1. Discrete Linear Canonical Transforms An Operator Theory Approach
Aykut Koç and Haldun M. Ozaktas
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2. Outline Fractional Fourier Transform (FRT)
Linear Canonical Transforms (LCTs)
Previous Definitions
Hyperdifferential Operator Theory Based DLCT
The Iwasawa Decomposition
The Hyperdifferential Forms
The Operator Theory based DLCT
Properties of Discrete Transform
Results & Conclusions

3. Fractional Fourier Transform (FRT)
Generalization to FT
Fourier transform: π/2+2nπ
Inverse Fourier transform: -π/2+2nπ
Parity: –π+2nπ
Identity: 2nπ
𝑎𝜋/2
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4. Canonical Transform
Change of variables from one set of canonical coordinates to another
What is the canonical coordinates?
Set of coordinates that can describe a physical system at any given point in time
Locates the system within phase space
For quantum mechanics: position and momentum
Thermodynamics: entropy-temperature, pressure-volume
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5. Linear Canonical Transforms (LCTs)
Given a generic input function f(u), LCT output g(u) is given by
where α, β, and γ are LCT parameters
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6. Applications Signal processing [3]
Computational and applied mathematics [5], [6], including
fast and efficient optimal filtering [7]
radar signal processing [8], [9]
speech processing [10]
image representation [11]
image encryption and watermarking [12], [13], [14]
LCTs have also been extensively studied for their applications in optics [2], [15], [16], [17], [18], [19], [20], electromagnetics, and classical and quantum mechanics [3], [1], [21], [22].
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7. Previous Approaches 1) Computational Approach
Rely on sampling
Two sub-classes:
Methods that directly convert the LCT integral to a summation, [55], [56], [57], [58]
Decomposing into more elemantary building blocks, [59], [60], [61], [23], [62].
2) Defining a DLCT and then directly use it, [63], [64], [65], [66], [67], [68], [69], [70].
No single definition has been widely established
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8. Fast FRT Algorithm Objective: to get O(NlogN) algorithms
Divide and Conquer
FRT can be put in the form:
Then,
This form is a Chirp Multiplication + Chirp Convolution + Chirp Multiplication
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9. Fast LCT Algorithm Iwasawa Decomposition – Again Divide and Conquer
Chirp
Multiplication
Scaling
FRT
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10. Linear Canonical Tranform – Special Cases
Scaling
FRT
Chirp Multip.
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11. The Iwasawa Decomposition
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12. The Hyperdifferential Forms
where
DUALITY!
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13. The Operator Theory based DLCT
Discrete Manifestation of Iwasawa Dec.
Discrete Manifestation of Operators
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14. The Operator Theory based DLCT
DLCT Output:
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15. Challenge: Deriving U and D
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16. Challenge: Deriving U and D
We turn our attention to the task of defining U_h.
It is tempting to define the discrete version of U by simply forming a diagonal matrix
with the diagonal entries being equal to the coordinate values.
However, it violates the duality and elagance of our approach.
We pursue a similar approach in deriving U_h to the one in D_h
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17. Challenge: Deriving U and D
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18. Properties of a Discrete Transform
Unitarity
Preservation of Group Structure
Concatenation Property
Reversibility (Special case of above)
Satisfactory approximation of continuous transform
However, a theorem from Group Theory states:
‘It is theoretically impossible to discretize all LCTs with a finite number of samples such that they are both unitary and they preserve the group structure.’
- K. B. Wolf, Linear Canonical Transforms: Theory and Applications. New York, NY: Springer New York, 2016, ch. Development of Linear
Canonical Transforms: A Historical Sketch, pp. 3–28.
- A. W. Knapp, Representation theory of semisimple groups: An overview based on examples. Princeton University Press, 2001.
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19. Proofs on Unitarity
U is real diagonal, so it is Hermitian.
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20. Proofs on Unitarity
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21. Proofs on Unitarity
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22. Proofs on Unitarity
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23. Numerical Experiments
As mentioned before, we cannot satisfy Group Properties analytically.
So, we study numerically.
Inputs: F1 F2
Transforms: T1 T2
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24. Numerical Experiments
APPROXIMATION OF CONT. TRANSFORM
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25. Numerical Experiments
CONCATENATION
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26. Numerical Experiments
REVERSIBILITY
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27. Conclusions
To our knowledge, the first application of Operator Theory to discrete transform domain
A pure, elegant, anlytical approach
Uses only DFT, differentiation and coordinate multiplication operations
Fully compatible with the circulant and dual structure of DFT theory
Important properties of DLCT are satisfied
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28. References
A. Koç, B. Bartan, and H. M. Ozaktas, “Discrete Linear Canonical Transform Based on Hyperdifferential Operators,” arXiv preprint arXiv: , 2018.
A. Koç, B. Bartan, and H. M. Ozaktas, “Discrete Scaling Based on Operator Theory, ” arXiv preprint arXiv: , 2018.
and the references there in.
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29. References
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30. References
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31. THANK YOU
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