Direction-of-Arrival Estimation in Time-Modulated Linear Arrays Based on the MT-BCS Approach

Tis article proposes a novel approach for the estimation of the direction-of-arrival (DoA) of multiple signals impinging on time-modulated arrays (TMAs). Te algorithm transforms DoA estimation into compressive sensing (BCS) formulation to tackle the sparse signal problem. Based on the voltage outputs of the TMAs at multiple times instants, a strategy using multitasks BCS (MT-BCS) is applied to recover the DoA and improve the accuracy. Te comparison with the existing algorithms of DoA estimation in TMAs verifes the efectiveness and feasibility of the proposed method.


Introduction
Due to the sideband radiation characteristics, the timemodulated arrays (TMAs) are extensively used in various applications, such as future cognitive radio systems [1], electronic zero scanning [2], multibeam mode [3], wireless power transmission [4], and communication applications [5]. Te direction-of-arrival (DoA) estimation has been an essential problem in array signal processing that has attracted signifcant attention and many studies have attempted to solve this problem based on the traditional phased array. On the contrary, few studies have been done on the DoA estimation in the TMAs. A method of DoA estimation based on the MUSIC algorithm in the TMAs was proposed [6], the sidebands were pointed in diferent directions, and the received data space could be formed through the corresponding received signals. In [7], the target of DoA could be recovered by comparing the carrier frequency of the echo signal with that of the transmitted signal. However, limited to the small number of snapshots, low signal-to-noise ratio (SNR), and correlated signals [8], the experimental results cannot meet the requirements.
Recently, compression sensing (CS) [9] has drawn signifcant attention due to its accuracy, computational efciency, and robustness. Terefore, CS-based methods have already been applied to the DoA estimation of traditional antenna arrays [10].
Te authors have previously estimated the DoA and bandwidth of unknown signals through the MT-BCS method based on the traditional phased array [11]. For the TMAs, few works have used the CS algorithm to estimate the azimuth information of the target signals. In [8], a weighted L1-norm with singular value decomposition operation (W-L1-SVD) method has been proposed for the DoA estimation in TMLA. Compared with the MUSIC algorithm, although the W-L1-SVD algorithm is improved and enhances the sparsity of the reconstructed coefcient vector, it is inefcient.
To overcome the above-mentioned drawbacks and satisfy the need for accurate and efcient estimation, this article provides a novel and efective approach based on MT-BCS for the DoA estimation in TMLA. Te proposed algorithm based on Laplace prior [12] is used to recover the DoA in TMLA with a unidirectional phase center motion (UPCM) scheme [6], and it can perfectly cope with coherent signals.
Te numerical results show that the proposed method has better estimation accuracy and efciency compared with the MUSIC, L1-SVD, and W-L1-SVD methods.
Notations, vectors, and matrices are denoted with lowercase and capital letters in bold, respectively. Te operators (·) T , (·) H , and (·) − 1 represent transpose, Hermitian transpose, and inverse, respectively. ⊗ denotes the Kronecker product. I a is an a × a identity matrix and 0 a×b is an a × b zero matrix. di ag · { } denotes the diagonalization operation. Re · { } and Im · { } return the real and the imaginary parts of a variable, respectively.

Model of the TMLA with UPCM Scheme
Consider an isotropic TMLA consisting of N-element equidistant and narrowband far-feld signals with the same carrier frequency f 0 . Array elements are numbered 1 to N from left to right. First, the leftmost M(M < N) elements open a time step τ, which is controlled as follows [6]: where T p is the time modulation period. Te array factor of TMLAs is expressed as follows: where s k (t) is the kth narrowband far-feld signal, d is the array element spacing, β � 2πf 0 /c, η n (t) is the additive Gaussian white noise, and U n (t) is the switching function of the of time of the nth element. Due to the fact that high-speed RF switches periodically switch on and of according to a specifc time sequence to realize the time modulation in TMAs, the received signals in some channels are forced to be zero during a certain time interval within one modulation period, which will deteriorate or invalidate conventional DoA estimation algorithms. Several strategies have been proposed to solve this problem, M. Pesavento, A. Gershman, and M. Haardt proposed the unitary root-MUSIC approach. M. Haardt and J. A. Nossek proposed a unitary ESPRIT approach. A novel approach for estimating DoAs in TMLAs with a unidirectional phase center motion (UPCM) scheme is proposed in this paper. With the UPCM scheme, the beams at diferent sidebands in TMLAs are capable of pointing in diferent directions [13], and the corresponding received signals can be used to compose a received data space [14]. Terefore, the UPCM scheme has been adopted in this article. According to the UPCM scheme, U n (t) is defned as follows [6]: where To explain in detail, an example of a TMLA with N � 24 and M � 2 is shown in Figure 1. Since U n (t) is a periodic function of time, the spatial and frequency responses of (1) can be obtained by decomposing it into Fourier series, and each frequency component has a frequency of f 0 + q/T p (q � 0, ± 1, ± 2, ... ± ∞). Te Fourier component of qth order can be written as follows: where η n ′ (t) is the additive noise of the qth sideband and b q,n is the complex excitation of the qth-order sideband of the nth element and is expressed as follows [6]: where f p � 1/T p , sin cx � sin x/x. Assuming the number of maximum orders sidebands is Q. Ten, the received signal can be expressed as follows: Τ is the noise vector. Based on the above analysis, Section 3 introduces the DoA estimation method in TMLAs with the UPCM scheme.
Using the received data Y, a sparse reconstruction model of DoA estimation based on CS is defned as follows: where ε is the noise level parameter and ‖ * ‖ p is l p -norm.

MT-BCS Model.
Te MT-BCS model is expressed as follows:

International Journal of Antennas and Propagation
where L is the number of snapshots. Te sparse signal vector is determined as follows: where S l ∧ , l � 1, 2, . . . L associates the hyperparameter vectors of diferent snapshots through appropriate "sharing." p r is the prior probability function. Te best values of the signal hyperparameter vector c are computed through the following RVM formula [11]: With where ψ 1 , ψ 2 are the user-defned parameters, while Re * { } and Im * { } are the real and the imaginary parts, respectively. Finally, the solution estimated by the MT-BCS is equal to

Numerical Results
In this section, the proposed MT-BCS algorithm is used for DoA estimation in TMLA with 24 elements UPCM timing sequence and compared with the MUSIC and SVD algorithms. Te following experiments are conducted to determine the appropriate value of Q. Te root-mean-square error (RMSE) is used to assess diferent DoA estimation methods. Te RMSE is defned as follows: where θ k ∧ (t) is the estimate of θ k in the t−th experiment and t is the number of Monte Carlo runs. Te statistical results of RMSE are obtained through an average of T �100 simulations.
Suppose that there are three uncorrelated signals with random codes and equal power, arriving from θ 1 � −8 0 , θ 2 � 0 0 and θ 3 � 14 0 . To select the appropriate value of Q, Figure 2 shows the RMSE of diferent methods versus the Q with 15 dB SNR and 50 snapshots. It can be seen that Q has no signifcant impact on the RMSEs of the three algorithms, and the proposed MT-BCS is stable after Q ≥ 5. In the MUSIC algorithm, the RMSE decreases with the increase in Q and stabilizes after Q ≥ 8. Tus, Q � 8 is adopted in the following study.
For the uncorrelated sources, the RMSEs in diferent SNRs and snapshots are shown in Figures 3 and 4, respectively. Figure 5 shows the corresponding spatial spectrums at L � 50 and SNR � 15 dB.

International Journal of Antennas and Propagation
Te results indicate that the proposed method has superior performance, especially under the circumstance of the low SNR, as well as the small number of snapshots. According to the results shown in Figure 3, the proposed method can estimate DoA accurately with a very small error when the SNR is greater than 10 dB. It can be seen from Figure 5 that the MUSIC algorithm has sharper spectral peaks while having large false peaks, and if the parameters are inappropriate, the estimation results turn inaccurate, according to the results shown in Figures 3  and 4. Te false peaks of the W-L1-SVD algorithm are signifcantly suppressed, and the spectral peaks of the W-L1-SVD algorithm are sharper than those of the L1-SVD algorithm. Te proposed algorithm does not need to consider false peaks because it mainly estimates the amplitude of the angle with strong signal energy, such as shown in Figure 5(d). Te uninterested estimated magnitude of DoA is smaller. Figure 6 shows the RMSEs of diferent algorithms for correlated sources. Te results show that the proposed method has better estimation results than the other three algorithms even in the correlated source. Table 1 compares the computational times of four diferent algorithms in the same scenario. Te time consumption of the two SVD algorithms is close. Te time consumption of the proposed method is longer than that of the MUSIC method but less than that of the two SVD methods. It shows that the proposed MT-BCS method has better efciency than the SVD-based algorithms.

Conclusion
Tis paper proposes a novel and efective method called MT-BCS to deal with the direction-of-arrival estimation problem in time-modulated linear arrays. Te simulation results show that the proposed method can obtain a more accurate DoA estimation compared with the MUSIC and SVD algorithms, even in the case of low SNR, small snapshots, and signal coherence. Moreover, the computational time of the proposed algorithm is also less than the SVD-based algorithms.

Data Availability
Te data are available upon request from the corresponding authors.

Conflicts of Interest
Te authors declare that they have no conficts of interest.