Tomographic and Entropic Analysis of Modulated Signals
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Tomographic and Entropic Analysis of Modulated Signals A. S. Mastiukovaa, b, *, M. A. Gavreeva, b, **, E. O. Kiktenkoa, b, c, ***, and A. K. Fedorova, b, **** a
Russian Quantum Center, Skolkovo, Moscow, 143025 Russia Moscow Institute of Physics and Technology, Dolgoprudny, Moscow oblast, 141700 Russia c Department of Mathematical Methods for Quantum Technologies, Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, 119991 Russia *e-mail: [email protected] **e-mail: [email protected] ***e-mail: [email protected] ****e-mail: [email protected] b
Received January 27, 2020; revised January 30, 2020; accepted February 27, 2020
Abstract—We study an application of the quantum tomography framework for the time-frequency analysis of modulated signals. In particular, we calculate optical tomographic representations and Wigner–Ville distributions for signals with amplitude and frequency modulations. We also consider time-frequency entropic relations for modulated signals, which are naturally associated with the Fourier analysis. A numerical toolbox for calculating optical time-frequency tomograms based on pseudo Wigner–Ville distributions for modulated signals is provided. Keywords: quantum tomography, signal processing, Wigner–Ville distribution DOI: 10.1134/S0030400X20070127
1. INTRODUCTION Time-frequency analysis is a powerful tool of modern signal processing [1–4]. Complementary to the information that can be extracted from the frequency domain via Fourier analysis, time-frequency analysis provides a way for studying a signal in both time and frequency representations simultaneously. This is useful, in particular, for signals of a sophisticated structure that change significantly over their duration, for example, music signals [5]. Existing approaches to time-frequency analysis use linear canonical transformations preserving the symplectic form [1]. Geometrically this can be illustrated as follows: the Fourier transform can be viewed as a π/2 rotation in the associated time-frequency plane, whereas other time-frequency representations allow arbitrary symplectic transformations in the time-frequency plane. There is a number of ways for defining a time-frequency distribution function with required properties (for a review, see [4]). Transformations between various distributions in time-frequency analysis are quite well-understood [1]. The idea behind time-frequency analysis is very close to the motivation for studying phase-space representations in quantum physics. As it is well known, the relation between position and momentum representations of the wave function is given by the Fourier transform, which is similar to the relation between sig-
nals in time and frequency domains. This analogy becomes even more transparent in the framework of analytic signals, which are complex as well as wave functions. One of the possible ways to characterize a quantum state in the phase space is to use the Wigner quasiprobability distribution [6]. The Wigner quasiprobability distribution resembles classical p
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