# 臺灣博碩士論文加值系統

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 格雷序列在通訊系統已有許多應用，如同步及通道估測。其中使用最廣的為長度是二的冪次方的BPSK格雷序列。二的冪次方長度的格雷序列字集已由實數的BPSK字集擴大到H-PSK與QAM等複數字集。其建構主要基於1999年Davis, Jedwab提出的廣義布林函數相加建構法。這個方法是將H-PSK格雷序列的相位對應到mod-H整數序列，並表為特定序列相加合併的結果。Budisin, Spasojevic於2013年提出了乘積建構法，使用乘積的方式產生格雷序列的元素值。不同於以往廣義布林函數建構法只描述相位，乘積建構法同時描述大小及相位，可以生成更多複數格雷序列，但序列長度仍限於二的冪次方。本研究探討Budisin, Spasojevic提出的乘積建構法，觀察乘積建構法與相加建構法之間的關係，與乘積建構法對應之相位相加廣義布林函數表示式。由此表示式可得知非標準格雷序列無法由乘積建構法生成。本研究也提出以乘積建構法生成有唯一性無重複的所有標準格雷序列的方法，並說明Budisin, Spasojevic研究所需做的一些更正。
 Golay complementary sequences have many applications in communications systems, including synchronization and hannel estimation. The most widely used are power-of-two length BPSK Golay complementary sequences. The alphabet of power-of-two length Golay complementary sequences has been extended to complex alphabets such as H-PSK and QAM. The constructions are mainly based on the generalized Boolean function sum method proposed by Davis and Jedwab in 1999. In their method, mod-H sequences corresponding to the phase of H-PSK Golay complementary sequences were obtained by the addition of specific base sequences. In 2013, Budisin and Spasojevic proposed a product construction, where the elements of Golay complementary sequences are obtained from a product operation. Unlike the generalized Boolean function construction which only describes the phase, the product construction describes both the magnitude and the phase, and can generate more complex Golay sequences with the restriction that the sequence lengths still being powers of two. This research aims to study Budisin and Spasojevic’s product construction and observe its relation with the sum construction and the corresponding generalized Boolean function expression. From this expression, it is concluded that non-standard Golay complementary sequences cannot be generated from the product construction. A method to generate unique standard Golay complementary sequences from the product construction is proposed, together with some minor corrections for the product construction.
 中文摘要 iABSTRACT ii誌謝 iii目 錄 iv符號表 v第一章 序論 11.1 研究動機與目的 11.2 研究成果 2第二章 二的冪次方長度格雷互補序列建構法 42.1 格雷互補序列的定義 42.2 二的冪次方長度PSK序列的碼字序列與廣義布林函數表示法 52.2.1 PSK序列與碼字序列 52.2.2 二的冪次方長度PSK序列廣義布林函數表示法 72.3 標準格雷序列廣義布林函數相加建構法 112.4 二的冪次方長度複數格雷序列乘積建構法 122.4.1 Unitary matrix 132.4.2 The Boolean Exponent Form 162.4.3 BooleanCS程式流程架構 17第三章 複數格雷序列乘積建構法的特性探討 223.1乘積建構法無法生成的格雷序列 223.2以乘積建構法生成無重複之PSK標準格雷序列 243.3 Budisin , Spasojevic[BuS 13]研究之訂正 263.4 特例說明 283.4.1 Permutation序列P的影響 283.4.2 互補對 32第四章 結論 37參考文獻 38
 [Bud 13]S. Z. Budišin and P. Spasojević, "Paraunitary generation/correlation of QAM com-mplementary sequence pairs, " Cryptography and Communications, (DOI 10.1007/s12095-013-0087-9), 2013.[BuS 13]S. Z. Budišin, and P. Spasojević. "Universal Generator for Complementary Pairs of Sequences Based on Boolean Functions," arXiv preprint arXiv:1311.4782, 2013.[Cha 06]張美雯, "OFDM序列PAPR, " 元智大學碩士論文, 2006.[CLH 10]C. Y. Chang, Y. Li and J. Hirata, "New 64-QAM Golay Complementary Sequences," IEEE Trans. Inf. Theory, vol. 56, no. 5, pp. 2479-2485, 2010.[ DaJ 99]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, 1999.[ FiJ 06 ]F. Fiedler and J. Jedwab, "How do more Golay sequences arise?," IEEE Trans. Inf. Theory, vol. 52 ,no. 9, pp. 4261-4266, 2006.[FJPa 08]F. Fiedler, J. Jedwab and M.G. Parker, "A multi-dimensional approach to the construction and enumeration of Golay complementary sequences," J. Comb. Theory, vol. 115, pp. 753-776, 2008.[FJPb 08]F. Fiedler, J. Jedwab and M.G. Parker, "A framework for the construction of Golay sequences," IEEE Trans. Inf. Theory, vol. 54, no. 7, pp. 3114-3129, 2008.[FJW 10]F. Fiedler, J. Jedwab and A. Wiebe, "A new source of seed pairs for Golay sequences of length 2m, " J. Comb. Theory, vol. 117, pp. 589–597, 2010.[Gol 61]M.J.E. Golay, "Complementary Series", IRE Trans. Inf. Theory, vol. IT-7,no. 2, pp. 82-87, 1961.[Hwa 06]黃國倫, "格雷互補序列遞迴建構探討, " 元智大學碩士論文, 2006.[ Li 08 ]Y. Li, "Comments on "A new construction of 16-QAM Golay complementary sequences" and extension for 64-QAM Golay sequences" IEEE Trans. Inf. Theory, vol. 54, no. 7, pp. 3246-3251, 2008.[LiC 05] Y. Li and W.B. Chu, "More Golay Sequences, " IEEE Trans. Inf. Theory, vol. 51, no. 3, pp. 1141-1145, 2005.
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 1 格雷互補序列遞迴建構探討 2 以Reed-Muller碼降低正交分頻多工訊號之功率峰均比

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