Cramer-Rao Research Articles

Hyperparameter free sparse estimation for wideband multiple‐input‐multiple‐output radar direction finding without secondary data

AbstractThe estimation of target directions of arrival (DOA) for wideband multiple‐input‐multiple‐output (MIMO) radar is investigated in this article. First, the authors establish a signal model for wideband MIMO radar systems. Then, an algorithm is proposed to estimate the target angles without secondary data (i.e. the training data from slow‐time is not required).The proposed algorithm unitises the spatial sparsity of target signals and it is derived under the Bayesian framework. It can estimate the spatial pseudo‐spectra of the targets through cyclic optimisation. Additionally, it is hyperparameter‐free and guarantees convergence. To analyse the performance of the proposed algorithm, the Cramér‐Rao bound (CRB) is derived for DOA estimation with wideband MIMO radar. Numerical examples demonstrate that even without secondary data, the proposed algorithm can accurately estimate the DOA of the targets.

Exploring charge sharing compensation using inter-pixel coincidence counters for photon counting detectors by deep-learning from local information

Photon counting detectors (PCDs) have well-acknowledged advantages in computed tomography (CT) imaging. However, charge sharing and other problems prevent PCDs from fully realizing the anticipated potential in diagnostic CT. PCDs with multi-energy inter-pixel coincidence counters (MEICC) have been proposed to provide particular information about charge sharing, thereby achieving lower Cramér-Rao Lower Bound (CRLB) than conventional PCDs when assessing its performance by estimating material thickness or virtual monochromatic attenuation integrals (VMAIs). This work explores charge sharing compensation using local spatial coincidence counter information for MEICC detectors through a deep-learning method.
Approach: By analyzing the impact of charge sharing on photon count detection, we designed our network with a focus on individual pixels. Employing MEICC data of patches centered on POIs as input, we utilized local information for effective charge sharing compensation. The output was VMAI at different energies to address real detector issues without knowledge of primary counts. To achieve data diversity, a fast and online data generation method was proposed to provide adequate training data. A new loss function was introduced to reduce bias for training with high-noise data. The proposed method was validated by Monte Carlo (MC) simulation data for MEICC detectors that were compared with conventional PCDs. 
Main-Results: For conventional data as a reference, networks trained on low-noise data yielded results with a minimal bias (about 0.7%) compared with > 3% for the polynomial fitting method. The results of networks trained on high-noise data exhibited a slightly increased bias (about 1.3%) but a significantly reduced standard deviation (STD) and normalized root mean square error (NRMSE). The simulation study of the MEICC detector demonstrated superior compared to the conventional detector across all the metrics. Specifically, for both networks trained on high-noise and low-noise data, their biases were reduced to about 1% and 0.6%, respectively. Meanwhile, the results from a MEICC detector were of about 10% lower noise than a conventional detector. Moreover, an ablation study showed that the additional loss function on bias was beneficial for training on high-noise data.
Significance: We demonstrated that a network-based method could utilize local information in PCDs effectively by patch-based learning to reduce the impact of charge sharing. MEICC detectors provide very valuable local spatial information by additional coincidence counters. Compared with MEICC detectors, conventional PCDs only have limited local spatial information for charge sharing compensation, resulting in higher bias and standard deviation in VMAI estimation with the same patch strategy.&#xD.

Precision comparison of intensity ratios and area ratios in spectral analysis

The long-debated question in analytical chemistry of which of the area ratio or the intensity ratio is the more precise has yielded no definitive analytical conclusion. To address this issue theoretically, we derived analytical solutions for the lower limits of estimation precision for spectral parameters, including the intensity ratio and area ratio, based on the Cramér–Rao lower bound (CRLB) framework for a Gaussian spectrum. The precisions of spectral parameter estimations from the analytical solutions were consistent with results obtained from Monte Carlo simulations. Our theoretical and simulation results revealed that the precision of estimating the area ratio surpassed that of the intensity ratio by a factor of 2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sqrt{2}$$\\end{document}. Additionally, our experimental results aligned well with both theoretical predictions and simulation outcomes, further validating our approach. This increased precision of the area ratio is due to negative covariance between intensity and bandwidth, rather than the area containing more intensity information, as often misinterpreted. Consequently, and quite counter intuitively, prior bandwidth and intensity related information does not improve the area ratio precision: it worsens it. The analytical solution we derived represents the fundamental limits of spectral parameter measurement precision. Thus, it can be used as an alternative method for estimating the minimum error when experimental measurement uncertainty cannot be determined.

Frequency and damping factor estimation of damped sinusoid by using DFT and DTFT

An algorithm for estimating the frequency and damping factor of damped complex sinusoidal signal is presented. After the rough estimation of frequency by discrete Fourier transform (DFT), the fine estimation of frequency and damping factor is achieved by utilizing the DFT bin with the maximum magnitude and two arbitrarily spaced discrete time Fourier transform (DTFT) spectral lines located on the same side of the maximum DFT spectrum bin. Then the theoretical mean square error (MSE) in additive white noise is analyzed. Simulation results show that the proposed algorithm tracks the Cramer-Rao lower bound (CRLB), and the parameters estimation accuracy is higher than the competing algorithms in most situations. The computational requirements are almost the same as several existing methods.

An Improved Unfolded Coprime Linear Array Design for DOA Estimation with No Phase Ambiguity

In this paper, the direction of arrival (DOA) estimation problem for the unfolded coprime linear array (UCLA) is researched. Existing common stacking subarray-based methods for the coprime array are invalid in the case of its subarrays, which have the same steering vectors of source angles. To solve the phase ambiguity problem, we reconstruct an improved unfolded coprime linear array (IUCLA) by rearranging the reference element of the prototype UCLA. Specifically, we design the multiple coprime inter pairs by introducing the third coprime integer, which can be pairwise with the other two integers. Then, the phase ambiguity problem can be solved via the multiple coprime property. Furthermore, we employ a spectral peak searching method that can exploit the whole aperture and full DOFs of the IUCLA to detect targets and achieve angle estimation. Meanwhile, the proposed method avoids extra processing in eliminating ambiguous angles, and reduces the computational complexity. Finally, the Cramer–Rao bound (CRB) and numerical simulations are provided to demonstrate the effectiveness and superiority of the proposed method.

An analytic, efficient and optimal readout algorithm for compact interferometers based on deep frequency modulation

Compact laser interferometers with large dynamic range are one of the core emerging tools to improve low frequency performance in gravitational wave detectors by providing local displacement sensing with sub 1 pmHz-0.5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {pm Hz}^{-0.5}$$\\end{document} precision. Strong sinusoidal frequency modulations are used in such laser interferometers to create heterodyne-like photodetector signals from which the phase and other parameters, such as the absolute distance, can be extracted. The nested sinusoidal function in such signals is a challenge for the real-time parameter estimation in low-noise applications. In this article, we present an algorithm to calculate exact signal parameters in a non-iterative way from such interferometric signals. The algorithm makes use of a recurrence relation between Bessel functions to enable a direct extraction of modulation parameters from the signal. Additionally, the algorithm is capable of dealing with high phase dynamics where the Doppler-shift of the signal becomes relevant and can limit the range and precision of the parameter estimation, if not accounted for. Simulations show that the algorithm is computationally efficient, can be well parallelised and the phase estimation is close to optimal precision given by the Cramer–Rao lower bound of the signal parameters.

Robust Direction Estimation of Terrestrial Signal via Sparse Non-Uniform Array Reconfiguration under Perturbations

DOA (Direction of Arrival), as an important observation parameter for accurately locating the Signals of Opportunity (SOP), is vital for navigation in GNSS-challenged environments and can be effectively obtained through sparse arrays. In practical application, array perturbations affect the estimation accuracy and stability of DOA, thereby adversely affecting the positioning performance of SOP. Against this backdrop, we propose an approach to reconstruct non-uniform arrays under perturbation conditions, aiming to improve the robustness of DOA estimation in sparse arrays. Firstly, we theoretically derive the mathematical expressions of the Cramér–Rao Bound (CRB) and Spatial Correlation Coefficient (SCC) for the uniform linear array (ULA) with perturbation. Then, we minimize CRB as the objective function to mitigate the adverse effects of array perturbations on DOA estimation, and use SCC as a constraint to suppress sidelobes. By doing this, the non-uniform array reconstruction model is formulated as a high-order 0–1 optimization problem. To effectively solve this nonconvex model, we propose a polynomial-time algorithm, which can converge to the optimal approximate solution of the original model. Finally, through a series of simulation experiments utilizing frequency modulation (FM) signal as an example, the exceptional performance of this method in array reconstruction has been thoroughly validated. Experimental data show that the reconstructed non-uniform array excels in DOA estimation accuracy compared to other sparse arrays, making it particularly suitable for estimating the direction of terrestrial SOP in perturbed environments.

Joint beamforming design for STAR-RIS-assisted integrated sensing, communication, and power transfer systems

In the sixth-generation network, many low-power devices are predicted to be integrated into tasks incorporating communication and sensing. The integrated sensing and communication (ISAC) technology reuses a wireless signal for data transfer and radar sensing. Furthermore, wireless signals have the capacity to transmit energy, allowing for the simultaneous wireless information and power transfer (SWIPT). To enhance spectrum utilization, the deeper integration of SWIPT and ISAC has opened up novel investigate directions for integrated sensing, communication, and power transfer (ISCPT). This paper investigates a simultaneously transmitting and reflecting reconfigurable intelligent surface (STAR-RIS) supported ISCPT system that solves the problem of traditional RIS not achieving full space wireless signal coverage. In particular, since the serious path loss issue of sensing, we propose a novel architecture for installing dedicated sensors on STAR-RIS. Under this setting, we jointly optimized the passive beamforming at STAR-RIS and the active beamforming at base station to minimize the Cramér-Rao bound (CRB) used to estimate the sensing target's two-dimensional direction of arrivals, while being constrained by the lowest signal-to-interference-plus-noise-ratio (SINR) of communication users (CUs), the lowest energy harvesting (EH) of energy users (EUs), and the maximum transmission power at base station. For complex non-convex problems, a proposed two-layer cyclic algorithm utilizes penalty dual decomposition (PDD) and block coordinate descent (BCD). Finally, the numerical outcomes verify the efficacy of our suggested design, which reveals the performance trade-off between communication, power transmission and sensing. Furthermore, compared to traditional RIS, the estimated CRB of this design is lower.

Asynchronous localization of dynamic node through underwater acoustic sensor networks

In most cases, the operation of underwater acoustic sensor networks (UASNs) relies on accurate sensor location information. However, UASNs present more challenges than terrestrial sensor networks, such as the propagation speed variation with depth (i.e., stratification effect), asynchronous clock, node mobility, etc. In this paper, an efficient method of asynchronous localization is proposed by taking into account the stratification effect and node mobility. Firstly, a network is constructed that consists of surface buoys, AUVs, and target sensors. Secondly, an iterative least squares (LS) method is developed to locate AUV by establishing the relationship between propagation delay and location estimation. Thirdly, the passive mobility velocity of AUV is used to develop and solve the mobility model of the target sensors. Then, with the support of AUV, an asynchronous localization method is developed to estimate the target position, which improves the localization accuracy through motion and stratification compensation strategies. Moreover, the proposed method is validated through Cramer Rao lower bound (CRLB) analysis. Finally, simulation is performed to show that the proposed method is more efficient in reducing the estimation errors.

On the characterization of reflective surfaces using dual-polarization GNSS-R

Global navigation satellite systems reflectometry (GNSS-R) is a technique to extract information from reflecting surfaces by the reflected GNSS signals. GNSS-R has garnered increasing attention in the scientific literature due to its continuous global coverage and its superior spatial resolution. Moreover, operating in the L-band renders GNSS-R relatively immune to adverse weather conditions and affords high sensitivity to soil electrical properties. This work introduces a new approach with a dual-polarization antenna, left-hand circular polarized (LHCP) and right-hand circular polarized (RHCP), receiving the reflected signal from a sufficiently smooth surface so that all reflected energy arrives from the specular reflection point. The objective is to characterize the reflecting surface by extracting the relative permittivity and conductivity from the reflected signal. In contrast to other studies found in the literature, the reflection of the GNSS signal on different materials, including dielectric and conductive materials is considered. We derive a maximum likelihood estimator (MLE) for estimating the dielectric parameters of the reflective surface and other parameters of the reflected signal. We also derive the respective Cramer–Rao Lower Bound (CRLB) evaluating the performance of the MLE. The attained results are assessed based on the signal-to-noise ratio (SNR) and the angle of reflection of the reflected signal, which are the parameters that predominantly influence the proposed approach. Lower elevation angles, for instance, lead to higher estimation accuracy, while for reflective surfaces composed of metallic materials a higher SNR is needed to yield favorable estimation performance. Regarding dielectric materials, the estimation results are encouraging and thus enable diverse remote sensing applications by GNSS-R using the proposed setup.

Data-Aided Maximum Likelihood Joint Angle and Delay Estimator Over Orthogonal Frequency Division Multiplex Single-Input Multiple-Output Channels Based on New Gray Wolf Optimization Embedding Importance Sampling.

In this paper, we propose a new data-aided (DA) joint angle and delay (JADE) maximum likelihood (ML) estimator. The latter consists of a substantially modified and, hence, significantly improved gray wolf optimization (GWO) technique by fully integrating and embedding within it the powerful importance sampling (IS) concept. This new approach, referred to hereafter as GWOEIS (for "GWO embedding IS"), guarantees global optimality, and offers higher resolution capabilities over orthogonal frequency division multiplex (OFDM) (i.e., multi-carrier and multi-path) single-input multiple-output (SIMO) channels. The traditional GWO randomly initializes the wolfs' positions (angles and delays) and, hence, requires larger packs and longer hunting (iterations) to catch the prey, i.e., find the correct angles of arrival (AoAs) and time delays (TDs), thereby affecting its search efficiency, whereas GWOEIS ensures faster convergence by providing reliable initial estimates based on a simplified importance function. More importantly, and beyond simple initialization of GWO with IS (coined as IS-GWO hereafter), we modify and dynamically update the conventional simple expression for the convergence factor of the GWO algorithm that entirely drives its hunting and tracking mechanisms by accounting for new cumulative distribution functions (CDFs) derived from the IS technique. Simulations unequivocally confirm these significant benefits in terms of increased accuracy and speed Moreover, GWOEIS reaches the Cramér-Rao lower bound (CRLB), even at low SNR levels.

Robust algorithms for spherical angle-of-arrival source localization

The performance of traditional algorithms for spherical angle-of-arrival (AOA) source localization will be significantly degraded when there are outliers in the angle measurements. By using the symmetric α-stable (SαS) distribution to describe the measurement noise containing outliers and constructing the cost function using the lp-norm, we propose a robust algorithm for spherical AOA source localization: the spherical iteratively reweighted pseudolinear estimator (SIRPLE). The SIRPLE is similar to the iteratively reweighted least squares (IRLS), with the difference that a homogeneous least squares (HLS) problem is solved in each iteration. The SIRPLE suffers from bias problems owing to the nature of the pseudolinear estimators. To overcome this problem, the instrumental variable (IV) method is introduced and the spherical iteratively reweighted instrumental variable estimator (SIRIVE) is proposed. Theoretical analysis shows that the SIRIVE is asymptotically unbiased and it can achieve the theoretical error covariance of the constrained least lp-norm estimation. Extensive simulation analyses demonstrate the better performance of the SIRIVE compared to the conventional spherical AOA source localization methods and the SIRPLE under SαS noise environment. The performance of the SIRIVE is similar to that of the Nelder–Mead algorithm (NM), but the SIRIVE are computationally more efficient. In addition, the SIRIVE is nearly unbiased and the root mean square error (RMSE) performance is close to the Cramér–Rao lower bound (CRLB).

Closed-Form Method for Unified Far-Field and Near-Field Localization Based on TDOA and FDOA Measurements

When the near-field and far-field information of a target is uncertain, it is necessary to choose a suitable localization method. The modified polar representation (MPR) method integrates the two scenarios and achieves a unified localization with direction of arrival (DOA) estimation in the far field and position estimation in the near field. Previous studies have only proposed solutions for stationary environments and have not considered the motion factor. Therefore, this paper proposes a new unified positioning algorithm using multi-sensor time difference of arrival (TDOA) and frequency difference of arrival (FDOA) measurements without prior target source information. The method represents the position of the target source using MPR and describes the localization problem as a weighted least squares (WLS) problem with two constraints. We first obtain the initial estimates by WLS without considering the constraints and then investigate a two-step error correction method based on the constraints. The first step corrects the initial estimate using the Taylor series expansion technique, and the second step corrects the DOA estimate in the previous step using the direct error compensation technique based on the properties of the second constraint. Simulation experiments show that the method is effective for the unified positioning of moving targets and can achieve the Cramer–Rao lower bound (CRLB).

Measurement precision bounds on aberrated single-molecule emission patterns

Single-molecule localization microscopy (SMLM) has revolutionized the study of biological phenomena by providing exquisite nanoscale spatial resolution. However, optical aberrations induced by sample and system imperfections distort the single-molecule emission patterns (i.e. PSFs), leading to reduced precision and resolution of SMLM, particularly in three-dimensional (3D) applications. While various methods, both analytical and instrumental, have been employed to mitigate these aberrations, a comprehensive analysis of how different types of commonly encountered aberrations affect single-molecule experiments and their image formation remains missing. In this study, we addressed this gap by conducting a quantitative study of the theoretical precision limit for position and wavefront distortion measurements in the presence of aberrations. Leveraging Fisher information and Cramér-Rao lower bound (CRLB), we quantitively analyzed and compared the effects of different aberration types, including index mismatch aberrations, on localization precision in both biplane and astigmatism 3D modalities as well as 2D SMLM imaging. Furthermore, we studied the achievable wavefront estimation precision from aberrated single-molecule emission patterns, a pivot step for successful adaptive optics in SMLM through thick specimens. This analysis lays a quantitative foundation for the development and application of SMLM in whole-cells, tissues and with a large field of view, providing in-depth insights into the behavior of different aberration types in single-molecule imaging and thus generating theoretical guidelines for developing highly efficient aberration correction strategies and enhancing the precision and reliability of 3D SMLM.

Joint estimation of IQ imbalance and PA nonlinearity: An iterative scheme

Power amplifier (PA) nonlinearity and in-phase and quadrature-phase (IQ) imbalance in the transmitter are of critical concern in modern communication systems due to their significant impact on the transmitter performance. In this paper, in order to eliminate their effects without increasing the complexity at the transmitter, we propose an iterative structure for IQ imbalance and PA nonlinearity joint estimation at the receiver side. The strategic design aims to mitigate the influence of one distortion before estimating the other, ultimately improving the accuracy of estimating each distortion. The proposed approach comprises two primary modules: an IQ imbalance estimation module and a PA nonlinearity estimation module. In addition, there are two supporting modules, namely the PA nonlinearity compensation module and the IQ imbalance reconstruction module, positioned before these two primary modules respectively. Furthermore, we have derived the theoretical Cramér–Rao lower bound (CRLB) for the joint distortion model of IQ imbalance and PA nonlinearity. Finally, extensive experiments are conducted to demonstrate the proposed algorithm's capability in estimating hardware distortions and mitigating them.

Sparse reconstruction methods for wideband source localization in spherical harmonics domain

Wideband acoustic source localization is a challenging task especially in the presence of noise. Generally wideband source localization is done by averaging the Direction of Arrival (DOA) estimates obtained over multiple frequencies or narrow subbands by the method of frequency smoothing. Localization of multiple wideband sources which are correlated is even more challenging. In this work acoustic source localization under these challenging conditions is addressed. A sparse reconstruction framework is developed for wideband source localization in the Spherical Harmonics (SH) domain. The proposed framework jointly computes both the azimuth and elevation. In contrast to earlier methods of source localization in the SH domain, this work utilizes the expression for the SH coefficients of the amplitude density of the plane wave, to develop the sparse reconstruction framework. An expression for Cramér Rao Lower Bound (CRLB) on the DOA using this framework is also derived for the wideband sources. This CRLB is shown to easily generalize for narrowband sources as well. The performance of the proposed method is compared with MUltiple Signal Classification in SH domain (MUSIC-SH), Steered Response Power in SH domain (SRP-SH), MUSIC with Direct Path Dominance test in SH domain (MUSIC-DPD-SH) and Pressure based Compressive Sensing in SH domain (PCS-SH). The performance is evaluated for correlated narrowband and wideband sources through simulations, conducting experiments in an anechoic chamber and under reverberant conditions. Using the proposed framework, it is shown that correlated narrowband and wideband sources can be resolved reasonably well when compared to conventional methods of acoustic source localization in SH domain.

A DPD Algorithm of Maximum Likelihood Based on DOAs and Doppler

Direct position determination (DPD) is a single step method that estimates the position of target by received signals. Localization in 3D scene is a complicated problem, because the signal propagation channel is complex, and it has many kinds of interference and noise. This paper establishes the Over-the-horizon (OTH) shortwave signals and line-of-sight propagation of ultra short wave model, and proposes a joint localization algorithm. We compare the Cramér-Rao Bound(CRB) and the root mean square error (RMSE) between the proposed algorithm and DPD used single information like DOA and Doppler. The simulation results verify the superiority of the proposed algorithm, and prove it has higher accuracy.

Rapid Initialization Method of Unmanned Aerial Vehicle Swarm Based on VIO-UWB in Satellite Denial Environment

In environments where satellite signals are blocked, initializing UAV swarms quickly is a technical challenge, especially indoors or in areas with weak satellite signals, making it difficult to establish the relative position of the swarm. Two common methods for initialization are using the camera for joint SLAM initialization, which increases communication burden due to image feature point analysis, and obtaining a rough positional relationship using prior information through a device such as a magnetic compass, which lacks accuracy. In recent years, visual–inertial odometry (VIO) technology has significantly progressed, providing new solutions. With improved computing power and enhanced VIO accuracy, it is now possible to establish the relative position relationship through the movement of drones. This paper proposes a two-stage robust initialization method for swarms of more than four UAVs, suitable for larger-scale satellite denial scenarios. Firstly, the paper analyzes the Cramér–Rao lower bound (CRLB) problem and the moving configuration problem of the cluster to determine the optimal anchor node for the algorithm. Subsequently, a strategy is used to screen anchor nodes that are close to the lower bound of CRLB, and an optimization problem is constructed to solve the position relationship between anchor nodes through the relative motion and ranging relationship between UAVs. This optimization problem includes quadratic constraints as well as linear constraints and is a quadratically constrained quadratic programming problem (QCQP) with high robustness and high precision. After addressing the anchor node problem, this paper simplifies and improves a fast swarm cooperative positioning algorithm, which is faster than the traditional multidimensional scaling (MDS) algorithm. The results of theoretical simulations and actual UAV tests demonstrate that the proposed algorithm is advanced, superior, and effectively solves the UAV swarm initialization problem under the condition of a satellite signal rejection.

On the Unique Channel Identification in Ambient Backscatter Communication Systems

Channel coefficient estimation is a challenging problem in ambient backscatter communication systems. In this paper, unique identification of channel coefficients using the Expectation-Maximization (EM) method is investigated. It is proven that the EM method is not able to always yield unique solutions to channel coefficients and therefore the tag symbol estimation accuracy can be compromised. A correct derivation of the hybrid Cramer-Rao Bound (HCRB) is also provided.

Joint Antenna Scheduling and Power Allocation for Multi-Target Tracking under Range Deception Jamming in Distributed MIMO Radar System

The proliferation of electronic countermeasure (ECM) technology has presented military radar with unprecedented challenges as it remains the primary method of battlefield situational awareness. In this paper, a joint antenna scheduling and power allocation (JASPA) scheme is put forward for multi-target tracking (MTT) in the distributed multiple-input multiple-output (D-MIMO) radar. Aiming at radar resource scheduling in the presence of range deception jamming (RDJ), the false target discriminator is designed based on the Cramer–Rao lower bound (CRLB) in terms of the spoofing range, and the predicted conditional CRLB (PC-CRLB) plays a role in evaluating tracking accuracy. The JASPA scheme integrates the quality of service (QoS) principle to develop an optimization model based on false target discrimination, with the objective of enhancing both the discrimination probability of false targets and the tracking accuracy of real targets concurrently. Since the optimal variables can be separated in constraints, a four-step optimization cycle (FSOC)-based algorithm is developed to solve the multidimensional non-convex problem. Numerical simulation results are provided to illustrate the effectiveness of the proposed JASPA scheme in dealing with MTT in the RDJ environment.