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Linear Canonical Domains and Degrees of Freedom of Signals and Systems

  • Chapter
Linear Canonical Transforms

Part of the book series: Springer Series in Optical Sciences ((SSOS,volume 198))

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Abstract

We discuss the relationships between linear canonical transform (LCT) domains, fractional Fourier transform (FRT) domains, and the space-frequency plane. In particular, we show that LCT domains correspond to scaled fractional Fourier domains and thus to scaled oblique axes in the space-frequency plane. This allows LCT domains to be labeled and monotonically ordered by the corresponding fractional order parameter and provides a more transparent view of the evolution of light through an optical system modeled by LCTs. We then study the number of degrees of freedom of optical systems and signals based on these concepts. We first discuss the bicanonical width product (BWP), which is the number of degrees of freedom of LCT-limited signals. The BWP generalizes the space-bandwidth product and often provides a tighter measure of the actual number of degrees of freedom of signals. We illustrate the usefulness of the notion of BWP in two applications: efficient signal representation and efficient system simulation. In the first application we provide a sub-Nyquist sampling approach to represent and reconstruct signals with arbitrary space-frequency support. In the second application we provide a fast discrete LCT (DLCT) computation method which can accurately compute a (continuous) LCT with the minimum number of samples given by the BWP. Finally, we focus on the degrees of freedom of first-order optical systems with multiple apertures. We show how to explicitly quantify the degrees of freedom of such systems, state conditions for lossless transfer through the system and analyze the effects of lossy transfer.

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Acknowledgements

This chapter is based on [33–37]. H.M. Ozaktas acknowledges partial support of the Turkish Academy of Sciences.

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Authors and Affiliations

  1. Department of Electrical and Electronics Engineering, Middle East Technical University, 06800, Cankaya, Ankara, Turkey

    Figen S. Oktem

  2. Department of Electrical Engineering, Bilkent University, 06800, Bilkent, Ankara, Turkey

    Haldun M. Ozaktas

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  1. Figen S. Oktem
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Correspondence to Figen S. Oktem .

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Editors and Affiliations

  1. School of Electrical and Electronic Engineering, University College Dublin, Dublin, Ireland

    John J. Healy

  2. The Scientific and Technological Research Council of Turkey, Ankara, Ankara, Turkey

    M. Alper Kutay

  3. Department of Electrical Engineering, Bilkent University, Ankara, Ankara, Turkey

    Haldun M. Ozaktas

  4. School of Electrical and Electronic Engineering, University College Dublin, Dublin, Dublin, Ireland

    John T. Sheridan

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Oktem, F.S., Ozaktas, H.M. (2016). Linear Canonical Domains and Degrees of Freedom of Signals and Systems. In: Healy, J., Alper Kutay, M., Ozaktas, H., Sheridan, J. (eds) Linear Canonical Transforms. Springer Series in Optical Sciences, vol 198. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-3028-9_7

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