimal scheme. * The optimal bit LLR under a Rayleigh fading channel is also given, wherein we assume that the statistical characteristic of the fading amplitude is available. In this context, the variance estimation for both additive white Gaussian noise (AWGN) and the statistical characteristic of the fading amplitude is involved, and extensive exponential operation is also unavoidable. A simple calculation scheme is then developed. * We find that the decision statistic is exactly the same under both phase noncoherent channels and the Rayleigh fading channel when no coding is considered, and no contribution from the fade amplitude and the variance of the AWGN noise is observed in this decision statistic. We also find that the CSI can be perfectly avoided in the bit  
Figure 1: Wireless smart framework for transmission of home healthcare information.