Fractional Signal Processing and Applications

[[abstract]]最近幾年，分數運算子和度量已在科學和工程上廣泛被探討，典型的範例如下：分數微積分、分數傅立葉轉換、分數延遲、分數維度、分數矩量、分數統計、分數希爾伯特轉換和分數圖形等，因此如何使用這些分數運算子和度量來發展新的數位信號處理的方法便是一有趣且新鮮的研究題目，也就是所謂的分數信號處理這個新領域，本計畫是一個三年的長期計畫，第一年將探討分數延遲運算子的設計，第二年將研究分數微分和積分運算子的設計與應用，第三年將探討分數希爾伯特轉換器的設計與應用，每一子計畫的摘要分別說明如下。第一年(2007 年) 題目：使用離散信號轉換設計分數延遲濾波器 (Design of fractional delay filters using discrete signal transforms) 摘要：在許多信號處理的應用，分數延遲取樣週期的數位信號是想要的，應用包括數位接收器的時間調整、天線陣列的波束方向調整、語音編碼和合成、樂器的模型化、取樣頻率轉換、時間延遲估計、梳型濾波器設計和類比至數位轉換器設計等，迄今，已有許多方法被提出來設計分數延遲濾波器，包括法諾法(Farrow)、權重最小平方法、限制性最小-最大誤差法和埃根設計法等，在本計畫中，我們將使用離散信號轉換來設計分數延遲濾波器，主要的好處是，公式解設計，快速又容易，不須複雜的遞迴步驟，所欲執行的工作共包括下列五大項目： 1. 使用離散傅立葉轉換設計分數延遲濾波器。 2. 使用離散哈特萊轉換設計分數延遲濾波器。 3. 使用離散餘弦轉換設計分數延遲濾波器。 4. 使用離散正弦轉換設計分數延遲濾波器。 5. 使用離散哈得馬德轉換設計分數延遲濾波器。以上五種設計均會和傳統的的拉格藍治最大平坦設計法和加窗法等作比較，來顯示所提方法的優越性，希望此計畫的實行可為科學的進展貢獻一點心力。第二年(2008 年) 題目：分數微積分在數位信號與系統上的應用 (Applications of fractional calculus in the digital signals and systems) 摘要：分數微積分在數學理論上已經被探討了300 多年，但在工程上的應用卻只有30 幾年的研究，這些應用包括流體力學、自動控制、電路設計、電磁理論、導熱現象和影像處理等，主要的研究包括：分數微分運算子的設計、分數積分運算子的設計和各種應用方法推導等。在本計畫中，我們將探討分數微積分在數位信號與系統上的應用，所欲執行的工作共包括下列五大項目： 1. 設計分數階FIR 和IIR 數位微分器。 2. 設計分數階FIR 和IIR 數位積分器。 3. 利用分數階導數限制設計最大平坦數位濾波器。 4. 使用分數微分運算子偵測影像中的邊緣。 5. 使用分數微分運算子抽取心電圖信號的辨識特徵。以上五個應用均會和傳統的的方法做比較，來顯示分數微分和積分運算子的用途與好處，希望此計畫的實行可為微積分的進展貢獻一點力量。第三年(2009 年) 題目：分數階希爾伯特轉換器的設計與應用 (Design and applications of fractional order digital Hilbert transformers) 摘要：希爾伯特轉換器是構建單旁波帶通訊系統的重要工具，而且二維希爾伯特轉換器亦可用來抽取數位影像中各種方向的邊緣，因此設計出優越的希爾伯特轉換器將可增進這些應用的性能，本計畫的目的便是要發展和設計分數階希爾伯特轉換器來進一步拓展傳統希爾伯特轉換器的功能，例如：分數值是建構安全性單旁波帶通訊系統的重要參數，主要的工作共包括下列五大項目： 1. 設計FIR 和IIR 數位分數希爾伯特轉換器。 2. 利用凱譜設計可調式分數階希爾伯特轉換器。 3. 利用分數階希爾伯特轉換器構建安全性單旁波帶通訊系統，並加以模擬和測試。 4. 設計二維分數階希爾伯特轉換器，並利用它來偵測影像中各種方向的邊緣。 5. 使用FPGA 實現數位分數階希爾伯特轉換器。希望上述五個工作的實行，可對信號處理的進展貢獻一點力量。[[abstract]]In recent years, the concepts of fractional operator and measure have been investigated in many engineering applications and science. Some typical examples are described as follows: fractional calculus, fractional Fourier transform, fractional delay, fractional dimension, fractional moment, fractional statistics, fractional Hilbert transform and fractional graph etc. Thus, it is an interesting topic to use these fractional operator and measure to develop new digital signal processing algorithms, that is, so-called fractional signal processing. The proposed project is a three-year plan. In the first year, the design of fractional delay filter will be studied. In the second year, the design and application of fractional differentiator and integrator will be investigated. In the third year, the design and application of fractional Hilbert transformer will be studied. Now, the abstract of each sub-project is described below: First year (2007) Title: Design of fractional delay filters using discrete signal transforms Abstract: In many applications of signal processing, there is a need for a delay which is fraction of the sampling period. These applications include time adjustment in digital receivers, beam steering of antenna array, speech coding and synthesis, modeling of music instruments, sampling rate conversion, time delay estimation, comb filter design and A/D conversion etc. So far, there have been several methods to design fractional delay filters. These include Farrow method, weighted least squares method, min-max optimization method and eigen-approach etc. In this sub-project, the discrete signal transforms will be used to design fractional delay filters. The main advantage of the proposed approach is that it is a closed-form design without using complicated iterative procedure. Five tasks are described below: 1. Use discrete Fourier transform to design fractional delay filter. 2. Use discrete Hartley transform to design fractional delay filter. 3. Use discrete cosine transform to design fractional delay filter. 4. Use discrete sine transform to design fractional delay filter. 5. Use discrete Hadamard transform to design fractional delay filter. The above five designs will be compared with conventional Lagrange maximally flat design and window-based method. Finally, we hope this project can make an effort for the progress of science and engineering. Second year (2008) Title: Applications of fractional calculus in digital signals and systems Abstract: The theory of fractional calculus has being studied before 300 years, but it is only 30 years to apply fractional calculus to solve engineering problems. These applications include fluid flow, automatic control, electrical network, electromagnetic theory, heat conduction, and image processing etc. Research topics contain the design of fractional differential and integral operators and the derivation of algorithm when fractional calculus is applied to solve practical engineering problems. In this sub-project, we will study the applications of fractional calculus in digital signals and systems. Five main tasks are described below: 1. Design of fractional order FIR and IIR differentiators. 2. Design of fractional order FIR and IIR integrators. 3. Use fractional derivative constraints to design maximally flat digital filters. 4. Use fractional differential operator to detect the edges in digital images. 5. Use fractional differential operator to extract the features of electrocardiogram. The above five applications will be compared with conventional methods to demonstrate the effectiveness of fractional calculus. Finally, we hope this project can make an effort for the progress of mathematics and engineering. Third year (2009) Title: Design and applications of fractional digital Hilbert transformer Abstract: The Hilbert transformer is an important device to construct the single sideband communication systems. Moreover, two-dimensional Hilbert transformer can be also used to detect edges in digital image with different directions. Thus, the performance of these applications can be improved if the better Hilbert transformer can be designed. The purpose of this sub-project is to develop fractional Hilbert transformer such that the capability of conventional Hilbert transformer can be extended. Five tasks are described below: 1. Design of FIR and IIR fractional Hilbert transformers. 2. Use complex cepstrum to design adjustable fractional Hilbert transformer. 3. Use fractional Hilbert transformer to construct secure single sideband communication system and use simulation to test its performance. 4. Use optimization method to design two-dimensional fractional Hilbert transformer and apply it to detect the edges in digital images. 5. Use FPGA technology to implement fractional Hilbert transformer. The above five applications will be compared with conventional methods to demonstrate the effectiveness of fractional Hilbert transformer. Finally, we hope this project can make an effort for the progress of engineering.[[note]]NSC96-2221-E327-007-MY