本文中定義了一套新式的Rudin-Shapiro序列，稱為廣義Rudin-Shapiro序列。有別於傳統的Rudin-Shapiro序列元素侷限於二進制（±１），我們定義的廣義Rudin-Shapiro序列將其元素組成擴充到複數平面中單位圓上的等分點，亦即序列可被推廣到q進制（q為大於1的任意正整數），使此序列在利用上更為廣泛。此外，我們也將此提出的新序列透過電腦進行模擬（C, MATLAB），觀察並比較其性質與傳統二進制序列的差異。
　　我們主要針對以下三項特性做討論：遞迴特性、上下限存在性及其數量級、上下限值。首先，在遞迴特性上，我們證明了在廣義q進制序列時擁有和二進制時相似的性質，並推導其通式，可利用於擴建已知的Generalized Rudin-Shapiro序列；再者，我們也透過分析數據，說明了q進制序列的上下限存在於 √n 的數量級；最後，我們以模擬結果估計了q=3 到q=8 時的上下限值，並說明其增長趨勢。
　　在分析模擬數據的過程中，我們在對數刻度為橫軸的序列和除以√n 圖形中發現了其週期性，當長度越長時，各週期內形成的圖案趨近某種特殊形狀，且在不同的進制下所得圖形相似，除了可用來說明上下限外，我們也將其推算出區間極值發生位置的通式。

In this thesis, we generalized the Rudin-Shapiro sequences from binary to q-ary case (where q>1). Different from the original binary Rudin-Shapiro sequences, our generalized sequences take values from the unit circle on the complex plane. Therefore, our generalized sequences are more practical than original ones. We simulated our proposed q-ary sequences using C and MATLAB and compared their properties with original binary ones. We focused our discussion on the following three aspects: recursive properties, existence of upper/lower bounds as well as their magnitudes.
Regarding their recursive properties, we derived their general construction formula in the q-ary case. Second, we showed that the existence of bound and found that its order locates at √n by analyzing simulation data. Furthermore, we estimated their upper/lower bounds where 3≤q≤8, and observed and analyzed their corresponding values. We found a very unusual periodic property when vertical axis was divided by √n and the horizontal axis representing the sum of sequence elements was on log scale. Such settings lead to certain unusually similar patterns when the sequence length becomes longer. We observed this property and derived the general form of the location where local maximum/minimum happens.

第一章 緒論
第二章 背景知識及符號介紹
2.1 符號說明
2.2 Rudin-Shapiro序列
第三章 Generalized Rudin-Shapiro序列
3.1 Generalized Rudin-Shapiro序列建造
3.2 Generalized Rudin-Shapiro序列之特性與比較
3.2.1 遞迴特性
3.2.2 序列和上下限
3.2.3 〖|S〗_n^q |/√n 圖形討論
3.2.4 序列和極值
第四章 結論及未來展望
參考文獻

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