The Values and Applications of Ramanujan Sum in Signal Processing

本論文有兩個主要部份：第二部份和第三部份。在第二部份我們介紹了Ramanujan sum以及它在信號處理的一些應用。在第三部份我們介紹一些關於Pade展開式及Prony分析的研究和應用，並提出一套新的方法來做阻尼振蕩的諧波分析。 Ramanujan sum是一個數論領域中的算術函數，近年來才開始被應用到信號處理中。在這些應用之中，最主要的是Ramanujan傅利葉轉換，它利用Ramanujan sum獨特的週期特性，產生一個類似傅利葉轉換的頻率轉換，而這個工具可以成功分析出許多傅利葉轉換無法分析的整數頻率成份。 在本論文的第二部份，我們主要在探討這個數學轉換的物理意義。從前人的相關研究中，我們觀察到幾個Ramanujan傅利葉轉換的優點和缺點，從這些觀點出發，我們認為Ramanujan傅利葉轉換有需要修正的地方，因而提出純粹整數週期函數的概念，並加上了時間平移，定義了一種新的RS圖。純粹整數週期函數可以說是頻率成份最單純的整數週期離散訊號，而我們又用數學證明出，RS圖可以將一個普通的週期訊號之各個純粹整數週期函數成份，呈現在其各個欄位上，也就是說，RS圖等同於純粹整數週期函數成份分解的數學運算。以這個RS圖的成份分解的觀點出發，我們成功解釋了Ramanujan傅利葉轉換的物理意義，並修正了其中的缺點。 在本論文的第三部份，我們從Z轉換的Pade展開式與Prony分析之間的等價性質出發，提出一種利用Pade展開式來改進Prony分析的演算法。由Pade展開式，我們可以分析一個離散訊號的極點和零點，而根據上面所提到的等價性質，那些極點就是Prony分析所得到的振蕩基底。在極點零點圖中，大部份極點和零點都會成雙成對出現，但我們知道主要的振蕩基底之極點不會和零點成對出現，所以我們引用了一篇文章的想法，將成對的極點和零點從複數平面上移除，再利用所剩的極點，用原有的Prony分析方式，把訊點重建出來。我們設計了幾個判定成對極點和零點的方法，也進行了許多實驗，結果顯示我們的方法比Prony分析更能對抗雜訊。This thesis contains two main parts: Part I and Part II. In Part I we give a summary of an arithmetic function, Ramanujan sum, and its applications in signal processing. In Part II we introduce a novel method for harmonic analysis of damped oscillators. Ramanujan sum is an arithmetic function, which is recently applied to signal processing. Among its applications, the most important one is the Ramanujan Fourier transform, which uses the unique periodic property of Ramanujan sum to create a Fourier-transform-like frequency transform. This transform can successfully extract some integer periodic components which Fourier transform cannot extract from the signals. In Part I, we mainly discuss about the physical meaning of this mathematical transform. According to previous works, we observed some advantages and disadvantages of Ramanujan Fourier transform. Based on these points, we conclude that there is something to improve on Ramanujan Fourier transform. Then, we introduce the concept of intrinsic integer-periodic functions. We also define a new RS map by considering time shift in the Ramanujan Fourier transform. The intrinsic integer-periodic functions can be regarded to be pure in periodic component, and we prove in mathematics that RS map presents the intrinsic integer-periodic components of a general signal on the column of the map. That is to say, RS map is equivalent to the intrinsic integer-periodic function decomposition, and with this decomposition by RS map, we can explain the physical meaning of Ramanujan Fourier transform. In Part II, we start with the equivalence between Pade approximation of Z-transform and Prony analysis and end up to propose a new algorithm to improve Prony analysis. By Pade approximation, we can analysis the poles and zeros of a discrete signal, and by the equivalent property mentioned above, those poles analyzed are the bases obtained by Prony analysis. In the pole-zero plot, most poles appear together with zeros, but we know that for the main bases of oscillators, their poles do not come up with zeros. Thus, we based on the idea in a previous work that we can remove pole-zero pairs from the complex plane so that we can use Prony analysis with the remaining poles to recover the original signal. We design several methods to recognition paired poles and zeros and do many experiments. The results show that our method work better than Prony analysis in noisy environment.誌謝 i 中文摘要 iii ABSTRACT v I Introduction 1 Chapter 1 Introduction 3 1.1 Introduction 3 1.2 Notation and Convention 6 1.2.1 Notation 6 1.2.2 Convention 7 Chapter 2 Background of Number Theory 11 2.1 Roots of Unity 12 2.2 Chinese Remainder Theorem 13 2.3 Arithmetic Function 15 2.3.1 Arithmetic Functions 15 2.3.2 Additive and Multiplicative Properties 17 2.3.3 Euler Totient Function 20 2.4 Dirichlet Convolution 22 2.4.1 Dirichlet Convolution 23 2.4.2 Mӧbius Inversion 26 2.5 Generating Function 28 II The Values and Applications of Ramanujan Sum in Signal Processing 33 Chapter 3 The Values and Applications of Ramanujan Sum in Signal Processing 35 3.1 The Ramanujan Sum 36 3.2 The Ramanujan Fourier Transform 47 3.2.1 Definition 47 3.2.2 Related Works 47 3.3 Discussion 52 3.4 Summary 54 Chapter 4 The Intrinsic Integer-Periodic Functions 57 4.1 Motivation 58 4.1.1 Periodicity 58 4.1.2 Intrinsic Integer Periods 62 4.2 The IIPFs 65 4.2.1 Definition 65 4.2.2 Properties 66 4.3 IIPF Decomposition 73 4.3.1 Existence and Uniqueness 73 4.3.2 Properties of IIPF Decomposition 77 4.3.3 Experimental Results 84 4.4 Summary 87 Chapter 5 The RS Map 89 5.1 Motivation 90 5.1.1 The First Definition 90 5.1.2 The Second Definition 92 5.2 The Relationship Between RS Map and IIPFs 95 5.3 Proof of Theorems 97 5.3.1 Proof of Main Theorem, Part 1 97 5.3.2 Proof of Main Theorem, Part 2 98 5.4 Other Issues on RS Map 106 5.4.1 Implementation Issues and Computational Complexity 106 5.4.2 Norms of Columns of RS Map 108 5.5 Applications 110 5.5.1 IIPF Decomposition 110 5.5.2 Frequency Estimation 114 5.5.3 Harmonic Analysis 114 5.6 Summary 117 III Exponential Fitting by Pade Approximation 119 Chapter 6 Z-transform, Pade Approximation and Prony Analysis 121 6.1 Pade Approximation 122 6.1.1 Motivation 122 6.1.2 Formulation 127 6.2 Pade-Z Transform 132 6.2.1 Pade-Z Transform of Exponential Functions 132 6.2.2 Distribution of Poles and Zeros 134 6.3 Prony Analysis 138 6.3.1 Harmonic Analysis by Fourier Series and Fourier Transform 139 6.3.2 Harmonic Analysis by Prony Analysis 142 6.4 The Equivalence between Pade-Z Transform and Prony Analysis 148 6.5 Summary 149 Chapter 7 The Proposed Exponential Fitting Method 151 7.1 System Model 152 7.1.1 Formulation 152 7.1.2 Z-transform of Damped Oscillators 153 7.2 Proposed Method 158 7.2.1 Previous Works 158 7.2.2 Framework of Proposed Method 163 7.3 P/Z Matching Algorithms 164 7.3.1 Distance Ranking 165 7.3.2 Hungarian Algorithm 166 7.4 P/Z Recognition and Recovering 176 7.4.1 P/Z Recognition 177 7.4.2 P/Z Recovering 181 7.5 Experimental Results 183 7.5.1 Performance Analysis for Each Stage 184 7.5.2 Comparison with Prony Analysis 190 7.5.3 Overall Accuracy Analysis 193 7.6 Summary 193 IV Conclusion and References 197 Chapter 8 Conclusion 199 References 20