## 1 Introduction
Phase-Shift Keying (PSK) and Quadrature Amplitude Modulation (QAM) are the most popular modulation techniques in LPWANs [5]. Higher-order PSK and QAM modulations increase data rate and spectral efficiency, with a compromise on bit error rate (BER). However, QAM is 400% less power efficient than PSK. LoRa uses Chirp Spread Spectrum (CSS) and Frequency-Shift Keying (FSK); NB-IoT uses binary phase-shift keying (BPSK) and quaternary phase-shift keying (QPSK) while Wi-Fi HaLow employs BPSK, QPSK and higher order QAM modulations [11]. If the error performance can be made consistent with the requirement, PSK is preferred over QAM. This can be achieved if a suitable diversity method is used with PSK modulation. Diversity techniques improve link reliability of LPWANs. The situations in which diversity helps include networks in remote areas, sparse location of nodes and hostile scattering environments [5]. They have moderate to high impact on overall power consumption, range, cost, and interference management of the system. Diversity methods can be broadly categorized into frequency, time, and spatial diversity. One of the design goals of the LPWANs is power and spectrum efficiency. Industrial entities are shifting to unlicensed spectrum that offers small bandwidth but with cheaper overall deployment and maintenance cost. Therefore, frequency diversity alone cannot offer desired redundancy. Time diversity is a better approach, but it is not power efficient because of retransmission [5]. Spatial diversity (SD) is the predominant diversity method with the rise of multiple-input multiple-output (MIMO) systems but not suitable for the lower frequency industrial, scientific, and medical (ISM) frequency bands. The size of the MIMO antenna structure can be prohibitive and increase the power consumption considerably. Specifically, SD based on MIMO requires large antenna spacing to maintain orthogonality. Thus, the antennas at the base station should be distanced tens of wavelengths apart, while separation comparable to a wavelength at mobile station is required [12]. However, using multiple diversity techniques together can optimize the performance of communication link. Polarization diversity (PD) is regarded as a spatial efficient and effective alternative to SD in Non-Line-of-Sight (NLoS) environment because the signals transmitted through two orthogonal polarizations are independent [13]. [PERSON] et al. [14] have investigated the performance of PD for a massive MIMO system in industrial environment to address the problem of cross-polarization discrimination. [PERSON] et al. [15] have reported significant improvement in spectral efficiency in PD based massive MIMO system for Industry 4.0 applications. There is a gap, however, in existing literature on exploring PD for M2M and industrial communication. PD should be further investigated to improve link reliability in industrial scenarios. Several channel models have been proposed for PD systems [16]. They can be classified as physical or analytical models. The physical models are mostly developed using the exact ray-tracing or the geometric approach. However, they do not offer flexibility to design polarization-based technologies for more general scenarios. Analytical models on the other hand allow for mathematical representation of dual-polarized systems. Most of the models belonging to this class are based on correlation that characterize the channel matrix statistically. Analytical models are more suitable for Rayleigh fading channels that correspond to NLoS scenarios. However, they do not account for actual propagation effects such as scattering and channel depolarization. The models that incorporate these effects are cumbersome and lack analytical tractability. To solve these problems, [PERSON] et al. [17] have proposed a quaternion-based model for dual-polarized channel. The model offers a way to differentiate the two orthogonal polarizations and generate fading channels with cross-polar scattering and channel depolarization.