Reliable Low Power Wide Area Networks-Aided Polar Code

LPWAN has been highly developed in areas such as AI, machine learning, data analytics, and blockchain technologies. There is tremendous potential for exponential growth in regard to the deployment in several applications among different sectors of society, professions, and industries. The application of the IoT requires energy-efficient and low-complexity nodes for a variety of uses that are to be deployed on scalable networks.

As the IoT continues to advance, the need for a long-range, low-power connectivity standard has started to hinder its progress. While technologies like Bluetooth Low Energy and Zigbee have successfully met the demands of the short-range, low-power device market, the current ecosystem offers limited solutions designed for devices requiring extended range alongside strict low-power operation. Recent advancements in communication technologies have highlighted the pivotal role of Long Range Wide Area Network (LoRaWAN) in bridging the energy harvesting gap by providing comprehensive long-range communication solutions. LoRaWAN is recognized for its low power consumption, extensive range, cost-effectiveness, security, and scalability, making it an ideal connectivity solution for both public operators and private networks, particularly in the context of IoT applications. Prior studies have underscored its effectiveness in various IoT scenarios, yet it is important to note that LoRaWAN's bandwidth limitations stem from challenges in its coding techniques. Specifically, its architecture employs spread spectrum modulation based on the LoRa physical layer, which, while beneficial for reducing power consumption and extending range, imposes constraints on bandwidth. Nonetheless, research indicates that devices leveraging LoRaWAN can operate on battery power for up to one year, maintaining robust performance even in indoor environments. The security used in LoRa WAN is based on the 128-bit Advanced Encryption Standard (128- AES) encryption. Nevertheless, it enables free location GPS applications related to its spread spectrum and timestamp.

LPWAN, which was introduced as a solution to the predicaments of limited range and high energy expenditure endemic to conventional IoT technologies, strives to augment the existing IoT framework. It proposes a unique proposition in the IoT landscape, ensuring wide-area connectivity without the burden of excessive power consumption or prohibitive costs [2]. Contrary to the traditional short-range, non-cellular IoT platforms such as Zigbee, Bluetooth, and Wi-Fi, LPWAN infrastructure is capable of delivering coverage that is superior in scope through a single gateway.

LPWAN protocol in the IoT environment has been unique for its ultra-wide connectivity for link capacity enhancement by +20dB [29]. The enhancement is capable of reaching a range of tens kilometers [15]. As stated before, LPWAN advantages come in their low-cost connectivity and large area coverage [30].

Moving forward with the system architecture and protocols are developed by LoRa Alliance [24]. LoRa Alliance is an open, nonprofit association that focus on spanning a wide range of players from chip makers to cloud provider such as Amazon. LoRa Alliance is considered one of the fast-ever-growing ecosystems. The LoRa Alliance facilitates a certification program and maintain interoperability specification in public.

The foundational architecture of LoRaWAN begins with its link-layer specification. Positioned above the LoRa physical layer and below the application layer, the link layer functions as the intermediary between end devices and the network. Acting as an over-the-air transport layer, it ensures that end devices can both transmit and receive application-layer payloads to and from the network. This layer includes key physical layer parameters, such as channel frequency, data rate, and transmission power.

Before delving deeper into the specifics of the link layer, it is important to highlight one of LoRaWAN’s primary objectives: creating low-cost devices capable of operating for 5–10 years without requiring maintenance. This is achieved through cost-effective hardware design and maintaining minimal transmission power.

LoRaWAN networks are typically designed in a star-of-stars topology, where gateways serve as intermediaries, relaying packets between end devices and a central network server. Unlike conventional systems, this design does not rely on fixed links between devices and gateways, enabling uplink transmissions to be received by multiple gateways simultaneously.

Communication between end devices and gateways utilizes frequency shift keying (FSK) and LoRa modulation, which disperses transmissions across multiple frequency channels and data rates. In LoRaWAN, data rates range from 0.3 to 50 kbps, striking a balance between communication range and packet transmission time. Lower data rates enhance communication range but require more airtime, whereas higher data rates decrease airtime usage at the cost of increased power consumption, leading to reduced battery life.

Devices operating at varying data rates, however, do not interfere with each other. To enhance both battery performance and overall network efficiency, the LoRaWAN infrastructure dynamically manages the data rate and output power for each device through an adaptive data rate (ADR) mechanism.

The adaptive data rate (ADR) mechanism primarily relies on the distribution of gateways within the network. ADR allows gateways to capture uplinks from any device, enhancing the network’s connectivity level and increasing receive redundancy. Recent studies, however, have explored the use of higher-order modulation techniques [31]. In essence, devices can transmit over any available channel at any time using any permitted data rate, provided the following conditions are met:

Devices switch channels in a pseudo-random manner for each transmission, improving robustness against interference.

Each device adheres to the maximum transmit duty cycle specific to its sub-band.

Devices comply with the maximum transmission duration allowed for their sub-band.

LoRaWAN devices typically follow an ALOHA-like communication protocol and may function in one of three operating classes.

Class A Devices: These devices are bidirectional and operate using an ALOHA-type protocol, where transmission slots are allocated based on individual communication demands, with minor random adjustments. After each uplink transmission, two brief downlink receive windows are available. This system operates with low power, making it suitable for applications that need downlink communication from the Network Server (NS) immediately following an uplink transmission. For other downlink communications, the next scheduled uplink must be awaited.

Class B Devices: Also bidirectional, these devices provide additional receive slots compared to Class A. Alongside the random receive windows, Class B devices feature scheduled receive slots enabled by time-synchronized beacons sent from a gateway.

Class C Devices: These bidirectional devices maximize receive opportunities by keeping receive windows nearly always open, only closing during transmissions. Although they consume more power than Class A or B devices, they significantly reduce latency for communication from the server to the device. Devices can alternate between Class A and Class B operation modes as required. Each one of these classes has advantages among battery lifetime and downlink network communication latency. For instance, class A is the most energy efficient and is supported by all devices; however, downlink is only available after sensor transmission. Concerning class B, the battery powered actuators are also energy efficient, and latency-controlled downlink and slotted communication is synchronized with end devices. Lastly, Class C are also powered actuators with devices that can listen continuously; however, there is no latency for downlink communication.

Compared to cellular networks that have their communication coding scheme, like turbo codes used in the third generation, LPWAN has lower cost and less complexity [32, 33]. We can find LPWAN has its own coding scheme but with a low data rate due to the coding technique. We aim to enhance LPWAN by using other coding schemes such as Polar Codes that can improve the data rate compared to a cellular network, by using high order coding. The majority of the LPWAN technologies use different bands from 1 sub GHz [34] up to 2.4 sub GHz and is predominantly used by 2.4 sub GHz for less attenuation and multipath fading [14]. The majority used IoT technologies for environment evaluation study are LoRaWAN either indoor [30] or outdoor [35].

Polar Code has an explicit construction. The benefit of using Polar Code is that it is suitable for the control channel. The performance is poor to Low-Density Parity Check (LDPC) and turbo coding techniques [36]. Polar Code can achieve near high- performance capacity. To achieve a valuable performance, the codes attain to achieve the capacity by using a binary-input memoryless and symmetric channels. The use of the inputs must follow with efficient encoding and decoding algorithms. Other research adapted error control coding approaches for wireless communication systems in Polar Codes [37]. The construction relies on the recursive model and application type. The construction can combine successive-cancellation decoding and the binary-input symmetric channel like other coding techniques such as Convolutional Coding and LDPC. The goal of the Polar Code is to control the errors in data transmission within an unreliable or noisy channel. This technique works by redundantly encoding the transmitted message. This mechanism allows the receiver to detect the error bits and remove it from the original message. The order to perform Polar Coding is to have any information that can be formalized by the length of information sequence N that shall be transmitted as bits. The bits will be submitted in two forms, $u 1$ and $u 2$, for the structure encoder when N = 4. To obtain the output of our codeword, it shall be denoted as X where it is simply expressed mathematically as $u 1 \oplus u 2 \oplus u 3 \oplus u 4$ and can be further represented as: $X^{\prime}=u^{\prime} G$, where G is the generator matrix for N = 4, which we explained earlier. Polar code has its recursive structure as these are two techniques introduces for the encoder structures, which will be studied further in the future for performance enhancement by using these techniques. To achieve the goal of implementing Polar Coding we shall have a generator matric for the encoder part. In the construction of encoding, we shall have one matrix for $N=4$ and the bits are in $u_1 u_2 u_3 u_4$ where the data bit vector is explained in $u_1^N \ldots u_i^N$ and for a codeword to achieve the polar coding, we must have it as follow $X_1^N=u_1^N G_n$ [28]. Where the generator matric can be obtained by $G_n=B_n F \oplus$ $N G_n=B_n F^{\wedge} X O R n$ where is the $B_n$ can be calculated as $B_n=$ $R_n\left(I_2 \oplus B_n\right)$ where the initial $B_n$ denotes the $N \times N$ reverse shuffle permutation matrix defined. Other authors have a different opinion in working with metrics permutation, which will necessitate further investigation and study. Polar Codes was recognized for finding the probability of bit error in split memoryless channels, as opposed to using algebraic methods. Arikan proposed using Bhattacharyya Parameter in finding the error decision in a certain bit called the determination of a frozen bit location in the Polar Code. Bhattacharyya Parameter in Polar Codes aims to find the erasure probability channel. The Bhattacharyya parameter can also be computed recursively, as demonstrated in the study by Gazi [27]:

$Z\left(W_N^{N K-1}\right)=2 Z\left(W_{\frac{N}{2}}^k\right)-\left[Z\left(W_{\frac{N}{2}}^k\right)\right]^2$                (1)

where the initial value to preform recursion to be

$Z\left(W_1^1\right)=\alpha$                (2)

whereas, $\alpha$ is the erasure probability of the channel and calculated for the Bhattacharyya parameters:

$Z\left(W_N^k\right)=1-Z\left(W_N^k\right)$                 (3)

The goal of calculating the parameter is to get the maximum value of the probability of channel transmission error. Assume N=4 where N is the transmitted channel numbers of the bits transmitted and $\alpha=0.5$ by applying the equation our erasure probability will be:

$Z\left(W_1^1\right)=0.5$              (4)

For our assumption N=4 will follow the results of the applied as we mentioned in the Eq. (1) for the evaluation at $k=1$, and $k=2$ where K is the iteration for calculating the probabilities of the transmitted bits as follow:

$\begin{gathered} \text{For} \,k=1\\Z\left(W_4^1\right)=2 Z\left(W_2^1\right)-\left[Z\left(W_2^1\right)\right]^2=2 \times 0.75- \\{[0.75]^2=0.9375}\end{gathered}$              (5)

$Z\left(W_4^2\right)=\left[Z\left(W_2^1\right)\right]^2=[0.75]^2=0.5625$              (6)

$\begin{gathered}k=2 \\ Z\left(W_4^3\right)=2 Z\left(W_2^2\right)-\left[Z\left(W_2^2\right)\right]^2 \\ =2 \times 0.25-[0.25]^2=0.4375\end{gathered}$               (7)

$Z\left(W_4^2\right)=\left[Z\left(W_2^2\right)\right]^2=[0.25]^2=0.0625$                (8)

The resultant:

$Z\left(W_4^i\right)=\left[\begin{array}{llll}0.9375 & 0.5625 & 0.4375 & 0.0625\end{array}\right]$

which implies that the least probability and largest error in $Z\left(W_4^4\right)$. For our assumption, the code rate is 0.5 this requires us to froze 2 bits to utilize the transmission.

Another approach compared to the Bhattacharyya parameter is Reed-Muller codes, which are like polar codes in the sense of generator matrices. Both codes are constructed by selecting rows from a matrix. The selection minimizes the error probability under successive codes. Both codes achieve the capacity of BEC using SC. The goal of using Reed-Muller Codes is to improve error correction performance [38].

The construction of Polar codes using RM can be described in polar using a length of $\mathrm{N}=2^n$ with $k$ and $\mathrm{R}=\frac{k}{N}$ following the generator matrix multiplication of $\mathrm{x}=u G^{* n}$ Both approaches can be used in Polar Codes and effectively Arikan proposed using Bhattacharyya Parameter for the ease of calculation.

Polar Codes are used based on the polarization of the channel [36]. The concept of the Polar Codes is to ascertain the set of unreliable channels that will cause an increase in the bit error rate. These sets are called the frozen bits [39]. A variant of successive cancellation decoder is utilized in Polar Code base on a folding channel [40]. Defining multiple folding operations or using the same SCD will result in the same performance of the multiple successive cancellations [41]. For better performance compared to other techniques, systematic Polar Codes were proposed by Arikan to improve the bit error rate (BER) [42]. For efficient encoders in the implementation of Polar Codes, systematic polar codes are used with (N log N) for the complexity of the implementation [43].

In section 1.2.3, we elaborated on determining of the frozen bit method; however, this technique will be utilized for encoding process for an example of both encoding and decoding process, using N = 8 Encoder happens in M where it is the encoded bits and happened when we multiply our bits with matrix generator.

$x^{\prime}=u^{\prime} \times G$

which is the information sequence and the arrangement of the frozen bit is K = 4 and the information bits are [1001] to get encoded bits M. The code rate used here is 1/2 there for polar encoder structure is depicted as shown in Figure 2.

$\begin{gathered}x 1=u 1 \oplus u 2 \oplus u 3 \oplus u 4 \oplus u 5 \oplus u 6 \oplus u 7 \oplus u 8 \\ x 2=u 5 \oplus u 6 \oplus u 7 \oplus u 8 \\ x 3=u 3 \oplus u 4 \oplus u 7 \oplus u 8 \\ x 4=\oplus u 7 \oplus u 8 \\ x 5=u 2 \oplus u 4 \oplus u 6 \oplus u 8 \\ x 6=u 6 \oplus u 8 \\ x 7=\oplus u 4 \oplus u 8 \\ x 8=\oplus u 8\end{gathered}$

The output for M = 8 which are the encoded bits [00001111]. Mainly these encoded bits with the air transmission noise can be affected during transmission; however, we will assume the channel has no noise in it and we will perform decoding using successive cancelation decoding technique.

We know our frozen bits; therefore, we need to calculate the LR of non-frozen bits. This can be done by using formula to determine xx by finding the Likely LR LLR’s. To perform decoding let’s summaries the following like the value of $\alpha=0.5$ and $N=8$ from our example so we determine and use the sorting vector.

$\pi 8=\left[\begin{array}{lllllll}8 & 7 & 6 & 5 & 4 & 3 & 2\end{array}\right]$

As we explained in chapter of determination the frozen bits, we know location of the frozen bits as it’s can be used as parity bits and error correction

$d=\left[\begin{array}{llllllll}0^* & 0^* & 0^* & 1 & 0^* & 1 & 1 & 1\end{array}\right]$

where, 0* are the frozen bits. Data vectors are none zeros.

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Figure 2. Eight bit polar code encoder

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Figure 3. Eight bit polar code decoder

$d=\left[\begin{array}{lllllllll}0^* & 0^* & 0^* & 1 & 0^* & 1 & 1 & 1\end{array}\right]$

Therefore, the reverse of encoding shall be the same for decoding, illustrated in Figure 3.

 $u^{\prime}=d^{\prime} G=\left[\begin{array}{lllllll}0 & 0 & 0 & 1 & 0 & 1 & 1&1\end{array}\right]$

Assume the received vector for an Additive White Gaussian Noise (AWGN)AWGN is the following

 $y^{\prime}=\left[\begin{array}{lllll}0 & 1 & 1-1-1 & 0 & 0-1\end{array}\right]$

We shall decode it by using the decoding tree. To decode, the first step is to calculate the l LR. So, for the first bits, let’s decode u4

$\begin{aligned} & L R\left(x_1 \mid y_1\right)=10 \quad L R\left(x_2 \mid y_2\right)=0\quad L R\left(x_3 \mid y_3\right)=0 \quad L R\left(x_4 \mid y_4\right)=1 \\ & L R\left(x_5 \mid y_5\right)=1 \quad L R\left(x_6 \mid y_6\right)=0\quad L R\left(x_7 \mid y_7\right)=10\quad L R\left(x_8 \mid y_8\right)=1\end{aligned}$

which can be written as

$L_3=\left[\begin{array}{llllllll}10 & 0 & 0 & 1 & 1 & 10 & 10 & 1\end{array}\right]$

As we know, the first 3 bits are frozen bits

$\left(u^{\prime}\right)=\left[u_1=u_2=u_3=0\right]$

We can distribute the first decoded to the tree, then we can decode the data in u4, and the likelihood in level 2 can be calculated as follows which can be written as:

$\begin{aligned} & L R\left(g_1\right)=\frac{1+L R\left(x_1\right) L R\left(x_2\right)}{L R\left(x_1\right)+L R\left(x_2\right)} \quad L R\left(g_2\right)=\frac{1+L R\left(x_3\right) L R\left(x_4\right)}{L R\left(x_3\right)+L R\left(x_4\right)} \\ & L R\left(g_3\right)=\frac{1+L R\left(x_5\right) L R\left(x_6\right)}{L R\left(x_5\right)+L R\left(x_6\right)} \quad L R\left(g_4\right)=\frac{1+L R\left(x_7\right) L R\left(x_8\right)}{L R\left(x_7\right)+L R\left(x_8\right)}\end{aligned}$

Moreover, likelihoods for level one will be calculated as follows:

$L R\left(g_1\right)=0.1\quad L R\left(g_2\right)=1 \quad L R\left(g_3\right)=1 \quad L R\left(g_4\right)=1$

Leading to the first value of likelihood of 1 and 2, we can write the vector as follows:

\begin{gathered} L_2=[0.1\quad1\quad1\quad1] \\ L R\left(h_1\right)=\left(L R\left(g_1\right)\right)^{1-2 \times 0} L R\left(g_2\right) \rightarrow L R\left(h_1\right)=L R\left(g_1\right) L R\left(g_2\right) \\ L R\left(g_1\right)=\left(L R\left(g_3\right)\right)^{1-2 \times 0} L R\left(g_4\right) \rightarrow L R\left(h_2\right)=L R\left(g_3\right) L R\left(g_4\right) \\ L R\left(h_1\right)=0.1 L R\left(h_2\right)=1 \\ L_2=[0.1\quad1] \end{gathered}

Finally, to the last stage of level zero, which can be calculated as follows:

$\left(u_4\right)=L R\left(h_1\right) L R\left(h_2\right)$

Leading to the value

$L R\left(u_4\right)=0.1$

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Figure 4. Eight bit polar code decoder resultant

Now, the last step is using the decoding logic operation as shown in Figure 4. We can decide that, if the first decoded information bit for our example u4 is 1 or 0, based on our last value with the frozen bits, the decoded bit can now be written as follows:

$\left(u_i\right)=\left\{\begin{array}{c}0 \text { if } L\left(u_i\right) \geq 1 \\ 1 \text { otherwise }\end{array}\right.$

The process will be continued on all data vectors of ui$~$which received from $y^{\prime}$.

The resultant output for the case with the arrangement of frozen bits is used to set our set of M = 8 encoded LLRs into the K=4 we recovered information bits [1001]. The LLRs obtained using the f function.

$\left(x^{\prime}\right)=f\left(x_a, x_b\right)$

and g functions are shown above each connection. The bits obtained using the partial sum computations of (4) and (5) are shown below each connection. The accompanying numbers in pare paper identify the step of the SC decoding process where the corresponding LLR or bit becomes available.

A major aspect of developing the Polar Coding rate in the non-orthogonal multiple access has been proved to increase the multiple-access achievable rate [44] compared to turbo code and LDPC. The Polar code showed an efficient construction in high-order modulation. This approach was estimated in the LM-rate while using AWGN in the demapping technique [28]. This can be considered a new area of research.

To the best of our knowledge, two research papers have shown the performance of using FEC in LPWAN. The first work has evaluated the latency performance using a rete-less polar code with a fixed rate in an LPWAN [45]. The work does not focus on enhancing error rate performance and data rate transmission. The other work used FSK-Turbo codes in LPWAN [46] to enhance the error rate performance. However, FSK-Turbo has high complexity. FSK Turbo uses Frequency Shift Keying, which helps in making high transmission by using two frequencies to transmit two bits but consumes more bandwidth. On the other hand, Polar Code uses BPSK, which uses phase-shifting carrier frequency, which is not affect the bandwidth as FSK [47].

The accompanying diagram shown in Figure 5 and the table detail the implementation process of our novel method, from the initial bit intended for transmission to the final reception of bits at the receiver's end.  As depicted, we append a Cyclic Redundancy Check (CRC) to detect any errors, and this is succeeded by the Polar Code encoder that encodes the bits meant for transmission. To minimize the potential for logical errors associated with the polar encoder, we employ Grey coding to represent the bits. Furthermore, prior to transmission, the bits undergo modulation via Chirp Spread Spectrum (CSS) technology, as elaborated in section 1.2.4, which finds application in Low-Power Wide-Area Network (LPWAN) technologies. To emulate the process of transmission through the air, an AWGN channel is utilized in our simulation. Upon receiving the bits, the demodulation is carried out using CSS, followed by the remapping and decoding of the bits with the Polar Code decoder. Finally, we strip away the CRC to retrieve the original transmitted bits.

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