16- 與 64-QAM 格雷序列

16- and 64-QAM Golay Complementary Sequences

張仲逸(Chung-Yi Chang)

指導教授 ： 李穎

元智大學/電機通訊學院/通訊工程學系/碩士(2008年)

https://doi.org/10.6838/YZU.2008.00210

摘要

格雷互補序列具有「成對序列之非週期自相關函數和為脈衝函數」的特性。長度2m的 H-PSK、16-QAM、64-QAM 格雷序列的建構已先後被提出，1999年，Davis & Jedwab 提出 H-PSK 格雷序列的直接建構法，產生的序列稱為「GDJ 格雷序列」。16- 和 64-QAM 格雷序列是使用二或三個 QPSK GDJ 格雷序列所合成，這些 QPSK GDJ 格雷序列間的關係可以使用廣義布林函數差異描述，稱為 offset 或 offset pair。Offsets 組合是 QAM 格雷序列建構法的核心。2007年，Li 修正了 16- 與 64-QAM 格雷序列建構法，並推論提出兩種 offset pairs，可產生新的 64-QAM 格雷序列，為 64-QAM 格雷序列新建構法。
本論文首先提出 64-QAM 格雷序列新建構法的證明，新建構法可產生的 64-QAM格雷序列與原建構法並無交集，64-QAM 格雷序列個數因而增加。在序列長度增長時，新建構法能產生的 64-QAM 格雷序列將會多過於原建構法。
其次，本論文探討 16- 與 64-QAM 格雷序列之互補對象序列。在序列長度大於等於4時，16-QAM 格雷序列的同格雷共集對象序列數有兩種可能：8個與16個；64-QAM 格雷序列的同格雷共集對象序列數有三種可能：8個、16個以及24個。我們說明如何由 offset 或 offset pair 判斷各個 16-QAM 或 64-QAM 格雷序列之對象序列及其數目。由此我們得出 16- 與 64-QAM 同格雷共集互補對(Golay pairs)建構法及個數公式。這也是既有文獻中沒有的結果。

關鍵字

格雷序列 ； 正交振幅調變

並列摘要

Golay complementary sequences have the property that the sum of aperiodic auto-correlation functions for pairing sequences is an impulse function. There exist constructions of H-PSK, 16-QAM, and 64-QAM Golay sequences of length n??2m. In 1999 Davis and Jedwab gave a direct construction for H-PSK Golay sequences known as “GDJ Golay sequences.” 16- and 64-QAM Golay sequences can be constructed as the weighted sum of two or three QPSK GDJ Golay sequences differing by an offset or an offset pair. The combinations of offsets is the core in the description of QAM Golay sequence constructions. In 2007, Li modified previous constructions of 16- and 64-QAM Golay sequences, and proposed two cases of new offset pairs in a conjecture. These offset pairs can be used to construct additional new 64-QAM Golay sequences.
In this thesis, we first present the proof for the new construction of 64-QAM Golay sequences. There are no intersections between the 64-QAM Golay sequences produced by the new construction and the original construction. When the sequence length is very long, the new construction yields more 64-QAM Golay sequences compare to the original construction.
Secondly, we study the pairing sequences for 16- and 64-QAM Golay sequences of length n ?d 4. The number of pairing sequences for each 16-QAM Golay sequence is 8 or 16, and is determined by the offset. The number of pairing sequences for each 64-QAM Golay sequence is 8, 16, or 24, and is determined by the offset pair. We demonstrate how the number of pairing sequences can be determine by the offset or offset pair, and give the constructions and enumerations of 16- and 64-QAM Golay pairs where the pairing sequences are consisted of QPSK Golay sequences in the same Golay coset.

並列關鍵字

Golay complementary sequences ； QAM

參考文獻

[Hua06] 黃國倫。「格雷互補序列遞迴建構探討」，元智大學通訊工程所碩士論文，2006。

[CVT03] C.V. Chong, R.Venkataramani, and V. Tarokh, “A new construction of 16-QAM Golay complementary sequences,” IEEE Trans. Inform. Theory, vol. 49, no. 11, pp. 2953-2959, Nov. 2003.

[CLH08] C. Y. Chang, Y. Li, J. Hirata, “New offset pairs for 64-QAM Golay complementary sequences,” submitted to IEEE Trans. Inform. Theory, 2008.

[DJP98] J.A. Davis, J Jedwab and K. G. Paterson, “Codes, correlations and power control in OFDM,” HP Laboratories Bristol, HPL-98-199, 1998.

[DJ99] J. A. Davis, J. Jedwab, “Peak-to-mean power control in OFDM, Golay complementary sequences, and Reed-Muller codes,” IEEE Trans. Inform. Theory, vol. 45, no. 7, pp. 2397-2417, Nov. 1999.