Time Reversal Operator Research Articles

Artificial neural network-based sparse channel estimation for V2V communication systems

Abstract Artificial neural networks (ANNs) have gained a lot of attention from researchers in the past few years and have been employed on a large scale. They have also been gaining momentum in wireless communication systems. For efficient vehicle-to-vehicle (V2V) channel communication, a sparse multipath channel issue must be studied. To minimize the multipath effect, a time reversal (TR) operation and time division synchronization orthogonal frequency division multiplexing (TDS-OFDM) have been appealing because of their fast synchronization and active spectral efficiency. To improve the transceiver's execution in a frequency-selective fading channel environment, an OFDM system is used to reduce inter- symbol interference (ISI). Simultaneous Orthogonal Matching Pursuit (SOMP) channel state estimator algorithm suffer from high computational cost and high computational complexity. The ANN algorithm has better performance than SOMP algorithm. The proposed neural network technologies have lower complexity than the SOMP algorithm. The application of ANN is capable of solving complex problems, such as those encountered in image, signal processing and have been implemented for channel estimation in OFDM. The proposed ANN outperformed the SOMP algorithm with regard to signal compensation. Overall, the ANN algorithm achieved the best performance. This study proposes an ANN-based sparse channel state estimator. Regarding the bit error rate (BER) metric, the proposed estimator outperforms the channel estimation approach based on the SOMP. The simulation results confirm the efficacy of the proposed approach.

Sym4state.jl: An efficient computation package for magnetic materials

Exploring magnetic configurations of magnets often involves utilizing the four-state method to obtain the magnetic interaction matrix, and Monte Carlo method to simulate spin textures and phase transition processes. However, computing the interaction matrix between magnetic atoms using the four-state method requires plenty of individual calculations. Despite manual simplifying the number of individual calculations based on material's symmetry is possible, there remains a necessity for an automated approach to streamline the process for high-throughput screening of magnetic materials. Meanwhile, the traditional sequential Monte Carlo simulation encounters challenges of low efficiency and long time consuming in dealing with large systems. Furthermore, the prior parallelism in the Heisenberg model was limited to parallel computation of the system's energy or run several replicas in parallel. Hence, in our pursuit of comprehensive parallelization for the Heisenberg model, we have introduced a novel adaptation of the checkerboard algorithm, enabling a fully parallelizable simulation of the Heisenberg model. To address these problems, we have developed Sym4state.jl, a program specifically designed to simplify the computation of magnetic interaction matrix and simulate spin textures under various environmental conditions. This program, available as a Julia package, can be freely accessed at https://github.com/A-LOST-WAPITI/Sym4state.jl. Program summaryProgram title: Sym4state.jlCPC Library link to program files:https://doi.org/10.17632/s6dkmgrjfw.1Developer's repository link:https://github.com/A-LOST-WAPITI/Sym4state.jlLicensing provisions: MITProgramming language: JuliaNature of problem: Employing the four-state method to calculate magnetic interaction matrix for magnetic materials can be simplified based on material symmetry, however, there is a lack of automated approach to streamline the simplification. Additionally, the commonly used Metropolis method for simulating magnetic texture can only make parallel computation of the system's energy or run several replicas in parallel, which could hardly boost the performance when simulating the large-scale magnetic textures.Solution method: We simplify the four-state method calculations by utilizing the principles of energy invariance under symmetry operations and time reversal operations. To enhance the efficiency of the Metropolis algorithm, we have designed a strategy to divide the entire 2D lattice into multiple domains. We then execute the Metropolis algorithm in parallel for each individual domain, thereby improving the overall computational efficiency.Additional comments including restrictions and unusual features: While the methods aimed at simplifying the four-state method and parallelizing the Metropolis algorithm are applicable to both 2D and 3D systems, the current program is specifically designed for the calculation and simulation of magnetism in 2D materials. As a result, compatibility with 3D systems has not yet been implemented.

Detection of Buried Nonlinear Targets Using DORT

Ground-penetrating radars (GPR) based on a variety of techniques have been proposed to improve the performance of buried target (e.g., landmines, threat devices) detection. However, the small radar cross section (RCS) of small electronic devices poses difficulties for target detection, especially when they are buried in lossy and inhomogeneous media. This paper presents a novel buried nonlinear target detection method based on the decomposition of the time-reversal operator (DORT) that uses a multistatic system to overcome the limitations of conventional GPR. Using harmonic radar, which detects the harmonic responses scattered from electronic devices, and DORT processing, which enables focusing/imaging of the detected target, the detection performance is verified by conducting simulation and measurements. The overall results demonstrate that the proposed method achieves accurate detection of buried targets with small RCS.

Sound Source Localization Method Based on Time Reversal Operator Decomposition in Reverberant Environments

Predicting sound sources in reverberant environments is a challenging task because reverberation causes reflection and scattering of sound waves, making it difficult to accurately determine the position of the sound source. Due to the characteristics of overcoming multipath effects and adaptive focusing of the time reversal technology, this paper focuses on the application of the time reversal operator decomposition method for sound source localization in reverberant environments and proposes the image-source time reversal multiple signals classification (ISTR-MUSIC) method. Firstly, the time reversal operator is derived, followed by the proposal of a subspace method to achieve sound source localization. Meanwhile, the use of the image-source method is proposed to calculate and construct the transfer matrix. To validate the effectiveness of the proposed method, simulations and real-data experiments were performed. In the simulation experiments, the performance of the proposed method under different array element numbers, signal-to-noise ratios, reverberation times, frequencies, and numbers of sound sources were studied and analyzed. A comparison was also made with the traditional time reversal method and the MUSIC algorithm. The experiment was conducted in a reverberation chamber. Simulation and experimental results show that the proposed method has good localization performance and robustness in reverberant environments.

Chiral symmetry breaking and topological charge of zigzag graphene nanoribbons

Interacting quasi-one-dimensional zigzag graphene nanoribbons display gapped edge excitations. Although the self-consistent Hartree–Fock fields break chiral symmetry, our work demonstrates that zigzag graphene nanoribbons maintain their status as short-range entangled symmetry-protected topological insulators. The relevant symmetry involves combined mirror and time-reversal operations. In undoped ribbons displaying edge ferromagnetism, the band gap edge states with a topological charge form on the zigzag edges. An analysis of the anomalous continuity equation elucidates that this topological charge is induced by the gap term. In low-doped zigzag ribbons, where the ground state exhibits edge spin density waves, this topological charge appears as a nearly zero-energy edge mode. Our system is outside the conventional classification for topological insulators.

Depiction of Hamiltonian PT-symmetry

The theory of PT-symmetry describes the non-hermitian Hamiltonian with real energy levels, which means that the Hamiltonian <i>H</i> is invariant neither under parity operator <i>P</i>, nor under time reversal operator <i>T</i>, <i>PTH</i> = <i>H</i>. Whether the Hamiltonian is real and symmetric is not a necessary condition for ensuring the fundamental axioms of quantum mechanics: real energy levels and unitary time evolution. The theory of PT-symmetry plays a significant role in studying quantum physics and quantum information science, Researchers have paid much attention to how to describe PT-symmetry of Hamiltonian. In the paper, we define operator <i>F</i> according to the PT-symmetry theory and the normalized eigenfunction of Hamiltonian. Then we first describe the PT-symmetry of Hamiltonian in dimensionless cases after finding the features of commutator and anti-commutator of operator <i>CPT</i> and operator <i>F</i>. Furthermore, we find that this method can also quantify the PT-symmetry of Hamiltonian in dimensionless case. <i>I</i>(<i>CPT</i>, <i>F</i>) = ||[<i>CPT</i>, <i>F</i>]||<i>CPT</i> represents the part of PT-symmetry broken, and <i>J</i>(<i>CPT</i>, <i>F</i>) = ||[<i>CPT</i>, <i>F</i>]||<i>CPT</i> represents the part of PT-symmetry. If <i>I</i>(<i>CPT</i>, <i>F</i>) = ||[<i>CPT</i>, <i>F</i>]||<i>CPT</i> = 0, Hamiltonian <i>H</i> is globally PT-symmetric. Once <i>I</i>(<i>CPT</i>, <i>F</i>) = ||[<i>CPT</i>, <i>F</i>]||<i>CPT</i> ≠ 0, Hamiltonian <i>H</i> is PT-symmetrically broken. In addition, we propose another method to describe PT-symmetry of Hamiltonian based on real and imaginary parts of eigenvalues of Hamiltonian, to judge whether the Hamiltonian is PT symmetric. Re<i>F</i> = 1/4||(<i>CPTF</i>+<i>F</i>)||CPT represents the sum of squares of real part of the eigenvalue <i>E<sub>n</sub></i> of Hamiltonian <i>H</i>, Im<i>F</i> = 1/4||(<i>CPTF</i>–<i>F</i>)||CPT is the sum of imaginary part of the eigenvalue <i>E<sub>n</sub></i> of a Hamiltonian <i>H</i>. If Im<i>F</i> = 0, Hamiltonian <i>H</i> is globally PT-symmetric. Once Im<i>F</i> ≠ 0, Hamiltonian <i>H</i> is PT-symmetrically broken. Re<i>F</i> = 0 implies that Hamiltonian <i>H</i> is PT-asymmetric, but it is a sufficient condition, not necessary condition. The later is easier to realize in the experiment, but the studying conditions are tighter, and it further requires that <i>CPT</i> <inline-formula><tex-math id="Z-20240108115351">\begin{document}$\phi_n $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20230458_Z-20240108115351.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20230458_Z-20240108115351.png"/></alternatives></inline-formula>(<i>x</i>) = <inline-formula><tex-math id="Z-20240108115401">\begin{document}$\phi_n $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20230458_Z-20240108115401.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20230458_Z-20240108115401.png"/></alternatives></inline-formula>(<i>x</i>). If we only pay attention to whether PT-symmetry is broken, it is simpler to use the latter method. The former method is perhaps better to quantify the PT-symmetrically broken part and the part of local PT-symmetry.

Cable Fault Location in Distribution Network Based on Decomposition of Time Reversal Operator

Cable Fault Location in Distribution Network Based on Decomposition of Time Reversal Operator

Low Complexity Time Reversal Imaging Methods Based on Truncated Time Reversal Operator

Low Complexity Time Reversal Imaging Methods Based on Truncated Time Reversal Operator

Ferrorotational Selectivity in Ilmenites.

Unlike what happens in conventional ferroics, the ferrorotational (FR) domain manipulation and visualization in FR materials are nontrivial as they are invariant under both space-inversion and time-reversal operations. FR domains have recently been observed by using the linear electrogyration (EG) effect and X-ray diffraction (XRD) diffraction mapping. However, ferrorotational selectivity, such as the selective processing of the FR domains and direct visualization of the FR domains, e.g., under an optical microscope, would be the next step to study the FR domains and their possible applications in technology. Unexpectedly, we discovered that the microscopic FR structural distortions in ilmenite crystals can be directly coupled with macroscopic mechanical rotations in such a way that FR domains can be visualized under an optical microscope after innovative rotational polishing, a combined ion milling with a specific rotational polishing, or a twisting-induced fracturing process. Thus, the FR domains could be a unique medium to register the memory of a rotational mechanical process due to a novel selective coupling between its microscopic structural rotations and an external macroscopic rotation. Analogous to the important enantioselectivity in modern chemistry and the pharmaceutical industry, this newly discovered ferrorotational selectivity opens up opportunities for FR manipulation and new FR functionality-based applications.

Conjugations on L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document} Space on the Real Line

Influenced by PT operators (parity and time reversal operators) the general conjugations acting in the L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\varvec{L}}^\ extbf{2}$$\\end{document} space on the real line with the Lebesgue measure are studied. The behaviour of conjugations with respect to multiplication operators and translations is investigated. We consider conjugations defined as the convolution of the Fourier transform of a given unimodular (even) function and test functions. We characterize all conjugations commuting (intertwining) with all translation operators on the real line. The conjugations leaving kernels of Toeplitz operators (a generalization of model spaces) on H2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\varvec{H}}^\ extbf{2}$$\\end{document} space on the upper half plane invariant are also characterized. Analogous results are obtained for conjugations and kernels of Wiener–Hopf operators.

VASP2KP: k⋅p Models and Landé g-Factors from ab initio Calculations

The k⋅p method is significant in condensed matter physics for the compact and analytical Hamiltonian. In the presence of magnetic field, it is described by the effective Zeeman’s coupling Hamiltonian with Landé g-factors. Here, we develop an open-source package VASP2KP (including two parts: vasp2mat and mat2kp) to compute k⋅p parameters and Landé g-factors directly from the wavefunctions provided by the density functional theory (DFT) as implemented in Vienna ab initio Simulation Package (VASP). First, we develop a VASP patch vasp2mat to compute matrix representations of the generalized momentum operator , spin operator , time reversal operator , and crystalline symmetry operators on the DFT wavefunctions. Second, we develop a python code mat2kp to obtain the unitary transformation U that rotates the degenerate DFT basis towards the standard basis, and then automatically compute the k⋅p parameters and g-factors. The theory and the methodology behind VASP2KP are described in detail. The matrix elements of the operators are derived comprehensively and computed correctly within the projector augmented wave method. We apply this package to some materials, e.g., Bi2Se3, Na3Bi, Te, InAs and 1H-TMD monolayers. The obtained effective model’s dispersions are in good agreement with the DFT data around the specific wave vector, and the g-factors are consistent with experimental data. The VASP2KP package is available at https://github.com/zjwang11/VASP2KP.

Asymmetric particle-antiparticle Dirac equation: first quantization

We derive a Dirac-like equation, the asymmetric Dirac equation, where particles and antiparticles sharing the same wave number have different energies and momenta. We show that this equation is Lorentz covariant under proper Lorentz transformations (boosts and spatial rotations) and also determine the corresponding transformation law for its wave function. We obtain a formal connection between the asymmetric Dirac equation and the standard Dirac equation and we show that by properly adjusting the free parameters of the present wave equation we can make it reproduce the predictions of the usual Dirac equation. We show that the rest mass of a particle in the theoretical framework of the asymmetric Dirac equation is a function of a set of four parameters, which are relativistic invariants under proper Lorentz transformations. These four parameters are the analog to the mass that appears in the standard Dirac equation. We prove that in order to guarantee the covariance of the asymmetric Dirac equation under parity and time reversal operations (improper Lorentz transformations) as well as under the charge conjugation operation, these four parameters change sign in exactly the same way as the four components of a four-vector. The mass, though, being a function of the square of those parameters remains an invariant. We also extensively study the free particle plane wave solutions to the asymmetric Dirac equation and derive its energy, helicity, and spin projection operators as well as several Gordon’s identities. The hydrogen atom is solved in the present context after applying the minimal coupling prescription to the asymmetric Dirac equation, which also allows us to appropriately obtain its non-relativistic limit.

Asymmetric particle-antiparticle Dirac equation: second quantization

We build the fully relativistic quantum field theory related to the asymmetric Dirac fields first presented in a prequel to this work. These fields are solutions of the asymmetric Dirac equation, a Lorentz covariant Dirac-like equation whose positive and ‘negative’ frequency plane wave solutions’ dispersion relations are no longer degenerate. At the second quantization level, we show that this implies that particles and antiparticles sharing the same wave number have different energies and momenta. In spite of that, we prove that by properly fixing the values of the relativistic invariants that define the asymmetric Dirac free field Lagrangian density, we can build a consistent, fully relativistic, and renormalizable quantum electrodynamics (QED) that is empirically equivalent to the standard QED. We discuss the reasons and implications of this non-trivial equivalence, exploring qualitatively other scenarios in which the asymmetric Dirac fields may lead to beyond the standard model predictions. We give a complete account of how the asymmetric Dirac fields and the corresponding annihilation and creation operators transform under improper Lorentz transformations (parity and time reversal operations) and under the charge conjugation operation. We also prove that the present theory respects the CPT theorem.

Valley contrasting bulk photovoltaic effect in a PT -symmetric MnPSe3 monolayer

Valleytronics that uses the inequivalent electronic states at the band extrema in semiconductors have been considered to play a vital role in the future information read/write technology. In the current work, we theoretically show that sizable valley contrasting bulk photovoltaic (BPV) effect could emerge, even when the total photocurrent is symmetrically forbidden. We illustrate our theory by using a prototypical two-dimensional antiferromagnetic semiconductor, MnPSe3 monolayer, that is PT-symmetric (P and T refer to spatially inversion and time-reversal operators, respectively). We show that the Neel vector well controls the magnetic point group at the $\Gamma$ point, and the BPV current direction. In addition, $\mathbf{k}$-dependent photocurrent generally arises due to the reduction of little group constraints at the valley. This leads to hidden valley-polarized photoconductivity which could reach a magnitude of 1350 $\mu$A/V&^2&, observable experimentally. We further predict that the MnPSe$_3$ monolayer could be two-dimensional ferrotoroidic, again depending on its Neel vector direction, which can be determined via magnetoelectric response measurements. Our work provides an exemplary platform for paving the route to future opto-spintronic and opto-valleytronic devices in a single antiferromagnetic nanomaterial.

Gauge equivalent structures of the integrable (2+1)-dimensional nonlocal nonlinear Schrödinger equations and their applications

In this paper, we are concerned with the gauge equivalent structures for the integrable (2+1)-dimensional nonlocal nonlinear Schrödinger (NLS) equations. Through constructing the gauge transformation, we prove that these (2+1)-dimensional nonlocal equations, both focusing and defocusing, are gauge equivalent to two types of coupled (2+1)-dimensional Heisenberg ferromagnet equations and two types of coupled (2+1)-dimensional modified Heisenberg ferromagnet equations. As an appropriate extension, we further illustrate that the nonlocal NLS equation is gauge equivalent to two types of coupled Heisenberg ferromagnet equations and two types of coupled modified Heisenberg ferromagnet equations, while its discrete version is gauge equivalent to two types of coupled discrete Heisenberg ferromagnet equations and two types of coupled discrete modified Heisenberg ferromagnet equations, respectively. From its invariance with the combined parity-reflection and time-reversal operators, we can observe that there exist significant differences and intimate connections between standard and nonlocal equations. On the other hand, by using the Darboux transformation and some limit techniques, two types of deformed soliton solutions, namely, the deformed exponential solitons and the deformed rational solitons for the (2+1)-dimensional nonlocal defocusing NLS equation are given explicitly. With no loss of generality, two deformed soliton interactions and their various degenerate cases are discussed and illustrated through some figures.

Nonlinear Through-the-Wall Detection Using DORT Applied With LFM Pulses

The detection of visually obstructed small nonlinear targets in a through-the-wall (TTW) environment is difficult because of linear scatterers and strong wall reflections. In this letter, harmonic-based nonlinear TTW detection is introduced as a method to effectively detect small nonlinear targets behind the wall. Nonlinear TTW detection allows for the suppression of linear scatterer responses and wall reflections by detecting only the harmonics generated from nonlinear targets. Here we apply the decomposition of the time-reversal operator (DORT) for the detection and separation of multiple nonlinear targets behind the wall. DORT extracts the eigenvalues and eigenvectors of nonlinear targets and allows for selective focusing (imaging) of each detected target. In addition, we utilize linear frequency modulated (LFM) pulses, which allow for the compression gain in the detection processing. To validate the proposed concept, we conducted simulation and measurements of various configurations of nonlinear TTW detection. The results demonstrate that nonlinear targets behind the wall can be effectively detected and located.

Ferrorotational domain walls revealed by electric quadrupole second harmonic generation microscopy

Domain walls are ubiquitous in materials that undergo phase transitions driven by spontaneous symmetry breaking. Domain walls in ferroics and multiferroics have received tremendous attention recently due to their emergent properties distinct from their domain counterparts, for example, their high mobility and controllability, as well as their potential applications in nanoelectronics. However, it is extremely challenging to detect, visualize and study the ferro-rotational (FR) domain walls because the FR order, in contrast to ferromagnetism (FM) and ferroelectricity (FE), is invariant under both the spatial-inversion and the time-reversal operations and thus hardly couple with conventional experimental probes. Here, an FR candidate $\mathrm{NiTiO_{3}}$ is investigated by ultrasensitive electric quadrupole (EQ) second harmonic generation rotational anisotropy (SHG RA) to probe the point symmetries of the two degenerate FR domain states, showing their relation by the vertical mirror operations that are broken below the FR critical temperature. We then visualize the real-space FR domains by scanning EQ SHG microscopy, and further resolve the FR domain walls by revealing a suppressed SHG intensity at domain walls. By taking local EQ SHG RA measurements, we show the restoration of the mirror symmetry at FR domain walls and prove their unconventional nonpolar nature. Our findings not only provide a comprehensive insight into FR domain walls, but also demonstrate a unique and powerful tool for future studies on domain walls of unconventional ferroics, both of which pave the way towards future manipulations and applications of FR domain walls.

A Lamb wave time-reversal field reconstruction method for damage detection with automatic focusing determination

This study proposes a wavefield reconstruction method with a time-reversal operation (WR-TR) based on the Lamb wave to detect the damage in the plate. Currently, it is challenging to implement the wavefield reconstruction method for damage detection because of two issues. One is how to quickly simulate the Lamb wavefield. The other is how to determine the focusing time for searching the desired frame from wavefield animation, which can show the location and size of damage. In response, this study introduces a multi-modal superposition finite difference time domain method (MS-FDTD) to simulate the Lamb waves propagation with little calculation cost, which facilitates damage imaging output quickly. In addition, a maximum energy frame method (MEF) is presented to automatically determine the focusing time from wavefield animation, enabling the detection of multiple damage points. The simulations and experiments have demonstrated good noise robustness, anti-distortion ability, and broad applicability with dense or sparse array layouts. Moreover, a detailed comparison between the proposed method and other four Lamb wave-based damage detection methods is evaluated in this paper.

Reconfigurable refraction manipulation at synthetic temporal interfaces with scalar and vector gauge potentials

Photonic gauge potentials, including scalar and vector ones, play fundamental roles in emulating photonic topological effects and for enabling intriguing light transport dynamics. While previous studies mainly focus on manipulating light propagation in uniformly distributed gauge potentials, here we create a series of gauge-potential interfaces with different orientations in a nonuniform discrete-time quantum walk and demonstrate various reconfigurable temporal-refraction effects. We show that for a lattice-site interface with the potential step along the lattice direction, the scalar potentials can yield total internal reflection (TIR) or Klein tunneling, while vector potentials manifest direction-invariant refractions. We also reveal the existence of penetration depth for the temporal TIR by demonstrating frustrated TIR with a double lattice-site interface structure. By contrast, for an interface emerging in the time-evolution direction, the scalar potentials have no effect on the packet propagation, while the vector potentials can enable birefringence, through which we further create a "temporal superlens" to achieve time-reversal operations. Finally, we experimentally demonstrate electric and magnetic Aharonov-Bohm effects using combined lattice-site and evolution-step interfaces of either scalar or vector potential. Our work initiates the creation of artificial heterointerfaces in synthetic time dimension by employing nonuniformly and reconfigurable distributed gauge potentials. This paradigm may find applications in optical pulse reshaping, fiber-optic communications, and quantum simulations.

Resolution-Enhanced and Accurate Cascade Time-Reversal Operator Decomposition (C-DORT) Approach for Positioning Radiated Passive Intermodulation Sources

Attaining a high-resolution and accurate location for a radiated passive intermodulation source (R-PIMS) has been an increasingly interesting problem in modern multi-carrier wireless communication systems. For precisely positioning multiple closely spaced R-PIMSs, a novel imaging method called cascade decomposition of time-reversal operator (C-DORT) was developed. C-DORT constructs a new spectrum calculation by normalizing and multiplying the pseudo-spectrum at each sampled frequency together. The cascade process focuses the pseudo-spectrum at R-PIMS positions to form a highly brightened spectrum peak and to suppress the remained pseudo-spectrum to approximately zero, contributing to distinguishing the closely spaced R-PIMSs. The positioning performance of the positioning resolution, pseudo-spectrum width, positioning accuracy, and imaging robustness are analyzed by numerical simulations. Compared with the conventional central frequency decomposition of time-reversal operator (CF-DORT) and the time domain decomposition of time-reversal operator (TD-DORT) methods, the multiple R-PIMSs, spaced at a distance of diffraction limit, which is the spacing of 1/2 of a wavelength, are distinguished effectively in C-DORT. Additionally, the cross-range pseudo-spectrum full width at half maxima (CRPS-FWHM) is suppressed to the width of 1/4 of a wavelength by multiplication to improve the cross-range resolution in C-DORT. In addition, accurate positioning is obtained by providing the approximately zero positioning root mean square estimation (RMSE) at an SNR ranging from 0 dB to 10 dB. The results show that the proposed C-DORT improves the positioning accuracy and enhances the positioning resolution for locating an R-PIMS.