Tomographic and entropic analysis of modulated signals

T omographic and en tropic analysis of mo dulated signals A.S. Mastiuk ov a, 1, 2 M.A. Ga vreev, 1, 2 E.O. Kiktenk o, 1, 2, 3 and A.K. F edoro v 1, 2 1 Russian Quantum Center, Skolkovo, Mosc ow 143025, Russia 2 Mosc ow Institute of Physics and T e chnolo gy, Dolgoprudny, Mosc ow R e gion 141700, Russia 3 Dep artment of Mathematic al Metho ds for Quantum T e chnolo gies, Steklov Mathematic al Institute of Russian A c ademy of Sciences, Mosc ow 119991, Russia (Dated: Septem b er 29, 2020) W e study an application of the quantum tomography framew ork for the time-frequency analysis of mo dulated signals. In particular, we calculate optical tomographic represen tations and Wigner-Ville distributions for signals with amplitude and frequency modulations. W e also consider t ime-frequency en tropic relations for mo dulated signals, which are naturally associated with the F ourier analysis. A n umerical to olbox for calculating optical time-frequency tomograms based on pseudo Wigner-Ville distributions for mo dulated signals is provided. I. INTR ODUCTION Time-frequency analysis is a pow erful to ol of modern signal processing [ 1 – 4 ]. Complemen tary to the informa- tion that can be extracted from the frequency domain via F ourier analysis, time-frequency analysis pro vides a wa y for studying a signal in b oth time and frequency repre- sen tations simultaneously . This is useful, in particular, for signals of a sophisticated structure that change signifi- can tly ov er their duration, for example, m usic signals [ 5 ]. Existing approaches to time-frequency analysis use lin- ear canonical transformations preserving the symplectic form [ 1 ]. Geometrically this can b e illustrated as follows: the F ourier transform can b e view ed as a π / 2 rotation in the asso ciated time-frequency plane, whereas other time-frequency representations allo w arbitrary symplec- tic transformations in the time-frequency plane. There is a num b er of wa ys for defining a time-frequency distribu- tion function with required properties (for a review, see Ref. [ 4 ]). T ransformations b et ween v arious distributions in time-frequency analysis are quite well-understoo d [ 1 ]. The idea behind time-frequency analysis is v ery close to the motiv ation for studying phase-space representa- tions in quantum physics. As it is w ell known, the re- lation b et ween position and momentum represen tations of the wa v e function is giv en b y the F ourier transform, whic h is similar to the relation betw een signals in time and frequency domains. This analogy b ecomes ev en more transparen t in the framework of analytic signals, which are complex as well as wa v e functions. One of the possible w ays to characterize a quantum state in the phase space are to use the Wigner quasiprobabilit y distribution [ 6 ]. The Wigner quasiprobability distribution resem bles clas- sical phase space probability distributions that is used in statistical mec hanics. How ev er, it cannot b e fully inter- preted as a probability distribution since it takes nega- tiv e v alues [ 6 – 8 ]. W e also note the Wigner quasiprobabil- it y distribution is successfully used for analyzing v arious phenomena in quantum optics and quantum statistical ph ysics [ 8 ]. In the field of signal pro cessing, the Wigner distribution function often referred to as the Wigner– Ville distribution [ 9 , 10 ]. The application of the Wigner distribution mak es an in teresting connection b et ween the metho ds in quantum physics and signal pro cessing, es- p ecially in the context of time-frequency and p osition- momen tum uncertaint y relations. Recen t decades, the link b et ween phase space form ula- tion of quan tum mec hanics and time-frequency analysis in tensively studied in the context of quan tum tomogra- ph y [ 11 ]. Quan tum tomography appears as a tec hnique for the reconstruction of the Wigner function (densit y matrix) in quantum-optical exp erimen ts [ 12 ]. The re- sults of tomographic measurements in principle contain all the information ab out the measured system, so they can b e considered as a quantities for the description of quan tum states [ 11 ]. This is the core idea b ehind the omographic representation of quantum states. In par- ticular, symplectic tomography proto cols use a marginal probabilit y distribution of shifted and squeezed p osition and momentum v ariables. This approac h has b een used in the con text of time-frequency analysis [ 13 ] and time- frequency en tropic analysis [ 14 ] for v arious t ypes of sig- nals, suc h as complex Gaussian signals [ 13 – 18 ] and reflec- tometry data [ 19 , 20 ]. A general analysis of the relation b et w een time-frequency tomograms and other transfor- mations (including w av elets) is presented in Ref. [ 15 ]. Ho wev er, the considered examples of signals lack ana- lyzing mo dulated signals, which are intensiv ely used in telecomm unication signals. Moreov er, the Wigner-Ville distribution has been considered in the con text of diag- nostics of features of mo dulated signals [ 21 ], so one can exp ect that tomograms are helpful for such an analysis. In this work, we consider tomographic representations for mo dulated signals. W e calculate optical tomographic represen tations and Wigner-Ville distributions for signals with amplitude and frequency m odulations. In particu- lar, we study the method of the optical time-frequency tomograms via pseudo Wigner-Ville distributions, and discuss adv antages of suc h an approach. W e also consider time-frequency en tropic relations for mo dulated signals. Our work is organized as follo ws. In Sec. I I , w e in tro- duce general relations for tomographic analysis of ana- lytic signals. In Sec. 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( 13 ): (a) the Wigner-Ville distribution is presen ted (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 b etw een 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 b etw een tomograms is illustrated. mo dulations. In Sec. IV , we analyze time-frequency en- tropic relations. W e conclude in Sec. V . I I. TOMOGRAPHIC ANAL YSIS Here we introduce basic to ols for the tomographic anal- ysis of signals. W e consider a time-dep endent signal s ( t ), whose representation in the frequency domain ˜ s ( ω ) can b e obtained via the F ourier transform. Conv en tionally , w e use an analytical representation of signals in the fol- lo wing form: S ( t ) = s ( t ) + iH [ s ( t )] , (1) where H [ s ( t )] = 1 π Z R dp s ( p ) t − p (2) is the Hilbert transform of the signal. T he adv antage of using the analytic signal is that in the frequency do- main the amplitude of negative frequency comp onents are zero. This satisfies mathematical completeness of the problem b y accoun ting for all frequencies, yet do es not limit the practical application since only p ositive fre- quency components hav e a practical in terpretation. The metho d based on the use of analytic signals also mak es a clear analogy b etw een time-frequency distributions in signal pro cessing and phase-space distributions in quan- tum mec hanics [ 1 ]. The family of marginal distributions, which contains complete information on the analytical signal, has been in tro duced in Ref. [ 11 ]. It has the following form: T ( X , θ ) = 1 2 π | sin θ | |I ( X, θ ) | 2 , (3) where I ( X, θ ) = Z R dt S ( t ) exp  it 2 cos θ 2 sin θ − itX sin θ  . (4) Here X = t cos θ + ω sin θ is the dimensionless quadrature v ariable. This representation is referred to as the optical time-frequency tomogram of the signal S ( t ). The inte- gral transformation in Eq. ( 4 ) is the fractional F ourier transform. The tomogram is normalized as follows: Z R dX T ( X, θ ) = 1 . (5) It also giv es the distribution of the signal in time and frequency domains, corresp ondingly: T ( X = t, 0) = |S ( t ) | 2 , T ( X = ω, π / 2) = | ˜ S ( ω ) | 2 . (6) The optical time-frequency tomogram is a particular case of the symplectic time-frequency tomogram T ( X , µ, ν ) 3 of the signal S ( t ), where µ = cos θ and ν = sin θ . In some cases, another v ariations of tomographic represen- tation, suc h as time-scale tomograms, frequency-scale to- mograms, and time-conformal tomograms, are used [ 20 ]. W e restrict ourselv es to the consideration of optical time- frequency tomograms only . It seems to be quite straigh tforw ard to calculate opti- cal time-frequency tomograms using Eq. ( 3 ). Ho wev er, there are well-kno wn problems in the field of signal pro- cessing, such as, for example, aliasing, whic h give rise to distortions during the signal reconstruction and com- putational difficulties. These problems also o ccur dur- ing the calculation of the Wigner-Ville distribution for analytic signals [ 10 ]. W e remind that the Wigner-Ville distribution of the signal has the following form: W ( t, ω ) = Z R dτ S  t + τ 2  S ∗  t − τ 2  e − iω τ . (7) The Wigner-Ville distribution is normalized as follows: 1 2 π Z R 2 dtdω W ( t, ω ) = 1 . (8) When the Wigner-Ville distribution is applied to a sig- nal with multi frequency components, cross-terms app ear due to its quadratic nature. In order to a v oid the effect of cross-terms the windo w ed version of the Wigner-Ville distribution, which is known as pseudo Wigner-Ville dis- tribution, is used. The pseudo Wigner-Ville distribution is defined as follows: W p ( t, ω ) = Z R dτ h ( τ ) S  t + τ 2  S ∗  t − τ 2  e − iω τ , (9) where h ( τ ) is the window function in the time domain. The windo w function h ( τ ) can b e used, for example, in the Hamming window form: h M ( τ )=0 . 54 − 0 . 46 cos  2 π τ M − 1  , 0 ≤ τ ≤ M − 1 . (10) Pseudo Wigner-Ville distributions [ 22 – 25 ] and their mo difications are actively used in v arious fields, such as the disp ersion analysis of wa veguides [ 26 ] and study- ing oil-in-water flow patterns [ 27 ]. W e note that v ari- ous time-frequency filters are employ ed for eliminating cross-terms in the Wigner-Ville distribution, whic h is of high imp ortance for the analysis of non-stationary sys- tems. Another approach for reducing cross-terms in the Wigner-Ville distribution uses a tunable-Q w av elet trans- form [ 28 ]. In our consideration b elo w, w e use the sim- plest case of the pseudo Wigner-Ville distribution with the simplest form of the window function. Using the relation betw een Wigner-Ville distributions and optical time-frequency tomograms, whic h is given b y the Radon transform, one can reconstruct the optical tomogram of the signal as follows: T ( X , θ ) = Z R 3 dk dtdω (2 π ) 2 W ( t, ω ) e − ik ( X − t cos θ − ω sin θ ) . 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Considered modulated signals: (a) AM signal in the time domain and (b) in the frequency domain; (c) FM signal in the time domain and (d) in the frequency domain. Then in order to reduce the complexit y of calculating op- tical time-frequency tomograms it is p ossible to redefine it via pseudo Wigner-Ville distributions as follows: T p ( X, θ ) = Z R 3 dk dtdω (2 π ) 2 W p ( t, ω ) e − ik ( X − t cos θ − ω sin θ ) . (12) This relation giv e rise to the modification of the in- tegral relation betw een analytic signal and its time- frequency optical tomogram, which is given b y Eq. ( 3 ). In our w ork, we use the pseudo time-frequency optical tomogram T p ( X, θ ) for analyzing prop erties of signals. W e note that this consideration is related to the estab- lishing a corresp ondence b etw een the fractional F ourier transform and the Wigner distribution [ 29 , 30 ]. In order to see a difference in calculating original and pseudo time-frequency optical tomograms, w e consider an example of a chirp signal of the following form: s ( t ) = A sin ( ω t + φ 0 ) e − α ( t − t 0 ) 2 , (13) where A is the fixed amplitude, φ 0 , α , and t 0 are fixed constan ts. F or this c hirp signal in the analytic form giv en b y Eq. ( 1 ) we calculate first original Wigner-Ville distribution (Fig. 1 a) and pseudo Wigner-Ville distri- bution (Fig. 1 b). One can capture a difference b e- t ween D ( t, ω ) = | W ( t, ω ) − W p ( t, ω ) | original Wigner- Ville distribution and pseudo Wigner-Ville distribution (see Fig. 1 c). This difference manifests in calculat- ing original and pseudo time-frequency optical tomo- grams (Fig. 1 d, Fig. 1 e, and Fig. 1 f ), where D ( X , θ ) = |T ( X , θ ) − T p ( X, θ ) | . The differences D ( t, ω ) and D ( X , θ ) are non-zero. 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AAAB/3icbVDLSgMxFM3UV62vUcGNm2ARKkiZqYIui4K4ESrYB3TGkkkzbWiSGZKMUMYu/BU3LhRx62+4829M21lo64ELh3Pu5d57gphRpR3n28otLC4tr+RXC2vrG5tb9vZOQ0WJxKSOIxbJVoAUYVSQuqaakVYsCeIBI81gcDn2mw9EKhqJOz2Mic9RT9CQYqSN1LH3mp34PvUkh1c3o5I+9iJOeuioYxedsjMBnCduRoogQ61jf3ndCCecCI0ZUqrtOrH2UyQ1xYyMCl6iSIzwAPVI21CBOFF+Orl/BA+N0oVhJE0JDSfq74kUcaWGPDCdHOm+mvXG4n9eO9HhuZ9SESeaCDxdFCYM6giOw4BdKgnWbGgIwpKaWyHuI4mwNpEVTAju7MvzpFEpuyflyu1psXqRxZEH++AAlIALzkAVXIMaqAMMHsEzeAVv1pP1Yr1bH9PWnJXN7II/sD5/AMYylUs= (a) (b) (d) (c) Figure 3. Results for mo dulated signals: (a) Pseudo Wigner- Ville distribution and (b) optical time-frequency tomograms for AM signals are presented; (a) Pseudo Wigner-Ville dis- tribution and (b) optical time-frequency tomograms for FM signals are illustrated. I II. TOMOGRAPHIC REPRESENT A TION FOR MODULA TED SIGNALS Sev eral types of signals hav e b een considered in the con text of tomographic analysis [ 13 – 18 ]. Ho wev er, exist- ing examples lack of analyzing mo dulated signals, which are intensiv ely used in telecommunication tasks. A gen- eral mo del of signals that we are in terested in has the follo wing form: s ( t ) = A ( t ) cos( ω ( t ) t + φ 0 ) , (14) where A ( t ) is the amplitude of the signal, ω ( t ) is the frequency and φ 0 is the phase. The fact that the am- plitude and frequency are time-dep enden t indicates that they can b e used for mo dulation purposes. Most radio systems in the 20th century used frequency mo dulation (FM) or amplitude mo dulation (AM) for radio broadcast. W e start from the simplest case of signals with AM. A signal with amplitude mo dulation is as follo ws: s AM ( t ) = A ( t ) cos( ω t + φ 0 ) , (15) where A ( t ) is a law of change of amplitude with time. In amplitude mo dulation, the amplitude (signal strength) of the carrier w av e is v aried in prop ortion to that of the message signal b eing transmitted. W e consider the fol- lo wing case: A ( t ) = 1 + ms m ( t ) , s m ( t ) = cos(Ω t ) , (16) where m is the mo dulation co efficien t and Ω is the fre- quency of the signal. W e illustrate mo dulated signals and their F ourier transforms in Fig. 2 a and Fig. 2 b. 0 0 . 2 0 . 4 0 2 4 6 8 1 0 10 15 20 25 0 0 . 2 0 . 4 0 2 4 6 8 1 0 10 15 20 25 ✓ AAAB7XicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Ae0oWy2m3btZhN2J0IJ/Q9ePCji1f/jzX/jts1BWx8MPN6bYWZekEhh0HW/ncLa+sbmVnG7tLO7t39QPjxqmTjVjDdZLGPdCajhUijeRIGSdxLNaRRI3g7GtzO//cS1EbF6wEnC/YgOlQgFo2ilVg9HHGm/XHGr7hxklXg5qUCORr/81RvELI24QiapMV3PTdDPqEbBJJ+WeqnhCWVjOuRdSxWNuPGz+bVTcmaVAQljbUshmau/JzIaGTOJAtsZURyZZW8m/ud1Uwyv/UyoJEWu2GJRmEqCMZm9TgZCc4ZyYgllWthbCRtRTRnagEo2BG/55VXSqlW9i2rt/rJSv8njKMIJnMI5eHAFdbiDBjSBwSM8wyu8ObHz4rw7H4vWgpPPHMMfOJ8/pUWPLA== ✓ AAAB7XicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Ae0oWy2m3btZhN2J0IJ/Q9ePCji1f/jzX/jts1BWx8MPN6bYWZekEhh0HW/ncLa+sbmVnG7tLO7t39QPjxqmTjVjDdZLGPdCajhUijeRIGSdxLNaRRI3g7GtzO//cS1EbF6wEnC/YgOlQgFo2ilVg9HHGm/XHGr7hxklXg5qUCORr/81RvELI24QiapMV3PTdDPqEbBJJ+WeqnhCWVjOuRdSxWNuPGz+bVTcmaVAQljbUshmau/JzIaGTOJAtsZURyZZW8m/ud1Uwyv/UyoJEWu2GJRmEqCMZm9TgZCc4ZyYgllWthbCRtRTRnagEo2BG/55VXSqlW9i2rt/rJSv8njKMIJnMI5eHAFdbiDBjSBwSM8wyu8ObHz4rw7H4vWgpPPHMMfOJ8/pUWPLA== m AAAB6HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48t2FpoQ9lsJ+3a3STsboQS+gu8eFDEqz/Jm//GbZuDtj4YeLw3w8y8IBFcG9f9dgpr6xubW8Xt0s7u3v5B+fCoreNUMWyxWMSqE1CNgkfYMtwI7CQKqQwEPgTj25n/8IRK8zi6N5MEfUmHEQ85o8ZKTdkvV9yqOwdZJV5OKpCj0S9/9QYxSyVGhgmqdddzE+NnVBnOBE5LvVRjQtmYDrFraUQlaj+bHzolZ1YZkDBWtiJD5urviYxKrScysJ2SmpFe9mbif143NeG1n/EoSQ1GbLEoTAUxMZl9TQZcITNiYgllittbCRtRRZmx2ZRsCN7yy6ukXat6F9Va87JSv8njKMIJnMI5eHAFdbiDBrSAAcIzvMKb8+i8OO/Ox6K14OQzx/AHzucP1tuM9Q== S AM ( m, ✓ ) AAAB/XicbVDLSgNBEJyNrxhf6+PmZTAIESTsRkGPUS9ehIjmAdkYZieTZMjM7jLTK8Ql+CtePCji1f/w5t84Sfag0YKGoqqb7i4/ElyD43xZmbn5hcWl7HJuZXVtfcPe3KrpMFaUVWkoQtXwiWaCB6wKHARrRIoR6QtW9wcXY79+z5TmYXALw4i1JOkFvMspASO17Z2bu8RTEp9djQry0IM+A3LQtvNO0ZkA/yVuSvIoRaVtf3qdkMaSBUAF0brpOhG0EqKAU8FGOS/WLCJ0QHqsaWhAJNOtZHL9CO8bpYO7oTIVAJ6oPycSIrUeSt90SgJ9PeuNxf+8Zgzd01bCgygGFtDpom4sMIR4HAXucMUoiKEhhCpubsW0TxShYALLmRDc2Zf/klqp6B4VS9fH+fJ5GkcW7aI9VEAuOkFldIkqqIooekBP6AW9Wo/Ws/VmvU9bM1Y6s41+wfr4Bix8lGU= ! d AAAB73icbVDLSgNBEJyNrxhfUY9eBoPgKexGQY9BLx4jmAckS5id7U2GzGOdmRVCyE948aCIV3/Hm3/jJNmDJhY0FFXddHdFKWfG+v63V1hb39jcKm6Xdnb39g/Kh0ctozJNoUkVV7oTEQOcSWhaZjl0Ug1ERBza0eh25refQBum5IMdpxAKMpAsYZRYJ3V6SsCA9ON+ueJX/TnwKglyUkE5Gv3yVy9WNBMgLeXEmG7gpzacEG0Z5TAt9TIDKaEjMoCuo5IIMOFkfu8UnzklxonSrqTFc/X3xIQIY8Yicp2C2KFZ9mbif143s8l1OGEyzSxIuliUZBxbhWfP45hpoJaPHSFUM3crpkOiCbUuopILIVh+eZW0atXgolq7v6zUb/I4iugEnaJzFKArVEd3qIGaiCKOntErevMevRfv3ftYtBa8fOYY/YH3+QMIMI/2 (b) (d) S FM ( ! d , ✓ ) AAACBHicbVBNS8NAEN34WetX1WMvi0VQkJKooEdREC9CRatCU8NmO22X7iZhdyKU0IMX/4oXD4p49Ud489+4bXPw68HA470ZZuaFiRQGXffTmZicmp6ZLcwV5xcWl5ZLK6tXJk41hzqPZaxvQmZAigjqKFDCTaKBqVDCddg7HvrXd6CNiKNL7CfQVKwTibbgDK0UlMoXt5mvFT05G2z6sYIOC1rbPnYB2VZQqrhVdwT6l3g5qZActaD04bdiniqIkEtmTMNzE2xmTKPgEgZFPzWQMN5jHWhYGjEFppmNnhjQDau0aDvWtiKkI/X7RMaUMX0V2k7FsGt+e0PxP6+RYvugmYkoSREiPl7UTiXFmA4ToS2hgaPsW8K4FvZWyrtMM442t6INwfv98l9ytVP1dqs753uVw6M8jgIpk3WySTyyTw7JKamROuHknjySZ/LiPDhPzqvzNm6dcPKZNfIDzvsXl4aXaw== (c) (a) S FM ( ! d , ✓ ) AAACBHicbVBNS8NAEN34WetX1WMvi0VQkJKooEdREC9CRatCU8NmO22X7iZhdyKU0IMX/4oXD4p49Ud489+4bXPw68HA470ZZuaFiRQGXffTmZicmp6ZLcwV5xcWl5ZLK6tXJk41hzqPZaxvQmZAigjqKFDCTaKBqVDCddg7HvrXd6CNiKNL7CfQVKwTibbgDK0UlMoXt5mvFT05G2z6sYIOC1rbPnYB2VZQqrhVdwT6l3g5qZActaD04bdiniqIkEtmTMNzE2xmTKPgEgZFPzWQMN5jHWhYGjEFppmNnhjQDau0aDvWtiKkI/X7RMaUMX0V2k7FsGt+e0PxP6+RYvugmYkoSREiPl7UTiXFmA4ToS2hgaPsW8K4FvZWyrtMM442t6INwfv98l9ytVP1dqs753uVw6M8jgIpk3WySTyyTw7JKamROuHknjySZ/LiPDhPzqvzNm6dcPKZNfIDzvsXl4aXaw== S AM ( m, ✓ ) AAAB/XicbVDLSgNBEJyNrxhf6+PmZTAIESTsRkGPUS9ehIjmAdkYZieTZMjM7jLTK8Ql+CtePCji1f/w5t84Sfag0YKGoqqb7i4/ElyD43xZmbn5hcWl7HJuZXVtfcPe3KrpMFaUVWkoQtXwiWaCB6wKHARrRIoR6QtW9wcXY79+z5TmYXALw4i1JOkFvMspASO17Z2bu8RTEp9djQry0IM+A3LQtvNO0ZkA/yVuSvIoRaVtf3qdkMaSBUAF0brpOhG0EqKAU8FGOS/WLCJ0QHqsaWhAJNOtZHL9CO8bpYO7oTIVAJ6oPycSIrUeSt90SgJ9PeuNxf+8Zgzd01bCgygGFtDpom4sMIR4HAXucMUoiKEhhCpubsW0TxShYALLmRDc2Zf/klqp6B4VS9fH+fJ5GkcW7aI9VEAuOkFldIkqqIooekBP6AW9Wo/Ws/VmvU9bM1Y6s41+wfr4Bix8lGU= Figure 4. Entropies as function of the angle θ and modulation parameter for (a) AM analytic signal and (b) FM analytic signal. Another case is to consider a signal with FM, whic h has the following form: s FM ( t ) = A cos  ω 0 t + ω d Z R dts m ( t ) + φ 0  . (17) In this case, the frequency v aries as follows: ω ( t ) = ω 0 + ω d s m ( t ) , (18) where ω d is frequency deviation, i.e. an analog of the mo dulation parameter m for the amplitude mo dulation signal. W e illustrate the signal and its F ourier transform in Fig. 2 c and Fig. 2 d. F or the signals with AM and FM giv en by Eq. ( 15 ) and Eq. ( 17 ), correspondingly , we calculate mo dified optical time-frequency tomograms based on pseudo Wigner-Ville distributions. These results are presen ted in Fig. 3 . IV. ENTR OPIC RELA TIONS Another interesting p oin t of view on the link b etw een signal pro cessing and quan tum physics originates from uncertain ty relations and related entropic relations. The idea b ehind this consideration is the fact that for analytic signals with normalized energy of the sp ectrum, Z R dt |S ( t ) | 2 = Z R dω | ˜ S ( ω ) | 2 = 1 , (19) one can think of the introduction of the differen tial en- trop y (also known as the con tinuous Shannon entrop y) in the time domain as follows: S t = − Z R dt |S ( t ) | 2 ln |S ( t ) | 2 . (20) 5 The differential entrop y on the analytic signal in the fre- quency domain has the following form: S ω = − Z R dω | ˜ S ( ω ) | 2 ln | ˜ S ( ω ) | 2 . (21) One can see that these expressions for differential en- tropies are equiv alent to those for the p osition | ψ ( q ) | 2 and momentum | ψ ( p ) | 2 represen tations of the probabil- it y distribution function, whic h are c alculated via the corresp onding w a vefunction. Therefore, there the follow- ing entropic inequalit y holds for the differential en tropies of analytic signals [ 31 ]: S t + S ω ≥ ln( π e ) . (22) The considered b elo w tomographic approach to signal analysis allo ws in tro ducing the time-frequency en trop y as follo ws: S ( θ ) = − Z R dX T p ( X, θ ) ln T p ( X, θ ) . (23) W e note that the optical time-frequency tomogram T p ( X, θ ) is calculated on the basis of the pseudo Wigner- Ville distribution. W e use this form ula for the analysis of AM and FM signals. In this case, S ( θ ) also b ecomes a function of the mo dulation parameter, so w e hav e S AM ( θ , m ) and S FM ( θ , ω d ) for signals given by Eq. ( 15 ) and Eq. ( 17 ), corresp ondingly . W e present the results of calculations time-frequency en tropies based on optical time-frequency tomograms in Fig. 4 . W e also can c heck entropic relations giv en by Eq. ( 23 ) for tomograms formulated as follo ws: S ( θ ) + S ( θ + π / 2) ≥ ln( π e ) . (24) V. CONCLUSION W e hav e considered applications of quantum tomog- raph y framework for the time-frequency analysis of sig- nals with amplitude and frequency modulations. W e ha ve demonstrated an efficient wa y for calculating opti- cal time-frequency tomograms for analytic signals based on the pseudo Wigner-Ville distribution, which seems to b e imp ortan t for signals of a sophisticated structure that c hange significantly ov er their duration. W e also ha ve analyzed differential entropies of the signals calculated via optical time-frequency tomograms and discussed cor- resp onding entropic relations. Our approach can b e extended and generalized in a n umber of wa ys. First, one can think of studying time- frequency (symplectic or optical) tomograms based on other types of mo dified Wigner-Ville distributions, such Wigner-Ville distributions with windows b oth in time and frequency domains. Second, an imp ortan t is to un- derstand the nature of the difference b etw een original time-frequency tomograms and mo dified time-frequency tomograms. In particular, it is imp ortant whereas mo di- fied time-frequency tomograms are able to capture some feature of highly non-stationary signals that are impor- tan t. Finally , an in teresting task to analyse how time- frequency tomograms can b e measured in v arious appli- cations, suc h as analysis of reflectometry data [ 19 , 20 ]. A CKNOWLEDGMENTS The work was supp orted by the gran t of the President of the Russian F ederation (pro ject MK- 923.2019.2). [1] L. Cohen, time-fre quency Analysis (Prentice-Hall, New Y ork, 1995). [2] A. P apandreou-Suppapp ola, Applic ations in time- fr e quency Signal Pr o c essing (CRC Press, Boca Raton, 2002). [3] D. Dragoman, Applications of the Wigner distribution function in signal pro cessing, EURASIP J. Adv. Signal Pro cess. 10 , 1520 (2005) . [4] E. Sejdi´ c, I. Djuro vi´ c, and J. 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