Discrete Linear Canonical Transforms An Operator Theory Approach

1. DiscreteLinear Canonical TransformsAn OperatorTheoryApproach Aykut Koçand Haldun M. Ozaktas

2. Outline • Fractional Fourier Transform (FRT) • Linear Canonical Transforms (LCTs) • PreviousDefinitions • HyperdifferentialOperatorTheoryBased DLCT • TheIwasawaDecomposition • TheHyperdifferential Forms • TheOperatorTheorybased DLCT • Properties of DiscreteTransform • Results & Conclusions

3. FractionalFourierTransform (FRT) • Generalizationto FT • Fourier transform: π/2+2nπ • Inverse Fouriertransform:-π/2+2nπ • Parity: –π+2nπ • Identity: 2nπ

4. CanonicalTransform • 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

5. LinearCanonicalTransforms (LCTs) • Given a generic input function f(u), LCT output g(u) is given by where α, β, and γ are LCT parameters

6. Applications • Signal processing [3] • Computationaland 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 alsobeen extensively studied for their applications in optics [2],[15], [16], [17], [18], [19], [20], electromagnetics, and classicaland quantum mechanics [3], [1], [21], [22].

7. PreviousApproaches 1) ComputationalApproach • Rely on sampling • Twosub-classes: • Methodsthat directly convert the LCT integralto a summation, [55], [56], [57], [58] • Decomposingintomoreelemantarybuildingblocks, [59], [60], [61], [23], [62]. • 2) Defining a DLCT andthendirectlyuse it, [63], [64], [65], [66],[67], [68], [69], [70]. No singledefinition has beenwidelyestablished

8. Fast FRT Algorithm • Objective: togetO(NlogN)algorithms • DivideandConquer • FRT can be put in the form: • Then, • This form is a ChirpMultiplication + ChirpConvolution + ChirpMultiplication

9. Fast LCT Algorithm • IwasawaDecomposition – AgainDivideandConquer Chirp Multiplication FRT Scaling

10. LinearCanonicalTranform – Special Cases Scaling FRT ChirpMultip.

11. TheIwasawaDecomposition

12. TheHyperdifferential Forms where DUALITY!

13. TheOperatorTheorybased DLCT DiscreteManifestation of IwasawaDec. DiscreteManifestation of Operators

14. TheOperatorTheorybased DLCT DLCT Output:

15. Challenge: Deriving U and D

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

17. Challenge: Deriving U and D

18. Properties of a DiscreteTransform • Unitarity • Preservation of GroupStructure • ConcatenationProperty • Reversibility (Special case of above) • Satisfactoryapproximation of continuoustransform However, a theoremfromGroupTheorystates: ‘It is theoreticallyimpossible to discretize all LCTs with a finite number ofsamplessuch that they are both unitary and they preservethe 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 overviewbased on examples. Princeton University Press, 2001.

19. Proofs on Unitarity U is realdiagonal, so it is Hermitian.

20. Proofs on Unitarity

21. Proofs on Unitarity

22. Proofs on Unitarity

23. Numerical Experiments As mentionedbefore, wecannotsatisfyGroupPropertiesanalytically. So, westudynumerically. Inputs: F1 F2 T1 Transforms: T2

24. Numerical Experiments APPROXIMATION OF CONT. TRANSFORM

25. Numerical Experiments CONCATENATION

26. Numerical Experiments REVERSIBILITY

27. Conclusions • Toourknowledge, thefirstapplication of OperatorTheorytodiscretetransform domain • A pure, elegant, anlyticalapproach • Usesonly DFT, differentiationandcoordinatemultiplicationoperations • Fullycompatiblewiththecirculantanddualstructure of DFT theory • Importantproperties of DLCT aresatisfied

28. References • A. Koç, B. Bartan, and H. M. Ozaktas, “Discrete Linear Canonical Transform Based on Hyperdifferential Operators,”arXivpreprint arXiv:1805.11416, 2018. • A. Koç, B. Bartan, and H. M. Ozaktas, “Discrete Scaling Based on Operator Theory, ”arXivpreprintarXiv:1805.03500, 2018. andthereferencesthere in.

29. References

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31. THANK YOU