Session B5: Receiver Design, Signal Processing, and Antennas

Novel Shortened Polar Codes for GNSS L1C Subframes 2 and 3
Nuwan J. G. Kankanamge, Nghi H. Tran, The University of Akron; Khanh Pham, Air Force Research Laboratory; Dan Shen and Genshe Chen, Intelligent Fusion Technology Inc.
Location: Beacon B

Modern Global Navigation Satellite Systems (GNSSs) are equipped with several forward-error control coding techniques, including Low-Density Parity-Check (LDPC) codes, convolutional codes, and Bose-Chaudhuri-Hocquenghem (BCH) codes. For example, L1C - the fourth civilian global positioning system (GPS) signal, designed to enable interoperability between GPS and international satellite navigation systems has Subframe 2 and Subframe 3 protected by rate 1/2 irregular LDPC codes with frame sizes of 1,200 coded bits and 548 coded bits, respectively. These LDPC coding schemes have been shown to provide superior error performance over other conventional coding counterparts. However, despite their widespread use, it is of great importance to explore new coding schemes that have the potential to bypass these LDPC codes and achieve ultra-reliable performances for high-precision GNSS.
Over the last decade, significant efforts and interest have been dedicated to polar codes, a class of capacity-achieving codes. With the aid of an outer cyclic redundancy check (CRC) code, CRC-aided polar codes using successive cancellation list (SCL) decoding have been shown to perform exceptionally well in the short and very-short block length regimes, e.g., a thousand or less bits. In fact, polar codes have been selected in various 5G New Radio (NR) deployment scenarios. Due to the nature of the Kronecker construction of polar codes, a conventional polar code must have a code length of a power of two. Thus, it is not readily available for use in practical applications, including GNSS. To overcome this limitation, rate matching schemes such as traditional puncturing or repetition can be applied to achieve an arbitrary code length. Another interesting rate-matching technique for polar codes is referred to as shortening. In shortening, shortened bits are chosen from the frozen set. Their values are known to the decoder and, therefore, beneficial to the decoding process. One of the ways to do that is to select shortened bits from the end of each frame, and it is considered in 5G NR. However, since later bits in each frame generally correspond to the most reliable positions, this shortening method, while being simple, might not fully utilize the benefits of channel polarization. Other shortening techniques have also been proposed in the literature. However, these approaches are either heuristic or excessively complex, with a lack of structured shortening patterns.
In this paper, we propose new CRC-aided shortened polar coding schemes that can be directly used in GNSS L1C Subframes 1 and 2 for superior error correction capability. We also make a direct comparison to the existing LDPC codes in terms of both error correction capabilities as well as computational complexity. Toward these goals, we exploit the detailed structures of a mother polar code to propose a novel shortening algorithm that systematically produces a shortening pattern to achieve a non-power-of-two code length. The method can thus be applied directly to GNSS Subframes 2 and 3. Because each of the columns of the generator matrix has a Hamming weight of the power of two, our idea is to first decompose the number of shortening positions into a sequence of integers representing its powers of two. Then, for a given reliable sequence, columns of the mother code are sequentially selected based on these integer values, starting from the greatest value, to produce the least reliable bit positions suitable for being shortened. Explicit shortening patterns are obtained for L1C Subframe 2 and Subframe 3. Specifically, for Subframe 2, starting from the mother polar code of length 2,048 bits, 848 shortening positions are generated from four columns of the mother code, resulting in 1,200 coded bits for transmission. In the case of Subframe 3, 476 shortening positions are created using six columns of the mother code of size 1024, leaving 548 remaining positions for Subframe 3’s coded bits.
Extensive simulations are finally conducted to demonstrate the competitiveness of the proposed polar coding schemes over the current LDPC codes used in GNSS L1C. Specifically, it is shown that the proposed scheme provides significant coding gains, ranging from 0.35 dB to 0.55 dB, at the frame-error-rate (FER) level of 10e-5 over GNSS LDPC codes in Subframes 2 and 3 at a comparable decoding complexity. These notable coding gains are achieved over both a static channel as well as a three-state fading channel, which accurately characterizes the significant dynamic fading effects observed in land mobile-satellite services (LMSS) expected in urban GNSS applications. The FER advantage of the proposed shortening scheme over the traditional shortening scheme adopted in 5G NR is also clearly demonstrated. With their large FER performance improvement, the proposed polar codes can therefore serve as an attractive alternative for the LDPC codes currently used in GNSS L1C protocols.