Abstract
Sparse recovery is one of the most important methods for single snapshot DOA estimation. Due to fact that the original -minimization problem is a NP-hard problem, we design a new alternative fraction function to solve DOA estimation problem. First, we discuss the theoretical guarantee about the new alternative model for solving DOA estimation problem. The equivalence between the alternative model and the original model is proved. Second, we present the optimal property about this new model and a fixed point algorithm with convergence conclusion are given. Finally, some simulation experiments are provided to demonstrate the effectiveness of the new algorithm compared with the classic sparse recovery method.
1. Introduction
The problem of estimating the direction of arrival (DOA) of signals impinging on an array of sensors is widely applied in radar, sonar, and wireless communication systems [1–9]. For fast-moving sources and multipath propagation problems, snapshots are limited, so high resolution adaptive DOA estimation approaches such as MVDR [10], MUSIC [11], and covariance matching methods [12, 13] fail due to inaccurate estimation of the spatial covariance matrix.
As one of the most important methods designed for single snapshot DOA estimation, sparse recovery has its own advantage for single snapshot case [14–17]. By dividing the angle range into grid points, the number of source is much less than that of grid points. By matching the grid points, these methods usually consider to solve the following -minimization:where stands for the number of nonzero elements. In recent years, many sparse algorithms such as OMP and -minimization [18–22] have been applied to solve this problem. Although a lot of work has given the rationality of sparse recovery algorithms [23–25], these conditions not only require the measurement matrix to meet the RIP condition, but also the corresponding RIC constant to meet certain conditions. However, verifying RIP conditions for a given matrix is itself an NP-hard problem, and the current RIP estimation conclusion is only valid for random matrices. However, DOA measurement matrix does not have such a random structure, so it is difficult to directly verify its RIP conditions. Therefore, there is no sufficient guarantee of model theory. In order to overcome these difficulties, we use the following fraction function:to replace the original 0-norm. In Figure 1, it is easy to get that the new alternative function tends to 0-norm when . Therefore, it is reasonable to believe this function is a good choice.
1.1. The Existed Main Problems of Classic Sparse DOA Estimation Methods
As the main idea of these methods, the real DOA is recovered by matching the real solution with the grid points. In order to deal with off-grid cases, we have to reduce the spacing between these points and increase the number of grid points. If we do not care about the hardware, the main problem caused by a big number of grid points is the measurement matrix with high coherence which leads to invalidation of the existed sparse methods, such as -minimization and OMP.
For a given matrix , the coherence is defined as
In [24], OMP can recover the real sparse solution as long as the number of sources satisfies the following inequalities:
Meanwhile, it is obvious that the coherence will increase as long as the number of grid points increases. In Figure 2, the coherence of measurement matrices changes as the number of grid points increases, and it is obvious that OMP only can guarantee one source when the number of grid points is more than ten.
Besides, increasing the number of grid points also lead to the RIP condition deterioration of the measurement matrix. A matrix is said to satisfy RIP of order if and only if there exists a constant such thatfor any sparse vector . It is obvious that increases as the number of grid points increases until RIP is no longer satisfied. For the alternative method, [25] has proved that sources can be recovered as long as , and this condition is theoretical optimal.
Therefore, both greedy algorithms and -minimization are difficult to deal with situation when the number of grid points increases.
1.2. The Main Contribution of This Paper
To summarize, the main contribution of this paper can be expressed as follows:(1)In order to design a reasonable sparse recovery model for solving DOA estimation, we use the alternative function to replace 0-norm and give some theoretical analysis about the new alternative function(2)With theoretical guarantee about the new alternative model, we design a fixed iterative algorithm and the convergence conclusion is also given
This paper is organized as follows. In Section 2, we review the model processing of DOA estimation problem and give the upper number of sources that the sparse method can recover. We give a new alternative function for -minimization and prove the equivalence between the new models and the original sparse model both in noiseless and noise cases. By analysing the optimal property about this new model in Section 3, an algorithm designed for minimizing the fraction function and its convergence are presented. Some simulation experiments are given. Compared to some classic methods, the proposed method has a better result than others.
1.3. Symbols
Through this paper, we use ,which stands for the real DOA solution, and use ,which stands for the grid point sets. For convenience, for , its support is defined by and the cardinality of set is denoted by . Let be the null space of the matrix . We define subscript notation to be such a vector that is equal to on the index set and zero everywhere else and use the subscript notation to denote a submatrix whose columns are those of the columns of that are in the set index . Let be the complement of . For any positive integer , we denote .
2. Sparse DOA Estimation Model and Some Theory Analysis
2.1. Data Model
Assume that far-field stationary and narrowband signals impinge on an -element uniform linear array with DOAs of . For a given , define a vector ,where and where is the distance between adjacent sensors and is the wavelength of the incident signals. Then, the array outputs of snapshots can be expressed aswhere stands for far-field signals and stands for the noise vector. is an array manifold matrix, whose elements
We consider to recover from a grid points set . If , then there exist two mappings and such that
It is obvious thatwhere
Once and , we can recover via the following -minimization:
Then, we can recover by .
2.2. Sparse DOA Estimation via Minimizing Fraction Function
As one of the most important methods designed for single snapshot DOA estimation, the following theorem shows the upper bound on the number of sources by the sparse method.
Theorem 1. For the measurement matrix defined in (10), if , and there is no noise during the measurement; then, the real solution of DOA estimation can be recovered by -minimization (14), i.e.,where is the solution of model (14).
Proof. Without loss of generality, we consider the following vector :It is obvious that and . Therefore, it is enough to prove that is the sparsest solution of .
If there exists another solution such thatTherefore, we can get that andsince .
However, for the given Vandermonde matrix , it is obvious that any of its submatrix of order is a full-rank matrix; i.e., for , we have thatsince , which is contradict conclusion (18).
By Theorem 1, the performance of the sparse recovery method is clearly demonstrated. In practices, we usually consider the following model because of noise:where stand for a certain norm. Similar to discussion above, in this paper, we use the following -minimization model instead of (20):Before, we prove the equivalence between (20) and (21), some lemmas are needed. The following lemma is easy to get by the definition of , and we leave the proof to the readers.
Lemma 1. If is the solution of -minimization (22), then the column vectors belong to are linearly independent.
Lemma 2. If is the solution of -minimization (23), then the column vectors belong to are linearly independent.
Proof. If the submatrix is not full rank, then there exists a vector such that . For such and , letTherefore, it is easy to get thatfor with .
Since is a concave function when , it is easy to get thatTherefore, we can get thatand it is easy to get thatwhich contradicts the assumptions.
In order to extend of application of in sparse recovery, we consider the following models, -minimization and -minimization. Furthermore, (20) and (21) can be treated as special cases of these two models:Next, the following theorem shows the equivalence between -minimization and -minimization.
Theorem 2. For any , and , there exists a constant such that the solution of -minimization (28) also solves -minimization (27) whenever .
Proof. It is easy to get the constraint region in model (27) and model (28) are polygons which is a convex combination of its limited extreme points.
Define a set , and is a concave function for a given quadrant so the solution of the following problem must be contained in the extreme points of the convex polygon :Since the number of quadrants in is limited, so there exists a limited point set such that model (28) is equal toFor such limited points set , define its subset asSince is a continuous function and the element number of is limited, we can define a constant such thatfor any , , and . Finally, it is obvious that the elements of also solve -minimization (27) since is a continuous function.
The proof is completed.
Corollary 1. For the noiseless cases and the measurement matrix defined in (10), if and , then there exists a constant such that the real solution of DOA estimation can be recovered by both model (20) and model (21) whenever .
Proof. By Lemma 2, it is obvious that both of model (20) and model (21) are equal to the themselves with a bounded constrained , whereSince the solution of with is limited, it is impossible to calculate for given and .
By Theorem 1, we can conclude the equivalence between model (27) and model (28), and the proof is completed.
Corollary 2. Let or , then there exists a constant such that the solution of -minimization (28) also solves -minimization (27) whenever .
Proof. By the prove in Theorem 1, it is enough to prove the constraint zone in (27) and (28) are polytopes. By Lemmas 1 and 2 and (33), it is easy to find that the solutions of (27) and (28) are contained in a bounded zone.
When or , the constraint zone can be rewritten aswhere the matrix with stands for the whole permutations by .
The proof is completed.
3. A Sparse Recovery Algorithm Designed for DOA Estimation
In Section 2, we show the theoretical performance of DOA sparse methods and the equivalence between the alternative function and the original 0-norm. In this section, we will focus on the algorithm designed for DOA estimation. As the theoretical basis for new algorithm, the following theorem shows us the local property of -minimization.
Theorem 3. If , , and , then is the solution of model (14):and satisfies the following equalities:where is a diagonal matrix with .
Proof. Without loss of generality, we assume that and consider the following problem:where . It is obvious that is the solution of model (37) and there exists a constant small enough such that the function is differentiable at the point when . Therefore, KKT condition can be applied in such area. Define the Lagrange function as follows:Therefore, must be the solution of the following equations:To solve equation (39), we can get thatSince , it is easy to get thatBy analysis expression (36), a fixed point iterative algorithm is presented in Algorithm 1. Next, the following theorem shows the convergence conclusion of this new algorithm.
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Theorem 4. The sequence produced bysatisfies the following equality:and the limit point satisfies equality (36).
Proof. Since , it is obvious thatTherefore, we can conclude that is the solution of the following problem:i.e.,By the expression of , it is easy to get thatBy (46) and (47), we can conclude thatSince and if and only if , the sequence is convergent and the limit point is the solution ofBy the Lagrange function of (49), we can get the conclusion of this theorem.
Next, we will give some experiment results to show the effective of the proposed method. In Figure 3, we consider the case when . In this experiments, we take and , and the range of angle is . Under different SNR , , , , , and , it is obvious that the proposed method has a better result than other classic algorithms and it should be emphasized that the proposed method can recover 10 sources which are much closer to theoretical optimal value in Theorem 1.
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For off-grid case, we give a reasonable estimation of the real solution by FFT [26] and then divide the responding zone to match the real solution by a more nuanced division. In Figure 4, the real DOA solution , the left one shows us a rough estimation of real solution by FFT, then we divide 200 grid points around and with a interval, the right one shows the result of a more precise segmentation by the estimation.
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4. Conclusion
In this paper, we consider the alternative function to replace 0-norm. Furthermore, the equivalence relationship between these two models is presented. Although is not a smooth function, we give an analysis expression of its local optimal solution and a fixed point algorithm. Finally, we use this new alternative function to solve DOA estimation problem. Compared to some classic algorithms, the result of our method is better than the classic algorithms. In conclusion, the authors hope that in publishing this paper, a brick will be thrown out and be replaced with a gem.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This paper was funded in part by the China Postdoctoral Science Foundation, grant numbers 2017M613076 and 2016M602775; in part by the National Natural Science Foundation of China, grant numbers 61801347, 61801344, 61522114, 61471284, 61571349, 61631019, 61871459, 61801390, and 11871392; by the Fundamental Research Funds for the Central Universities, grant numbers XJS17070, NSIY031403, and 3102017jg02014; in part by the Aeronautical Science Foundation of China, grant number 20181081003; and by the Science, Technology, and Innovation Commission of Shenzhen Municipality, grant number JCYJ20170306154716846.