The Fractional Fourier Transform and Its Applications Presenter: Pao-Yen Lin Research Advisor: Jian-Jiun Ding, Ph. D. Assistant professor Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University

Outlines Introduction Fractional Fourier Transform (FrFT) Linear Canonical Transform (LCT) Relations to other Transformations Applications 2016/1/7 2 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University

Introduction Generalization of the Fourier Transform Categories of Fourier Transform a) Continuous-time aperiodic signal b) Continuous-time periodic signal (FS) c) Discrete-time aperiodic signal (DTFT) d) Discrete-time periodic signal (DFT) 2016/1/7 3 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University

Fractional Fourier Transform (FrFT) Notation is a transform of 2016/1/7 4 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University

Fractional Fourier Transform (FrFT) (cont.) Constraints of FrFT Boundary condition ‚Additive property 2016/1/7 5 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University

Definition of FrFT Eigenvalues and Eigenfunctions of FT Hermite-Gauss Function 2016/1/7 6 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University

Definition of FrFT (cont.) Eigenvalues and Eigenfunctions of FT 2016/1/7 7 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University

Definition of FrFT (cont.) Eigenvalues and Eigenfunctions of FrFT Use the same eigenfunction but α order eigenvalues 2016/1/7 8 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University

Definition of FrFT (cont.) Kernel of FrFT 2016/1/7 9 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University

Definition of FrFT (cont.) 2016/1/7 10 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University

Properties of FrFT Linear. The first-order transform corresponds to the conventional Fourier transform and the zeroth-order transform means doing no transform. Additive. 2016/1/7 11 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University

Linear Canonical Transform (LCT) Definition where 2016/1/7 12 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University

Linear Canonical Transform (LCT) (cont.) Properties of LCT 1.When, the LCT becomes FrFT. 2.Additive property where 2016/1/7 13 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University

Relation to other Transformations Wigner Distribution Chirp Transform Gabor Transform Gabor-Wigner Transform Wavelet Transform Random Process 2016/1/7 14 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University

Relation to Wigner Distribution Definition Property 2016/1/7 15 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University

Relation to Wigner Distribution 2016/1/7 16 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University

Relation to Wigner Distribution (cont.) WD V.S. FrFT Rotated with angle 2016/1/7 17 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University

Relation to Wigner Distribution (cont.) Examples 2016/1/7 18 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University slope=

Relation to Chirp Transform for Note that is the same as rotated by 2016/1/7 19 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University

Relation to Chirp Transform (cont.) Generally, 2016/1/7 20 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University

Relation to Gabor Transform (GT) Special case of the Short-Time Fourier Transform (STFT) Definition 2016/1/7 21 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University

Relation to Gabor Transform (GT) (cont.) GT V.S. FrFT Rotated with angle 2016/1/7 22 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University

Relation to Gabor Transform (GT) (cont.) Examples (a)GT of (b)GT of (c)GT of (d)WD of 2016/1/7 23 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University

GT V.S. WD GT has no cross term problem GT has less complexity WD has better resolution Solution: Gabor-Wigner Transform 2016/1/7 24 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University

Relation to Gabor-Wigner Transform (GWT) Combine GT and WD with arbitrary function 2016/1/7 25 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University

Relation to Gabor-Wigner Transform (GWT) (cont.) Examples 1.In (a) 2.In (b) 2016/1/7 26 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University

Relation to Gabor-Wigner Transform (GWT) (cont.) Examples 3.In (c) 4.In (d) 2016/1/7 27 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University

Relation to Wavelet Transform The kernels of Fractional Fourier Transform corresponding to different values of can be regarded as a wavelet family. 2016/1/7 28 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University

Relation to Random Process Classification 1.Non-Stationary Random Process 2.Stationary Random Process Autocorrelation function, PSD are invariant with time t 2016/1/7 29 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University

Relation to Random Process (cont.) Auto-correlation function Power Spectral Density (PSD) 2016/1/7 30 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University

Relation to Random Process (cont.) FrFT V.S. Stationary random process Nearly stationary 2016/1/7 31 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University

Relation to Random Process (cont.) FrFT V.S. Stationary random process for 2016/1/7 32 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University

Relation to Random Process (cont.) FrFT V.S. Stationary random process PSD: 2016/1/7 33 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University

Relation to Random Process (cont.) FrFT V.S. Non-stationary random process Auto-correlation function PSD rotated with angle 2016/1/7 34 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University

Relation to Random Process (cont.) Fractional Stationary Random Process If is a non-stationary random process but is stationary and the autocorrelation function of is independent of, then we call the -order fractional stationary random process. 2016/1/7 35 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University

Relation to Random Process (cont.) Properties of fractional stationary random process 1.After performing the fractional filter, a white noise becomes a fractional stationary random process. 2.Any non-stationary random process can be expressed as a summation of several fractional stationary random process. 2016/1/7 36 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University

Applications of FrFT Filter design Optical systems Convolution Multiplexing Generalization of sampling theorem 2016/1/7 37 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University

Filter design using FrFT Filtering a known noise Filtering in fractional domain 2016/1/7 38 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University signal noise

Filter design using FrFT (cont.) Random noise removal If is a white noise whose autocorrelation function and PSD are: After doing FrFT Remain unchanged after doing FrFT! 2016/1/7 39 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University

Filter design using FrFT (cont.) Random noise removal Area of WD ≡ Total energy 2016/1/7 40 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University signal

Optical systems Using FrFT/LCT to Represent Optical Components Using FrFT/LCT to Represent the Optical Systems Implementing FrFT/LCT by Optical Systems 2016/1/7 41 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University

Using FrFT/LCT to Represent Optical Components 1.Propagation through the cylinder lens with focus length 2.Propagation through the free space (Fresnel Transform) with length 2016/1/7 42 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University

Using FrFT/LCT to Represent the Optical Systems 2016/1/7 43 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University input output

Implementing FrFT/LCT by Optical Systems All the Linear Canonical Transform can be decomposed as the combination of the chirp multiplication and chirp convolution and we can decompose the parameter matrix into the following form 2016/1/7 44 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University

Implementing FrFT/LCT by Optical Systems (cont.) 2016/1/7 45 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University input output The implementation of LCT with 1 cylinder lens and 2 free spaces input output The implementation of LCT with 2 cylinder lenses and 1 free space

Convolution Convolution in domain Multiplication in domain 2016/1/7 46 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University

Convolution (cont.) 2016/1/7 47 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University

Multiplexing using FrFT 2016/1/7 48 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University TDM FDM

Multiplexing using FrFT 2016/1/7 49 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University Inefficient multiplexing Efficient multiplexing

Generalization of sampling theorem If is band-limited in some transformed domain of LCT, i.e., then we can sample by the interval as 2016/1/7 50 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University

Generalization of sampling theorem (cont.) 2016/1/7 51 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University

Conclusion and future works Other relations with other transformations Other applications 2016/1/7 52 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University

References [1] Haldun M. Ozaktas and M. Alper Kutay, “Introduction to the Fractional Fourier Transform and Its Applications,” Advances in Imaging and Electron Physics, vol. 106, pp. 239~286. [2] V. Namias, “The Fractional Order Fourier Transform and Its Application to Quantum Mechanics,” J. Inst. Math. Appl., vol. 25, pp , [3] Luis B. Almeida, “The Fractional Fourier Transform and Time-Frequency Representations,” IEEE Trans. on Signal Processing, vol. 42, no. 11, November, [4] H. M. Ozaktas and D. Mendlovic, “Fourier Transforms of Fractional Order and Their Optical Implementation,” J. Opt. Soc. Am. A 10, pp , [5] H. M. Ozaktas, B. Barshan, D. Mendlovic, L. Onural, “Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms,” J. Opt. Soc. Am. A, vol. 11, no. 2, pp , Feb [6] A. W. Lohmann, “Image Rotation, Wigner Rotation, and the Fractional Fourier Transform,” J. Opt. Soc. Am. 10,pp , [7] S. C. Pei and J. J. Ding, “Relations between Gabor Transform and Fractional Fourier Transforms and Their Applications for Signal Processing,” IEEE Trans. on Signal Processing, vol. 55, no. 10, pp , Oct /1/7 53 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University

References [8] Haldun M. Ozaktas, Zeev Zalevsky, M. Alper Kutay, The fractional Fourier transform with applications in optics and signal processing, John Wiley & Sons, [9] 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, pp , May [10] J. J. Ding, “Research of Fractional Fourier Transform and Linear Canonical Transform,” Doctoral Dissertation, National Taiwan University, [11] C. J. Lien, “Fractional Fourier transform and its applications,” National Taiwan University, June, [12] Alan V. Oppenheim, Ronald W. Schafer and John R. Buck, Discrete-Time Signal Processing, 2 nd Edition, Prentice Hall, [13] R. N. Bracewell, The Fourier Transform and Its Applications, 3 rd ed., Boston, McGraw Hill, /1/7 54 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University

Chenquieh! 2016/1/7 55 Digital Image and Signal Processing Lab Graduate Institute of Communication Engineering National Taiwan University