在此篇論文中，我們提出了一種使用極化碼之全新機制來處理正交分頻多工系統中的高尖峰平均功率比問題。本文所提出的方法利用了極化碼中的凍結位元來創建偏移量，而此偏移量將被添加至已編碼的碼字中，使得碼字在傳送時具有較低的尖峰平均功率比。此外，對於給定的維度和碼字長度 ，我們提出了一種演算法來協助尋找適當的凍結位元組合。在某些情況下，運用了所提出的方法來傳送的碼字將會是特定的格雷互補碼集合中的一個組成序列。而格雷互補碼集合具有一個相當好的特性，即其尖峰平均功率比有著理論上限，因此傳送的碼字之尖峰平均功率比也具有上限。因為所提出的方法使用的凍結位元的值是預先決定的，所以會比選擇性映射機制具有更低的計算複雜度。模擬結果顯示，本文提出的演算法並不影響極化碼解碼性能，且具有良好的降低尖峰平均功率比。

In this thesis, a novel scheme utilizing polar codes is presented to deal with the high peak-to-average power ratio (PAPR) problem in the orthogonal frequency division multiplexing (OFDM) system. The proposed method uses the frozen bits in polar codes to create the offset which is added to the encoded codeword such that the transmitted codeword can have lower PAPR. We also propose an algorithm to help find the appropriate combinations of the frozen bits for given dimension and codeword length. In some cases, the transmitted codeword by the proposed scheme is a constituent sequence of a certain Golay complementary set (GCS). Since the GCS has the well-known upper-bounded PAPR property, the transmitted codewords have bounded PAPRs as well. In comparison with the selected mapping (SLM) scheme, the proposed scheme has lower computational complexity since the values of the frozen bits are determined in advance. Simulation results show that the proposed algorithm offers good PAPR reduction without affecting the decoding performance of the polar code.

[1] E. Arikan, “Channel polarization: A method for constructing capacity-achieving codes for symmetric binary-input memoryless channels,” IEEE Trans. Inf. Theory, vol. 55, no. 7, pp. 3051–3073, Jun. 2009.
[2] K. Niu, K. Chen, J. Lin, and Q. T. Zhang, “Polar codes: Primary concepts and practical decoding algorithms,” IEEE Commun. Mag., vol. 52, no. 7, pp. 192–203, Jul. 2014.
[3] S. A. Hashemi, N. Doan, M. Mondelli, and W. J. Gross, “Decoding Reed-Muller and polar codes by successive factor graph permutations,” in 2018 IEEE Int. Symp. on Turbo Codes & Iterative Information Processing (ISTC), Dec. 2018, pp. 1–5.
[4] E. Arikan, “A survey of Reed-Muller codes from polar coding perspective,” in IEEE Inf. Theory Workshop, Cairo, Egypt, Jan. 2010, pp. 1–5.
[5] H. Vangala, E. Viterbo, and Y. Hong, “Permuted successive cancellation decoder for polar codes,” in Int. Symp. Inf. Theory and Applications (ISITA), Victoria, BC, Canada, Oct. 2014, pp. 438–442.
[6] A. Elkelesh, M. Ebada, S. Cammerer, and S. ten Brink, “Belief propagation decoding of polar codes on permuted factor graphs,” in Proc. IEEE Wireless Commun. Netw. Conf. (WCNC), Barcelona, Spain, Apr. 2018, pp. 1–6.
[7] N. Hussami, S. B. Korada, and R. Urbanke, “Performance of polar codes for channel and source coding,” in Proc. IEEE Int. Symp. on Information Theory, Jun. 2009, pp. 1488–1492. 63
[8] B. Li, H. Shen, and D. Tse, “A RM-polar codes,” arXiv:1407.5483, 2014, [Online]. Available: https://arxiv.org/ftp/arxiv/papers/1407/1407.5483.pdf.
[9] 3GPP TS 38.212, “5G; NR; multiplexing and channel coding,” V15.2.0 , Jul. 2018.
[10] E. Arikan, “A performance comparison of polar codes and Reed-Muller codes,” IEEE Commun. Lett., vol. 12, no. 6, pp. 447–449, Jun. 2008.
[11] R. Mori and T. Tanaka, “Performance of polar codes with the construction using density evolution,” IEEE Commun. Lett., vol. 13, no. 7, pp. 519–521, Jul. 2009.
[12] P. Trifonov, “Performance of polar codes with the construction using density evolution,”IEEE Trans. Commun., vol. 60, no. 11, pp. 3221–3227, Aug. 2012.
[13] I. Tal and A. Vardy, “List decoding of polar codes,” IEEE Trans. Inf. Theory, vol. 61, no. 5, pp. 2213–2226, May. 2015.
[14] A. Balatsoukas-Stimming, M. B. Parizi, and A. Burg, “LLR-based successive cancellation list decoding of polar codes,” IEEE Trans. Signal Process., vol. 63, no. 19, pp. 5165–5179, Oct. 2015.
[15] S. H. Han and J. H. Lee, “An overview of peak-to-average power ratio reduction techniques for multicarrier transmission,” IEEE Wireless Commun., vol. 12, no. 2, pp. 56–65, Apr. 2005.
[16] R. O’Neill and L. Lopes, “Envelope variations and spectral splatter in clipped multicarrier signals,” in Proc. 6th Int. Symp. Pers., Indoor Mobile Radio Commun., vol. 1, Sep. 1995, pp. 71–75.
[17] S. H. Muller and J. B. Huber, “OFDM with reduced peak-to-average power ratio by optimum combination of partial transmit sequences,” Electron. Lett., vol. 33, no. 5, pp. 368–369, Feb. 1997.
[18] R. W. Bauml, R. F. H. Fischer, and J. B. Huber, “Reducing the peak-to-average power ratio of multicarrier modulation by selected mapping,” Electron. Lett., vol. 32, no. 22, pp. 2056–2057, Oct. 1996.
[19] Y.-c. Tsai, S.-k. Deng, K.-c. Chen, and M.-c. Lin, “Turbo coded OFDM for reducing PAPR and error rates,” IEEE Trans. Wireless Commun., vol. 7, no. 1, pp. 84–89, Jan. 2008.
[20] M. Sabbaghian, Y. Kwak, B. Smida, and V. Tarokh, “Near shannon limit and low peak to average power ratio turbo block coded OFDM,” IEEE Trans. Commun., vol. 59, no. 8, pp. 2042–2045, Aug. 2011.
[21] J. A. Davis and J. Jedwab, “Peak-to-mean power control in OFDM, Golay complementary sequences, and Reed-Muller codes,” IEEE Trans. Inf. Theory, vol. 45, no. 7, pp. 2397–2417, Nov. 1999.
[22] K. G. Paterson, “Generalized Reed-Muller codes and power control in OFDM modulation,” IEEE Trans. Inf. Theory, vol. 46, no. 1, pp. 104–120, Jan. 2000.
[23] C.-Y. Chen, C.-H. Wang, and C.-C. Chao, “Complementary sets and Reed-Muller codes for peak-to-average power ratio reduction in OFDM,” in Proc. 16th Int. Symp. AAECC, LNCS 3857, Las Vegas, NV, Feb. 2006, pp. 317–327.
[24] C.-Y. Chen, C.-H. Wang, and C.-C. Chao, “Complete complementary codes and generalized Reed-Muller codes,” IEEE Commun. Lett., vol. 12, pp. 849–851, Nov. 2008.
[25] M. Mondelli, S. H. Hassani, and R. L. Urbanke, “From polar to Reed-Muller codes: A technique to improve the finite-length performance,” IEEE Trans. Commun., vol. 62, no. 9, pp. 3084–3091, Aug. 2014.
[26] H. Li and J. Yuan, “A practical construction method for polar codes in AWGN channels,” in TENCON Spring Conference, Sydney, NSW, Apr. 2013, pp. 223–226.
[27] H. Vangala, E. Viterbo, and Y. Hong, “A comparative study of polar code constructions for the AWGN channel,” arXiv:1501.02473, 2015, [Online]. Available: https://arxiv.org/pdf/1501.02473.pdf.
[28] M. J. E. Golay, “Complementary series,” IRE Trans. Inf. Theory, vol. IT-7, pp. 82–87, Apr. 1961.
[29] T. Matsumine and H. Ochiai, “A novel PAPR reduction scheme for polar-coded OFDM systems,” IEEE Commun. Lett., vol. 23, no. 12, pp. 2372–2375, Dec. 2019.
[30] S.-Y. Zhang and B. Shahrrava, “A SLM scheme for PAPR reduction in polar coded OFDM-IM systems without using side information,” IEEE Trans. Broadcast., vol. 67, no. 2, pp. 463–472, Jun. 2021.
[31] S.-C. Lim, N. Kim, and H. Park, “Polar coding-based selective mapping for PAPR reduction without redundant information transmission,” IEEE Commun. Lett., vol. 24, no. 8, pp. 1621–1625, Aug. 2020.