Compressive Sensing (CS) theory, deduced from signal processing and information theories   , has been widely applied in radar, image processing, wireless communication and many other engineering fields     . The CS theory indicates that the solution of a norm optimization problem can rebuild a sparse signal with comparatively high accuracy by adopting finite nonadaptive random projected measure value .
The strong scatter centers of target in interested area only occupy finite angle resolution cells and the target is sparse in space-domain, so that CS theory has been widely applied in Direction-of-Arrival (DOA) estimation -. A major advantage of CS-based algorithms over conventional super-resolution algorithms  is that the CS-based algorithm can offer higher resolution with reduced antenna elements and Radio Frequency (RF) channels. For example,  presents a CS-based DOA estimation method, which reduces the sampling number by making use of the sparsity of radar echo signals to perform compressive sampling in time-domain.  adopts an array element randomly distributed antenna to perform compressive sampling in space domain to reduce the number of RF channels of the system. However, these two algorithms treat the over-complete based matrices as the redundant dictionaries, obtained from the angle interval of uniform quantization interested area, which cannot ensure that the corresponding sensing matrix meets the Restricted Isometry Property (RIP) condition .  uses random Gauss matrix to perform compressive sampling in space-domain and adopts Regularized Multi-vectors Focal Undetermined System Solver (RMFOCUSS) algorithm to achieve high-resolution DOA estimation. However, the algorithm has computational complexity increasing dramatically with the increasing of snapshots, while at the same time is unsuitable for low signal-to-noise ratio (SNR) situations. Furthermore, the authors in  investigate the CS-based DOA estimation in the presence of sensing model mismatching errors, proving that the performance of CS-based DOA estimation algorithm degrades dramatically in the presence of sensing model mismatching.    present a DOA estimation model under sensing model mismatching, and then use Bayesian method to realize DOA estimation.  proposes a joint Least-Absolute Shrinkage and Selection Operator (LASSO) algorithm to achieve DOA estimation in the presence of mismatching.
In addition, all the aforementioned algorithms utilize multi-channel data so that the estimation performance degrades seriously in the presence of inconsistent gain between RF channels of the array.
In this paper, we derive a single-channel CS-based DOA estimation algorithm via sensing mode optimization to solve the above mentioned problems. Firstly, a DOA estimation model is set up considering mismatch error between system sensing model and target angle information. Secondly, a single-channel array system, which can avoid gain inconsistency between RF channels, is introduced. Meanwhile, it can be proved that the sensing matrix of the single-channel array system meets the RIP condition. Finally, on the basis of Robust Smooth L0 (RSL0) algorithm  and LASSO algorithm , a new DOA estimation algorithm is presented to achieve high resolution DOA estimates.
The paper is organized as follows. Section II formulates the problem of interest. Section III develops the proposed algorithm. Section IV provides the simulation results that demonstrate the efficacy of the proposed algorithm. Section V concludes the paper.
2. Signal Model
2.1. Space Signal Model
Consider K far-field narrow-band signals impinging upon a uniform linear array (ULA) of L elements. The receive signal can be represented as
where is steering vector of the kth signal, , d is the distance between two adjacent elements, is the wavelength of carrier wave, is array noise vector, and is waveform of the kth signal.
For CS processing, the angle field-of-view of the interested area is sampled uniformly at. Denoting as angle information vector, where, is defined as quantization angle, N is the length of vector, and is the angle resolution cell, then if target angle matches with one of the quantization angles, Equation (1) can be rewritten as
where is denoted as the target waveform information vector, and is the steering vector matrix of all sampled angles. In practical, targets in interested area only occupy finite angle resolution cells, so that, with denoting norm. Therefore, the receive signal vector is K sparse signal, is the sparsity based matrix, and K is the sparsity of target angle information vector.
2.2. DOA Estimation Model under Sensing Model Mismatching
Obviously, since N is finite, the target angle might not match exactly with one of the quantization angles. This phenomenon is called mismatching between sensing model and target angle information. According to CS theory, sensing model mismatching will lead to the angle information vector failing to represent target angle precisely, increasing the estimation error of target angles through conventional CS-based DOA estimation method.
Let be the quantization angle that is nearest to the target angle. Approximating the steering vector of the kth target by its first-order Taylor series expansion with respect to the variable, about, we have
where. Expressing in matrix form, we have
where, , is a vector representing the sensing model mismatching errors, and
is denoted as angular quantization error.
Considering the quantization error, Equation (2) can be modified as
3. Proposed Model and Algorithm
3.1. Compressive Sensing Model Based on RF Single-Channel Array
The introduced RF single-channel array is shown in Figure 1. Because the array has only one RF channel, it is characterized by low-power and small-size compared with those RF multi-channel arrays. Moreover, the RF single-channel array can effectively avoid gain inconsistency between RF channels along with the influence on subsequent signal processing caused by imbalance of amplitude and phase, and hence, it plays an important role in practical applications .
Unlike previously developed multi-channel CS-based algorithms, this paper will, for the first time, derive a single-channel CS based algorithm for DOA estimation. First, a phase shifter is connected to each array element and random sampling in space-domain is accomplished through randomly adjusting the phase of phase shifters. Second, an L-combiner is used to combine the signals from L paths through phase-shifter to one signal. Finally, the digital signal is obtained through single RF channel and A/D converter. The channel output signal can be expressed as
where denotes random weighting vector, which is generated by L sets of phase shifters. The value of is either +1 or −1 for.
Figure 1. Single-channel array system.
As the target is sparse in space, we assume that the target does not cross angle resolution cells within M snapshots. Then, the M snapshots measured value of the target can be denoted as
where is weighting coefficient matrix. Since the element of is randomly generated, the elements of, are independent identically distributed Bernoulli random variables. is an sensing matrix and is noise vector.
By observing (8), we can conclude that sampling of space-domain signals through single channel array can be regarded as performing random projection of measurement matrix on receive signal, thus converting the multiple measurement vectors (MMV) problem to a single measurement vector (SMV) problem . In addition, the sensing matrix is the product of matrix, whose elements are Bernoulli distributed, and sparsity based matrix, which can be generated by discrete Fourier transform (DFT) matrix of space-domain signal. Therefore, meets the RIP condition with great probability, thus ensuring the effectiveness and robustness of using compressive sensing reconstruction algorithm to perform DOA estimation.
3.2. Derivation of the Proposed Algorithm
It is found from (8) that the influences of measurement noise and sensing model mismatching error on DOA estimation can be summed up to two parts: “additive” disturbance and “productive” disturbance. Conventional CS-based algorithms only have constraint on “additive” disturbance, but do not consider the influence of “productive” disturbance on the accuracy of target angle information reconstruction. Therefore, these algorithms are not robust in the presence of sensing model mismatching since they cannot effectively reduce the effect of quantization errors.
To overcome these problems, we present a novel CS-based DOA estimation algorithm using single-channel array. The insight of the proposed algorithm is to combine RSL0 algorithm and LASSO algorithm to achieve valid DOA estimates by performing alternative iterative optimization separately on target angle information vector and sensing model quantization error. The basic step of the proposed algorithm can be summarized as follows. The parameters to be optimized are separated into two sets: target angle information set and quantization error set. Each time, a CS cost function that depends only on one set is minimized. With the solution of this CS problem, the subsequent stages of the proposed algorithm consist of applying the same principle on another set of parameter. The algorithm iterates, changing from one set to the next, until the variation of the cost function or of the parameters is less than a predefined convergence criterion.
To initiate the algorithm, we set the sensing model quantization error. Then, according to CS theory, by solving the optimization problem expressed in (8) with norm optimization, we can obtain the estimate of target waveform information vector. Mathematically,
where constant is relevant to noise variance. This optimization problem can be perfectly solved by RSL0 algorithm.
Insert the estimate obtained from (9) into (8), we can get
where. Equation (11) can be transformed to
From (5), we know that the sensing model quantization error vector should have the same sparsity as that of target waveform information vector. Therefore, Equation (12) can be treated as another CS optimization problem by considering the sensing model quantization error as sparse signal. This CS optimization problem can be donated as
which can be perfectly solved by LASSO algorithm.
Finally, inserting the estimate obtained from (13) into (9), we perform the repetition of the above two CS optimization until the difference of two target waveform information vectors’ Frobenius norm is less than a certain predefined threshold, i.e.,
where denotes the estimate of the target waveform information vector obtained at the pth iteration, and is a predefined small value. The K largest values of give the estimates of the target angles.
3.3. Implementation of the Proposed Algorithm
Assuming that the number of signals is known or correctly estimated, the proposed algorithm can be summarized as follows.
1) Initialize with and.
2) Solve (9) to get the estimate.
3) Solve (13) to get the estimate, and then set.
4) Stop the iteration if expression (14) is satisfied. Otherwise go back to step 2).
Performance of the proposed algorithm is evaluated by comparing to the CS-based algorithm in    and RMFOCUSS algorithm in . A ULA with elements, separated by is considered. The angle resolution cell is set as. Three independent narrowband signals are impinge upon the array from angles, and. The additive noise is assumed to be spatial white complex Gaussian, and the SNR is defined relative to each signal. The number of snapshots for each trial is set to be. The performance metrics used are the root mean squared error (RMSE), which for the unknown target is computed as
where is the number of independent Monte Carlo trials.
In the first simulation, we study the performance of the proposed algorithm in the presence of sensing model mismatching. Figure 2 shows the RMSE of the proposed algorithm and its three competitors as functions of SNR varying from −15 dB to 10 dB. From the figure, we see that the proposed algorithm has an estimation accuracy that is higher than those of the other algorithms. The superiority of the proposed algorithm over the other algorithms is due to the fact that the proposed algorithm performs sensing model optimization by taking the quantization errors into account.
Next, we consider the presence of gain inconsistency between RF channels. The gains of RF channels are assumed to be of Gaussian distribution with mean value and variance. The remaining parameters used are the same as those in plotting Figure 2. The RMSE of the algorithms versus SNR are shown in Figure 3. It is seen from the figure that the proposed algorithm does not suffer from gain inconsistency between RF channels due to the usage of single-channel array system, thus, providing better DOA estimation performance than the other multi-channel CS-based algorithms.
Figure 2. RMSE angle error performance versus SNR.
Figure 3. RMSE angle error performance versus SNR in the presence of inconsistent gain between RF channels.
Then, we study the performance on randomly generated DOAs. Suppose that the directions of the input three signals are uniformly generated within the intervals, the remaining parameters used are the same as those in plotting Figure 2. Figure 4 shows the RMSE of different DOA estimation algorithms versus different SNR. From the figure, we see that the performance of the proposed method increases with the decrease of the certain predefined threshold, and when the certain predefined threshold, the proposed method can achieve higher estimation accuracy compared with other off-grid CS-DOA methods.
In the last simulation, we consider the ability of the proposed method to represent the true signals with different angle resolution cells. The angle resolution cells are set as. Figure 5 shows the RMSE versus different angle resolution cells. It is seen from the figure that the performance of the proposed method increases with the decrease of the angle resolution cell.
We have proposed a novel CS-based DOA estimation algorithm via sensing model optimization using single-channel array to solve the problems of sensing model mismatching and channel gain inconsistency, from which most conventional multi-channel CS-based algorithms would suffer. The key idea of the proposed algorithm is to iteratively solve two CS optimizations with respect to target angle information vector and sensing model quantization error vector, respectively. Simulation results have also been presented to verify the efficacy of the proposed algorithm.
Figure 4. RMSE of the DOA estimates versus input SNR.
Figure 5. RMSE of the DOA estimates versus input SNR with different angle resolution cells.
This work was supported by the National Natural Science Foundation of China (Grant no. 61401204), Postdoctoral Science Foundation of Jiangsu Province (Grant no. 1501104C), Technology Research and Development Program of Jiangsu Province (Grant no. BY2015004-03), and the Fundamental Research Funds for the Central Universities (Grant no. 30916011319).
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