- 學位論文
分數傅利葉轉換與時頻分析及其在聲音訊號上的應用
Fractional Fourier Transform and Time-Frequency Analysis and Apply to Acoustic Signals
摘要
傅立葉分析在信號處理中佔有很重要的角色，我們常拿它來分解訊號中各頻率的成分，然而在時頻分析上仍尚嫌不足，因為傅立葉分析只能適用穩態的訊號，無法處理時變的訊號，故而發展出許多的時頻分析的工具。固然發展時頻分可以幫助我們處理時變的訊號，而擴展傅立葉這個工具本身也是另一種方式—分數傅立葉轉換。 本篇論文中主要分成兩個部份: 第一個部份為擴充傅立葉轉換的數學理論，包括了分數傅立葉轉換和線性完整轉換的理論介紹，還有分數傅立葉轉換結合時頻分析的應用，也可以推廣至光學、雷達的應用。我們推導了測不準原理在分數傅立葉轉換和線性標準轉換的不等式，還有將連續的分數傅立葉轉換從轉換成離散的方法。一維的分數傅立葉轉換可做出傳統傅立葉做不出來的濾波器、降低取樣頻率、加密等的應用，而二維的分數傅立葉轉換可應用在影像、光學上。 第二個部份是系統性的介紹時頻分析的工具，整理各種時頻分析的優缺點，同時提出三種方法可以降低時頻分析的計算，其中包含了可適性的時頻分析。將這些時頻分析的工具實際應用在音樂和人聲上，將自動簡譜產生器的構想進一步的實現，包含了去除倍頻的訊號、根據不同的訊號特性將計算量降低，還有模擬與實驗的成果。
並列摘要
Fourier analysis takes an important role in signal processing, and we often use it to decompose frequencies for further applications. However, in the time-frequency analy-sis, the Fourier transform is not good enough. Since the Fourier transform can only deal with the stationary signals but can not deal with non-stationary or time-varying signals. Although we can develop time-frequency analysis to help us deal with time-varying signals, we also can extend the math of Fourier transform itself – fractional Fourier transform. This thesis mainly has two parts: the first part is to extend the math of the Fourier transform. This includes the theorem of the fractional Fourier transform and the linear canonical transform. The fractional Fourier transform can combine time-frequency analysis, and extend to applications of optics and Radar systems. We also derive the uncertainty principles of the fractional Fourier transform and the linear canonical trans-form, and discrete the continuous fractional Fourier transform with ei-gen-decomposition. One dimensional fractional Fourier transform can produce filters which traditional Fourier transform cannot and also reduce the sampling rate and en-cryption…. Two dimensional fractional Fourier transform can deal with image and op-tics. The second part will introduce the time-frequency distribution systematically. We list pros and cons of each time-frequency distribution and propose three methods to re-duce the computation, including adaptive time-frequency distribution. We will apply these tools to music and acoustics signals. We are going to realize some parts of the auto-transcription and discuss the problems we face and the solutions, including exceed the harmonics and computation. Finally, there are our results and simulations.