Browsing by Subject "Sampling theorem"

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    Linear canonical domains and degrees of freedom of signals and systems
    (Springer Verlag, 2016) Öktem, F. S.; Özaktaş, Haldun; Healy, J. J.; Kutay, Mehmet Alper; Özaktaş, Haldun; Sheridan, J. T.
    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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    Sampling and discrete linear canonical transforms
    (Springer, 2016) Healy, J. J.; Özaktaş, Haldun M.; Healy, J. J.; Kutay, M. A.; Özaktaş, Haldun M.; Sheridan, J. T.
    A discrete linear canonical transform would facilitate numerical calculations in many applications in signal processing, scalar wave optics, and nuclear physics. The question is how to define a discrete transform so that it not only approximates the continuous transform well, but also constitutes a discrete transform in its own right, being complete, unitary, etc. The key idea is that the LCT of a discrete signal consists of modulated replicas. Based on that result, it is possible to define a discrete transform that has many desirable properties. This discrete transform is compatible with certain algorithms more than others.