Parallel Factor Decomposition Research Articles

Tensor‐based mobile channel tracking for IRS‐aided transmission systems

AbstractIn communication and sensing integrations, an intelligent reflective surface (IRS) will provide an auxiliary wireless link to enhance coverage and information transmission quality. For IRS‐aided millimetre wave multi‐input multi‐output systems, the instantaneous cascaded channel is highly dimensional, resulting in the difficulty of acquiring channel state information. This letter proposed a tensor‐based Kalman filtering method. Under a parallel factor decomposition condition, a distributed sub‐filter algorithm is established. This method benefits from low‐dimensional tensor modes and known IRS phase shifts. It can track the channel with a few pilot overheads.

3-D Positioning Method for Anonymous UAV Based on Bistatic Polarized MIMO Radar

The Angle-of-Arrival (AoA)-based approach is an appealing solution for unmanned aerial vehicle (UAV) positioning, and has received significant interest recently. In this article, we propose a novel framework for UAV three-dimensional (3-D) positioning, the core of which is to measure the two-dimensional (2-D) Angle-of-Departure (2D-AoD) and 2D-AoA via a bistatic multiple-input multiple-output (MIMO) radar. Unlike the existing positioning architectures, the MIMO radar is equipped with polarized array antennas. An estimator based on the parallel factor (PARAFAC) decomposition is developed. It first obtains the direction matrices via performing the PARAFAC decomposition of the array data. Thereafter, the rotational invariance characteristic is utilized to form a normalized polarization response vector, from which the 2D-AoD, 2D-AoA, and polarization status of the UAVs are achieved via incorporating the vector cross-product method and the least squares (LSs) technique. Finally, the 3-D positions of the UAVs are easily calculated via the location relationship between the 2D-AoD, 2D-AoA, and the coordinates of transmitting/receiving (Tx/Rx) array. The proposed framework is computationally friendly, and is capable of positioning anonymous UAV. Moreover, it is insensitive to the geometry of the Tx/Rx array, indicating that the proposed framework supports configurable Tx/Rx antennas. Simulation results are provided to verify our theoretical advantages.

Accelerated PARAFAC-Based Channel Estimation for Reconfigurable Intelligent Surface-Assisted MISO Systems

To achieve fast and accurate channel estimation of reconfigurable intelligent surface (RIS)-assisted multiple-input single-output (MISO) systems, we propose an accelerated bilinear alternating least squares algorithm (ABALS) based on parallel factor decomposition. Firstly, we build a tensor model of the received signal, and expand it to obtain the unfolded forms of the model. Secondly, we derive the expression of the estimation problem of two channels based on the unfolded forms to transform the problem into a cost function problem. Furthermore, we solve the cost function problem by introducing a simpler iterative optimization constraint and linear interpolation. Finally, we provide a strategy on the receiver design based on the feasibility conditions discussed in this paper, which can guarantee the uniqueness of the channel estimation problem. Simulation results show that the proposed algorithm can obtain a faster estimation speed and less iteration steps than the alternating least squares (ALS) algorithm, and the accuracy of the two algorithms is very close.

Search-Free Angle, Range, and Velocity Estimation for Monostatic FDA-MIMO

The monostatic frequency diverse array multiple-input multiple-output (FDA-MIMO) has attracted much attention recently. However, much research is concentrated on the estimation of angle-range parameters based on the FDA-MIMO radar, and the velocity has not been considered. In this study, we propose a search-free method to estimate these parameters. To overcome the problem of the high computational complexity associated with the searching estimation algorithms, the parallel factor (PARAFAC) decomposition is introduced to estimate the space-time steering vector. Next, we can utilize the least square method to solve the angle, range, and velocity of each target. In addition, the Cramér–Rao bounds (CRBs) of angle, range, and velocity are derived. Besides, the other performance analysis consists of the root mean square error, and complexity is derived. We compare the PARAFAC decomposition algorithm with the estimation of signal parameters via the rotational invariance techniques (ESPRIT) algorithm, and our method owns a superior performance. Finally, the proposed method is verified by simulations and has the ability to achieve greater estimation accuracy than existing algorithms.

Double IRSs Aided Massive MIMO Channel Estimation and Spectrum Efficiency Maximization for High-Speed Railway Communications

Intelligent reflecting surface (IRS)-enhanced wireless communication is one of the critical technologies of the sixth generation mobile communication, which has recently emerged as an innovative solution to the realization of intelligent transportation. It can meet the increasing demand for high-speed railway (HSR) communications services. In this paper, we consider a train-ground TDD wireless mobile communication paradigm to deploy two IRSs on the roadside and the train roof to assist the wireless transmission for the HSR communication system. To achieve the more significant performance gain, in the uplink transmission phase, we propose the practical channel estimation framework of the MIMO system aided by double IRSs to acquire reliable channel state information (CSI) of each channel. To facilitate the processing of the three-dimension tensor, we leverage the trilinearity of the parallel factor decomposition to decompose the received signal into a two-dimensional matrix. The Doppler frequency offset compensating scheme and channel tracking scheme based on Kalman-Filter are proposed to further decrease the pilot overheads. Then, in the downlink phase, a block coordinate descent (BCD) algorithm is proposed to handle the data rate maximization problem of double IRSs communication system with the estimated CSI. We reformulate the original problem and decouple the double IRSs phase optimization problem into two single IRS phase optimization non-convex problems under the unit-modulus constraint. The Majorization-Minimization (MM) algorithm is proposed to approximate the optimal solution of the two optimization subproblems. Simulation results corroborate the effectiveness of the proposed channel estimation and phase shifts optimization scheme.

Multi-Channel High-Dimensional Data Analysis with PARAFAC-GA-BP for Nonstationary Mechanical Fault Diagnosis

Conventional signal processing methods such as Principle Component Analysis (PCA) focus on the decomposition of signals in the 2D time–frequency domain. Parallel factor analysis (PARAFAC) is a novel method used to decompose multi-dimensional arrays, which focuses on analyzing the relevant feature information by deleting the duplicated information among the multiple measurement points. In the paper, a novel hybrid intelligent algorithm for the fault diagnosis of a mechanical system was proposed to analyze the multiple vibration signals of the centrifugal pump system and multi-dimensional complex signals created by pressure and flow information. The continuous wavelet transform was applied to analyze the high-dimensional multi-channel signals to construct the 3D tensor, which makes use of the advantages of the parallel factor decomposition to extract feature information of the complex system. The method was validated by diagnosing the nonstationary failure modes under the faulty conditions with impeller blade damage, impeller perforation damage and impeller edge damage. The correspondence between different fault characteristics of a centrifugal pump in a time and frequency information matrix was established. The characteristic frequency ranges of the fault modes are effectively presented. The optimization method for a PARAFAC-BP neural network is proposed using a genetic algorithm (GA) to significantly improve the accuracy of the centrifugal pump fault diagnosis.

PARAFAC Estimators for Coherent Targets in EMVS-MIMO Radar with Arbitrary Geometry

In the past few years, multiple-input multiple-output (MIMO) radar with electromagnetic vector sensor (EMVS) array, or called EMVS-MIMO radar, has attracted extensive attention in target detection. Unlike the traditional scalar sensor-based MIMO radar, an EMVS-MIMO radar can not only provides a two-dimensional (2D) direction finding of the targets but also offers 2D polarization parameter estimation, which may be important for detecting weak targets. In this paper, we investigate into multiple parameter estimations for a bistatic EMVS-MIMO radar in the presence of coherent targets, whose transmitting EMVS and receiving EMVS are placed in an arbitrary topology. Three tensor-aware spatial smoothing estimators are introduced. The core of the proposed estimators is to de-correlate the coherent targets via the spatial smoothing technique and then formulate the covariance matrix into a third-order parallel factor (PARAFAC) tensor. After the PARAFAC decomposition of the tensor, the factor matrices can be obtained. Thereafter, the 2D direction finding can be accomplished via the normalized vector cross-product technique. Finally, the 2D polarization parameter can be estimated via the least squares method. Unlike the state-of-the-art PARAFAC estimator, the proposed estimators are suitable for arbitrary sensor geometries, and they are robust to coherent targets as well as sensor position errors. In addition, they have better estimation performance than the current matrix-based estimators. Moreover, they are computationally efficient than the current subspace methods, especially in the presence of a large-scale sensor array. In addition, the proposed estimators are analyzed in detail. Numerical experiments coincide with our theoretical findings.

Joint Angle and Range Estimation in Monostatic FDA-MIMO Radar via Compressed Unitary PARAFAC

In this paper, we study the joint range and angle estimation problem based in monostatic frequency diverse-array multiple-input multiple-output (FDA-MIMO) radar, and propose a method for range and angle estimation base on compressed unitary parallel factor (PARAFAC). First, the received complex signal matrix is stacked into a third-order complex signal tensor. Then, we can transform the obtained third-order complex signal tensor into a third-order real-valued signal tensor by employing forward–backward and unitary transformation techniques. Next, a smaller third-order real-valued signal tensor is composed by using compressing the third-order real-valued signal tensor. After that, PARAFAC decomposition is applied to obtain the direction matrix. Lastly, the angle and range are estimated by employing the least square (LS) fitting. The estimation error of the proposed method is about 10% lower than that of the traditional PARAFAC method under the low number of snapshots. When the number of snapshots is high, the performance of the two methods is close. Moreover, the computational complexity of the proposed method is nearly 96% less than those of the traditional PARAFAC methods in the case of low snapshots, while the gap is larger in the case of high snapshots. The superiority and effectiveness of the method are proved by complexity analysis and simulation experiments.

Operational modal identification based on parallel factor decomposition with the presence of harmonic excitation

One of the main difficulties of the operational modal analysis is to deal with underdetermined problems inwhich the number of sensors is less than the number of active modes. In the last decade, methods based on the PARAllel FACtor (PARAFAC) decomposition have attracted a lot of attention in the field of modal analysis because it has been proven that these methods can deal with underdetermined cases, as well as the presence of harmonic excitations. Moreover, in combination with kurtosis value as a harmonic indicator, this makes them more efficient in distinguishing between harmonic and structural components. However, it can lead to distorted results as it does not take into account the variation in the length of the covariance functions of the modal coordinates. Since the kurtosis values are estimated from these covariance functions, the length of the latter directly affects the kurtosis. To overcome this limit, the present study proposes to introduce the choice of the length of these functions based on their frequency and damping coefficient. This change improves the existing method by more efficient separating between harmonics and modal components. The proposed procedure is validated using numerical simulations, followed by ambient vibration measurements. © 2021 Academie des sciences, Paris and the authors.

Novel tensor subspace system identification algorithm to identify time-varying modal parameters of bridge structures

Subspace-based system identification algorithms have been developed as an advanced technique for performing modal analysis. We introduce a novel tensor subspace-based algorithm to identify the time-varying modal parameters of bridge structures. A new time dimension is introduced in the traditional Hankel matrix, and a mathematical model of tensor subspace decomposition is established. Combined with the stabilization diagram, tensor parallel factor decomposition is used to estimate the frequencies, mode shapes, and modal damping ratios. The effectiveness of the proposed algorithm is validated by comparing it with the classical sliding-window–based stochastic subspace algorithm on a model cable-stayed bridge dynamic test. The proposed algorithm is further applied to process the dynamic responses of a real bridge health monitoring system to identify its time-varying modal frequencies. Our results demonstrated that the proposed algorithm significantly reduces computational efforts and extends the range of solution ideas for future out-only time-varying system identification problems.

Angle Estimation for MIMO Radar in the Presence of Gain-Phase Errors with One Instrumental Tx/Rx Sensor: A Theoretical and Numerical Study

Ideal transmitting and receiving (Tx/Rx) array response is always desirable in multiple-input multiple-output (MIMO) radar. In practice, nevertheless, Tx/Rx arrays may be susceptible to unknown gain-phase errors (GPE) and yield seriously decreased positioning accuracy. This paper focuses on the direction-of-departure (DOD) and direction-of-arrival (DOA) problem in bistatic MIMO radar with unknown gain-phase errors (GPE). A novel parallel factor (PARAFAC) estimator is proposed. The factor matrices containing DOD and DOA are firstly obtained via PARAFAC decomposition. One DOD-DOA pair estimation is then accomplished from the spectrum searching. Thereafter, the remainder DOD and DOA are achieved by the least squares technique with the previous estimated angle pair. The proposed estimator is analyzed in detail. It only requires one instrumental Tx/Rx sensor, and it outperforms the state-of-the-art algorithms. Numerical simulations verify the theoretical advantages.

Bioavailable DOC: reactive nutrient ratios control heterotrophic nutrient assimilation\u2014An experimental proof of the macronutrient-access hypothesis

We investigate the "macronutrient-access hypothesis", which states that the balance between stoichiometric macronutrient demand and accessible macronutrients controls nutrient assimilation by aquatic heterotrophs. Within this hypothesis, we consider bioavailable dissolved organic carbon (bDOC), reactive nitrogen (N) and reactive phosphorus (P) to be the macronutrients accessible to heterotrophic assimilation. Here, reactive N and P are the sums of dissolved inorganic N (nitrate-N, nitrite-N, ammonium-N), soluble-reactive P (SRP), and bioavailable dissolved organic N (bDON) and P (bDOP). Previous data from various freshwaters suggests this hypothesis, yet clear experimental support is missing. We assessed this hypothesis in a proof-of-concept experiment for waters from four small agricultural streams. We used seven different bDOC:reactive N and bDOC:reactive P ratios, induced by seven levels of alder leaf leachate addition. With these treatments and a stream-water specific bacterial inoculum, we conducted a 3-day experiment with three independent replicates per combination of stream water, treatment, and sampling occasion. Here, we extracted dissolved organic matter (DOM) fluorophores by measuring excitation-emission matrices with subsequent parallel factor decomposition (EEM-PARAFAC). We assessed the true bioavailability of DOC, DON, and the DOM fluorophores as the concentration difference between the beginning and end of each experiment. Subsequently, we calculated the bDOC and bDON concentrations based on the bioavailable EEM-PARAFAC fluorophores, and compared the calculated bDOC and bDON concentrations to their true bioavailability. Due to very low DOP concentrations, the DOP determination uncertainty was high, and we assumed DOP to be a negligible part of the reactive P. For bDOC and bDON, the true bioavailability measurements agreed with the same fractions calculated indirectly from bioavailable EEM-PARAFAC fluorophores (bDOC r2 = 0.96, p < 0.001; bDON r2 = 0.77, p < 0.001). Hence we could predict bDOC and bDON concentrations based on the EEM-PARAFAC fluorophores. The ratios of bDOC:reactive N (sum of bDON and DIN) and bDOC:reactive P (equal to SRP) exerted a strong, predictable stoichiometric control on reactive N and P uptake (R2 = 0.80 and 0.83). To define zones of C:N:P (co-)limitation of heterotrophic assimilation, we used a novel ternary-plot approach combining our data with literature data on C:N:P ranges of bacterial biomass. Here, we found a zone of maximum reactive N uptake (C:N:P approx. > 114: < 9:1), reactive P uptake (C:N:P approx. > 170:21: < 1) and reactive N and P co-limitation of nutrient uptake (C:N:P approx. > 204:14:1). The “macronutrient-access hypothesis” links ecological stoichiometry and biogeochemistry, and may be of importance for nutrient uptake in many freshwater ecosystems. However, this experiment is only a starting point and this hypothesis needs to be corroborated by further experiments for more sites, by in-situ studies, and with different DOC sources.

Channel Estimation for RIS-Empowered Multi-User MISO Wireless Communications

Reconfigurable Intelligent Surfaces (RISs) have been recently considered as an energy-efficient solution for future wireless networks due to their fast and low-power configuration, which has increased potential in enabling massive connectivity and low-latency communications. Accurate and low-overhead channel estimation in RIS-based systems is one of the most critical challenges due to the usually large number of RIS unit elements and their distinctive hardware constraints. In this paper, we focus on the uplink of a RIS-empowered multi-user Multiple Input Single Output (MISO) uplink communication systems and propose a channel estimation framework based on the parallel factor decomposition to unfold the resulting cascaded channel model. We present two iterative estimation algorithms for the channels between the base station and RIS, as well as the channels between RIS and users. One is based on alternating least squares (ALS), while the other uses vector approximate message passing to iteratively reconstruct two unknown channels from the estimated vectors. To theoretically assess the performance of the ALS-based algorithm, we derived its estimation Cramér-Rao Bound (CRB). We also discuss the downlink achievable sum rate computation with estimated channels and different precoding schemes for the base station. Our extensive simulation results show that our algorithms outperform benchmark schemes and that the ALS technique achieves the CRB. It is also demonstrated that the sum rate using the estimated channels always reach that of perfect channels under various settings, thus, verifying the effectiveness and robustness of the proposed estimation algorithms.

Mixed-Timescale Channel Estimation for MIMO Relay Multi-User Systems Based on the PARAFAC Decomposition

This letter presents novel mixed-timescale channel estimation algorithms for uplink multiple-input multiple-output (MIMO) relay multi-user systems, considering the mixed-timescale property of practical dual-hop channel. With the one-stage training (OST) scheme, we formulate the channel estimation problem based on the parallel factor (PARAFAC) decomposition using the received pilots at multiple time slots, where the second-hop channel is viewed as a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">long-timescale</i> variable and the first-hop channel is <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">short-timescale</i> . Then, the alternating least-square (LS) fitting method in batch mode (Batch-ALS) and online recursive alternating LS (Online-RALS) algorithms are proposed for mixed-timescale channel estimation. Simulation results show that the proposed algorithms improve the estimation performance significantly compared to the existing single-timescale channel estimation algorithms.

Fast Logistics Vehicle Localizing Based on EMVS-MIMO Radar and Edge Computing

This article focus on logistics vehicle active localizing using multiple-input multiple output (MIMO) radar, and a fast localizing architecture that integrates edge computing and cloud platform is proposed. The core of the proposed methodology is to measure the angles of logistics vehicle using bistatic MIMO radars, which are configured with coprime electromagnetic vector sensors (EMVS). Unlike the existing localization systems, the proposed localizing architecture provides two-dimensional (2D) angle estimation, i.e., 2D direction-of-departure (DOD) estimation and 2D direction-of-arrival (DOA) estimation, and it offers additional polarization status of the logistics vehicle. A parallel factor (PARAFAC) estimator is developed. Firstly, it estimate the factor matrices via PARAFAC decomposition. Thereafter, the elevation angle estimation is accomplished via least squares (LS) technique, which are ambiguous due to the coprime property of the EMVS. The azimuth angle estimation are followed via vector cross-product. Besides, the polarization information of the targets can be obtained estimated via LS approach. The above process is accomplished at cloud platform. Finally, the localization of the logistics vehicle is achieved with the estimated 2D angles, which is computationally friendly and achieved via edge computing. Detailed analyses concerning degree-of-freedom, identifiability, complexity as well as Cramér-Rao bound (CRB) are provided. To show the effectiveness of the proposed architecture, numerical simulations have been designed.

Joint DOD and DOA Estimation in Slow-Time MIMO Radar via PARAFAC Decomposition

We develop a new tensor model for slow-time multiple-input multiple-output (MIMO) radar, and apply it for joint direction-of-departure (DOD), and direction-of-arrival (DOA) estimation. This tensor model aims to exploit the independence of phase modulation matrix, and receive array in the received signal for slow-time MIMO radar. Such tensor can be decomposed into two tensors of different ranks, one of which has identical structure to that of the conventional tensor model for MIMO radar, and the other contains all phase modulation values used in the transmit array. We then develop a modification of the alternating least squares algorithm to enable parallel factor decomposition of tensors with extra constants. The Vandermonde structure of the transmit, and receive steering matrices (if both arrays are uniform, and linear) is then utilized to obtain angle estimates from factor matrices. The multi-linear structure of the received signal is maintained to take advantage of tensor-based angle estimation algorithms, while the shortage of samples in Doppler domain for slow-time MIMO radar is mitigated. As a result, the joint DOD, and DOA estimation performance is improved as compared to existing angle estimation techniques for slow-time MIMO radar. Simulation results verify the effectiveness of the proposed method.

Joint 2D-DOD, 2D-DOA, and Polarization Angles Estimation for Bistatic EMVS-MIMO Radar via PARAFAC Analysis

Multiple-input multiple-output (MIMO) is a technical hotspot in physical layer with numerous applications in wireless communications, radars, sonars, and well beyond. In this paper, we focus on the multi-dimensional angle estimation problem in a bistatic electromagnetic vector sensors (EMVS) MIMO system. Namely, we need to simultaneously estimate two-dimensional (2D) direction-of-arrival (DOA), 2D direction-of-departure (DOD), 2D receive polarization angle (RPA) and 2D transmit polarization angle (TPA). To tackle this issue, a parallel factor (PARAFAC) analysis-based estimator is proposed. Firstly, a third-order PARAFAC analysis data model is established, which can efficiently exploit the tensor structure of the array measurement. After performing PARAFAC decomposition on the tensor measurement, the factor matrices are achieved. By combining the estimation method of signal parameters via rotational invariance technique (ESPRIT) with the vector cross-product method, joint estimates of 2D-DOD, 2D-DOA, 2D-TPA and 2D-RPA are obtained without further pairing calculation. Compared with the state-of-the-art ESPRIT-Like approach, the proposed method can achieve better performance by enforcing the third-order structure information, and it is suitable for arbitrary array manifolds. Theoretical analyses are given and numerical results corroborate our analysis.

Direction finding in MIMO radar with large antenna arrays and nonorthogonal waveforms

As an important research branch in multiple-input multiple-output (MIMO) radar, orthogonal waveforms are always desirable. In practice, however, the orthogonality may not be always guaranteed. In this paper, a computationally efficient parallel factor (PARAFAC) estimator is proposed, which is suitable for direction-of-arrival (DOA) estimation in colocated MIMO radar with large antenna arrays as well as nonorthogonal waveforms. Firstly, the spatially colored noise caused by the nonorthogonal waveforms is eliminated via temporal cross-correlation of the measurement. Thereafter, a third-order PARAFAC (or called trilinear decomposition) model is established by exploiting the multidimensional structure as well as the low rank property of the cross covariance matrix. The DOA estimation problem is then linked to PARAFAC decomposition and finally obtained via least squares technique. Compared with the existing matrix completion-based algorithm, the proposed estimator is attractive from the perspective of computational complexity and estimation accuracy, especially with large antenna arrays scenario. In addition, the stochastic Cramér-Rao bound on DOA estimation with waveform imperfectly is derived. Numerical simulations are provided to verify the effectiveness of the proposed estimator.

Two-Dimensional Direction Finding Estimation for Uniform Rectangular Array with Unknown Mutual Coupling via Real-Valued PARAFAC Decomposition

Two-dimensional direction-of-arrival (2D-DOA) estimation for uniform rectangular array (URA) is a canonical problem with numerous applications, e.g., wireless communications, sonar and radar systems. The conventional 2D-DOA estimators usually are derived with the assumption of ideal arrays. However, in practice, the arrays may not be well calibrated and suffer from unknown mutual coupling. Using the conventional estimators may lead to low accuracy estimation and high computational complexity in the condition of large number of array elements. In this paper, a novel real-valued parallel factor (PARAFAC) decomposition algorithm is proposed to tackle this problem. The proposed algorithm has better angle estimation performance than the multiple signal classification (MUSIC) algorithm, estimation of signal parameters via rotational invariance techniques (ESPRIT) algorithm and conventional PARAFAC algorithm. But it has lower complexity than MUSIC algorithm. Moreover, the proposed algorithm can obtain automatically paired 2D-DOA estimation, and it is suitable to coherent or closely spaced signals and can eliminate the mutual coupling. Simulation results verify the effectiveness of the proposed algorithm.

Joint 2-D DOA and Doppler Estimation for L-Shaped Array via Dual PARAFAC With Triple Matching Implementation

A dual parallel factor (PARAFAC)-based approach to jointly estimating the two-dimensional direction of arrival (2-D DOA) and Doppler is proposed in this paper, where an L-shaped array consisting of acoustic vector-sensor is used. First, we apply the PARAFAC decomposition to the data model formed by concatenating the outputs of multi-level delays of the observations, and we get the parameter matrix H, which accomplish the 2-D DOA estimation and pairing automatically, then the dual PARAFAC decomposition is applied to the achieved composite steering matrix from the first PARAFAC decomposition, and thus, the same permutation matrices link the estimates of steering matrices and delay matrices from X-subarray and Y-subarray, respectively. Following this, the Doppler and 2-D DOA matching information are obtained via triple matching implementation, e.g. 2-D DOA and frequency matching. Finally, Doppler is estimated by delay matrices. The proposed algorithm is computationally effective for both uniform and non-uniform L-shaped array as SNR exceeds 15dB, and its performance outperforms the joint angle and Doppler shift ESPRIT (JAD-ESPRIT) algorithm and the joint angle and Doppler shift PM (JAD-PM) algorithm. The simulation results justified the effectiveness of the proposed algorithm.