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研究生:黃彥文
研究生(外文):Yen-Wen Huang
論文名稱:複數格雷互補序列新建構法
論文名稱(外文):New Construction of General Complex Golay Complementary Sequences
指導教授:李穎李穎引用關係
指導教授(外文):Ying Li
學位類別:博士
校院名稱:元智大學
系所名稱:通訊工程學系
學門:工程學門
學類:電資工程學類
論文種類:學術論文
論文出版年:2011
畢業學年度:99
語文別:中文
論文頁數:225
中文關鍵詞:格雷互補序列格雷序列建構法參數化多項式複數字集上行探測序列16-QAM 格雷序列計算機搜尋
外文關鍵詞:Golay complementary sequencesGolay sequence constructionparameter polynomialcomplex-valueduplink sounding sequences16-QAM Golay sequencescomputer search
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格雷序列具有「成對序列之非週期自相關函數和為脈衝函數」的特性。此性質使格雷序列在通訊領域中有許多應用。自 Golay 提出元素值為 ( +1, -1) 的實數格雷序列的遞迴建構與直接建構後, PSK 與 QAM 字集之複數值格雷序列的建構法相繼被提出。

本論文提出一種適用於所有複數字集的參數化格雷序列建構法, 並介紹由計算機搜尋取得的新 16-QAM 格雷對, 根據搜尋結果我們提出 new interleaving 建構法合併二組長度 $N+1$ 和 $N$ 的特定格雷對, 生成長度 $2N+1$ 的格雷對。

我們提出的參數化建構法 (parameterized construction) 合併二組長度較短的格雷對與三個參數, 建構出另一組長度較長的包含任何 PSK 與 QAM 字集的複數值格雷對。長度較短的格雷對可為各種長度與任意字集, 並可藉由調整參數置入額外零值元素於長度較長的格雷對, 而不影響互補性質。部份已知格雷序列建構法可視為本建構法的特例。本建構法並解決了含零格雷序列建構法無法預知零值元素置放位置, 以及零值元素數目隨序列長度增加而上升的問題。應用於 IEEE 802.16 上行探測序列設計, 生成的探測訊號功率峰均比上限為 3 dB 且具低相關值, 優於既有結果。

過去已知的 QAM 格雷序列長度均為 2 的冪次方, 且均由 QPSK 格雷序列所組成。此外, 過去文獻指出長度 7, 9, 14, 15 的 QPSK 格雷對不存在, 且不知是否存在其他字集的長度 7, 9, 14, 15 的格雷對。我們以計算機搜尋長度至 8 的 16-QAM 格雷對, 發現存在長度 7 的格雷序列, 以參數化建構法得出長度 14 的 16-QAM 格雷對。基於搜尋結果, 以我們提出 new interleaving 建構法, 合併二組長度分別為 N+1 和 N 的特定 16-QAM 格雷對, 可生成長度 $2N+1$ 的 16-QAM 格雷對, 並得出存在長度 9, 15 的 16-QAM 格雷對。本研究成果顯示長度 1 至 16 均存在格雷對, 也指出長度 4, 6, 8 存在 primitive 格雷對, 以及長度 3 至 8 的某些格雷對具有不等序列功率性質。

Golay complementary sequences have the property that the sum of aperiodic autocorrelation functions for pairing sequences is an impulse function. They have application in the communications engineerings. Since Golay introducted recursive and direct constructions of real-valued ( +1, -1) Golay complementary sequences, several constructions of complex-valued, including PSK and QAM alphabets, Golay complementary sequences were proposed.

In this dissertation, we propose a new construction of general complex alphabets Golay complementary sequences, namely parameterized construction, and also introduce new 16-QAM Golay complementary sequences obtained by computer search. Moreover, we provide a new interleaving construction to combine two lengths $N+1$ and $N$ particular Golay pairs to generate a length $2N+1$ Golay pair.

Our construction, parameterized construction, is combining two shorter Golay pairs and three parameters to construct longer complex alphabets, including arbitrary PSK and QAM, Golay pairs. Two shorter Golay pairs can can employ arbitrary lengths and any alphabet. Three parameters in this construction allow for flexible extra zero insertions. Some of known constructions can be seem the special cases. This construction solved that constructions of Golay sequences with zeros can not predict the places of extra zeros, and the number of zeros increased with sequence lengths. For IEEE 802.16 uplink sounding sequence design, we proposed the QPSK Golay sequences that variation maintain 3 dB maximum PAPR and low cross correlation for various multiplex options in all FFT size with full band sounding.

Existing lengths of QAM Golay sequences are power of 2, and constructed from QPSK Golay sequences. Previous studies showed that length 7, 9, 14, 15 Golay sequences over QPSK alphabet do not exist, and in other alphabets do not know. Using computer search for 16-QAM Golay sequences of lengths up to 8, we found the length 7 16-QAM Golay sequences exist, then we obtained length 14 16-QAM Golay sequences via parameterized construction. Based on computer search results, we provided a new interleaving construction from two length N+1, N particular Golay pairs to construct a length $2N+1$ Golay pair, then we knew length 9, 15 16-QAM Golay sequences exist. Our studies stated that Golay sequences exist of lengths up to 16, some of length 4, 6, 8 16-QAM Golay pairs are primitive Golay pairs, and some of 16-QAM Golay pairs have unequal sequence power property.

List of Figures xi
List of Tables xiii
Notation xvii
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Outline of this dissertation . . . . . . . . . . . . . . . . . . . . . 4
2 Background 5
2.1 General Notation and Basic Definitions . . . . . . . . . . . . . . . 5
2.2 Aperiodic Correlation Function . . . . . . . . . . . . . . . . . . . 7
2.3 Boolean Function and Generalized Boolean Function . . . . . . . 9
2.4 Golay Complementary Sequences . . . . . . . . . . . . . . . . . . 10
2.5 The power of OFDM signals . . . . . . . . . . . . . . . . . . . . . 15
3 A Parameterized Construction of General Complex Golay Com-
plementary Sequences 17
3.1 Introduction and Chapter Overview . . . . . . . . . . . . . . . . . 17
3.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2.1 A Short Review of Golay Complementary Sequence Con-
structions . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2.2 Notation, Definitions, and Terminologies . . . . . . . . . . 21
3.3 Parameterized Construction . . . . . . . . . . . . . . . . . . . . . 22
3.3.1 Parameterized Construction of Full Golay Complementary
Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3.2 Special Cases of Parameterized Construction of Full Golay
Complementary Sequences . . . . . . . . . . . . . . . . . . 32
3.3.3 Parameterized Construction of length 2^n H-PSK Golay Com-
plementary Sequences in terms of Generalized Boolean Func-
tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3.4 Parameters and Sequence Elements . . . . . . . . . . . . . 41
3.3.5 Comparsion with Fiedler, Jedwab and Parker’s Construction 48
3.4 Zero Insertion Patterns for the Parameterized Construction . . . . 50
3.4.1 Zero Insertion Patterns for Block Concatenation Construc-
tion without Symbol Collision . . . . . . . . . . . . . . . . 53
3.4.2 Zero Insertion Patterns for Block Interleaving Construction
without Symbol Collision . . . . . . . . . . . . . . . . . . . 55
3.4.3 Zero Insertion Patterns for Element Interleaving Construc-
tion without Symbol Collision . . . . . . . . . . . . . . . . 57
3.5 Uplink Sounding Sequence of IEEE 802.16 [LH08] [LHK09] [HL10] 63
3.5.1 New Uplink Sounding Sequence Construction . . . . . . . 64
3.5.2 PAPR and Crosscorrelation . . . . . . . . . . . . . . . . . 69
3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4 Computer Search Result for 16-QAM Golay Sequences and New
Interleaving Construction 75
4.1 Introduction and Chapter Overview . . . . . . . . . . . . . . . . . 75
4.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.2.1 A Short Review of QAM Golay Complementary Sequence
Construction . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.2.2 Notation, Definitions, and Terminologies . . . . . . . . . . 76
4.3 New Length 4 16-QAM Golay Sequences . . . . . . . . . . . . . . 78
4.3.1 Analysis of New Length Four 16-QAM Golay Complemen-
tary Sequences . . . . . . . . . . . . . . . . . . . . . . . . 83
4.3.2 The Peak Envelope Power (PEP) and Peak-to-Mean Enve-
lope Power Ratio (PMEPR) Upper Bounds . . . . . . . . . 87
4.3.3 Golay Complementary Pairs with Unequal Sequence Power 91
4.3.4 Extension to Golay Complementary Sequences with Larger
QAM Alphabets and Longer Sequence Lengths . . . . . . 93
4.4 Other New 16-QAM Golay Sequences . . . . . . . . . . . . . . . . 99
4.4.1 A Short Review of Golay Complementary Sequence Results
by Computer Search . . . . . . . . . . . . . . . . . . . . . 99
4.4.2 16-QAM Golay Complementary Pairs of Lengths up to 8 . . . . . 100
4.5 Golay Sequences of Odd Lengths . . . . . . . . . . . . . . . . . . . . 103
4.6 Existing Golay Sequence Lengths and Smallest Alphabets . . . . . 107
4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5 Summary 111
5.1 Results of this Dissertation . . . . . . . . . . . . . . . . . . . . . . 111
5.2 Further Possible Research Topics . . . . . . . . . . . . . . . . . . 113
A Length 18 QSPK Golay Complementary Pairs 115
B 16-QAM Golay Complementary Pairs of Lengths 2, 3, 5, 6, 7, 8 . . . . . . . . 119
B.1 Length 2 16-QAM Golay Complementary Pairs . . . . . . . . . . 119
B.2 Length 3 16-QAM Golay Complementary Pairs . . . . . . . . . . 120
B.3 Length 5 16-QAM Golay Complementary Pairs . . . . . . . . . . 122
B.4 Length 6 16-QAM Golay Complementary Pairs . . . . . . . . . . 125
B.5 Length 7 16-QAM Golay Complementary Pairs . . . . . . . . . . 140
B.6 Length 8 16-QAM Golay Complementary Pairs . . . . . . . . . . 148
C 16-QAM Golay Complementary Sequences of Odd Lengths 9, 11,
13, 15 201
C.1 Length 9 16-QAM Basic Golay Complementary Pairs . . . . . . . 202
C.2 Length 11 16-QAM Basic Golay Complementary Pairs . . . . . . 204
C.3 Length 13 16-QAM Basic Golay Complementary Pairs . . . . . . 211
C.4 Length 15 16-QAM Basic Golay Complementary Pairs . . . . . . 213
Bibliography 217
Index 224


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