高斯整數(Gaussian Integer)是一個實部與虛部皆為整數的複數，而完美序列(Perfect Sequence)是指具有理想的週期性自相關函數(Periodic Auto-Correlation
Function, PACF)。本篇論文針對複合數長度的稀疏完美高斯整數序列(Sparse Perfect Gaussian Integer Sequence, SPGIS)提出建構之方法，其不但具備了理想的
PACF 和元素值大部分為零之外，非零值皆為高斯整數。若序列長度為N=PQ，在P、Q皆為整數且大於1的情況下，SPGIS是透過P個基底序列(Base Sequences)，分別使用相對應的大小參數和循環位移(Cyclic Shift)參數線性組合產生；當中的大小參數皆為等振幅(Amplitude)的高斯整數，且每個基底序列皆有P 個非零值。本文也會使用建構SPGIS 的方法，將相同長度的兩相異SPGIS，利用週期性交相關函數(Periodic Cross-Correlation Function, PCCF)仍為SPGIS之特性，產生出更多的SPGIS。最後會將SPGIS應用在實體層安全(Physical Layer Security)上，建立能不被竊聽的安全性架構。

A Gaussian integer is a complex number whose real and imaginary parts are both integers. A sequence is defined as perfect if and only if it has an ideal periodic auto-correlation function. This thesis proposes a method to construct a sparse perfect Gaussian integer sequence (SPGIS) of composite length in which most of the sequence elements are zero. The proposed SPGISs are obtained by linearly combining P base sequences or their cyclic-shift equivalents by using nonzero Gaussian integer coefficients of equal magnitudes, in which the sequence length is N=PQ, where P and Q are both positive integers, and P, Q >1. Each base sequence consists of P nonzero
elements. The mathematical expression of the periodic cross-correlation function of any two proposed SPGISs of the same length is also a SPGIS. Consequently, this characteristic can be adopted to increase the number of proposed SPGISs.
Furthermore, an application of the proposed SPGIS is presented in this thesis for physical layer security to prevent eavesdroppers.

第一章 導論+1
1.1 研究動機+2
1.2 論文架構+3
第二章 複合數長度之稀疏完美高斯整數序列(SPGISs)設計+4
2.1 完美序列介紹+4
2.2 基底序列+5
2.3 SPGIS序列之結構+6
2.4 SPGIS的延伸+12
2.4.1 週期性交相關函數+12
2.4.2 新SPGISs+13
第三章 安全性架構+17
3.1 系統模型+17
3.2 安全性架構設計+18
3.2.1 建立私密鑰匙的方法+18
3.2.2 訊息交換流程+20
3.3 通道估測的方法+23
3.3.1估測步驟1+23
3.3.2估測步驟2+28
3.3.3估測步驟3+32
3.3.4估測步驟4+36
3.3.5估測步驟5+39
3.3.6估測步驟6+44
3.4 安全性說明+50
第四章 模擬結果+53
4.1 MSE 比較+53
4.2 BER 比較+55
第五章 結論+58
參考文獻+59
中英文對照表+63
英文縮寫對照表+66

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