1. 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

2. Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
Outlines
Introduction
Fractional Fourier Transform (FrFT)
Linear Canonical Transform (LCT)
Relations to other Transformations
Applications

3. Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
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)

4. Fractional Fourier Transform (FrFT)
Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
Fractional Fourier Transform (FrFT)
Notation
is a transform of
is a transform of

5. Fractional Fourier Transform (FrFT) (cont.)
Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
Fractional Fourier Transform (FrFT) (cont.)
Constraints of FrFT
Boundary condition
Additive property

6. Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
Definition of FrFT
Eigenvalues and Eigenfunctions of FT
Hermite-Gauss Function

7. Definition of FrFT (cont.)
Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
Definition of FrFT (cont.)
Eigenvalues and Eigenfunctions of FT

8. Definition of FrFT (cont.)
Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
Definition of FrFT (cont.)
Eigenvalues and Eigenfunctions of FrFT
Use the same eigenfunction but α order eigenvalues

9. Definition of FrFT (cont.)
Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
Definition of FrFT (cont.)
Kernel of FrFT
把An 代入上頁的結果

10. Definition of FrFT (cont.)
Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
Definition of FrFT (cont.)

11. Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
Properties of FrFT
Linear.
The first-order transform corresponds to the conventional Fourier transform and the zeroth-order transform means doing no transform.
Additive.

12. Linear Canonical Transform (LCT)
Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
Linear Canonical Transform (LCT)
Definition
where

13. Linear Canonical Transform (LCT) (cont.)
Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
Linear Canonical Transform (LCT) (cont.)
Properties of LCT
When , the LCT becomes
FrFT.
Additive property
where

14. Relation to other Transformations
Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
Relation to other Transformations
Wigner Distribution
Chirp Transform
Gabor Transform
Gabor-Wigner Transform
Wavelet Transform
Random Process

15. Relation to Wigner Distribution
Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
Relation to Wigner Distribution
Definition
Property

16. Relation to Wigner Distribution
Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
Relation to Wigner Distribution

17. Relation to Wigner Distribution (cont.)
Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
Relation to Wigner Distribution (cont.)
WD V.S. FrFT
Rotated with angle

18. Relation to Wigner Distribution (cont.)
Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
Relation to Wigner Distribution (cont.)
Examples
slope=

19. Relation to Chirp Transform
Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
Relation to Chirp Transform
for
Note that
is the same as rotated by

20. Relation to Chirp Transform (cont.)
Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
Relation to Chirp Transform (cont.)
Generally,

21. Relation to Gabor Transform (GT)
Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
Relation to Gabor Transform (GT)
Special case of the Short-Time Fourier Transform (STFT)
Definition

22. Relation to Gabor Transform (GT) (cont.)
Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
Relation to Gabor Transform (GT) (cont.)
GT V.S. FrFT
Rotated with angle

23. Relation to Gabor Transform (GT) (cont.)
Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
Relation to Gabor Transform (GT) (cont.)
Examples
(a)GT of (b)GT of (c)GT of (d)WD of

24. Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
GT V.S. WD
GT has no cross term problem
GT has less complexity
WD has better resolution
Solution: Gabor-Wigner Transform

25. Relation to Gabor-Wigner Transform (GWT)
Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
Relation to Gabor-Wigner Transform (GWT)
Combine GT and WD with arbitrary function

26. Relation to Gabor-Wigner Transform (GWT) (cont.)
Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
Relation to Gabor-Wigner Transform (GWT) (cont.)
Examples
In (a)
In (b)

27. Relation to Gabor-Wigner Transform (GWT) (cont.)
Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
Relation to Gabor-Wigner Transform (GWT) (cont.)
Examples
In (c)
In (d)

28. Relation to Wavelet Transform
Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
Relation to Wavelet Transform
The kernels of Fractional Fourier Transform corresponding to different values of can be regarded as a wavelet family.

29. Relation to Random Process
Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
Relation to Random Process
Classification
Non-Stationary Random Process
Stationary Random Process
Autocorrelation function, PSD are invariant with time t

30. Relation to Random Process (cont.)
Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
Relation to Random Process (cont.)
Auto-correlation function
Power Spectral Density (PSD)

31. Relation to Random Process (cont.)
Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
Relation to Random Process (cont.)
FrFT V.S. Stationary random process
Nearly stationary

32. Relation to Random Process (cont.)
Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
Relation to Random Process (cont.)
FrFT V.S. Stationary random process
for

33. Relation to Random Process (cont.)
Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
Relation to Random Process (cont.)
FrFT V.S. Stationary random process
PSD:

34. Relation to Random Process (cont.)
Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
Relation to Random Process (cont.)
FrFT V.S. Non-stationary random process
Auto-correlation function
PSD
rotated with angle

35. Relation to Random Process (cont.)
Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
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.

36. Relation to Random Process (cont.)
Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
Relation to Random Process (cont.)
Properties of fractional stationary random process
After performing the fractional filter, a white noise becomes a fractional stationary random process.
Any non-stationary random process can be expressed as a summation of several fractional stationary random process.

37. Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
Applications of FrFT
Filter design
Optical systems
Convolution
Multiplexing
Generalization of sampling theorem

38. Filter design using FrFT
Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
Filter design using FrFT
Filtering a known noise
Filtering in fractional domain
signal
noise

39. Filter design using FrFT (cont.)
Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
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!

40. Filter design using FrFT (cont.)
Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
Filter design using FrFT (cont.)
Random noise removal
Area of WD ≡ Total energy
signal
signal

41. Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
Optical systems
Using FrFT/LCT to Represent Optical Components
Using FrFT/LCT to Represent the Optical Systems
Implementing FrFT/LCT by Optical Systems

42. Using FrFT/LCT to Represent Optical Components
Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
Using FrFT/LCT to Represent Optical Components
Propagation through the cylinder lens with focus length
Propagation through the free space (Fresnel Transform) with length

43. Using FrFT/LCT to Represent the Optical Systems
Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
Using FrFT/LCT to Represent the Optical Systems
input
output

44. Implementing FrFT/LCT by Optical Systems
Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
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

45. Implementing FrFT/LCT by Optical Systems (cont.)
Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
Implementing FrFT/LCT by Optical Systems (cont.)
input
output
The implementation of LCT with 2 cylinder lenses and 1 free space
input
output
The implementation of LCT with 1 cylinder lens and 2 free spaces

46. Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
Convolution
Convolution in domain
Multiplication in domain

47. Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
Convolution (cont.)

48. Multiplexing using FrFT
Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
Multiplexing using FrFT
TDM
FDM

49. Multiplexing using FrFT
Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
Multiplexing using FrFT
Inefficient multiplexing
Efficient multiplexing

50. Generalization of sampling theorem
Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
Generalization of sampling theorem
If is band-limited in some transformed domain of LCT, i.e.,
then we can sample by the interval as

51. Generalization of sampling theorem (cont.)
Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
Generalization of sampling theorem (cont.)

52. Conclusion and future works
Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
Conclusion and future works
Other relations with other transformations
Other applications

53. Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
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

54. Digital Image and Signal Processing Lab
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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, 2nd Edition, Prentice Hall, [13] R. N. Bracewell, The Fourier Transform and Its Applications, 3rd ed., Boston, McGraw Hill, 2000.

55. Digital Image and Signal Processing Lab
Graduate Institute of Communication Engineering National Taiwan University
2018/12/31
Chenquieh!