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Polar Encoding Kernel

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Polar Encoding Kernel

Introduction

Polar coding is a technique used in digital communication systems to improve the reliability of the transmitted data. Polar codes are a type of channel coding scheme that uses a specific polar encoding kernel to transform the input data into a binary codeword that can be transmitted over a noisy channel. In this article, we will discuss the polar encoding kernel in detail and its technical aspects.

Overview of Polar Coding

Polar coding is a type of channel coding that uses polar transformations to convert a set of N input bits into a set of N encoded bits that can be transmitted over a noisy channel. Polar codes are based on the concept of channel polarization, which refers to the phenomenon of some channels becoming more reliable than others as the signal-to-noise ratio (SNR) increases.

In polar coding, the input data is first divided into blocks of equal size. Each block is then passed through a polar encoder, which applies a specific polar encoding kernel to transform the input data into a binary codeword. The codeword is then transmitted over the channel. The receiver receives the codeword, and the polar decoding process is applied to recover the original data from the received codeword.

Polar Encoding Kernel

The polar encoding kernel is a key component of the polar coding scheme. The polar encoding kernel is used to transform the input data into a binary codeword that can be transmitted over the channel. The polar encoding kernel is a binary matrix of size N x N, where N is the number of input bits.

The polar encoding kernel is constructed by applying a recursive algorithm called the Kronecker product to a basic transformation matrix. The basic transformation matrix is a 2x2 matrix that performs a simple transformation on two input bits. The Kronecker product recursively applies the basic transformation matrix to create a larger matrix of size N x N.

The polar encoding kernel can be represented as follows:

G_N = G_2⊗G_2⊗...⊗G_2

where G_2 is the basic transformation matrix, and the Kronecker product is applied N/2 times to create a matrix of size N x N.

The basic transformation matrix G_2 is defined as follows:

G_2 = [1 0; 1 1]

The matrix G_2 performs a simple transformation on two input bits. The first input bit is copied to the output, and the second input bit is XORed with the first input bit and then copied to the output. The matrix G_2 can be represented as follows:

y_1 = x_1 y_2 = x_1 ⊕ x_2

where x_1 and x_2 are the two input bits, and y_1 and y_2 are the two output bits.

The polar encoding kernel can be represented as a binary matrix of size N x N, where each row of the matrix represents a binary codeword that corresponds to an input block of size N. The polar encoding kernel matrix is defined as follows:

G_N = [g_1; g_2; ...; g_N]

where g_i is the i-th row of the matrix G_N, which represents the binary codeword that corresponds to the i-th input block of size N.

The polar encoding process can be summarized as follows:

  1. Divide the input data into blocks of size N.
  2. Apply the polar encoding kernel G_N to each input block to obtain a binary codeword.
  3. Transmit the binary codeword over the channel.

The input data is first divided into blocks of size N. Each block is then passed through the polar encoder, which applies the polar encoding kernel to transform the input data into a binary codeword. The binary codeword is then transmitted over the channel.

Advantages of Polar Coding

Polar coding has several advantages over other coding schemes, such as turbo coding and LDPC coding. Some of these advantages are:

  1. Low Complexity: Polar coding has a lower encoding and decoding complexity compared to other coding schemes. This is because polar coding uses a simple recursive algorithm to construct the polar encoding kernel, which requires less computational power and memory.
  2. High Error-Correction Capability: Polar coding has a high error-correction capability, especially at high SNR values. This is because polar coding exploits the phenomenon of channel polarization, which allows some channels to become more reliable than others as the SNR increases.
  3. Scalability: Polar coding is scalable, which means it can be used for different block sizes and channel conditions. This makes polar coding suitable for various applications, such as wireless communication systems, satellite communication systems, and optical communication systems.
  4. Low Latency: Polar coding has a low decoding latency, which makes it suitable for real-time applications such as video streaming and voice communication.

Conclusion

In conclusion, the polar encoding kernel is a crucial component of the polar coding scheme used in digital communication systems. The polar encoding kernel is a binary matrix of size N x N that transforms the input data into a binary codeword that can be transmitted over the channel. Polar coding has several advantages over other coding schemes, such as low complexity, high error-correction capability, scalability, and low latency, making it a popular choice for various communication systems. The polar encoding kernel is a powerful tool that can improve the reliability and efficiency of digital communication systems, especially in challenging channel conditions.