Tomographic and Entropic Analysis of Modulated Signals

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 information 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 reflectometry 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

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Figure 1. time-frequency analysis for the chirp signal for the following set of parameters (A ≈ 0.166, φ0 = 0, α = 0.03, t0 = 100, and ω = π) given by Eq. (13): (a) the Wigner-Ville distribution is presented (the 2D representation in the time-frequency plane is in insert); (b) the pseudo Wigner-Ville distribution is presented (the 2D representation in the time-frequency plane is in insert); (c) the diﬀerence between Wigner-Ville and pseudo Wigner-Ville distributions is illustrated; (d) the optical time- frequency tomogram is presented; (e) the optical time-frequency tomogram based on the calculation of the pseudo Wigner-Ville distributions is presented; (f) the diﬀerence between tomograms is illustrated.

modulations. In Sec.IV, we analyze time-frequency en- signal processing and phase-space distributions in quan- tropic relations. We conclude in Sec.V. tum mechanics [1]. The family of marginal distributions, which contains complete information on the analytical signal, has been II. TOMOGRAPHIC ANALYSIS introduced in Ref. [11]. It has the following form:

1 Here we introduce basic tools for the tomographic anal- (X, θ) = (X, θ) 2 , (3) T 2π sin θ |I | ysis of signals. We consider a time-dependent signal s(t), | | whose representation in the frequency domains ˜(ω) can where be obtained via the Fourier transform. Conventionally, we use an analytical representation of signals in the fol- Z it2 cos θ itX lowing form: (X, θ) = dt (t) exp . (4) I R S 2 sin θ − sin θ (t) = s(t) + iH [s(t)] , (1) Here X = t cos θ+ω sin θ is the dimensionless quadrature S variable. This representation is referred to as the optical where time-frequency tomogram of the signal (t). The inte- gral transformation in Eq. (4) is the fractionalS Fourier 1 Z s(p) H [s(t)] = dp (2) transform. The tomogram is normalized as follows: π t p R − Z is the Hilbert transform of the signal. The advantage dX (X, θ) = 1. (5) T of using the analytic signal is that in the frequency do- R main the amplitude of negative frequency components It also gives the distribution of the signal in time and are zero. This satisﬁes mathematical completeness of frequency domains, correspondingly: the problem by accounting for all frequencies, yet does not limit the practical application since only positive fre- (X = t, 0) = (t) 2, (X = ω, π/2) = ˜(ω) 2. (6) quency components have a practical interpretation. The T |S | T |S | method based on the use of analytic signals also makes The optical time-frequency tomogram is a particular case a clear analogy between time-frequency distributions in of the symplectic time-frequency tomogram T (X, µ, ν) 3

of the signal (t), where µ = cos θ and ν = sin θ. In S s (t) s (!) some cases, another variations of tomographic represen- AAAB9HicbVBNSwMxEM36WetX1aOXYBHqpexWQY9VL16ECvYD2qVk02wbmmTXZLZQlv4OLx4U8eqP8ea/MW33oK0PBh7vzTAzL4gFN+C6387K6tr6xmZuK7+9s7u3Xzg4bJgo0ZTVaSQi3QqIYYIrVgcOgrVizYgMBGsGw9up3xwxbXikHmEcM1+SvuIhpwSs5Jtu2tESX99PSnDWLRTdsjsDXiZeRoooQ61b+Or0IppIpoAKYkzbc2PwU6KBU8Em+U5iWEzokPRZ21JFJDN+Ojt6gk+t0sNhpG0pwDP190RKpDFjGdhOSWBgFr2p+J/XTiC88lOu4gSYovNFYSIwRHiaAO5xzSiIsSWEam5vxXRANKFgc8rbELzFl5dJo1L2zsuVh4ti9SaLI4eO0QkqIQ9doiq6QzVURxQ9oWf0it6ckfPivDsf89YVJ5s5Qn/gfP4A0LWReA== AM AAAB+3icbVDLSsNAFJ34rPUV69LNYBHqpiRV0GXVjRuhgn1AE8JkOmmHzkzCzEQsIb/ixoUibv0Rd/6N0zYLbT1w4XDOvdx7T5gwqrTjfFsrq2vrG5ulrfL2zu7evn1Q6ag4lZi0ccxi2QuRIowK0tZUM9JLJEE8ZKQbjm+mfveRSEVj8aAnCfE5GgoaUYy0kQK7ooLMkxxe3eU1L+ZkiE4Du+rUnRngMnELUgUFWoH95Q1inHIiNGZIqb7rJNrPkNQUM5KXvVSRBOExGpK+oQJxovxsdnsOT4wygFEsTQkNZ+rviQxxpSY8NJ0c6ZFa9Kbif14/1dGln1GRpJoIPF8UpQzqGE6DgAMqCdZsYgjCkppbIR4hibA2cZVNCO7iy8uk06i7Z/XG/Xm1eV3EUQJH4BjUgAsuQBPcghZoAwyewDN4BW9Wbr1Y79bHvHXFKmYOwR9Ynz8OjpPM AM (a) (b) tation, such as time-scale tomograms, frequency-scale1 to- 1 0.4 mograms, and time-conformal tomograms, are used0.5 [20]. 0.5 0.4 We restrict ourselves to the consideration of optical time- 0 0 AAAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHbRRI+oF4+QyCOBDZkdemFkdnYzM2tCCF/gxYPGePWTvPk3DrAHBSvppFLVne6uIBFcG9f9dnJr6xubW/ntws7u3v5B8fCoqeNUMWywWMSqHVCNgktsGG4EthOFNAoEtoLR3cxvPaHSPJYPZpygH9GB5CFn1FipftMrltyyOwdZJV5GSpCh1it+dfsxSyOUhgmqdcdzE+NPqDKcCZwWuqnGhLIRHWDHUkkj1P5kfuiUnFmlT8JY2ZKGzNXfExMaaT2OAtsZUTPUy95M/M/rpCa89idcJqlByRaLwlQQE5PZ16TPFTIjxpZQpri9lbAhVZQZm03BhuAtv7xKmpWyd1Gu1C9L1dssjjycwCmcgwdXUIV7qEEDGCA8wyu8OY/Oi/PufCxac042cwx/4Hz+AJQrjMk= frequency tomograms only. 0.2 0.2 0.5 0.5 It seems to be quite straightforward to calculate opti- 1 1 0 0 0 50 0 100 50 150 100 0 1500.2 0.4 0.6 00.8 0.2 0.4 0.6 0.8

AAAB7XicbVDLSgNBEJyNrxhfUY9eBoPgKexGQY9BLx4jmAckS5id9CZj5rHMzAphyT948aCIV//Hm3/jJNmDJhY0FFXddHdFCWfG+v63V1hb39jcKm6Xdnb39g/Kh0cto1JNoUkVV7oTEQOcSWhaZjl0Eg1ERBza0fh25refQBum5IOdJBAKMpQsZpRYJ7V6SsCQ9MsVv+rPgVdJkJMKytHol796A0VTAdJSTozpBn5iw4xoyyiHaamXGkgIHZMhdB2VRIAJs/m1U3zmlAGOlXYlLZ6rvycyIoyZiMh1CmJHZtmbif953dTG12HGZJJakHSxKE45tgrPXscDpoFaPnGEUM3crZiOiCbUuoBKLoRg+eVV0qpVg4tq7f6yUr/J4yiiE3SKzlGArlAd3aEGaiKKHtEzekVvnvJevHfvY9Fa8PKZY/QH3ucPkX+PHw== cal time-frequency tomograms using Eq. (3). However, tAAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsN+3azSbsToQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IJHCoOt+O4W19Y3NreJ2aWd3b/+gfHjUNnGqGW+xWMa6E1DDpVC8hQIl7ySa0yiQ/CEY3878hyeujYjVPU4S7kd0qEQoGEUrNbFfrrhVdw6ySrycVCBHo1/+6g1ilkZcIZPUmK7nJuhnVKNgkk9LvdTwhLIxHfKupYpG3PjZ/NApObPKgISxtqWQzNXfExmNjJlEge2MKI7MsjcT//O6KYbXfiZUkiJXbLEoTCXBmMy+JgOhOUM5sYQyLeythI2opgxtNiUbgrf88ipp16reRbXWvKzUb/I4inACp3AOHlxBHe6gAS1gwOEZXuHNeXRenHfnY9FacPKZY/gD5/MH4XeM/A== ! there are well-known problems in the ﬁeld of signal pro- s (!) s (t) AAAB+3icbVDLSsNAFJ34rPUV69LNYBHqpiRV0GVREDdCBfuAJoTJdNIOnZmEmYlYQn7FjQtF3Poj7vwbp20W2nrgwuGce7n3njBhVGnH+bZWVtfWNzZLW+Xtnd29ffug0lFxKjFp45jFshciRRgVpK2pZqSXSIJ4yEg3HF9P/e4jkYrG4kFPEuJzNBQ0ohhpIwV2RQWZJzm8uctrXszJEJ0GdtWpOzPAZeIWpAoKtAL7yxvEOOVEaMyQUn3XSbSfIakpZiQve6kiCcJjNCR9QwXiRPnZ7PYcnhhlAKNYmhIaztTfExniSk14aDo50iO16E3F/7x+qqNLP6MiSTUReL4oShnUMZwGAQdUEqzZxBCEJTW3QjxCEmFt4iqbENzFl5dJp1F3z+qN+/Nq86qIowSOwDGoARdcgCa4BS3QBhg8gWfwCt6s3Hqx3q2PeeuKVcwcgj+wPn8AFlST0Q== (c) AAAB9HicbVBNSwMxEM36WetX1aOXYBHqpexWQY9FQbwIFewHtEvJptk2NMmuyWyhLP0dXjwo4tUf481/Y9ruQVsfDDzem2FmXhALbsB1v52V1bX1jc3cVn57Z3dvv3Bw2DBRoimr00hEuhUQwwRXrA4cBGvFmhEZCNYMhjdTvzli2vBIPcI4Zr4kfcVDTglYyTfdtKMlvr2flOCsWyi6ZXcGvEy8jBRRhlq38NXpRTSRTAEVxJi258bgp0QDp4JN8p3EsJjQIemztqWKSGb8dHb0BJ9apYfDSNtSgGfq74mUSGPGMrCdksDALHpT8T+vnUB45adcxQkwReeLwkRgiPA0AdzjmlEQY0sI1dzeiumAaELB5pS3IXiLLy+TRqXsnZcrDxfF6nUWRw4doxNUQh66RFV0h2qojih6Qs/oFb05I+fFeXc+5q0rTjZzhP7A+fwB2GKRfQ== FM (d) FM cessing, such as, for example, aliasing, which give1 rise 1 0.4 0.4 to distortions during the signal reconstruction and com- 0.5 0.5 0.3 0.3 putational diﬃculties. These problems also occur dur- 0 0 0.2 0.2 AAAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHbRRI+oF4+QyCOBDZkdemFkdnYzM2tCCF/gxYPGePWTvPk3DrAHBSvppFLVne6uIBFcG9f9dnJr6xubW/ntws7u3v5B8fCoqeNUMWywWMSqHVCNgktsGG4EthOFNAoEtoLR3cxvPaHSPJYPZpygH9GB5CFn1FipftMrltyyOwdZJV5GSpCh1it+dfsxSyOUhgmqdcdzE+NPqDKcCZwWuqnGhLIRHWDHUkkj1P5kfuiUnFmlT8JY2ZKGzNXfExMaaT2OAtsZUTPUy95M/M/rpCa89idcJqlByRaLwlQQE5PZ16TPFTIjxpZQpri9lbAhVZQZm03BhuAtv7xKmpWyd1Gu1C9L1dssjjycwCmcgwdXUIV7qEEDGCA8wyu8OY/Oi/PufCxac042cwx/4Hz+AJQrjMk= AAAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHbRRI+oF4+QyCOBDZkdemFkdnYzM2tCCF/gxYPGePWTvPk3DrAHBSvppFLVne6uIBFcG9f9dnJr6xubW/ntws7u3v5B8fCoqeNUMWywWMSqHVCNgktsGG4EthOFNAoEtoLR3cxvPaHSPJYPZpygH9GB5CFn1FipftMrltyyOwdZJV5GSpCh1it+dfsxSyOUhgmqdcdzE+NPqDKcCZwWuqnGhLIRHWDHUkkj1P5kfuiUnFmlT8JY2ZKGzNXfExMaaT2OAtsZUTPUy95M/M/rpCa89idcJqlByRaLwlQQE5PZ16TPFTIjxpZQpri9lbAhVZQZm03BhuAtv7xKmpWyd1Gu1C9L1dssjjycwCmcgwdXUIV7qEEDGCA8wyu8OY/Oi/PufCxac042cwx/4Hz+AJQrjMk= ing the calculation of the Wigner-Ville distribution for 0.5 0.5 0.1 0.1 analytic signals [10]. We remind that the Wigner-Ville 1 1 0 0 distribution of the signal has the following form: 0 50 0 100 50 150 100 0 1500.2 0.4 0.6 0 0. 0.2 0.4 0.6 0.

AAAB7XicbVDLSgNBEJyNrxhfUY9eBoPgKexGQY9BLx4jmAckS5id9CZj5rHMzAphyT948aCIV//Hm3/jJNmDJhY0FFXddHdFCWfG+v63V1hb39jcKm6Xdnb39g/Kh0cto1JNoUkVV7oTEQOcSWhaZjl0Eg1ERBza0fh25refQBum5IOdJBAKMpQsZpRYJ7V6SsCQ9MsVv+rPgVdJkJMKytHol796A0VTAdJSTozpBn5iw4xoyyiHaamXGkgIHZMhdB2VRIAJs/m1U3zmlAGOlXYlLZ6rvycyIoyZiMh1CmJHZtmbif953dTG12HGZJJakHSxKE45tgrPXscDpoFaPnGEUM3crZiOiCbUuoBKLoRg+eVV0qpVg4tq7f6yUr/J4yiiE3SKzlGArlAd3aEGaiKKHtEzekVvnvJevHfvY9Fa8PKZY/QH3ucPkX+PHw== tAAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsN+3azSbsToQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IJHCoOt+O4W19Y3NreJ2aWd3b/+gfHjUNnGqGW+xWMa6E1DDpVC8hQIl7ySa0yiQ/CEY3878hyeujYjVPU4S7kd0qEQoGEUrNbFfrrhVdw6ySrycVCBHo1/+6g1ilkZcIZPUmK7nJuhnVKNgkk9LvdTwhLIxHfKupYpG3PjZ/NApObPKgISxtqWQzNXfExmNjJlEge2MKI7MsjcT//O6KYbXfiZUkiJXbLEoTCXBmMy+JgOhOUM5sYQyLeythI2opgxtNiUbgrf88ipp16reRbXWvKzUb/I4inACp3AOHlxBHe6gAS1gwOEZXuHNeXRenHfnY9FacPKZY/gD5/MH4XeM/A== ! Z τ τ W (t, ω) = dτ t + ∗ t e−iωτ . (7) Figure 2. Considered modulated signals: (a) AM signal in the time domain and (b) in the frequency domain; (c) FM signal R S 2 S − 2 in the time domain and (d) in the frequency domain. The Wigner-Ville distribution is normalized as follows: 1 Z dtdωW (t, ω) = 1. (8) 2π 2 Then in order to reduce the complexity of calculating op- R tical time-frequency tomograms it is possible to redefine When the Wigner-Ville distribution is applied to a sig- it via pseudo Wigner-Ville distributions as follows: nal with multi frequency components, cross-terms appear due to its quadratic nature. In order to avoid the effect Z dkdtdω −ik(X−t cos θ−ω sin θ) of cross-terms the windowed version of the Wigner-Ville p(X, θ) = 2 Wp(t, ω)e . (12) T 3 (2π) distribution, which is known as pseudo Wigner-Ville dis- R tribution, is used. The pseudo Wigner-Ville distribution is defined as follows: This relation give rise to the modification of the in- tegral relation between analytic signal and its time- Z τ ∗ τ −iωτ frequency optical tomogram, which is given by Eq. (3). Wp(t, ω) = dτh(τ) t + t e , (9) R S 2 S − 2 In our work, we use the pseudo time-frequency optical tomogram p(X, θ) for analyzing properties of signals. where h(τ) is the window function in the time domain. We note thatT this consideration is related to the estab- The window function h(τ) can be used, for example, in lishing a correspondence between the fractional Fourier the Hamming window form: transform and the Wigner distribution [29, 30]. 2πτ In order to see a difference in calculating original and h (τ)=0.54 0.46 cos , 0 τ M 1. (10) M − M 1 ≤ ≤ − pseudo time-frequency optical tomograms, we consider − an example of a chirp signal of the following form: Pseudo Wigner-Ville distributions [22–25] and their modifications are actively used in various fields, such −α(t−t )2 s(t) = A sin (ωt + φ ) e 0 , (13) as the dispersion analysis of waveguides [26] and study- 0 ing oil-in-water flow patterns [27]. We note that various time-frequency filters are employed for eliminating where A is the fixed amplitude, φ0, α, and t0 are fixed cross-terms in the Wigner-Ville distribution, which is of constants. For this chirp signal in the analytic form high importance for the analysis of non-stationary sys- given by Eq. (1) we calculate first original Wigner-Ville tems. Another approach for reducing cross-terms in the distribution (Fig.1a) and pseudo Wigner-Ville distri- Wigner-Ville distribution uses a tunable-Q wavelet trans- bution (Fig.1b). One can capture a difference between (t, ω) = W (t, ω) W (t, ω) original Wigner- form [28]. In our consideration below, we use the sim- D | − p | plest case of the pseudo Wigner-Ville distribution with Ville distribution and pseudo Wigner-Ville distribution the simplest form of the window function. (see Fig.1c). This difference manifests in calculat- Using the relation between Wigner-Ville distributions ing original and pseudo time-frequency optical tomo- and optical time-frequency tomograms, which is given grams (Fig.1d, Fig.1e, and Fig.1f), where (X, θ) = (X, θ) (X, θ) . The differences (t, ω) andD (X, θ) by the Radon transform, one can reconstruct the optical |T −Tp | D D tomogram of the signal as follows: are non-zero. This can be a signature of the fact that the signal can be sensitive to the presence of the time- Z dkdtdω window, which is of importance for capturing properties (X, θ) = W (t, ω)e−ik(X−t cos θ−ω sin θ). (11) 3 2 of non-stationary signals. T R (2π) 4

(a) AM (b) AM (a) SAM(m, ✓) (b) SAM(m, ✓) W (t, !) (X, ✓) AAAB/XicbVDLSgNBEJyNrxhf6+PmZTAIESTsRkGPUS9ehIjmAdkYZieTZMjM7jLTK8Ql+CtePCji1f/w5t84Sfag0YKGoqqb7i4/ElyD43xZmbn5hcWl7HJuZXVtfcPe3KrpMFaUVWkoQtXwiWaCB6wKHARrRIoR6QtW9wcXY79+z5TmYXALw4i1JOkFvMspASO17Z2bu8RTEp9djQry0IM+A3LQtvNO0ZkA/yVuSvIoRaVtf3qdkMaSBUAF0brpOhG0EqKAU8FGOS/WLCJ0QHqsaWhAJNOtZHL9CO8bpYO7oTIVAJ6oPycSIrUeSt90SgJ9PeuNxf+8Zgzd01bCgygGFtDpom4sMIR4HAXucMUoiKEhhCpubsW0TxShYALLmRDc2Zf/klqp6B4VS9fH+fJ5GkcW7aI9VEAuOkFldIkqqIooekBP6AW9Wo/Ws/VmvU9bM1Y6s41+wfr4Bix8lGU= AAAB/XicbVDLSgNBEJyNrxhf6+PmZTAIESTsRkGPUS9ehIjmAdkYZieTZMjM7jLTK8Ql+CtePCji1f/w5t84Sfag0YKGoqqb7i4/ElyD43xZmbn5hcWl7HJuZXVtfcPe3KrpMFaUVWkoQtXwiWaCB6wKHARrRIoR6QtW9wcXY79+z5TmYXALw4i1JOkFvMspASO17Z2bu8RTEp9djQry0IM+A3LQtvNO0ZkA/yVuSvIoRaVtf3qdkMaSBUAF0brpOhG0EqKAU8FGOS/WLCJ0QHqsaWhAJNOtZHL9CO8bpYO7oTIVAJ6oPycSIrUeSt90SgJ9PeuNxf+8Zgzd01bCgygGFtDpom4sMIR4HAXucMUoiKEhhCpubsW0TxShYALLmRDc2Zf/klqp6B4VS9fH+fJ5GkcW7aI9VEAuOkFldIkqqIooekBP6AW9Wo/Ws/VmvU9bM1Y6s41+wfr4Bix8lGU=

AAAB/3icbVDLSgMxFM3UV62vUcGNm2ARKkiZqYIuq27cCBXsAzpjyaSZNjTJDElGKGMX/oobF4q49Tfc+Tem7Sy09cCFwzn3cu89Qcyo0o7zbeUWFpeWV/KrhbX1jc0te3unoaJEYlLHEYtkK0CKMCpIXVPNSCuWBPGAkWYwuBr7zQciFY3EnR7GxOeoJ2hIMdJG6th7zU58n3qSw4ubUUkfexEnPXTUsYtO2ZkAzhM3I0WQodaxv7xuhBNOhMYMKdV2nVj7KZKaYkZGBS9RJEZ4gHqkbahAnCg/ndw/godG6cIwkqaEhhP190SKuFJDHphOjnRfzXpj8T+vnejw3E+piBNNBJ4uChMGdQTHYcAulQRrNjQEYUnNrRD3kURYm8gKJgR39uV50qiU3ZNy5fa0WL3M4siDfXAASsAFZ6AKrkEN1AEGj+AZvII368l6sd6tj2lrzspmdsEfWJ8/vmKVRg== p AAACC3icbVDJSgNBEO1xjXEb9eilSRAiSJiJgh6jXrwIEbJBJg49nZ6kSc9Cd40Qhrl78Ve8eFDEqz/gzb+xsxw08UHB470qqup5seAKLOvbWFpeWV1bz23kN7e2d3bNvf2mihJJWYNGIpJtjygmeMgawEGwdiwZCTzBWt7weuy3HphUPArrMIpZNyD9kPucEtCSaxacgMCAEpHWs/vUkQG+vM3cNM5K7RMHBgzIsWsWrbI1AV4k9owU0Qw11/xyehFNAhYCFUSpjm3F0E2JBE4Fy/JOolhM6JD0WUfTkARMddPJLxk+0koP+5HUFQKeqL8nUhIoNQo83Tm+XM17Y/E/r5OAf9FNeRgnwEI6XeQnAkOEx8HgHpeMghhpQqjk+lZMB0QSCjq+vA7Bnn95kTQrZfu0XLk7K1avZnHk0CEqoBKy0TmqohtUQw1E0SN6Rq/ozXgyXox342PaumTMZg7QHxifP7dNmtI= p T 10 150 1 0. 100 0.8 25

50 0.6 0.

0.4 0 8

0.2 50 0.

0 0 50 100 150 20 0.2 6

0.2 XAAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3azSbsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgktsGW4EdhKFNAoEPgTj25n/8IRK81jem0mCfkSHkoecUWOlZqdfrrhVdw6ySrycVCBHo1/+6g1ilkYoDRNU667nJsbPqDKcCZyWeqnGhLIxHWLXUkkj1H42P3RKzqwyIGGsbElD5urviYxGWk+iwHZG1Iz0sjcT//O6qQmv/YzLJDUo2WJRmApiYjL7mgy4QmbExBLKFLe3EjaiijJjsynZELzll1dJu1b1Lqq15mWlfpPHUYQTOIVz8OAK6nAHDWgBA4RneIU359F5cd6dj0VrwclnjuEPnM8ftweM4A== mAAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3a3STsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgkfYMtwI7CQKqQwEPgTj25n/8IRK8zi6N5MEfUmHEQ85o8ZKTdkvV9yqOwdZJV5OKpCj0S9/9QYxSyVGhgmqdddzE+NnVBnOBE5LvVRjQtmYDrFraUQlaj+bHzolZ1YZkDBWtiJD5urviYxKrScysJ2SmpFe9mbif143NeG1n/EoSQ1GbLEoTAUxMZl9TQZcITNiYgllittbCRtRRZmx2ZRsCN7yy6ukXat6F9Va87JSv8njKMIJnMI5eHAFdbiDBrSAAcIzvMKb8+i8OO/Ox6K14OQzx/AHzucP1tuM9Q== 0.1 15 4

0.1

0.0 2 10

0 0 1 2

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100

0.8 25 50 0.6 0.08 0.4 0 8

0.2 50

0 0 50 100 150 0.06 20

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0.02

2 10 0 0 1 2 3

✓AAAB7XicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Ae0oWy2m3btZhN2J0IJ/Q9ePCji1f/jzX/jts1BWx8MPN6bYWZekEhh0HW/ncLa+sbmVnG7tLO7t39QPjxqmTjVjDdZLGPdCajhUijeRIGSdxLNaRRI3g7GtzO//cS1EbF6wEnC/YgOlQgFo2ilVg9HHGm/XHGr7hxklXg5qUCORr/81RvELI24QiapMV3PTdDPqEbBJJ+WeqnhCWVjOuRdSxWNuPGz+bVTcmaVAQljbUshmau/JzIaGTOJAtsZURyZZW8m/ud1Uwyv/UyoJEWu2GJRmEqCMZm9TgZCc4ZyYgllWthbCRtRTRnagEo2BG/55VXSqlW9i2rt/rJSv8njKMIJnMI5eHAFdbiDBjSBwSM8wyu8ObHz4rw7H4vWgpPPHMMfOJ8/pUWPLA== Figure 3. Results for modulated signals: (a) Pseudo Wigner- Ville distribution and (b) optical time-frequency tomograms Figure 4. Entropies as function of the angle θ and modulation for AM signals are presented; (a) Pseudo Wigner-Ville dis- parameter for (a) AM analytic signal and (b) FM analytic tribution and (b) optical time-frequency tomograms for FM signal. signals are illustrated.

Another case is to consider a signal with FM, which III. TOMOGRAPHIC REPRESENTATION FOR has the following form: MODULATED SIGNALS Z sFM(t) = A cos ω0t + ωd dtsm(t) + φ0 . (17) Several types of signals have been considered in the R context of tomographic analysis [13–18]. However, exist- In this case, the frequency varies as follows: ing examples lack of analyzing modulated signals, which are intensively used in telecommunication tasks. A gen- ω(t) = ω0 + ωdsm(t), (18) eral model of signals that we are interested in has the where ωd is frequency deviation, i.e. an analog of the following form: modulation parameter m for the amplitude modulation signal. We illustrate the signal and its Fourier transform s(t) = A(t) cos(ω(t)t + φ ), (14) 0 in Fig.2c and Fig.2d. where A(t) is the amplitude of the signal, ω(t) is the For the signals with AM and FM given by Eq. (15) and Eq. (17), correspondingly, we calculate modified optical frequency and φ0 is the phase. The fact that the amplitude and frequency are time-dependent indicates that time-frequency tomograms based on pseudo Wigner-Ville they can be used for modulation purposes. Most radio distributions. These results are presented in Fig.3. systems in the 20th century used frequency modulation (FM) or amplitude modulation (AM) for radio broadcast. IV. ENTROPIC RELATIONS We start from the simplest case of signals with AM. A signal with amplitude modulation is as follows: Another interesting point of view on the link between sAM(t) = A(t) cos(ωt + φ0), (15) signal processing and quantum physics originates from uncertainty relations and related entropic relations. The where A(t) is a law of change of amplitude with time. In idea behind this consideration is the fact that for analytic amplitude modulation, the amplitude (signal strength) signals with normalized energy of the spectrum, of the carrier wave is varied in proportion to that of the Z Z message signal being transmitted. We consider the fol- dt (t) 2 = dω ˜(ω) 2 = 1, (19) lowing case: R |S | R |S | one can think of the introduction of the differential en- A(t) = 1 + ms (t), s (t) = cos(Ωt), (16) m m tropy (also known as the continuous Shannon entropy) where m is the modulation coefficient and Ω is the fre- in the time domain as follows: quency of the signal. We illustrate modulated signals and Z S = dt (t) 2 ln (t) 2. (20) their Fourier transforms in Fig.2a and Fig.2b. t − R |S | |S | 5

The diﬀerential entropy on the analytic signal in the fre- tomograms in Fig.4. We also can check entropic relations quency domain has the following form: given by Eq. (23) for tomograms formulated as follows:

S(θ) + S(θ + π/2) ln(πe). (24) Z ≥ S = dω ˜(ω) 2 ln ˜(ω) 2. (21) ω − |S | |S | R V. CONCLUSION

One can see that these expressions for differential en- We have considered applications of quantum tomog- tropies are equivalent to those for the position ψ(q) 2 raphy framework for the time-frequency analysis of sig- and momentum ψ(p) 2 representations of the probabil-| | nals with amplitude and frequency modulations. We ity distribution function,| | which are calculated via the have demonstrated an efficient way for calculating opti- corresponding wavefunction. Therefore, there the follow- cal time-frequency tomograms for analytic signals based ing entropic inequality holds for the differential entropies on the pseudo Wigner-Ville distribution, which seems to of analytic signals [31]: be important for signals of a sophisticated structure that change significantly over their duration. We also have analyzed differential entropies of the signals calculated via optical time-frequency tomograms and discussed cor- St + Sω ln(πe). (22) ≥ responding entropic relations. Our approach can be extended and generalized in a number of ways. First, one can think of studying time- The considered below tomographic approach to signal frequency (symplectic or optical) tomograms based on analysis allows introducing the time-frequency entropy as other types of modified Wigner-Ville distributions, such follows: Wigner-Ville distributions with windows both in time and frequency domains. Second, an important is to un- derstand the nature of the difference between original Z time-frequency tomograms and modified time-frequency S(θ) = dX p(X, θ) ln p(X, θ). (23) − R T T tomograms. In particular, it is important whereas modified time-frequency tomograms are able to capture some feature of highly non-stationary signals that are impor- We note that the optical time-frequency tomogram tant. Finally, an interesting task to analyse how time- p(X, θ) is calculated on the basis of the pseudo Wigner- frequency tomograms can be measured in various appli- TVille distribution. cations, such as analysis of reflectometry data [19, 20]. We use this formula for the analysis of AM and FM signals. In this case, S(θ) also becomes a function of the modulation parameter, so we have SAM(θ, m) and ACKNOWLEDGMENTS FM S (θ, ωd) for signals given by Eq. (15) and Eq. (17), correspondingly. We present the results of calculations The work was supported by the grant of the President time-frequency entropies based on optical time-frequency of the Russian Federation (project MK- 923.2019.2).

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