Phase Matrices Research Articles

Ris-aided integrated sensing and communication: Beamforming design and antenna selection

This work considers an integrated sensing and communication system, where a reconfigurable intelligent surface (RIS) is utilized to manage interference and radar signals. The sensors are attached to the RIS to sense multiple targets. A joint design of the base station transmit beamforming and RIS phase shift matrix is proposed to minimize total interference and maximize the worst received signal power at the RIS sensors. Due to highly coupled transmit beamforming and RIS phase matrices, the optimization problem is decoupled into two subproblems and solved iteratively by semidefinite programming and a manifold-based Riemannian steepest descent algorithm. We further design energy-aware beamforming to eliminate the interference induced by radar probing signals. Antenna selection with the ℓ0 norm is introduced to exclude redundant antennas while maintaining the sufficient multiple beams for multiple users and targets with minimized required antennas. Due to the nonconvexity of the ℓ0-norm, we relax the number of active transmit antennas as a weighted ℓ1-norm and employ a concave approximation for the constraint on the radar beampattern. Numerical results illustrate that the proposed algorithms can effectively reduce interference and strengthen the received signal power for radar sensing, achieving mutual benefit for communication and sensing performance.

Terahertz Emission Modeling of Lunar Regolith

We investigate the terahertz (THz) scattering and emission properties of lunar regolith by modeling it as a random medium with rough top and bottom boundaries and a host medium situated beneath. The total scattering and emission arise from three sources: the rough boundaries, the volume, and the interactions between the boundaries and the volume. To account for these sources, we model their respective phase matrices and apply the matrix doubling approach to couple these phase matrices to compute the total emission. The model is then used to explore insights into lunar regolith scattering and emission processes. The simulations reveal that surface roughness is the primary contributor to total scattering, while dielectric contrasts between the volume and the boundaries dominate total emission. The THz emissivity is highly sensitive to the regolith dielectric constant, particularly its imaginary part, making it a promising alternative for identifying previously undetected water ice in the lunar polar regions. The THz emissivity model developed in this study can be readily applied to invert the surface parameters of lunar regolith using THz observations.

Rayleigh and Raman scattering cross-sections and phase matrices of the ground-state hydrogen atom, and their astrophysical implications

ABSTRACT We present explicit expressions for Rayleigh and Raman scattering cross-sections and phase matrices of the ground 1s state hydrogen atom based on the Kramers–Heisenberg–Waller dispersion formula. The Rayleigh scattering leaves the hydrogen atom in the ground-state while the Raman scattering leaves the hydrogen atom in either ns (n ≥ 2; s-branch) or nd (n ≥ 3; d-branch) excited state, and the Raman scattering converts incident ultraviolet (UV) photons around the Lyman resonance lines into optical-infrared (IR) photons. We show that this Raman wavelength conversion of incident flat UV continuum in dense hydrogen gas with a column density of NH > 1021 cm−2 can produce broad emission features centred at Balmer, Paschen, and higher level lines, which would mimic Doppler-broadened hydrogen lines with the velocity width of ≳1000 km s−1 that could be misinterpreted as signatures of active galactic nuclei, supernovae, or fast stellar winds. We show that the phase matrix of the Rayleigh and Raman s-branch scatterings is identical to that of the Thomson scattering while the Raman d-branch scattering is more isotropic, thus the Paschen and higher level Raman features are depolarized compared to the Balmer features due to the flux contribution from the Raman d-branch. We argue that observations of the line widths, line flux ratios, and linear polarization of multiple optical/IR hydrogen lines are crucial to discriminate between the Raman-scattered broad emission features and Doppler-broadened emission lines.

Thermal Decomposition of 2- and 4-Iodobenzyl Iodide Yields Fulvenallene and Ethynylcyclopentadienes: A Joint Threshold Photoelectron and Matrix Isolation Spectroscopic Study.

The thermal decomposition of 2- and 4-iodobenzyl iodide at high temperatures was investigated by mass-selective threshold photoelectron spectroscopy (ms-TPES) in the gas phase, as well as by matrix isolation infrared spectroscopy in cryogenic matrices. Scission of the benzylic C-I bond in the precursors at 850 K affords 2- and 4-iodobenzyl radicals (ortho- and para-IC6H4CH2•), respectively, in high yields. The adiabatic ionization energies of ortho-IC6H4CH2• to the X̃+(1A') and ã+(3A') cation states were determined to be 7.31 ± 0.01 and 8.78 ± 0.01 eV, whereas those of para-IC6H4CH2• were measured to be 7.17 ± 0.01 eV for X̃+(1A1) and 8.98 ± 0.01 eV for ã+(3A1). Vibrational frequencies of the ring breathing mode were measured to be 560 ± 80 and 240 ± 80 cm-1 for the X̃+(1A') and ã+(3A') cation states of ortho-IC6H4CH2•, respectively. At higher temperatures, subsequent aryl C-I cleavage takes place to form α,2- and α,4-didehydrotoluene diradicals, which rapidly undergo ring contraction to a stable product, fulvenallene. Nevertheless, the most intense vibrational bands of the elusive α,2- and α,4-didehydrotoluene diradicals were observed in the Ar matrices. In addition, high-energy and astrochemically relevant C7H6 isomers 1-, 2-, and 5-ethynylcyclopentadiene are observed at even higher pyrolysis temperatures along with fulvenallene. Complementary quantum chemical computations on the C7H6 potential energy surface predict a feasible reaction cascade at high temperatures from the diradicals to fulvenallene, supporting the experimental observations in both the gas phase and cryogenic matrices.

Modeling of Scattering by Dense Random Media Consisting of Particle Clusters With DMRT Bicontinuous

In this article, we developed the dense media radiative transfer (DMRT) model for random media with densely packed particle clusters. The dense media are computer-generated by applying the bicontinuous random media characterization. The microstructure of the media is controlled by two parameters: mean grain size <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\langle \zeta \rangle $ </tex-math></inline-formula> and aggregation parameter <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${b}$ </tex-math></inline-formula> . Phase matrices and extinction coefficients are computed with full-wave simulations. These are then substituted into radiative transfer (RT) equations to obtain backscattering coefficients of the dense media layer with particle clusters. The backscattering has a weaker frequency dependence (e.g., <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\sim $ </tex-math></inline-formula> 2.6 power for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${b}=\mathbf {0}.\mathbf {4}$ </tex-math></inline-formula> ) than the Rayleigh scattering of fourth power. Backscattering results exhibit strong co-, cross-polarization (pol), and ratio of cross- to co-polarization. For applications to remote sensing of terrestrial snow at C-band, the results of DMRT simulations show that cross-polarization of snow volume scattering can be larger than that of soil surface scattering. Recently, this property of strong cross-polarization has become a useful satellite remote sensing technique for global retrieval of snow depth. Comparisons of DMRT results are made with ESA Satellite Sentinel 1 C-band data and ground-based radar observations. The calculated backscatter is in good agreement with radar measurements for both the co- and cross-polarization.

Grid-adaptive Fourier pseudospectral time domain model for the light scattering simulation of atmospheric nonspherical particles.

PSTD (pseudospectral time domain) is recognized as one of the powerful models to accurately calculate the scattering properties of nonspherical particles. But it is only good at the computation in coarse spatial resolution, and large "staircase approximation error" will occur in the actual computation. To solve this problem, the variable dimension scheme is introduced to improve the PSTD computation, in which, the finer grid cells are set near the particle's surface. In order to ensure that the PSTD algorithm can be performed on non-uniform grids, we have improved the PSTD with the space mapping technique so that the FFT algorithm can be implemented. The performance of the improved PSTD (called "IPSTD" in this paper) is investigated from two aspects: for the calculation accuracy, the phase matrices calculated by IPSTD are compared with those well tested scattering models like Lorenz-Mie theory, T-matrix method and DDSCAT; for computational efficiency, the computational time of PSTD and IPSTD are compared for the spheres with different sizes. From the results, it can be found that, the IPSTD scheme can improve the simulation accuracy of phase matrix elements notably, especially in the large scattering angles; though the computational burden of IPSTD is larger than that of PSTD, its computational burden does not increase substantially.

Matrix-valued Allen–Cahn equation and the Keller–Rubinstein–Sternberg problem

In this paper, we consider the sharp interface limit of a matrix-valued Allen–Cahn equation, which takes the form: $$\begin{aligned} \partial _t{\textbf{A}}=\Delta {\textbf{A}}-\varepsilon ^{-2}({\textbf{A}}{\textbf{A}}^{\textrm{T}}{\textbf{A}}-{\textbf{A}})\quad \text {with}\quad {\textbf{A}}:\Omega \subset {\mathbb {R}}^m\rightarrow {\mathbb {R}}^{n\times n}. \end{aligned}$$ We show that the sharp interface system is a two-phases flow system: the interface evolves according to the motion by mean curvature; in the two bulk phase regions, the solution obeys the heat flow of harmonic maps with values in $$O^+(n)$$ and $$O^-(n)$$ (represent the sets of $$n\times n$$ orthogonal matrices with determinant $$+1$$ and $$-1$$ respectively); on the interface, the phase matrices on two sides satisfy a novel mixed boundary condition. The above result provides a solution to the Keller–Rubinstein–Sternberg’s problem in the O(n) setting. Our proof relies on two key ingredients. First, in order to construct the approximate solutions by matched asymptotic expansions, as the standard approach does not seem to work, we introduce the notion of quasi-minimal connecting orbits. They satisfy the usual leading order equations up to some small higher order terms. In addition, the linearized systems around these quasi-minimal orbits needs to be solvable up to some good remainders. These flexibilities are needed for the possible “degenerations” and higher dimensional kernels for the linearized operators on matrix-valued functions due to intriguing boundary conditions at the sharp interface. The second key point is to establish a spectral uniform lower bound estimate for the linearized operator around approximate solutions. To this end, we introduce additional decompositions to reduce the problem into the coercive estimates of several linearized operators for scalar functions and some singular product estimates which are accomplished by exploring special cancellation structures between eigenfunctions of these linearized operators.

Effect of Multiple Scattering on the Transmission Spectra and the Polarization Phase Curves for Earth-like Exoplanets

It is the most appropriate time to characterize the Earth-like exoplanets in order to detect biosignature beyond the Earth because such exoplanets will be the prime targets of big-budget missions like JWST, Roman Space Telescope, HabEx, LUVOIR, Thirty Meter Telescope, Extremely Large Telescope, etc. We provide models for the transmission spectra of Earth-like exoplanets by incorporating the effects of multiple scattering. For this purpose we numerically solve the full multiple-scattering radiative transfer equations instead of using Beer–Bouguer–Lambert’s law, which does not include the diffuse radiation due to scattering. Our models demonstrate that the effect of this diffuse transmission radiation can be observationally significant, especially in the presence of clouds. We also calculate the reflection spectra and polarization phase curves of Earth-like exoplanets by considering both cloud-free and cloudy atmospheres. We solve the 3D vector radiative transfer equations numerically and calculate the phase curves of albedo and disk-integrated polarization by using appropriate scattering phase matrices and integrating the local Stokes vectors over the illuminated part of the disks along the line of sight. We present the effects of the globally averaged surface albedo on the reflection spectra and phase curves as the surface features of such planets are known to significantly dictate the nature of these observational quantities. Synergic observations of the spectra and phase curves will certainly prove to be useful in extracting more information and reducing the degeneracy among the estimated parameters of terrestrial exoplanets. Thus, our models will play a pivotal role in driving future observations.

Stabilizing hydrogen-mediated sextuple bonds by quintuple superatomic bonding and a bond.

Multiple bond orders of four and five are frequently obtained for d-block elements. However, compounds with sextuple bonding (2σ, 2π, and 2δ), for instance, Cr2, Mo2 and W2, are less stable and trapped only in the gas phase or inert matrices, probably resulting from large repulsion of two σ-type (σs and ) orbitals within the same zone. Herein, a superatomic bonding model is proposed to describe experimentally synthesized bridging hydride compounds. We theoretically predicted four unprecedented quintuple bridging hydride species with large EHL values, among which [(Cp)2Sc2(μ-H)5]- and [(Cp)2Mn2(μ-H)5]- (Cp = cyclopentadienyl) contain quintuple superatomic bonding (σ, 2π, and 2δ). In particular, two other species, [(Cp)2Ti2(μ-H)5]- and [(Cp)2V2(μ-H)5]+, were found to feature stable hydrogen-mediated sextuple bonding comprising a quintuple superatomic delocalized bond and an extra localized bond. Thereinto, the superatomic σs bond disperses to the outer space for better overlap with the orbitals of bridging H atoms, thus leaving the inner regions with low electron densities and providing adequate space for the bond. These two quintuple bridging hydride compounds are verified to have thermal stability by molecular dynamics simulations and expected to provide an effective strategy to synthesize molecules with stable hydrogen-mediated sextuple bonding under normal experimental conditions.

Effects of Isothermal Temperature on Dislocation Density in Bainite Transformation of 4140 Steel

To relate the bainitic microstructures to the mechanical properties of steel, the average dislocation density needs to be determined. Using X-ray diffraction and diffraction line broadening analysis, this research quantifies the average dislocation density in the four bainite phase matrices, (upper bainite, upper and lower bainite mixture, lower bainite, lower bainite and martensite mixture), which are transformed in a wide range of isothermal temperatures. The effects of isothermal temperatures on the average dislocation density are assessed for different thermal dynamic driving forces in terms of activation energy and cooling rate. It is found that as isothermal holding temperature is increased, the dislocation density in the bainite matrix decreases from 1.55 × 1017 to 8.33 × 1015 (m−2) due to the reduction in the plastic deformation in the austenite in the transformation. At the same time, the activation energy required decreases only after passing the martensite and lower bainite mixed phase. A new method for better estimating the average dislocation density in bainitic steel is also proposed.

“Caught in the Act” @ disruption of A-ET-E process in the recognition of F− by a lamellar EuIII-MOF in heterogeneous manner with logic gate construction: From protagonist idea to implementation world

Development of cost-effective and reliable fluoride sensor for assessing water quality of natural water samples is of immense importance in developing countries as they can provide an easy platform for safeguarding human health. These sensors should be as simple as possible to be fabricated locally by layman. In this context, EuIII-based MOFs provide trustable platform with bright luminescence in the visible region due to their absorbance-energy transfer-emission (A-ET-E) process. Herein the designed synthesis of a 2D porous coordination polymer, Eu@CMERI, has been carried out following a solvothermal reaction route. The compound shows selective “turn-off“ sensing of fluoride in heterogeneous manner from purely aqueous phase and other biological matrices with a detection limit of 28.4 ppb and it carries enormous importance for drinking water analysis under internal regulations. Prohibition of A-ET-E cycle of the EuIII-MOF is proposed to be the prime reason for fluorescence quenching upon interaction with F−. DFT studies also revealed that lowest △EHOMO-LUMO and highest chemical potential value (μ) of F− are the driving forces for selectivity of EuIII-MOF towards the targeted anion. The high stability of the porous frameworks along with its interesting sensing features, including fast response and wide linear detection range etc. instigated us not to restrict the chemistry of EuIII-MOFs at protagonist idea rather to explore its application to real-world analysis. Based on the fluorescence signal exhibited by the targeted analyte, an integrated AND-OR logic gate has also been fabricated which depicts its applicability in molecular electronics. In view of the modular design principle of our polymeric probe, the proposed strategy could open a new horizon to construct powerful sensing materials for ultrafast detection of other important pollutants in the domain of supramolecular chemistry in coming days.

Multiple color image encryption based on cascaded quaternion gyrator transforms

In this paper, we propose a novel encryption algorithm using cascaded quaternion gyrator transforms that allows protecting multiple color images with efficiency at once. The originality is reflected in the integration of quaternion algebra with a cascaded encryption structure. First, image RGB channels are constituted into three imaginary parts of a quaternion matrix. A cascaded encryption structure is constructed to cooperate with the vortex phase mask, which is easy to transmit and saves storage. Then, a quaternion gyrator transform is conducted, after which the phase is extracted to serve as the decryption key while the amplitude is used as the real part of the following quaternion matrix. Finally, the amplitude of the last gyrator transform is scrambled to obtain the ciphertext. All phase matrices are further modulated by a secondary image phase mask for additional security. Numerical simulations have been conducted to verify the validity of the proposed algorithm. Experiments on several data test set demonstrate the robustness of the proposed method to various attacks.

A note on computing stokes phase matrices for macroscopically isotropic mirror symmetric scattering medium stokes scattering matrices

Equations used to compute Stokes phase matrices for macroscopically isotropic mirror symmetric scattering medium from Stokes scattering matrices are reformulated to eliminate numerical instability issues and to eliminate the need to treat positive and negative relative azimuth angles as separate cases.

Microstructural Transformation and Corrosion Property Improvement of CoCrFeNiTi‐Based Multi‐Principal Element Alloys Fabricated via Laser Powder Bed Fusion

Herein, CoCrFeNiTi‐based multiprincipal element alloys (MPEAs) are proposed to meet the demands of high mechanical strength and corrosion resistance in additively manufactured products. Two types of MPEAs with different chemical compositions are obtained by laser powder bed fusion (LPBF) using gas‐atomized prealloyed powders. Both MPEAs as‐built products have columnar crystal structures with face‐centered cubic (FCC) phase matrices. In the solution and aging heat treatment, columnar crystal grains transform into equiaxial recrystallized grains with precipitates dispersed in the FCC matrix. The alloys exhibit ultimate tensile strength over 1500 MPa and corrosion rates of less than 1 mm year−1 in boiling 10% H2SO4. The microcellular structures formed in the LPBF process, and the precipitates formed after heat treatments can affect the mechanical properties and corrosion behavior. The higher cobalt and chromium content of the proposed CoCrFeNiTi‐based MPEA and the microstructure control in heat treatments demonstrate their effectiveness in retarding acid corrosion.

Spider Silk Inspired Robust and Photoluminescent Soybean‐Protein‐Based Materials

AbstractDevelopment of mechanical robust and functional biomass‐based materials still remains challenging. Here, a design strategy inspired by spider silk structure is proposed to prepare strong, robust, and photoluminescent soybean protein isolate (SPI)‐based materials, by integrating epoxy soybean oil (ESO) and SPI as soft phase matrices and graphene oxide quantum dots (GQDs) as hard phase. The results show that the soft–hard coordination network can form a close covalent/hydrogen bond network. The tensile strength, elongation at break, and toughness of the SPI/ESO/GQD film are 13.22 MPa, 209%, and 22.54 MJ m−3, respectively. In addition, SPI/ESO/GQD has strong photoluminescence intensity due to the ring‐opening polymerization of amino structure with epoxy resin. The prepared SPI‐based materials are promising candidates for optical coatings and provide new ideas for the intelligent research of other protein‐based materials.

Low Complexity Equalization of Orthogonal Chirp Division Multiplexing in Doubly-Selective Channels.

Orthogonal Chirp Division Multiplexing (OCDM) is a modulation scheme which outperforms the conventional Orthogonal Frequency Division Multiplexing (OFDM) under frequency selective channels by using chirp subcarriers. However, low complexity equalization algorithms for OCDM based systems under doubly selective channels have not been investigated yet. Moreover, in OCDM, the usage of different phase matrices in modulation will lead to extra storage overhead. In this paper, we investigate an OCDM based modulation scheme termed uniform phase-Orthogonal Chirp Division Multiplexing (UP-OCDM) for high-speed communication over doubly selective channels. With uniform phase matrices equipped, UP-OCDM can reduce the storage requirement of modulation. We also prove that like OCDM, the transform matrix of UP-OCDM is circulant. Based on the circulant transform matrix, we show that the channel matrices in UP-OCDM system over doubly selective channels have special structures that (1) the equivalent frequency-domain channel matrix can be approximated as a band matrix, and (2) the transform domain channel matrix in the framework of the basis expansion model (BEM) is a sum of the product of diagonal and circulant matrices. Based on these special channel structures, two low-complexity equalization algorithms are proposed for UP-OCDM in this paper. The equalization algorithms are based on block LDL factorization and iterative matrix inversion, respectively. Numerical simulations are finally proposed to show the performance of UP-OCDM and the validity of the proposed low complexity equalization algorithms. It is shown that when the channel is doubly selective, UP-OCDM and OCDM have similar BER performance, and both of them outperform OFDM. Moreover, the proposed low complexity equalizers for UP-OCDM both show better BER performance than their OFDM counterparts.

Twenty-Vertex Model with Domain Wall Boundaries and Domino Tilings

We consider the triangular lattice ice model (20-Vertex model) with four types of domain-wall type boundary conditions. In types 1 and 2, the configurations are shown to be equinumerous to the quarter-turn symmetric domino tilings of an Aztec-like holey square, with a central cross-shaped hole. The proof of this statement makes extensive use of integrability and of a connection to the 6-Vertex model. The type 3 configurations are conjectured to be in same number as domino tilings of a particular triangle. The four enumeration problems are reformulated in terms of four types of Alternating Phase Matrices with entries $0$ and sixth roots of unity, subject to suitable alternation conditions. Our result is a generalization of the ASM-DPP correspondence. Several refined versions of the above correspondences are also discussed.

Methods and Devices of Speckle-Noise Suppression (Review)

Speckles in images formed by a laser beam are interference noise arising due to coherent nature of laser radiation. As the number of projection systems using laser light sources increases, new methods and devices appear for reducing the contrast of speckle-noise and thereby suppressing it. Taking into account the long-standing nature of the problem and the continuing relevance of its solution, this review discusses modern practical methods and devices—despecklers, used to reduce the speckle contrast and to achieve high quality of projected images. The review discusses the nature of the speckles appearance, considers their statistical properties, and describes the measurement of the speckle contrast. The requirements for despecklers are considered, the most important of which are speckle suppression efficiency, simplicity of design and compactness, low power consumption and low optical loss. The characteristics of different types of despecklers, including diffusers, devices based on electroactive polymers, optical waveguides, colloidal solutions and liquid crystals, as well as orthogonal phase matrices and diffraction gratings proposed as despeckers, are examined in detail. It is shown that despecklers with decorrelation of the phase front based on the mirror deformation and optical fiber have less light losses. Electro-optical-liquid crystal despecklers do not have mechanically deformable or moving elements that reduce the reliability and durability of operation, and are more compact and simple in design. A comparative table of characteristics of the most effective despecklers in which the speckle contrast is reduced to ten percent or less, with an indication of their advantages and disadvantages, is given.

Metal 3D-printed catalytic jet and flame ionization detection for in situ trace carbon oxides analysis by gas chromatography.

A gas chromatographic approach for the determination and quantification of trace levels of carbon oxides in gas phase matrices for in situ or near-line/at-line analysis has been successfully developed. Catalytic conversion of the target compounds to methane via the methanation process was conducted inside a metal 3D-printed jet that also acted as a hydrogen burner for the flame ionization detector. Modifications made to a field transportable gas chromatograph enabled the leveraging of advantaged microfluidic-enhanced chromatography capability for improved chromatographic performance and serviceability. The compatibility with adsorption chromatography technology was demonstrated with in-house constructed columns. Sustained reliable conversion efficiencies of greater than 99% with respectable peak symmetries were attained at 400°C. Quantification of carbon monoxide and carbon dioxide at a parts-per-million level over a range from 0.2ppm to 5% v/v for both compounds with a respectable precision of less than 3% relative standard deviation for peak area (n=10) and a detection limit of 0.1ppm v/v was achieved. Linearity with correlation coefficients of R2 greater than 0.9995 and measured recoveries of >99% for spike tests were achieved. The 3D-printed steel jet was found to be reliable and resilient against potential contamination from the matrices owing to the in situ backflushing capability.

3-D InISAR Imaging of the Ship Target Based on Joint Cross S-Method Algorithm

3-D interferometric inverse synthetic aperture radar (InISAR) imaging of ship targets has always been a hot and difficult issue in research, and how to preserve the interferometric phase while performing high-resolution 2-D ISAR imaging of the complex echoes is the key to 3-D InISAR imaging. In this letter, a novel 3-D InISAR imaging algorithm of the ship target based on joint cross S-method is proposed. First, the azimuth short-time Fourier transform (STFT) is applied to the 1-D range profiles after motion compensation and image coregistration. Then, extract the interferometric phase matrices after performing the joint cross S-method transformation of the STFT results along the two baselines. Besides, the S-method transformations are implemented recursively for the joint cross S-method transformed results to obtain high-resolution ISAR images. Finally, the 3-D InISAR imaging can be achieved by combining the phase matrices and ISAR images at the same time. Simulation results verify the effectiveness and superiority of the proposed algorithm.