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

­e 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 veried by simulations and has the ability to achieve greater estimation accuracy than existing algorithms.


Introduction
Since the frequency diverse array (FDA) was rst proposed in 2006, it has attracted much attention [1,2], and more and more research studies focus on it. e beampattern of the FDA radar is related to both range and angle because of the small o set between transmit array elements [3,4]. us, the FDA radar has the ability to attain additional information in range, and it can provide more exibility in array signal processing [5,6].
In recent years, a novel radar system called the MIMO radar attains great attention due to its special advantages [7]. Compared with the traditional phased array (PA) radar, MIMO radar brings increased degrees-of-freedoms (DoFs), and many parameter estimation algorithms have been researched in the existing literature [8,9]. Although the MIMO radar has many obvious advantages, the beampattern of it only depends on the angle rather than both range and angle. Considering this problem, a combination of the FDA radar with MIMO was proposed in [10,11], and it can form the beampattern which is related to range. FDA-MIMO is used in a variety of applications, including parameter estimation, deceptive jamming suppression [11,12], target tracking [13,14], beamspace design [15], and communication system [16].
Target multiparameter estimation, which is an important technology in array radar applications. Both angle and range can be estimated simultaneously in the FDA-MIMO radar. Multiple signal classi cation (MUSIC) and reduce-dimension algorithms were applied to the FDA-MIMO radar in [17,18]. A maximum likelihood estimator to obtain angle and range parameter estimation has been proposed in [11]. e sparse reconstruction-based algorithm was implemented in angle and range estimation [19]. However, these algorithms with peak-searching need a large amount of complexity, and the complexity increases rapidly as the number of estimated parameters increases. Searching-free estimation algorithms have been used in the FDA-MIMO radar. For example, the estimation of signal parameters via rotational invariance techniques (ESPRIT) has been used to attain the range-angle parameters of target in FDA-MIMO [20][21][22][23]. In [24], the real-valued subspace decomposition is used to estimate the angle and range. e parallel factor (PARAFAC) decomposition was considered in the parameter estimation [25,26] and has superior estimation accuracy and low computational complexity. PARAFAC decomposition was used to estimate the 2-D direction of arrival and frequency for an L-shaped array in [27,28]. Because of these advantages, the algorithm has been introduced to estimate the angle and range of targets based on the FDA-MIMO radar [29].
As for the moving target localization, the information about angle, range, and velocity is crucial. However, the existing estimation methods with FDA-MIMO are only utilized in angle-range estimation, and velocity estimation has not been considered. In this study, a search-free anglerange-velocity estimation algorithm for monostatic FDA-MIMO is proposed. We utilize the PARAFAC decomposition algorithm to estimate the space-time steering vector. Once the steering vector is acquired, angle, range, and velocity can be calculated.
e remaining sections are organized as follows: Section 2 introduces the data model of FDA-MIMO. e PARAFAC algorithm is used to offer an angle-range-velocity estimation approach for the FDA-MIMO radar in Section 3. Section 4 provides a performance benchmark for the Cramér-Rao Bounds (CRBs), computational complexity, and root mean square errors (RMSE). Simulations and findings are given in Section 5. We conclude this paper in Section 6. e PARAFAC algorithm can significantly reduce computational complexity. Results verify the effectiveness and superiority of our method. e main contributions are summarised as follows: (1) Based on the conventional FDA-MIMO model, we introduce the Doppler frequency to the data model and expand the dimension of parameter estimation. e angle-range-velocity estimation method of the FDA-MIMO radar is proposed based on the PAR-AFAC decomposition algorithm.
(2) e CRBs of angle, range, and velocity are derived based on the 3-dimension parameter model of the FDA-MIMO radar.
Notation: lower-case (upper-case) bold italic characters stand for vectors (matrices). (·) T , (·) H , and (·) † denote the transpose, conjugate transpose, and pseudo-inverse transformation. ⊗ and ∘ are defined as the Kronecker product and the Khatri-Rao product. diag(·) is used to build a diagonal matrix when the object is a vector or to extract diagonal elements as a vector when the object is a diagonal matrix. angle(·) denotes the phase of the complex value.

Data Model
e model of monostatic FDA-MIMO is shown in Figure 1. We assume that the transmitters and receivers are from a uniform linear array with an M-element transmit array and an N-element receive array. e frequency of the m-th element can be written as where f 0 denotes the carrier frequency and Δf denotes the frequency offset. Assume that the FDA-MIMO radar transmits K pulses in the CPI. e k-th transmitted pulse signal of the m-th element can be expressed as where φ m (t) denotes the transmitted waveform, E is the total transmit energy, and T denotes the pulse duration. According to the signal waveforms transmitted by each transmitting element being orthogonal, they are supposed to satisfy the condition [11].
e signal received by the n-th element of the FDA-MIMO radar is based on P targets, and ζ p is the complex reflection coefficient of the p-th target.
e k-th pulse received signal of the n-th received element can be written as τ p m,n,k denotes the propagation delay of the p-th (p � 1, 2, . . . , P) target between the m-th transmitted element and the n-th receive element at the k-th (k � 1, 2, . . . , K) pulse which is expressed as where c, d denote the light speed and the element spacing, respectively. r p , θ p denote the range and angle of the p-th target, respectively. d i denotes the distance between the i-th element and the reference element. T P denotes the pulse repetition time (PRF). Under the narrowband assumption, by using (5), (4) can be updated to where τ p 0 � 2r p /c denotes the transmit time between the p-th target and the radar. rough performing matched filtering with φ m (t)e − j2πmΔf , the output corresponding to the m-th transmit element, n-th received element, the k-th pulse, and the p-th target can be expressed as [12].
Introduce the 3-dimension data model, i.e., in the case of the moving target. Assume that the p-th target has a fixed velocity v p . Rearrange the outputs of matched filters into a MNK × 1 vector, and the 3-D vector corresponding to the p-th target is written as [22]. where denote the receive steering vector, transmit steering vector, and time steering vector, respectively. eir expressions can be written as where λ � c/f 0 denotes the wavelength. From (10), the transmit vector is dependent on both the range and angle. It shows that the FDA-MIMO radar has a range of information in the transmitter. Assume that the fast time sample is L, and at the l-th sample, the total signals of P targets can be expressed as with where s(l) denotes the complex reflection coefficient of P targets at the l-th sample. rough (12), the receive signal matrix Y can be written as where S(t) � [s(1), · · · , s(L)] ∈ C P×L , N(t) ∈ C MNK×L denote the signal reflection coefficient matrix and the Gaussian white noise matrix with zero mean and covariance matrix

Proposed Algorithm
To estimate the parameters, in general, spatial spectrum search algorithms such as the Capon and MUSIC algorithms can be utilized. However, as the dimension increases, so does the computing complexity. We presented a sequential PARAFAC algorithm in this section to estimate the angle, range, and velocity of objects, respectively. By (12), the new received model can be written as where N x � N. D i (A r ) stands for a diagonal matrix constructed out of the i-th row of A r . Referring to the PAR-AFAC decomposition model in [30], the received data (15) are denoted as a three-order tensor and can be written as where x i,j,k denotes the (i, j, e)-th element of the three-order tensor χ, i � 1, . . . , MK; j � 1, . . . , N; e � 1, . . . , L. Y N can be viewed by slicing the three dimension data into a series of slices along the spatial direction. e model of (16) can be written as simultaneous equations along three different dimensions, and two more matrices will be expressed as where W y , W z denote the rearrangement matrices of the three-order tensor and N y , N z denote the rearranged noise Transmit array θ Receive array International Journal of Antennas and Propagation slices. W y can be interpreted as a matrix of three-dimensional data slices arranged along the direction of the transmit array, and W z , W z can be expressed as a matrix of threedimensional data slices arranged along the direction of the received array. Both A t and A r are connected with W x , W y , W y , and W z , W z . A common algorithm named trilinear alternating least square (TALS) is adopted to solve the PARAFAC effectively [26,31]. e principle of the TALS algorithm can be expressed as follows: Step.1: set initialization update matrix Step.2: fitting one of W x , W x , W y , W y , W z , W z using the LS algorithm, where the other two matrixes are obtained in the last iteration Step.3: update the other two matrixes by step. 2 Step.4: repeat Step. 2 and Step.3 when the value of cost function is satisfied with the stop condition. e detailed steps of the TALS algorithm are shown as follows: e slice forms of the three-order tensor have been written in (15), (17), and (18). According to (15) where ‖ · ‖ F denotes the Frobenius norm. e LS estimate S can be expressed as where A r , A t are previously obtained estimates in the last iteration. e LS fitting of W y is e update for A T t can be expressed as where A r , S T are previously obtained estimates in the last iteration. In the same way, by (18), the LS fitting of W z is e update for A T t , A T t can be written as where A t , S T are previously obtained estimates in the last iteration, respectively. According to the (20), (22), and (24), S T , S T , A t , A t , and A r are calculated with the LS algorithm. e iteration can repeat until the method convergence or reaching the maximum number of iterations. e received and transmitted steering vectors of the p-th target are estimated as a r (θ p ), α(θ p , r p , v p ). e angle can be estimated first. e phase of the normalization version of a r (θ p ) can be obtained. g rp � − angle a r (θ) a r (θ) [1] , where a r (θ) [1] denotes the first element in the string vector a r (θ). To obtain the angle of targets, we construct a matrix as follows: e solution to LS fitting for u p is known as u p � U † 1 g rp . en, the angle of the p-th target is expressed as Secondly, the range of the p-th target is estimated. Obtaining the phase of the normalization version of where α(θ p , r p , v p )[1: M] presents a vector with the first element to the m-th element in α(θ p , r p , v p ). Similarly, a matrix will be constructed as follows: e solution to LS fitting for q p is known as q p � U † 2 g tp . e range of the p-th target can be expressed as To avoid the unambiguity phase of the range parameter, the constrained range can be given by r max < c/Δf.

Cramér-Rao
Bound. We will derive the formula of the Cramér-Rao bounds (CRBs) in this subsection for angle, range, and velocity estimation. Assume that there is just one target and that the reflection coefficient is constant when deriving the CRBs. By (12), the received signal can be written as e unknown parameter vector is e corresponding Fisher information matrix (FIM) can be derived as where R n � σ 2 I MNK×MNK denotes the noise covariance matrix and σ 2 and L denote the noise power and the number of snapshots, respectively. e partial derivative can be written as whose terms can be given by where Λ P � diag(0, 1, . . . , P − 1). e inverse of the FIM can be expressed as e diagonal elements of Q are the CRBs for angle (θ), range (r), and velocity (v) estimations, respectively.

Complexity Analysis.
e main aim is to reduce the complexity of the multiple parameter estimation with no searching estimation algorithm. ere is large complexity when the spectral peak searching algorithms are used. e 3D-MUSIC requires Ο n θ n r n v [MNK(MNK − P) e OMP algorithm requires O NMKn θ n r n v + NMKP 2 + NMKP + P 3 }. e computational complexity of the TALS algorithm for PARAFAC decomposition is

Root Mean Square Error.
To assess the estimation (angle, range, and velocity) performance, the root mean square of error (RMSE) is adopted. e RMSE can be expressed as where B denotes the number of the Monte Carlo trials.

e Selection of Frequency
Offset. According to radar theories, the maximum unambiguous range can be expressed as where f PRF denotes the PRF of the FDA-MIMO radar.
In addition, to avoid the multiple solution of estimated range, the frequency increment Δf should be satisfied as c 2Δf > r max .
International Journal of Antennas and Propagation   International Journal of Antennas and Propagation e maximum unambiguous velocity can be expressed as We proposed a selection criterion [11] of frequency offset, and it can be provided as

Simulation Result
B � 500 Monte Carlo trials are used to evaluate the accuracy of the proposed method. e simulation parameters of a monostatic FDA-MIMO radar are shown in Table 1.
We assume that there are P � 2 uncorrelated sources located at (θ 1 , r 1 ) � (25°, 5km), (θ 2 , r 2 ) � (35°, 20km). e  Figure 2. e proposed method is capable of accurately estimating angles, ranges, and velocities. e estimation results are close to the actual scenario. Figure 3 shows the RMSE θ , RMSE r , RMSE v comparisons of different methods such as MUSIC, ESPRIT, and Orthogonal Matching Pursuit (OMP) algorithms versus SNR. We can find that the proposed method attains a superior   International Journal of Antennas and Propagation angle, range, and velocity estimation performance to other algorithms. In addition, the range estimation accuracy is close to the CRB by the proposed method.
To evaluate the estimation performance of different number of snapshots, we calculate the RMSE of different estimation algorithms in different number of snapshots with a fixed SNR � 5, and the result is shown in Figure 4. In Figures 4(a)-4(c), we can find that the RMSEs of the angle, range, and velocity are slightly decreasing versus the increasing number of snapshots. e computational complexity is compared among PARAFAC, ESPRIT, and 3D-MUSIC in Figure 5. e simulation parameters are set as follows: P � 2, l � 30, the number of searching grids corresponding to the angle, range, and velocity is n θ � 300, n r � 1000, and n v � 100, respectively. According to the results, the proposed algorithm and the ESPRIT algorithm have much lower complexity than the 3D-MUSIC. Figures 5(a) and 5(b) show the complexity versus the number of transmit and receive elements, and we can find that the complexities of 3D-MUSIC and ESPRIT are rising faster  than PARAFAC. In Figure 5(c), the complexity versus the number of snapshots has no significant change. Figure 5(d) shows the complexity versus the number of targets, and we can find that the complexity of the proposed algorithm increases versus the number of targets, and the number of targets has a small impact on 3D-MUSIC and ESPRIT.

Conclusion
e angle and range estimation based on the FDA-MIMO radar has attracted much attention. However, the velocity of the moving target cannot be estimated directly. In this paper, the information in the angle-range-velocity domains is combined. We concentrate on the angle, range, and velocity estimation with search-free algorithms for monostatic FDA-MIMO. e PARAFAC algorithm is used to estimate the steering vectors, and we can calculate the angle, range, and velocity of the targets from the estimated steering vectors. Because of the velocity, the dimension of the steering vector is greatly improved, which leads to an increase in computational complexity. Both theoretical analysis and simulations indicate that the PARAFAC algorithm has much lower complexity than the 3D-MUSIC algorithm. e proposed algorithm can attain a higher estimation accuracy than ESPRIT, MUSIC, and OMP. [32][33][34][35][36].

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.