時頻分析與線性完整轉換

Time-Frequency Analysis and Linear Canonical Transform

黃世谷(Shih-Gu Huang)

指導教授 ： 貝蘇章

國立臺灣大學/電機資訊學院/電信工程學研究所/博士(2016年)

https://doi.org/10.6342/NTU201601090

摘要

一個理想時頻表示法(TFR)，要能顯示僅在該特定時刻出現的頻譜信息。因此，更為集中的時頻能量分佈且無交叉項(cross terms)，是設計時頻表示法時主要關注的項目。為了改善短時傅立葉轉換(STFT)的能量集中，我們開發了一種會隨著時頻改變寬度的適應性高斯窗 。而為了更進一步提高能量集中程度，我們將高斯窗加以旋轉，旋轉角度也是隨著時頻改變。我們推導出最佳的窗口寬度和最佳的旋轉角度。模擬結果顯示，我們的方法，在於能量集中程度以及瞬時頻率估計上，優於其他種適應性時頻表示法。至於瞬時頻率的估計，常見的方法是從時頻表示法振幅的脊來偵測。然而，相位的部分也可以用來估計瞬時頻率。在論文中，基於短時傅立葉轉換相位對時間微分及對頻率微分，我們開發出一個新的瞬時頻率估計法。
線性完整轉換(LCT)是一個線性積分轉換，會在時間頻率平面上對訊號造成扭曲。傅立葉轉換、菲涅爾轉換和分數傅立葉轉換(FRFT)都是它的特例。在論文中，我們提出一個離散線性完整轉換，其與取樣週期無關也不需提升取樣。它具有完美的可逆性，這特性並不存在於許多現有離散線性完整轉換，另外它有近似的可加性以及更高的準確度。此外，我們對線性完整轉換做了一些延伸。一是將解析訊號和線性完整轉換結合成低複雜度、可逆且無失真的轉換。結合的理由在於解析訊號比實數訊號更為實用，而線性完整轉換也更有彈性。另一個延伸則是將時間偏移及頻率偏移兩參數引進線性完整轉換，並推導出與其有交換關係的運算子，可用來產生特徵函數。在二維情況下，最一般化的形式是二維不可分線性完整轉換(2D NsLCT)。二維傅立葉轉換、二維分數傅立葉轉換、迴轉轉換(gyrator transform)和複數分數傅立葉轉換(complex FRFT)都是它的特例。我們開發了一個二維不可分線性完整轉換的離散實現方法，其使用了兩個二維啾聲信號(chirp)載波和兩個二維啾聲信號卷積。因為沒有使用到任何二維仿射變換，所以沒有內插誤差。對於二維不可分線性完整轉換的特例，迴轉轉換，我們提出四種低複雜度的實現方法，其中最後一個方法具有完美的可加性。對於另一特例，複數分數傅立葉轉換，我們也提出了一個高效率實現方法，並推導出一些特性和特徵函數的一般式。
最後，我們介紹了巴格曼轉換。它是使用複數參數的線性完整轉換的一個特例。我們提出一個更穩定的轉換，稱為正規化巴格曼轉換，並開發出數個實現方法。我們最後還結合了分數傅立葉轉換與四元數的概念，提出兩種一維四元數分數傅立轉換，和六種二維四元數分數傅立轉換。

關鍵字

適應性時頻分析 ； 短時傅立葉轉換 ； 瞬時頻率估計 ； 分數傅立葉轉換 ； 線性完整轉換 ； 迴轉轉換 ； 二維不可分線性完整轉換

並列摘要

An ideal time-frequency representation (TFR) should reveal only the spectral information about the signal occurring at any given time instant. Thus, more concentrated time-frequency energy distribution without cross terms is the major concern when designing a TFR. In order to improve the energy concentration of the short-time Fourier transform (STFT), we develop an adaptive Gaussian window with time-frequency-varying window width. And to further enhance the energy concentration, we rotate the window by a time-frequency-varying rotation angle. Optimal window width and optimal rotation angle are derived. Simulation results show that our methods outperform many other adaptive TFRs in energy concentration and instantaneous frequency (IF) estimation. IFs are commonly estimated from the ridges of the envelopes of the TFRs. However, the phase part can also be used for IF estimation. In this dissertation, we develop an IF estimation method based on the time derivative and frequency derivative of the phase part of the STFT.
Linear canonical transform (LCT) is a linear integral transform that produces twisting in the time-frequency plane. Fourier transform, Fresnel transform and fractional Fourier transform (FRFT) are all its special cases. In this dissertation, a discrete LCT (DLCT) irrelevant to the sampling periods and without oversampling operation is proposed. It has perfect reversibility property, which doesn't hold in many existing DLCTs, approximate additivity property, and higher accuracy. Besides, some extensions of the LCT are proposed. One is combining the analytic signal and the LCT into low-complexity, reversible and undistorted transforms, due to the practicality of analytic signals over real signals and the flexibility of the LCT. Another one is generalizing the LCT by using two more parameters, i.e. time offset and frequency offset. We also propose an operator that commutes with the offset LCT and can be used to generate the eigenfunctions. For the 2D case, the most general form is the 2D nonseparable LCT (2D NsLCT). The 2D Fourier transform, 2D FRFT, gyrator transform and complex FRFT are all its special cases. We develop a discrete implementation method of the 2D NsLCT based on two 2D chirp multiplications and two 2D chirp convolutions. There's no interpolation error because no 2D affine transforms are used. For its special case, gyrator transform, four kinds of low-complexity implementation methods are proposed, and the last one has perfect additivity property. For the complex FRFT, an efficient implementation is proposed. Some properties and the general form of the eigenfunctions of the complex FRFT are also derived.
At last, we introduce the Bargmann transform, which is a special case of the LCT with complex parameters. A more stable transform called normalized Bargmann transform is proposed, and several implementation methods are developed. We also combine the concept of quaternion with the FRFT, and propose two types of 1D quaternion FRFTs and six types of 2D quaternion FRFTs.

並列關鍵字

Adaptive time-frequency analysis ； short-time Fourier transform ； instantaneous frequency estimation ； fractional Fourier transform ； linear canonical transform ； gyrator transform ； 2D nonseparable linear canonical transform

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