3.1 Modulation and demodulation

The regression in error performance can be effectively mitigated by implementing a greater sampling rate at the receiver which allows for the receipt of signals across a broad spectrum of polarization angles. The increased variety of polarization angles offers a wider range of choices for selecting polarization throughout the process of detecting the message signal. The incorporation of higher order modulations involving message signal entails increased intricacy in the designs of both the transmitting and receiving systems. In comparison to PD systems, which require just one modulator, the design of a RPW transmitter is very complex which involves the utilization of two distinct and non identical Phase Shift Keying (PSK) modulators [16]. It is worth mentioning that RPW ensures signal integrity in varying propagation environments through precise control of polarization states. The previous literature on RPW has mostly focused on the investigation of RP-BPSK, RP-QPSK and RP-MPSK modulation schemes [13, 16]. RPW of a waveform is achieved by transmitting two baseband signals with a phase difference of π/2 through two orthogonal polarizations. This causes the polarization of the signal to rotate at a frequency of ω_r. On the receiver end the received RPW signal is captured by a separate DP antenna and is subjected to sampling at a sampling frequency f_s which is an integral multiple of f_r.

The foundational principles of classical modulation theory have been widely acknowledged for Quadrature Phase Shift Keying (QPSK) which exhibit superior performance compared to BPSK in terms of spectral efficiency. QPSK enables higher data rates while simultaneously ensuring an equivalent BER performance to that of BPSK as supported by reference [16]. The recognition of this knowledge led to the investigation of Rotating Polarization Quadrature Phase Shift Keying (RP-QPSK) a modulation technique designed to use the benefits of QPSK in the domain of RPW communication. Nevertheless, the adoption of RP-QPSK presented a substantial obstacle including both QPSK symbols and two orthogonal polarized signals with a 90° phase difference on the carriers simultaneously have proved to be a significant challenge. This limitation arose due to the inherent constraints associated with using two identical carriers for RP-BPSK. To achieve effective implementation of RP-QPSK, two QPSK transmitters were required to run in synchronization on the identical baseband frequency facilitating the production of the separate RPW signals. The exploration of the complexities involved in the advancement of RP-QPSK emerged as a crucial area of focus aiming to integrate the advantages of QPSK’s spectral efficiency with the distinguishing attributes of RPW communication. Each sample acquired through this method on the two antennas exhibits polarization at a distinct angle and serves as a duplicate of the broadcasted symbol. The desired sample is selected by choosing the signal with higher signal power between the two polarizations for each sample. Subsequently, every chosen sample is demodulated to retrieve an exact copy of the broadcasted symbol. The Eq (2) explains the horizontal and vertical polarized signals for RP-QPSK. (2) Where ψ₁ and ψ₂ are the orthogonal basis functions defined by: and

It should be noted that the positions of even and odd bits have been interchanged in the signals intended for transmission via horizontal and vertical polarizations. RP-QPSK can now be represented in a more comprehensive manner in Eq (3).

(3)

Examining the performance of BER in RPW communication brings up the noteworthy observation that the use of RP-QPSK instead of RP- BPSK does not have a negative impact on BER performance. The primary factor to be considered is that achieving an improved data rate is feasible albeit with a trade-off in terms of BER and energy efficiency however, this trade-off can be overcome. To augment the data rate and energy efficiency of communication based on RPW an approach known as Rotating Polarization Multilevel Phase Shift Keying (RP-MPSK) modulation has been reported in [13] which provides higher data rates and energy efficiency. While RPW offers advantages in terms of reliable data transmission in challenging environments, its energy consumption characteristics require further investigation. The potential for lower power operation compared to existing LPWAN solutions like LoRa stems from RPW’s utilization of polarization diversity. This diversity allows the signal to potentially overcome noise and fading effects, enabling transmission at lower power levels for similar communication ranges. Additionally, research suggests that employing higher-order modulation schemes like MPSK within the RPW framework holds promise for achieving higher data rates. However, this needs to be balanced against potential increases in bit error rate. If the receiver can sample the higher-order modulated signal at a sufficiently high rate, it might be possible to transmit more data with comparable BER performance, effectively translating to lower energy consumption per transmitted bit [13]. The primary factor to be considered is that achieving an improved data rate is feasible albeit with a trade-off in terms of BER and energy efficiency however, this trade-off can be overcome. Hence, the integration of RP-MPSK modulation with enhanced receiver sampling rates presents a potential solution for establishing a reliable and effective communication system. This approach enables the utilization of greater data rates while ensuring the preservation of integrity of the transmitted information.

The generation of RPW requires a setup within the transmitter, as illustrated in Fig 1(a). The proposed configuration involves equipping the transmitter with two separate baseband phase shift keying modulators each functioning at a frequency of ω_r = 2π = f_r = 2π/T_r. The modulators play a critical role in maintaining orthogonality between them, with one dedicated to horizontal polarization and the other to vertical polarization. The primary element of this procedure entails the concurrent entry of the data symbol s_m into both modulators. The process of simultaneous modulation yields two separate signals namely h_t(t) and v_t(t), where the subscript “t” denotes their association with the transmitted signal. The process involves the establishment of a fundamental framework for the encoding of information contained within the data symbol s_m onto orthogonal polarizations namely horizontal h_t(t)and vertical v_t(t). RPW communication employs polarized signals transmission leveraging spatial diversity from orthogonal polarizations, thus enhancing data transfer scope and communication system reliability with its utilization for forming the core of RPW’s distinctive features and capabilities [13]. The RP-MPSK transmitter is similar to the RP-QPSK transmitter however there are differences in the serial to parallel logic when using a higher level of modulation. Fig 1(b) is a block diagram illustrating the digital implementation of the RP-MPSK receiver. The received signals on the two elements of a dual polarized antenna experience significant degradation due to scattering multipath propagation. Both regular and irregular reflections are given in Eqs (4) and (5). (4)

Received base-band signals are sampled at frequency . Each sample is then coherently demodulated using Eq (5): (5)

In view of forgoing detailed exposition of RPW this research study employs altered RPW to describe polarization states of electromagnetic waves using quaternion mathematics involving the use of quaternion algebra to represent rotations in three-dimensional space. Quaternion symbols for dual polarized system are represented Q_m = s_H + s_Vj. Rotation of Q_m in polarization domain by an angle of φ₁ can be represented as Q_m,l = Q_me^−jφl. For Q-RPW transmission Q_m is rotated by N_l polarization angles to generate Q_m,l such that are obtained. From an implementation perspective this is equivalent to interpolating the transmitted symbol stream Q_m by factor of N_l.

3.2 Encoding and decoding of Q-RPW

In the constantly evolving world of modern communication systems the integration of emerging technologies like the IoT and LPWAN with novel signal modulation techniques particularly RPW offers a promising opportunity to enhance the effectiveness and dependability of information transmission. This study undertakes a thorough investigation of the encoding and decoding mechanisms of RPW specifically within the framework of LPWAN and IoT applications. This study probes into the synergies between the distinctive qualities of RPW the energy efficient capabilities of LPWAN, and the different requirements of IoT devices with a specific emphasis on the integration of Hamming codes as a robust error correction mechanism. Additionally, for applications primarily related with single-bit error correction and where codeword size is a limiting factor, Hamming codes offer superior efficiency. They achieve this correction with less redundancy compared to Reed-Solomon (RS) codes which are designed to handle multiple errors within a codeword [4]. The robust error correction capabilities of RS codes involve complexity to offer advantages in channels with significant noise or for applications transmitting large data packets where correcting multiple errors within a codeword is crucial. While RS codes offer potent error correction, Hamming codes present a compelling alternative in specific scenarios due to their inherent advantages. The well-defined decoding structure of Hamming codes, exemplified by syndrome decoding, translates into lower implementation complexity and low memory footprint compared to RS codes that rely on algorithms like Berlekamp-Massey [17]. This is particularly beneficial in resource-constrained environments like wireless sensor networks, where low-complexity decoding is essential [18, 19]. Besides, the simpler decoding process of Hamming codes results in lower latency, a crucial factor in real-time applications.

Since society increasingly relies on networked devices for various tasks, understanding the complexities of encoding and decoding in the RPW system is of utmost importance. This understanding is crucial for enhancing communication reliability and security inside LPWAN enabled IoT ecosystems. The utilization of Hamming codes in the encoding and decoding procedures within communication systems has a significant benefit in relation to power efficiency when compared to alternative error correction codes. Hamming codes are a type of block codes that exhibit the unique capability of detecting and correcting single bit errors [20]. This property renders them highly suitable for scenarios where power conservation is of vital importance.

In the same vein, the simple structure of Hamming codes reduces hardware complexity, this potentially lowering the power consumption [21]. The strategic use of Hamming codes in low-noise environments can achieve reliable data transmission with less power than more complex methods [22]. The study in [23] positioned Hamming codes as a potential standard due to their balance between power consumption and error correction capabilities. Conclusively, the combination of simple hardware implementation, strategic application in low-noise scenarios, and potential as a low-power design baseline makes Hamming codes an asset in achieving energy efficiency. The utilization of Hamming codes in IoT devices and LPWAN environments is appealing due to their capacity to offer fault resistance without significantly impacting computational resources, which is particularly advantageous considering the inherent limitations on power consumption [24]. The efficient application of power in Hamming codes corresponds to the larger objectives of sustainability and energy efficiency in contemporary communication systems. Hamming codes play an important role in reducing power consumption by decreasing the computational cost related to error correction. This contributes to the durability of battery powered devices and facilitates the seamless integration of IoT inside LPWAN architectures [25, 26]. The objective of this study is to examine the intricacies of power efficient encoding and decoding techniques by specifically focusing on Hamming codes within the domains of LPWAN and IoT, thereby contributing to the existing research on communication technologies that prioritize energy conservation. The Hamming code is widely employed as an ECC renowned for its ability to detect and correct single bit errors. The codeword of the Hamming code with parameters (n, k) is composed of n bits divided into two parts. The k bits whose indices are not powers of two bits represent the message bits while the remaining n—k bits are referred to as parity check bits. These additional bits serve the purpose of detecting and correcting single bit errors. The Hamming (7, 4) code which is a widely used Hamming code system for a 7-bit codeword of four message bits and three parity-check bits is applied in this research study for encoding and decoding of RP-MPSK and RP-QPSK. In the following illustration, a demonstration of the encoding process of the Hamming (7, 4) code is presented wherein four message bits are transformed into a single 7-bit codeword. As an example: one block of codeword made from 100,000 symbols with parity bits is shown in Table 1, where P1, P2, and P4 are the parity bits and D3, D5, D6, and D7 are the data bits.

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Table 1. Example of one block of codeword.

https://doi.org/10.1371/journal.pone.0304386.t001

Additionally, the code possesses the capability to rectify a single bit error within the codeword. To perform its operations, the Hamming (7, 4) code requires precomputation of two matrices: the code generator matrix, denoted as G, and the parity check matrix, denoted as H. An illustration of the two matrices adopted for this research study is presented as follows:

The mathematical expression to encapsulate the encoding procedure for Hamming (7,4) codes is as follows: CD = mod((data)*G′, 2). In this expression, “data” represents the original data block that is to be communicated, and G denotes the generator matrix of the Hamming (7,4) code. Within this particular scenario, the generating matrix G plays a pivotal role in enabling the conversion of the initial 4-bit data block into a 7-bit encoded block. Matrix multiplication involves the computation of the dot product between a data block and the transpose of a generating matrix G’. The codeword (CD), which is a 7-bit vector, is designed to include strategically placed redundant parity bits to enable the identification and rectification of errors. The modulo-2 operation is thereafter employed to ensure that the encoded block remains confined within the binary field. The encoding procedure has a twofold function: (1) to improve the resilience of the transmitted data against errors, and (2) to facilitate the subsequent decoding process in detecting and correcting possible inconsistencies. The Hamming (7,4) code, characterized by its systematic architecture, serves as a notable illustration of a proficient method for error detection and correction. Consequently, it offers significant benefits in scenarios where ensuring the integrity of data is of great importance. In order to retrieve the message from Hamming (7,4) code, syndrome decoding employed which is applicable to a wide range of Hamming code parameters. This eliminates the need for code specific decoding logic by being, particularly advantageous for variable parameter or dynamic code selection. Additionally, syndrome decoding possesses the potential to detect not only single-bit errors but also double errors in some code configurations a crucial feature for noisy channels susceptible to multiple errors. The systematic mathematical structure of syndrome decoding facilitates analysis, debugging, and potential optimization compared to custom logic tailored for specific codes. However, for fixed-parameter codes like (7,4), custom logic utilizing bit wise operations can achieve superior efficiency in terms of memory and computation compared to syndrome decoding, which might involve less efficient matrix operations. Custom logic can also offer greater simplicity and ease of implementation, especially for less complex codes. Ultimately, the selection between syndrome decoding and custom logic hinges on the specific application’s requirements. If flexibility and robust error detection are paramount, syndrome decoding remains the preferred choice which is why it has been considered in this research study. Conversely, for applications with fixed codes prioritizing efficiency and simplicity, custom logic can be a more suitable option [17].

The process of decoding messages encoded with the Hamming (7,4) code involves a meticulous series of steps aimed at identifying and rectifying errors introduced during transmission. The process involves dividing the encoded message into segments consisting of 7 bits each. For every segment, a syndrome is determined by performing matrix multiplication with the predetermined parity-check matrix H. The matrix shown above has been specifically designed for the Hamming (7,4) code, with the purpose of efficiently detecting errors occurring within the block. The derived syndrome is further examined to identify the precise location of any existing errors. If an error is identified, the algorithm rectifies it through the process of toggling the bit that corresponds to the defect. Afterwards, the relevant components (except data bits) are removed from the block and added to the developing decoded message. The iterative nature of this procedure guarantees a systematic approach to address errors throughout the whole encoded message. As a result, a decoded message is obtained to accurately reflect the original data, while also minimizing the effects of potential transmission errors.

The study conducted in [27] provides a detailed investigation into the frame error rate (FER) of Hamming Coded LoRa modulation when subjected to additive white AWGN and carrier frequency offset (CFO). The study used well established approximations to determine the BER of LoRa in the presence of AWGN as shown in Fig 2.

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Fig 2. LoRa PHY based Transmitter (Tx) and Receiver (Rx) [27].

https://doi.org/10.1371/journal.pone.0304386.g002

The analysis is expanded to incorporate the effects of channel coding, interleaving, and gray mapping in the LoRa physical layer. The authors also calculated the BER of LoRa under CFO and provided a related study of the FER. To verify the precision of its conclusions, the study utilized Monte Carlo simulations and contrasted the outcomes with the derived frame error rate expressions. The authors provided a concise summary of their main contributions which involved developing simplified methods for estimating the error rate of coded LoRa signals in the presence of AWGN. The second approximation showed a high degree of agreement, with a deviation of only 0.2 dB, when compared to Monte Carlo simulations conducted at different spreading factors and FERs as low as 10⁻⁵. The study further emphasized the resilience of LoRa to low residual CFO values, under scoring the beneficial effects of gray mapping, interleaving, and coding. The research finishes by presenting a simplified method for estimating the LoRa FER under CFO. This estimate demonstrates a minimal divergence of 0.5 dB or less compared to the results obtained from the equivalent Monte Carlo simulation. The research offers a thorough comprehension of the encoded FER performance of Hamming Coded LoRa in difficult communication conditions providing useful knowledge for future improvements in wireless communication and IoT applications.

Meanwhile [28] examines the utilization of Bit Interleaved Code Modulation (BICM) to improve the performance of LoRa communication. The paper specifically emphasized the error correction capabilities of BICM in LoRa communication. The utilization of Forward Error Correction (FEC) codes in the physical layer of LoRa, is predominantly based on the basic (7,4) Hamming code with coding rates ranging from 4/5 to 4/8. The study investigated the feasibility of utilizing BICM for LoRa signals, with a focus on its capacity to optimize the coding benefits which could be achieved using the current FEC codes in commercial LoRa chips. The inquiry entailed evaluating the BER performance in both AWGN and Rayleigh fading channels. In channels affected by AWGN using BICM with a coding rate (CR) of 1 demonstrated a 40% increase in data rates. This approach achieved a BER performance comparable to that of hard decision decoding with a coding rate of 3. BICM demonstrated significant advantages in Rayleigh fading channels resulting in performance enhancements ranging from 7.2dB to 8.0dB when compared to hard decision decoding with a coding rate of 3. Significantly, BICM also exhibited significant BER improvements of 25.5 decibels when compared to the BER performance without coding in Rayleigh channels. The schematic representation of Q-RPW system model shown in Fig 3, integrates Hamming encoding and decoding techniques consists of multiple interconnected modules that collaborate to achieve resilient transmission of information. The RP-MPSK model is designed to analyze the polarization properties of electromagnetic waves in a multipath environment at the receiving antenna. The model employs quaternion representation, in which complex symbols are communicated via horizontal and vertical polarized elements in dual polarized systems. Adaptations for RP-MPSK involve introducing a phase difference of π/2 between base-band modulation frequencies. Quaternion algebra is utilized for the purpose of rotations in the polarization domain, and the resultant quaternion stream is transformed to a higher frequency for radio frequency transmission. After being received, the RF signal undergoes down conversion. The quaternion channel, which is characterized by real random variables, represents the received signals from both horizontal and vertical polarized antennas. The quaternion model offers the advantage of reducing the required number of real random variables by half in comparison to the standard approach. Maximum Likelihood Detection (MLD) is used to retrieve the broadcast sequence, showcasing the effectiveness of the quaternion model for RPW communication and then received data is passed through decoder for decoding of errors and removal of parity bits. During the encoding phase, the original data is subjected to quaternion rotation to modulate the polarization wave efficiently. Afterward, the data undergoes Hamming encoding which introduces redundancy to facilitate error detection and correction. The encoded blocks, which have been enhanced with supplementary parity information are transmitted across the communication channel. After being received, the RP-MSPK system proceeds to the decoding phase during which Hamming decoding is employed to detect and correct errors in the received blocks. After the decoding process, quaternion inverse rotations are applied to get the original data. The utilization of this integrated methodology guarantees the preservation of polarized wave information, while simultaneously improving the dependability and consistency of the transmitted data through the implementation of Hamming encoding and decoding. As a result, this approach is highly suitable for applications that require resilient communication in quaternion-based polarization systems.

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Fig 3. Block diagram of Hamming coded Q-RPW.

https://doi.org/10.1371/journal.pone.0304386.g003

The use of Hamming codes in this research study for Q-RPW model is based on their ability to detect and correct singe bit errors in case of Hamming (7,4) whereas in case of Hamming (15,11) can detect up to 2 errors and correct single bit errors. Since the working principle of both the (7,4) and (15,11) is almost similar, we have used Hamming (7,4).

3.3 Development of Hamming coded RP-MPSK and RP-QPSK

For software development of Hamming Coded RP-MPSK communication systems, the theoretical developments are converted into software implementation. The software development carried out in MATLAB consists of many components that are specifically planned to enable smooth integration with the Q-RPW framework facilitating the encoding and decoding processes. The encoding module utilizes advanced techniques to coordinate quaternion rotations on the received data. This crucial stage corresponds to the distinct attributes of RP-MPSK in which the polarization data is manipulated by means of quaternion rotations guaranteeing effective information encoding. The Hamming (7,4) coding method is used in the process to intentionally introduce redundancy which aids in robust error detection and correction as shown in the block diagram in Fig 3. The process of encoding which consists of two layers serves the purpose of encapsulating both the theoretical principles outlined in this study and establishing a practical basis for the dependable conveyance of information. The decoding module is an essential part of the software architecture which complements the communication spectrum. Customized algorithms perform reverse quaternion rotations to extract the original data from the RPW modulated signals. The decoding process is enhanced by the implementation of Hamming decoding which utilizes the redundancy included during encoding to identify and rectify any faults that may have occurred during transmission effectively. The decoding process is carefully coordinated to reconstruct the original data properly assuring the fidelity of the communicated payload. The powerful encoding and decoding software framework enhance the capabilities of Q-RPW communication system to overcome the difficulties caused by noise, interference, and other impairments in consideration of real-world situations. In addition, the software development process includes the integration of different modulation schemes into the Q-RPW framework. The software architecture smoothly incorporates RP-8PSK, RP-16PSK, and RP-QPSK modulations. The ability of various modulation schemes to allow methodically assess the performance of different levels of modulation complexity in the RPW paradigm. The software modules are intricately patched together to provide seamless transition between modulation schemes, enabling thorough comparative analysis across different channel circumstances. Being an agile and dynamic communication paradigm Q-RPW can modify its modulation approach to suit the specific demands of various real-world circumstances.

The software development is specifically designed to effortlessly connect RPW transceivers and antennas as shown in Fig 1. This comprehensive method enables hardware in the loop simulations, enabling thorough testing and validation of the proposed communication system of Fig 3. The incorporation of encoding and decoding modules along with the integration of flexible modulation schemes establishes the foundation for a resilient and versatile communication paradigm. This software designed specifically for Q-RPW communication systems allows for immediate experimentation to demonstrate the theoretical improvements discussed in this research study. Additionally, it positions RPW as a promising solution for the changing field of wireless communication wherein resilience, adaptability, and versatility in the face of various and difficult real-world situations are required. Table 2 lists the simulation parameters along with modulation type and channel type, etc. The use simulation parameters and integration of building blocks guaranteed the efficient translation of theoretical developments into software implementations.

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Table 2. Simulation parameters used for Q-RPW model simulations.

https://doi.org/10.1371/journal.pone.0304386.t002