Tomographic and entropic analysis of modulated signals A.S. Mastiukova,1, 2 M.A. Gavreev,1, 2 E.O. Kiktenko,1, 2, 3 and A.K. Fedorov1, 2 1Russian Quantum Center, Skolkovo, Moscow 143025, Russia 2Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Region 141700, Russia 3Department of Mathematical Methods for Quantum Technologies, Steklov Mathematical Institute of Russian Academy of Sciences, Moscow 119991, Russia (Dated: September 29, 2020) 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. I. INTRODUCTION distribution makes an interesting connection between the methods in quantum physics and signal processing, es- pecially in the context of time-frequency and position- Time-frequency analysis is a powerful tool of modern momentum uncertainty relations. signal processing [1{4]. Complementary to the informa- tion that can be extracted from the frequency domain via Recent decades, the link between phase space formula- Fourier analysis, time-frequency analysis provides a way tion of quantum mechanics and time-frequency analysis for studying a signal in both time and frequency repre- intensively studied in the context of quantum tomogra- sentations simultaneously. This is useful, in particular, phy [11]. Quantum tomography appears as a technique for signals of a sophisticated structure that change signifi- for the reconstruction of the Wigner function (density cantly over their duration, for example, music signals [5]. matrix) in quantum-optical experiments [12]. The re- Existing approaches to time-frequency analysis use lin- sults of tomographic measurements in principle contain ear canonical transformations preserving the symplectic all the information about the measured system, so they form [1]. Geometrically this can be illustrated as follows: can be considered as a quantities for the description of the Fourier transform can be viewed as a π=2 rotation quantum states [11]. This is the core idea behind the in the associated time-frequency plane, whereas other omographic representation of quantum states. In par- time-frequency representations allow arbitrary symplec- ticular, symplectic tomography protocols use a marginal tic transformations in the time-frequency plane. There is probability distribution of shifted and squeezed position a number of ways for defining a time-frequency distribu- and momentum variables. This approach has been used tion function with required properties (for a review, see in the context of time-frequency analysis [13] and time- Ref. [4]). Transformations between various distributions frequency entropic analysis [14] for various types of sig- in time-frequency analysis are quite well-understood [1]. nals, such as complex Gaussian signals [13{18] and reflec- tometry data [19, 20]. A general analysis of the relation The idea behind time-frequency analysis is very close between time-frequency tomograms and other transfor- to the motivation for studying phase-space representa- mations (including wavelets) is presented in Ref. [15]. tions in quantum physics. As it is well known, the re- However, the considered examples of signals lack ana- lation between position and momentum representations lyzing modulated signals, which are intensively used in of the wave function is given by the Fourier transform, telecommunication signals. Moreover, the Wigner-Ville which is similar to the relation between signals in time distribution has been considered in the context of diag- and frequency domains. This analogy becomes even more nostics of features of modulated signals [21], so one can arXiv:2001.07707v2 [eess.SP] 27 Sep 2020 transparent in the framework of analytic signals, which expect that tomograms are helpful for such an analysis. are complex as well as wave functions. One of the possible ways to characterize a quantum state in the phase space In this work, we consider tomographic representations are to use the Wigner quasiprobability distribution [6]. for modulated signals. We calculate optical tomographic The Wigner quasiprobability distribution resembles clas- representations and Wigner-Ville distributions for signals sical phase space probability distributions that is used in with amplitude and frequency modulations. In particu- statistical mechanics. However, it cannot be fully inter- lar, we study the method of the optical time-frequency preted as a probability distribution since it takes nega- tomograms via pseudo Wigner-Ville distributions, and tive values [6{8]. We also note the Wigner quasiprobabil- discuss advantages of such an approach. We also consider ity distribution is successfully used for analyzing various time-frequency entropic relations for modulated signals. phenomena in quantum optics and quantum statistical Our work is organized as follows. In Sec. II, we intro- physics [8]. In the field of signal processing, the Wigner duce general relations for tomographic analysis of ana- distribution function often referred to as the Wigner{ lytic signals. In Sec.III, we calculate optical tomographic Ville distribution [9, 10]. The application of the Wigner representations for signals with amplitude and frequency 2 (t, !) W (t, !) 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