Polar Coding for Forward Error Correction in Space Communications with LDPC Comparisons Naveed Naimipour Haleh Safavi Harry Shaw NASA Goddard Space Flight Center NASA Goddard Space Flight Center NASA Goddard Space Flight Center Greenbelt, MD Greenbelt, MD Greenbelt, MD [email protected] [email protected] [email protected] Abstract—With the surging development of optical telecom- codes were seen as substantially more useful with their lower munications for space applications, the importance of error complexity and higher coding gains [3]. The discovery of correction has become more apparent than ever. Specifically, LDPC’s flexibility in terms of degrees of freedom led to even the exploration of forward error correction code (FEC) method- ologies will be instrumental in developing the standards for better performance after design inefficiencies were addressed optical communications in space. Despite the widespread use [3]. of low-density parity-check (LDPC) codes, alternate FEC codes The goal of LDPC coding is to implement a parity-check such as polar codes have shown immense promise in assisting matrix that is sparse and randomly generated. Introduced by space communications error correction with their ability to Gallagher in the 1960s, they were not fully explored until bypass the error floors that plague LDPC codes. Extremely promising techniques including cyclic redundancy checks (CRC), the 1990s when their lower complexity and ability to perform successive cancellation (SC), and successive cancellation lists near the theoretically achievable coding gain were seen as a (SCL) that assist polar coding in achieving the Shannon limit major advantage [4]. Along with their graphical representa- in a timely manner are evaluated. MATLAB simulations are tions, LDPC codes were developed to bypass the commonly conducted with AWGN and burst noise to test each technique’s used turbo codes for higher code rates [5] [6]. Additional ability to handle noise typically encountered in space and each technique’s ability to correct unexpected errors. Results of LDPC coding schemes, such as ones with Reed-Soloman outer simulations for different rates and message lengths are also coding, were developed for improved performance and better reported to determine each technique’s ability to handle large compatibility with lower code rates [7]. data volumes and fix errors. Similar simulations are conducted Recently, polar codes entered the fray with improved perfor- for LDPC codes with additional tests for convolutional and no mance and lower complexity stemming from its block structure interleavers. Finally, a discussion regarding the future ability of polar codes to satisfy current missions in the place of, or in design and recursive nature. Described by Arikan in his 2009 conjunction with, LDPC codes along with the merits of each work, polar codes garnered interest when it was proven they FEC technique’s ability to process data efficiently and handle had the ability to asymptotically achieve the Shannon capacity data while maintaining adequate performance will be provided. on many channels [8]. Their highly regular structure makes Preliminary recommendations will be made for each technique’s them particularly useful for real world applications and 3GPPs effectiveness for GEO related missions along with discussions regarding each technique’s ability to fit within the CCSDS preliminary adoption of polar codes for eMBB in 5G further standards for optical communications. exemplifies this. Polar coding’s ability to bypass the “error Index Terms—FEC, LDPC codes, polar codes, space commu- floors” that plague LDPC coding while also maintaining low nications, optical communications complexity and high performance make it extremely unique. In this paper, we report on our research into the viability I. INTRODUCTION of polar codes for optical communications and compare them The requirement for error free space communications has with their conventional LDPC counterparts. We have evaluated existed since the first NASA mission. Specifically, as de- extremely promising techniques including cyclic redundancy creased signal power became a major challenge in error checks (CRC), successive cancellation (SC), and successive free communications, FEC codes assisted in catching and cancellation lists (SCL) that assist polar coding in achieving correcting the resulting errors. In the earlier days of space the Shannon limit. These methodologies have the unique missions, Reed-Muller codes and convolutional codes were potential to overtake current LDPC standards as shown by the implemented as they were common for that time-period. reported results of MATLAB simulations to test the ability Convolutional codes carried onto future missions and even of each polar coding methodology to handle a variety of included Viterbi decoders for missions such as Voyager [1]. variables. In addition, simulations conducted with AWGN and Furthermore, the addition of Reed-Soloman codes resulted in a burst noise to test each technique’s ability to handle different concatenated convolutional and Reed-Soloman coding scheme types of noise are reported. Further simulations are run via that was necessary for multiple deep-space missions requiring MATLAB and C for LDPC codes with additional tests for more powerful codes [2]. Variations of such concatenated convolutional and no interleavers. codes become common place for many missions until turbo Finally, there is a brief discussion regarding the future ability of polar codes to satisfy current missions by itself or 2) Successive Cancellation (SC): SC codes eliminate in a hybrid coding scheme with LDPC codes. The merits of redundancies by cutting the polar codes into smaller pieces each FEC technique’s ability to process data efficiently and for processing. Again, based on [8], if we let the estimate of N N N handle data while maintaining adequate performance will also u1 be denoted as u^1 , then u^j can be found successively after N be provided. Preliminary recommendations will be made for y1 has been received using the following equation: each technique’s effectiveness for GEO related missions along ( N j−1 with discussions regarding each technique’s ability to fit within hj(y1 ; u^1 ) j 2 J u^j = (3) the CCSDS standards for optical communications. uj j 2 F where the bits are determined successively for j from 1 to II. CODING SCHEMES N and the decision function, hj is defined as A. Polar Coding 8 W (j)(yN ;u^j−1j0) <0 if N 1 1 ≥ 1 W (j)(yN ;u^j−1j1) As described in [8], the description of polar codes begins hj = N 1 1 (4) with letting W : X ! Y denote a binary-input discrete :1 otherwise memoryless channel with and representing the binary X Y It is further established that a decoder block error occurs if input alphabet and the output alphabet respectively. Then, the u^N 6= uN . channel transition probabilities are W (yjx); x 2 and y 2 . 1 1 X Y 3) Successive Cancellation List (SCL): Successive can- Specifically, polar codes utilize the polarization effect of the cellation lists (SCL) for polar codes were exhibited in [9] as a matrix G = 1 0 . A polar code with length N = 2n 1 1 1 means to improve the performance of SC decoding. Although will establish N virtual channels with the polarization effect SC performs extremely well and approaches the Shannon and the generator matrix, G , can be defined as the n-fold n capacity, it struggles with small and medium length codes in Kronecker product of G . In other words, the generator matrix 1 polar coding. SCLs operate by keeping a list of L survival code can be defined as: bits at each step instead of a single survival path implemented by SC. G = G ⊗n: (1) n 1 If we let u^i be a random bit, the decoder makes 2L candidates from the original L by keeping both paths with With this definition and a message length of K bits (re- u^i = 0 and u^i = 1. Moreover, L can be used as a threshold sulting in a code rate of R = K=N), the information bits (j) to determine when SCL discards the worst paths and, thus, are carried on the K most reliable polarized channels WN improves the performance of SC. with indices j 2 J. The remaining channels are then used to transmit the “frozen bits,” which is a fixed binary sequence B. Low-Density Parity-Check (LDPC) codes and can be also be represented as the frozen set F. It should Low-density parity-check (LDPC) codes implement a series be noted that F is the complement of the information set J. of parity check equations based on binary parity check block If we use a binary source block with K information bits, codes. Typically, the symbols in the code satisfy m parity then a code block can be mapped using N − K frozen bits check equations with block codes that consist of binary vectors using the following equation: of fixed length n. As a result, each codeword will contain (n− m) = k information digits and m check digits. Subsequently, N N a sparse parity matrix, H, is created whose dimensions are x = u · Gn; (2) 1 1 m × n. N N One of the more common ways to implement a LDPC where u1 is the binary input and x1 is the output code vector. code is to invert H to obtain the generator matrix G. Then, 1) Cyclic Redundancy Check (CRC): Cyclic error- matrix multiplication can be utilized to encode. Due to G typ- correcting (CRC) techniques typically add a fixed-length check ically being dense, the encoding complexity is thus quadratic value as their encoding scheme for messages. If there are k in codeword length. As a result, regular LDPC codes are information bits and m bit CRC sequence added, then there normally implemented with irregular codes being utilized will be K = k +m bits for a K bit input block.