Discrete Linear Canonical Transforms An Operator Theory Approach Aykut Koç and Haldun M. Ozaktas 30/06/2018

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

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 30/06/2018

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   30/06/2018

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 30/06/2018

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]. 30/06/2018

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 30/06/2018

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 12/13/2017

Fast LCT Algorithm Iwasawa Decomposition – Again Divide and Conquer Chirp Multiplication Scaling FRT 12/13/2017

Linear Canonical Tranform – Special Cases Scaling FRT Chirp Multip. 30/06/2018

The Iwasawa Decomposition 30/06/2018

The Hyperdifferential Forms where DUALITY! 30/06/2018

The Operator Theory based DLCT Discrete Manifestation of Iwasawa Dec. Discrete Manifestation of Operators 30/06/2018

The Operator Theory based DLCT DLCT Output: 30/06/2018

Challenge: Deriving U and D 30/06/2018

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 30/06/2018

Challenge: Deriving U and D 30/06/2018

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. 30/06/2018

Proofs on Unitarity U is real diagonal, so it is Hermitian. 30/06/2018

Proofs on Unitarity 30/06/2018

Proofs on Unitarity 30/06/2018

Proofs on Unitarity 30/06/2018

Numerical Experiments As mentioned before, we cannot satisfy Group Properties analytically. So, we study numerically. Inputs: F1 F2 Transforms: T1 T2 30/06/2018

Numerical Experiments APPROXIMATION OF CONT. TRANSFORM 30/06/2018

Numerical Experiments CONCATENATION 30/06/2018

Numerical Experiments REVERSIBILITY 30/06/2018

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 30/06/2018

References A. Koç, B. Bartan, and H. M. Ozaktas, “Discrete Linear Canonical Transform Based on Hyperdifferential Operators,” arXiv preprint arXiv:1805.11416, 2018. A. Koç, B. Bartan, and H. M. Ozaktas, “Discrete Scaling Based on Operator Theory, ” arXiv preprint arXiv:1805.03500, 2018. and the references there in. 30/06/2018

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THANK YOU 12/13/2017