臺灣博碩士論文加值系統

傅立葉轉換 (Fourier transform)，是大家耳熟能詳的一個數學工具。它被廣泛的應用在工程上，訊號處理上，以及其他許多方面。
在這本論文中，我將介紹傅立葉轉換的一般化，即分數傅立葉轉換 (Fractional Fourier Transform (FRFT)) 和線性完整轉換 (Linear Canonical Transform (LCT))。
分數傅立葉轉換有一個參數alpha。當alpha= R*pi/2，則代表此分數傅立葉轉換相當於我們連續作 R 次傅立葉轉換後的結果。因此，對分數傅立葉轉換而言，當alpha=pi/2，則相當於原本的傅立葉轉換；當alpha=pi，則相當於將時間反轉的運算；當alpha= 3pi/2，則相當於逆傅立葉轉換；當alpha= 0，則什麼也沒有做。那我們可以問一個有趣的問題：當alpha = R*pi/2，且 R 不為整數時，分數傅立葉轉換會變成什麼樣子呢? 在第一章我們將看到，此時，分數傅立葉轉換相當於輸入的方程式，先乘上一個啾聲 (chirp) 方程式，再作一個經放大或縮小的傅立葉轉換 (scaled Fourier transform)，最後再乘上一個啾聲方程式。
至於線性完整轉換，則比分數傅立葉轉換更加一般化。它共有四個參數 {a, b, c, d}。分數傅立葉轉換可視為線性完整轉換當 {a, b, c, d} = {cos(alpha), sin(alpha), -sin(alpha), cos(alpha)} 時的特例。
分數傅立葉轉換比傳統傅立葉轉換靈活，而線性完整轉換則比分數傅立葉轉換更靈活。由於它們比傅立葉轉換靈活，因此它們的效用，比傅立葉轉換更強。它們可以成功的處理一些無法用傅立葉轉換妥善處理的問題。
近年來，關於分數傅立葉轉換的研究，可以說是蓬勃發展。它被廣泛的用在各種不同的應用上 (關於分數與線性完整轉換的應用，在第七章的圖7-1，我們有作系統的整理)。至於關於線性完整轉換，雖然目前對它的研究相對上比較少，但由於它的靈活度高，因此在未來它具有很強的發展潛力。
在這本論文中，我將把目前學術界對於分數傅立葉轉換及完整傅立葉轉的研究結果，作一個有系統的整理與介紹。裏面包含幾年來，教授與我關於分數傅立葉轉換及完整傅立葉轉的研究成果。
在第一章，我將介紹分數及線性完整轉換的定義與基本概念。
在第二到五章，我將介紹分數及線性完整轉換的性質，尤其是它們與韋格納分布方程式 (Wigner distribution function) 與其他時頻分析工具的關係 (第三章)，以及它們的固有函數 (eigenfunctions) (第四，五章)。
第六章，我將說明如何裝置分數及線性完整轉換。第七章，我將介紹分數及線性完整轉換的各種應用。
在第八，九，十章，我將介紹離散分數及線性完整轉換。第八章是作通盤的介紹，第九章是詳加介紹其中一種，即可公式化的離散分數及線性完整轉換，第十章則討論有加乘性的離散分數傅立葉轉換的固有函數性質。
第十一章，我將討論簡化型態的分數傅立葉轉換。第十二章，我將討論二維的線性完整轉換。第十三章，我將討論分數、完整及簡化型態的分數正弦、餘弦、哈特里 (Hartley) 轉換。第十四章，我將討論分數希爾伯特 (Hilbert) 轉換。第十五章，我討論其他與分數及線性完整轉換相關的轉換與運算。
在第十六章，我作個結論。在參考資料 (Reference) 部分，我也對目前有關分數及線性完整轉換的論文，加以分類整理。
希望這本論文對您有幫助。

Fourier transform (FT) is a very popular mathematical tool. It has been widely applied in engineering, signal processing, etc.
In this thesis, we will introduce the generalization of FT, i.e., fractional Fourier transform (FRFT) and linear canonical transform (LCT).
FRFT has one parameter alpha. The FRFT with parameter alpha= R*pi/2 just means we do FT for R times. So when alpha= pi/2, FRFT becomes the conventional FT. When alpha= pi, FRFT be-comes the time-reverse operation. When alpha= 3pi/2, FRFT becomes the inverse Fourier transform (IFT). And when alpha= 0, FRFT becomes the identity operation. Now we can ask an interesting question: What does the FRFT becomes when alpha= R*pi/2 and R isn’t an integer number? In Chap. 1, we will see in this case FRFT will correspond to the input function multiplied by a chirp function, then transformed by a scaled Fourier transform, then multiplied by a chirp function.
LCT, however, is the further generalization of FRFT. LCT has 4 parameters {a, b, c, d}, and FRFT is the special case of LCT that {a, b, c, d} = {cos(alpha), sin(alpha), -sin(alpha), cos(alpha)}.
FRFT is more flexible than FT, and LCT is more flexible than FRFT. Since FRFT and LCT are more flexible than FT, so their utilities are stronger than FT. They can solve some problems that can’t be solved well by FT.
Recently, there are many research works about FRFT. FRFT has been used for many applications (We use Fig. 7-1 to list the applications of FRFT and LCT systematically). Until now, the research works about LCT are not as many as those of FTFT. But since LCT is very flexible, so it has a lot of potentiality in the future.
In this thesis, I will introduce the research works about FRFT and LCT systematically, including the research works of my professor and I.
In Chap. 1, I will introduce the definition and basic ideas of FRFT and LCT.
In Chap. 25, I will introduce the properties of FRFT and LCT, especially their relation with Wigner discuss function (WDF) and other time-frequency analysis tool (Chap. 3), and their eigenfunctions (Chaps. 4, 5).
In Ch. 6, I will illustrate how to implement FRFT and LCT. In Chap. 7, I will introduce all the applications of FRFT and LCT.
In Chaps. 8, 9, 10, I will introduce discrete FRFT and LCT. In Chap. 8 I will make an overview introduction. In Chap. 9 we will introduce one type of discrete FRFT, i.e., the closed form discrete FRFT in detail. In Chap. 10, discuss the eigenfunctions of the discrete FRFT that has additivity property.
In Chap. 11, I will discuss the simplified FRFT. In Chap. 12, I will discuss 2-D FRFT / LCT. In Chap. 13, I will discuss the fractional, canonical, and simplified fractional sine, cosine, and Hartley transforms. In Chap. 14, I will discuss the fractional Hilbert transform. In Chap. 15, I will discuss other transforms and operations related to FRFT and LCT.
In Chap. 16, I make a conclusion. In the Reference, I will list the papers and books related to FRFT and LCT, and classify them.
May this thesis be helpful for you.

Cover
Contents
Part 1: Introduction
Chap. 1 Introduction for Fractional Fourier Transform (FRFT) and Linear Canonical Transform (LCT)
Part 2: Properties of FRFT and LCT
Chap. 2 Basic Properties and Transform results of FRFT and LCT
Chap. 3 Relations between Fractional Operations and TimeFrequency Distributions, and Their Applications
Chap. 4 Eigenfunctions of Linear Canonical Transform
Chap. 5 Properties and Applications of the Eigenfunctions of LCT
Part 3: Implementation and Applications of FRFT / LCT
Chap. 6 Implementation Algorithm of FRFT and LCT
Chap. 7 Applications of FRFT and LCT
Part 4: Discrete FRFT and LCT
Chap. 8 Several Types of Discrete FRFT and LCT
Chap. 9 Closed Form Discrete FRFT and LCT
Chap. 10 SelfDiscrete Fractional Fourier Vectors
Part 5: The Transforms and Operations Related to FRFT and LCT
Chap. 11 Simplified Fractional Fourier Transform (SFRFT)
Chap. 12 Two-Dimensional Affine Generalized Fractional Fourier Transform
Chap. 13 Fractional, Canonical, and Simplified Fractional Cosine, Sine and Hartley Transforms
Chap. 14 On Fractional Hilbert Transform
Chap. 15 Other Transforms and Operations Related to FRFT and LCT
Part 6: Conclusions and Reference
Chap. 16 Conclusions and Future Works
Reference

A. Fractional Fourier transform
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D. Wigner distribution and other time-frequency analysis tools and their relations with FRFT / LCT
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E. Eigenfunctions of FRFT / LCT
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[E10] T. Alieva, ‘On the self-fractional Fourier functions’, J. Phys. A: Math. Gen., vol. 29, p 377-379, 1996.
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F. Implementation of FRFT and LCT
[F1] H. M. Ozaktas, O. Arikan, ‘Digital computation of the fractional Fourier transform’, IEEE Trans. on Signal Proc., vol. 44, no. 9, p. 2141-2150, Sep. 1996.
[F2] X. Deng, B. Bihari, J. Gan, F. Zhou, and R. T. Chen, ‘Fast algorithm for chirp trans-forms with zoomingin ability and its applications’, J. Opt. Soc. Am. A, vol. 17, no. 4, p. 762-771, Apr. 2000.
[F3] X. G. Xia, ‘On bandlimited signals with fractional Fourier transform’, IEEE Signal Processing Letters, vol. 3, no. 3, p. 72-74, Mar. 1996.
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G. Applications of FRFT / LCT for optical system, radar system, and GRIN medium analysis
[G1] L. M. Bernardo, ‘ABCD matrix formalism of fractional Fourier optics’, Optical Eng., vol. 35, no. 3, p 732-740, March 1996.
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[G3] A. W. Lohmann, D. Mendlovic, and Z. Zalevsky, ‘Fractional transformations in op-tics’, Progress in Optics, vol. 38, Chapter 4, Elsevier, 1998.
[G4] D. Mendlovic, H. M. Ozaktas, and A. W. Lohmann, ‘Graded-index fibers, Wigner distribution and the fractional Fourier transform’ Appl. Opt., vol. 33, p. 6188-6193, 1994.
[G5] L. Yu, M. Huang, L. Wu, Y. Lu, W. Huang, M. Chen, and Z. Zhu, ‘Fractional Fourier transform and the elliptic gradient-index medium’, Opt. Commun., vol. 152, p 23-25, Jun. 1998.
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H. Applications of FRFT and LCT for filter design
[H1] M. A. Kutay, H. M. Ozaktas, O. Arikan, and L. Onural, ‘Optimal filter in fractional Fourier domains’, IEEE Trans. Signal Processing, vol. 45, no. 5, p 1129-1143, May 1997.
[H2] Z. Zalevsky and D. Mendlovic, ‘Fractional Wiener filter’, Appl. Opt., vol. 35, no. 20, p. 3930-3936, Jul. 1996.
[H3] B. Barshan, M. A. Kutay, H. M. Ozaktas, ‘Optimal filters with linear canonical trans-formations’, Opt. Commun., vol. 135, p. 32-36, 1997.
[H4] L. L. Scharf and J. K. Thomas, ‘Wiener filters in canonical coordinates for transform coding, filtering, and quantizing’, IEEE Trans. Signal Processing, vol. 46, no. 3, p. 647-654, Mar. 1998.
[H5] M. F. Erden and H. M. Ozaktas, ‘Synthesis of general linear systems with repeated filtering in consecutive fractional Fourier domains’, J. Opt. Soc. Am. A., vol. 15, p. 1647-1657, no. 6, Jun. 1998.
[H6] D. Mendlovic, Z. Zalevsky, A. W. Lohmann, and R. G. Dorsch, ‘Signal spatial-filtering using the localized fractional Fourier transform’, Opt. Commun., vol. 126, p. 14-18, May 1996.
I. Fractional and canonical correlation
[I1] D. Mendlovic, H. M. Zalevsky, and A. W. Lohmann, ‘Fractional correlation’, Appl. Opt., vol. 34, no. 2, p 303-309, Jan. 1995.
[I2] A. W. Lohmann, Z. Zalevsky, and D. Mendlovic, ‘Synthesis of pattern recognition fil-ters for fractional Fourier processing’, Opt. Commun., vol. 128, p 199-204, Jul. 1996.
[I3] A. W. Lohmann and D. Mendlovic, ‘Fractional joint transform correlator’, Appl. Opt., vol. 36, no. 29, p. 7402-7407, Oct. 1997.
[I4] S. Granieri, R. Arizaga, and E. E. Sicre, ‘Optical correlation based on the fractional Fourier transform’, Appl. Opt., vol. 36, no. 26, p. 6636-6645, Sep. 1997.
[I5] A. M. Almanasreh and M. G. Abushagur, ‘Fractional correlation based on the modified fractional order Fourier transform’, Opt. Eng., vol. 37, no. 1, p 175-184, Jan. 1998.
[I6] M. A. Kutay and H. M. Ozaktas, ‘Optimal image restoration with the fractional Fourier transform’, J. Opt. Soc. Am. A, vol. 15, no. 4, p 825-833, Apr. 1998.
[I7] D. Mendlovic, ‘Fractional triple correlation and its applications’, J. Opt. Soc. Am. A, vol. 15, no. 6, p. 1658-1661, Jun. 1998.
J. Fractional and canonical convolution
[J1] H. M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, ‘Convolution, filtering, and multiplexing in fractional Fourier domains and their rotation to chirp and wavelet transform’, J. Opt. Soc. Am. A, vol. 11, no. 2, p. 547-559, Feb. 1994.
[J2] L. B. Almeida, ‘Product and convolution theorems for the fractional Fourier transform’, IEEE Signal Processing Letters, vol. 4, no. 1, p. 15-17, Jan. 1997.
[J3] A. I. Zayed, ‘A convolution and product theorem for the fractional Fourier transform’, IEEE Signal Processing Letters, vol. 5, no. 4, p. 101-103, Apr. 1998.
K. Fractional / canonical Hilbert transform
[K1] A. W. Lohmann, D. Mendlovic, and Z. Zalevsky, ‘Fractional Hilbert transform’, Opt. Lett., vol. 21, no. 4, p 281-283, Feb. 1996.
[K2] A. I. Zayed, ‘Hilbert transform associated with the fractional Fourier transform’, IEEE Signal Processing Letters, vol. 5, no. 8, p. 206-208, Aug. 1998.
[K3] C. C. Tseng and S. C. Pei, ‘Design and application of discrete-time fractional Hilbert transformer’, IEEE Trans. on Circuits and Systems, Part II : Analog and Digital Sig-nal Processing, vol. 47, no. 12, p. 1529-1533, Dec. 2000. .
[K4] S. C. Pei and M. H. Yeh, ‘Discrete fractional Hilbert transform’, Proc. of IEEE Int'l Symp. on Circuits and Systems, vol. 4, p. 506-509, Jun. 1998.
[K5] S. C. Pei and P. W. Wang, ‘Analytical design of digital nonrecursive maximally flat fractional Hilbert transformer’, Proc. of IEEE Int’l Symp. on Circuits and Systems, Orlando, vol. 3, p. 175-178, Jun. 1999.
[K6] C. C. Tseng and S. C. Pei, ‘Discrete-time Hilbert transformer’, Proc. of IEEE Int’l Symp. on Circuits and Systems, Geneva, Switzerland, May 2000.
[K7] A. W. Lohmann, J. Ojeda-Castañeda, and L. Diaz-Santana, ‘Fractional Hilbert trans-form: optical implementation for 1-D objects’, Opt. Mem. Neural Networks, vol. 5, p 131-135, 1996.
[K8] A. W. Lohmann, E. Tepichin, and J. G. Ramirez, ‘Optical implementation of the frac-tional Hilbert transform for twodimensional objects’, Appl. Opt., vol. 36, p 6620-6626, 1997.
[K9] J. A. Davis, D. E. McNamara, and D. M. Cottrell, ‘Analysis of the fractional Hilbert transform’, Appl. Opt., vol. 37, no. 29, p 6911-6913, Oct. 1998.
L. Other applications of FRFT and LCT
[L1] G. Z. Yang, B. Z. Dong, B. Y. Gu, J. Y. Zhuang, and O. K. Ersoy, ‘Gerchberg-Saxton and Yang-Gu algorithms for phase retrieval in a nonunitary transform system: a com-parison’, Appl. Opt., vol. 33, no. 2, p. 209-218, Jan. 1994.
[L2] Z. Zalevsky, D. Mendlovic, and R. G. Dorsch, ‘Gerchberg-Saxton algorithm applied in the fractional Fourier or the Fresnel domain’, Opt. Lett., vol. 21, no. 12, p. 342-344, Jun. 1996.
[L3] B. Z. Dong, Y. Zhang, B. Y. Gu, and G. Z. Yang, ‘Numerical investigation of phase retrieval in a fractional Fourier transform’, J. Opt. Soc. Am. A, vol. 14, no. 10, p. 2709-2713, 1997.
[L4] W. X. Cong, N, X. Chen, and B. Y. Gu, ‘Recursive algorithm for phase retrieval in the fractional Fourier transform domain’, Appl. Opt., vol. 37, no. 29, p 6906- 6910, Oct. 1998.
[L5] Y. Zhang, B. Z. Dong, B. Y. Gu, and G. Z. Yang, ‘Beam shaping in the fractional Fourier transform domain’, J. Opt. Soc. Am. A, vol. 15, no. 5, p. 1114-1120, May 1998.
[L6] S. Liu, L. Yu, and B. Zhu, ‘Optical image encryption by cascaded fractional Fourier transforms with random phase filtering’, Opt. Commun., vol. 187, p. 57-63, Jan. 2001.
[L7] F. H. Kerr, ‘Namias’ fractional Fourier transforms on L2 and applications to differenti-al equations’, Journal of Mathematical Analysis and Applications, vol. 136, p. 404-418, 1988.
[L8] I. S. Yetik, H. M. Ozaktas, B. Barshan, and L. Onural, ‘Perspective projections in the space-frequency plane and fractional Fourier transforms’, J. Opt. Soc. Am. A, vol. 17, no. 12, p. 2382-2390, Dec. 2000.
[L9] A. Bultan, ‘A four-parameter atomic decomposition of chirplets’, IEEE Trans. Signal Processing, vol. 47, no. 3, p. 731-745, Mar. 1999.
[L10] C. Mendlovic and A. W. Lohmann, ‘Space-bandwidth product adaption and its appli-cation to superresolution: fundamentals’, J. Opt. Soc. Am. A, vol. 14, no. 3, p. 558-562, Mar. 1997.
M. Other operations related to FRFT and LCT
[M1] Z. Zalevsky and D. Mendlovic, ‘Fractional Radon transform: definition’, Appl. Opt., vol. 35, no. 23, p. 4628-4631, Aug. 1996.
[M2] D. Dragoman, ‘Fractional Fourier-related functions’, Opt. Commun., vol.128, p 91-98, July 1996.
[M3] D. Dragoman and M. Dragoman, ‘Temporal implementation of Fourier-related trans-forms’, Opt. Commun., vol. 145, p 33-37, Jan. 1998.
[M4] D. Mendlovic, Z. Zalevsky, D. Mas, J. Garcia, and C. Ferreira, ‘Fractional wavelet transform’, Appl. Opt., vol. 36, no. 20, p. 4801-4806, Jul. 1997.
N. The transforms related to FRFT and LCT
[N1] S. C. Pei and J. J. Ding, ‘Simplified fractional Fourier transforms’, J. Opt. Soc. Am. A, vol. 17, no. 12, p. 2355-2367, Dec., 2000.
[N2] S. C. Pei and J. J. Ding, ‘Fractional, canonical, and simplified fractional cosine trans-forms’, to appear in ICASSP 2001.
[N3] S. C. Pei and J. J. Ding, ‘Fractional, canonical, and simplified fractional cosine, sine and Hartley transforms’, submitted to IEEE Trans. Signal Processing.
[N4] Y. Huang and B. Suter, ‘The fractional wave packet transform’, Multidimensional Systems and Signal Processing, vol. 9, p. 399-402, 1998.
[N5] Y. Zhang, B. Y. Gu, B. Z. Dong, and G. Z. Yang, ‘A new kind of windowed fractional transforms’, Opt. Commun., vol. 152, p. 127-134, Jun. 1998.
[N6] C. C. Shih, ‘Fractionalization of Fourier transform’, Opt. Commun., vol. 118, p. 495-498, Aug. 1995.
[N7] S. Liu, J. Jiang, Y. Zhang, and J. Zhang, ‘Generalized fractional Fourier transforms’, J. Phys. A: Math. Gen., vol. 30, p. 973-981, 1997.
[N8] G. Cariolaro, T. Erseghe, P. Kraniauskas, and N. Laurenti, ‘A unified framework for the fractional Fourier transform’, IEEE Trans. Signal Processing., v. 46, n. 12, p 3206-3219, Dec. 1998.
[N9] G. Cariolaro, T. Erseghe, P. Kraniauskas, and N. Laurenti, ‘Multiplicity of fractional Fourier transforms and their relationships’, IEEE Trans. Signal Processing, vol. 48, no. 1, p. 227-241, Jan. 2000.
[N10] A. W. Lohmann, D. Mendlovic, Z. Zalevsky, and R. G. Dorsch, ‘Some important fractional transforms for signal processing’, Opt. Commun., vol. 125, p 18-20, Apr. 1996.
[N11] S. Liu, J. Zhang, and Y. Zhang, ‘Properties of the fractionalization of a Fourier trans-form’, Opt. Commun., vol. 133, p. 50-54, Jan. 1997.
[N12] Y. Zhang, B. Y. Gu, B. Z. Dong, G. Z. Yang, H. Ren, X. Zhang, and S. Liu, ‘Frac-tional Gabor transform’, Opt. Lett., vol. 22, no. 21, p. 1583-1585, Nov. 1997.
O. Two-Dimensional FRFT and LCT
[O1] G. B. Folland, “Harmonic Analysis in Phase Space”, the Annals of Math. Studies vol. 122, Princeton University Press, 1989.
[O2] J. J. Ding and S. C. Pei, ‘2D affine generalized fractional Fourier transform’, IC-ASSP’99, vol. 6, p. 31813184, 1999.
[O3] S. C. Pei and J. J. Ding, ‘Two-dimensional affine generalized fractional Fourier trans-form’, to appear in IEEE Trans. Signal Processing, Apr. 2001.
[O4] A. Sahin, H. M. Ozaktas, and D. Mendlovic, ‘Optical implementation of two-dimensional fractional Fourier transforms and linear canonical transforms with arbi-trary parameters’, Appl. Opt., vol. 37, no. 11, p 2130-2141, Apr 1998.
[O5] A. Sahin, M. A. Kutay, and H. M. Ozaktas, ‘Nonseparable two-dimensional fractional Fourier transform`, Appl. Opt., vol. 37, no. 23, p 5444-5453, Aug 1998.
[O6] V. Namias, ‘Fractionalization of Hankel transforms’, J. Inst. Maths Applics, vol. 26, p. 187-197, 1980.
[O7] L. Yu, W. Huang, M. Huang, Z. Zhu, X. Zeng, and W. Ji, ‘The LaguerreGaussian series representation of two-dimensional fractional Fourier transform’, J. Phys A: Math. Gen., vol. 31, p. 9353-9357, 1998.
[O8] Z. Zalevsky, D. Mendlovic, and A. W. Lohmann, ‘The ABCDBessel transformation’, Opt. Commun., vol. 147, p. 39-41, Feb. 1998.
P. Discrete FRFT and LCT
[P1] S. C. Pei and J. J. Ding, ‘Closed form discrete fractional and affine Fourier transforms’, IEEE Trans. Signal Processing, vol. 48, no. 5, p. 1338-1353, May 2000.
[P2] S. C. Pei and M. H. Yeh, ‘Improved discrete fractional Fourier transform’, Opt. Lett., July 1997, p 1047-1049.
[P3] S. C. Pei, C. C. Tseng, and M. H. Yeh, ‘Discrete fractional Hartley and Fourier trans-form’, IEEE Trans. on Circuits and Systems, II: Analog and Digital Signal Processing, vol. 45, no. 6, p 665-675, June 1998.
[P4] S. C. Pei, M. H. Yeh, and C. C. Tseng, ‘Discrete fractional Fourier transform based on orthogonal projection’, IEEE Trans. Signal Processing., vol. 47, no. 5, p 1335-1348, May 1999.
[P5] S. C. Pei and C. C. Tseng, ‘New discrete fractional Fourier transform based on con-strained eigendecomposition of DFT matrix by Lagrange multiplier method’, IC-ASSP’97 v. 5, p 3965-3968, 1997.
[P6] B. Santhanam and J. H. McClellan, ‘The discrete rotational Fourier transform’, IEEE Trans. Signal Processing., vol. 42, p 994-998, Apr 1996.
[P7] M. S. Richman and T. W. Parks, ‘Understanding discrete rotations’, ICASSP’97, vol. 3, p 2057-2060, 1997.
[P8] S. C. Pei and M. H. Yeh, "Two-dimensional discrete fractional Fourier transform", Signal Processing, vol. 67, p 99-108, Feb. 1998.
[P9] S. C. Pei, M. H. Yeh and T. L. Luo, “Fractional Fourier Series expansion for finite sig-nals and dual extension to discrete-time fractional Fourier transform”, IEEE Trans. Signal Processing, vol. 47, no. 10, p. 2883-2888, Oct. 1999.
[P10] T. Alieva and A. Barbe, ‘Fractional Fourier and Radon-Wigner transforms of periodic signals’, Signal Processing, vol. 69, p. 183-189, 1998.
[P11] O. Arikan, M. A. Kutay, H. M. Ozaktas, and O. K. Akdemir, ‘The discrete fractional Fourier transformation’, Proceedings of IEEE International Symposium on Time-Frequency and Time-Scale Analysis, p 205-207, 1996.
Q. Others
[Q1] M. Abramowitz and I. A. Stegun, “Handbook of Mathematical Functions, with For-mula, Graphs and Mathematical Tables”, Dover Publication, New York, 1965.
[Q2] M. R. Spiegel, “Mathematical Handbook of Formulas and Tables”, McGraw-Hill, 1990.
[Q3] J. W. Goodman, “Introduction to Fourier Optics”, McGraw-Hill, 2nd ed., 1988.
[Q4] S. G. Lipson and H. Lipson, “Optical Physics”, 2nd Ed., Cambridge U. Press, Cam-bridge, 1981, p 190-192.
[Q5] J. T. Winthrop and C. R. Worthington, ‘Theory of Fresnel images. 1. Plane periodic objects in monochromatic light’, J. Opt. Soc. Am., vol. 55, p 373-381, 1965.
[Q6] A. W. Lohmann, ‘An array illuminator based on the Talbot effect’, Optik (Stuttgart), vol. 79, p 41-45, 1988.
[Q7] J. Leger and G. J. Swanson, ‘Efficient array illuminator using binary-optics phase plates as fractional Talbot planes’, Opt. Lett., vol. 15, p 288-290, 1990.
[Q8] G. W. Wronell, “Signal Processing with Fractals”, Prentice-Hall, Upper Saddle River, New Jersey, 1996.
[Q9] M. L. Curtis, “Matrix Groups”, 2nd ed., Springer-Verlag, 1979.
[Q10] J. J. Ding, “Derivation and Properties of Orthogonal transform”, Master Thesis, Na-tional Taiwan University, 1997.
[Q11] M. H. Yeh, “Research of Fractional Signal Transforms”, Chap. 8, Ph.D. Thesis, Na-tional Taiwan University, 1997.
[Q12] G. Mandyam and N. Ahmed, ‘The discrete Laguerre transform: derivation and appli-cations’, IEEE Trans. Signal Processing., vol. 44, no. 12, p. 2925-2931, Dec. 1996.