格雷序列(Golay sequence)用於OFDM訊號編碼具有低功率峰均比(Low PAPR)的特性，可將PAPR限制在2之內。Davis和Jedwab提出目前所知可產生最多格雷序列的方法，其格雷序列是Reed-Muller碼字，因此有較佳編解碼結構。但即使用了所有已知格雷序列進行降PAPR編碼仍有碼率(Code Rate)太低的問題。
本論文研究提升碼率的方法，及在802.11a系統實現的方式與效能。我們用「窮盡搜尋法」找出其他低PAPR(不大於2)的序列，包含非格雷序列。結果發現在長度較長時，其他序列佔全部低PAPR序列超過85%，遠多於過去使用的格雷序列。因此把這些序列與過去使用的格雷序列共同作為碼字可有效的提升碼率。對802.11a OFDM訊號編碼時，若避開「前導位元(Pilot tone)」會造成編碼區塊過多，無法發揮降低PAPR性能。本文提出將前導位元編入碼字的方法。並以電腦模擬評估其性能。觀察的指標包括，PAPR、編碼率、位元錯誤率、及OBO mask。
過去學者研究認為Davis和Jedwab提出的方法可能已產生所有的格雷序列。本研究搜尋結果顯示，長度16的QPSK-OFDM序列中存在「1024」個非Reed-Muller碼字的格雷序列，其存在推翻了「所有格雷序列皆可由Davis和Jedwab方式產生」的假說。

Golay sequences can be used as codewords for OFDM signals to limit the PAPR to 2. Davis and Jedwab propose a construction that produces the largest number of Golay sequences known previously. These Golay sequences have better structure for coding and decoding because they are Reed-Muller codewords. Even when all these Golay sequences are used as codewords, the code rate is still very low.
The purpose of this research is to explore methods to increase the code rate and to realize these methods in 802.11a. We use exhaustive search to find all low PAPR (no more than 2) sequences, including non-Golay sequences, and more than 85% of all low PAPR sequences are not Golay sequences. Therefore using both Golay and non-Golay sequences as codewords can increase the code rate. When coding 802.11a signals, avoiding pilot tones increases the numebr of coding blocks and degrades PAPR reduction performance. A new idea is introduced to include pilot tones in the codewords. Computer simulations are conducted and performance is evaluated through PAPR, code rate, bit error rate, and OBO mask.
Davis and Jedwab’s construction was previously believed to possibly generate all binary and quaternary Golay sequences of length 2m . Our exhaustive search reveals that there are “1024” new Golay sequences in 16 subcarrier QPSK-OFDM that are not Reed-Muller codewords. The hypothesis that “all Golay sequences can be generated form Davis and Jedwab’s construction ” is thus not true.

目錄
頁次
中文摘要……………………………………………………………… i
英文摘要………………………………………………………………ii
誌謝 ………………………………………………………………… iii
目錄…………………………………………………………………… iv
表目錄 ……………………………………………………………… vii
圖目錄………………………………………………………………… ix
符號表………………………………………………………………… xi
第一章 序論…………………………………………………………… 1
1.1研究動機與目的……………………………………………1
1.2研究成果……………………………………………………2
1.3各章節提要…………………………………………………4
第二章 研究背景………………………………………………………5
2.1正交分頻多工(OFDM)系統介紹……………………………5
2.1.1正交分頻多工(OFDM)系統架構…………………… 6
2.1.2 OFDM系統中PAPR的特性……………………………8
2.2非線性特性對OFDM系統之影響………………………… 10
2.2.1功率放大器的非線性特性…………………………10
2.3改善OFDM系統的非線性影響……………………………12
2.4編碼法之架構…………………………………………… 13
2.4.1格雷互補對(Golay complementary pair)………13
2.4.2布林函數(Boolean function)……………………16
2.4.3 Reed-Muller碼……………………………………17
2.4.4由Reed-Muller碼產生出格雷序列………………19
2.5 dd向量……………………………………………………21
2.5.1 dd向量與峰均比(PAPR)關係…………………… 23
第三章 降低PAPR之編碼與碼率提昇……………………………28
3.1使用分區編碼提升碼率的的性能與缺點……………… 29
3.2窮盡搜尋(Exhaustive Search)…………………………33
3.2.1利用dd向量與PAPR的關係減少搜尋量………… 33
3.2.2窮盡搜尋低PAPR的結果………………………… 34
3.2.3使用Non GDJ序列之優缺點---碼率增加、BER
性能下降………………………………………… 38
3.3 GDJ序列與Non GDJ序列之編解碼流程………………… 42
3.3.1利用dd方式解碼以增快解碼速度……………… 45
3.4將編碼方法應用在IEEE 802.11a中…………………… 48
3.4.1 IEEE 802.11a協定…………………………………49
3.4.2前導位元對編碼法的限制…………………………51
3.4.3將前導位元(Pilot tone)編入碼字中…………… 52
第四章 電腦模擬與性能分析 …………………………………… 55
4.1 802.11a系統發射訊號規範…………………………… 56
4.2 以編碼方式降低PAPR之模擬分析…………………… 59
4.2.1使用在802.11a的編碼架構(1)………………… 59
4.2.2使用在802.11a的編碼架構(2)………………… 61
4.3 改進編碼架構(2)之性能……………………………… 63
4.3.1 使用不同個數之保留頻帶降低PAPR性能………65
4.3.2利用保留頻帶降低PAPR之分析………………… 67
4.3.2 滿足頻譜規範的最佳反減性能表現…………… 70
4.3.3 位元錯誤率(Bit Error Rate)之性能評估…… 74
第五章 結論………………………………………………………… 76
參考文獻………………………………………………………………78
附錄A─非GDJ序列的新格雷序列……………………………… 80
附錄B─非格雷序列 ……………………………………………… 84

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