時頻分析與之實作快速演算法

傅立葉轉換是傳統上很常用來分析頻率的工具，不過它卻不能使用在非穏態訊號上。所以傅立葉轉換只適合用於穏態訊號且線性訊號。因此有了時頻域分佈函數針對非穩態訊號且線性訊號的分析工具。時頻分析的理論發展至今，已為訊號分析領域帶來極為深遠的影響，其運用範圍十分廣泛，涵蓋了語音、物理、心電圖、地球科學、音樂。為了得到訊號的時變頻譜特性，許多學者提出了各種形式的時頻分析函數，例如：短時間傅立葉轉換（STFT）、韋格納（WD）、加伯（GT），各種分佈多達幾十種。本篇論文主要分成三個部份：第一部份為時頻分析，包括了各種時頻分析的理論介紹，及其優缺點比較、模擬結果，應用範圍等。更提出多種方法降低時頻分析在實作上的運算量。第二部份特別介紹希爾伯－黃轉換（HHT），黃鍔院士於1998年發表提出。有別於傳統的時頻分析方法，傳統的方法是不夠的，因為它們均必須假設訊號為線性。因此，介紹了適合用於非穩態訊號且非線性訊號的分析工具。第三部份為時頻分析與隨機程序的關係，我們研究發現各種時頻分析的方法與隨機程序有著定理的關係存在。

關鍵字

時頻分析；短時間傅立葉轉換；韋格納；加伯；希爾伯-黃轉換；可適性時頻分析；隨機程序；傅立葉轉換

並列摘要

The Fourier transform is an important tool in frequency analysis, but it cannot use for non-stationary or time-varying signals. Since The Fourier transform only deal with the stationary and linear signals. The time-frequency distribution function deals with the non-stationary and linear signals. It describes signals in terms of joint time-frequency form and is a powerful tool for analyzing signals. It has been widely applied in much kind of fields, such as speech, physics, electrocardiography (ECG), earth science and music. It is particularly useful for people to analyze signals with continuously time-varying frequency way. A lot of the time-frequency analysis have been widely used and researched for a number of years, such as the short time Fourier transform, the Wigner distribution, the Gabor transform, and other distributions. This thesis mainly has three parts: The first part is the time-frequency analysis. We will introduce a lot of algorithm of the time-frequency distribution, including the theorem of algorithms, advantages and disadvantages, simulations, and applications. We will propose some fast implementation algorithms to reduce the computation. The second part will introduce a recently method, the Hilbert-Huang transform (HHT), by Huang (1998). Traditional data analysis methods are all based on linear and stationary assumptions. The HHT can to solve the problem that the data is non-linear and non-stationary. The third part we will discuss the relation between the random process (including the stationary and the non-stationary ones) and several well-known time-frequency distributions.

並列關鍵字

time-frequency analysis ； short time Fourier transform ； Wigner distribution ； Gabor transform ； Hilbert Huang transform ； adaptive time-frequency distribution ； random process ； Fourier transform

參考文獻

Hilbert Huang transform

[A1]N. E. Huang, Z. Shen and S. R. Long, et al., “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time Series Analysis＂, Proc. Royal Society, vol. 454, pp.903-995, London, 1998.

[A2]N. E. Huang, S. Shen, Hilbert-Huang Transform and its Applications, World scientific, Singapore, 2005.

[B1]C. H. Page, Instantaneous Power Spectra, National Bureau of Standards, Washington, D. C., 1951

[B2]S. C. Pei and J. J. Ding, “Relations between the fractional operations and the Wigner distribution, ambiguity function,” IEEE Trans. Signal Processing, vol. 49, no. 8, pp. 1638-1655, Aug. 2001.

國際替代計量

時頻分析與之實作快速演算法

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