Abstract
The sparse arrays using two uniform linear arrays have attracted considerable interest due to the capability of giving analytical expression of sensor location and owning robust direction-of-arrival (DOA) facing strong mutual coupling and sensor failure. In order to achieve the maximum consecutive virtual uniform linear array in difference coarray, in this paper, a design method of a novel sparse array using two uniform linear arrays (NSA-U2) is proposed. We first analyze the relationship between the values of displacement of two subarrays and difference coarray, and then we give the analytical expressions of the displacement and the number of consecutive lags. By discussing the selection of number of subarray sensors, the design of NSA-U2 is completed. Moreover, through choosing a proper compressed interelement spacing, NSA-U2 can be robust to mutual coupling effect. Numerical experiments prove the effectiveness and favorable performance of DOA estimation with mutual coupling.
1. Introduction
Direction-of-arrival (DOA) estimation is one of the key technologies in the field of passive location. It is widely used in seismic detection, military early warning, astronomical observation, radar monitoring, underwater targets positioning, and so on [1, 2]. Traditionally, the researchers apply the super resolution methods, such as subspace methods, to solve DOAs mainly in uniform nonsparse arrays, such as uniform linear arrays (ULAs). In recent years, many experts and scholars have focused on nonuniform sparse arrays, whose sensor spacing can be larger than the half of wavelength of impinging signals. Minimum redundancy arrays (MRAs) [3], coprime linear arrays (CLAs) [4, 5], and nested linear arrays (NLAs) [5, 6] are the typical sparse arrays. Compared with ULAs, the nonuniform sparse arrays can construct the larger array aperture with the same number of sensors, which also means the higher resolution. In order to solve DOAs without any ambiguous values, spatial smoothing multiple signal classification (SS-MUSIC) [7] and direct augmentation approach (DAA) [8] are proposed, respectively. Two algorithms both try to transform the physical array to a virtual nonsparse array, where they both use the difference coarray. [9] demonstrated the effectiveness of two algorithms and concluded that the two share the same performance.
The performance of DOA estimation is mainly decided by the array aperture. While in sparse array, it is changed to be decided by the aperture of consecutive ULA belonging to the difference coarray. Although MRA has the largest virtual ULA aperture in theory, they have no closed-form expression. Thus, a structure named generalized coprime linear array (GCLA) is proposed, where NLA and CLA are the typical arrays. The sensors location of GCLA has an analytical expression, which reduces the complexity of array design. And [10–13] proves that this structure can be much more robust facing sensors failures compared with other sparse arrays. Then, to enlarge the aperture of virtual ULA under the fixed number of sensors, coprime array with compressed interelement spacing (CACIS), coprime array with displaced subarrays (CADiS) [14, 15], coprime array with multiperiod subarrays (CAMpS) [16], and shifted coprime array (SCA) [17] are proposed. CACIS and CADiS show that setting one subarray with a smaller interelement spacing can have the larger aperture of virtual ULA. CAMpS demonstrate that, in order to expand the aperture of virtual ULA, only one subarray can have a compressed interelement spacing. SCA reveals that the displacement between two subarrays is the main factor to the aperture of virtual ULA. Based on sparse array using two ULAs, although many valuable researches have been proposed, the solution of how to maximize virtual ULA aperture corresponding to different parameters has not been answered and is still deserved to study.
Moreover, besides the array aperture, the mutual coupling is another factor affecting the performance of DOA estimation. The severe mutual coupling causes the degrade of accuracy. To tackle the problem, the researchers aim to enlarge the interelement spacing and propose super nested arrays (SNAs) [18, 19], the augmented nested array (ANA) [20], and the maximum interelement spacing constraint (MISC) [21]. Based on the GCLA, they divide the dense subarray into three or four sparse subarrays, which decreases the weight of interelement spacing equal to half of wavelength. However, when the mutual coupling is strong, the weight of interelement spacing bigger than half of wavelength should be considered.
Hence, in this paper, based on GCLA configuration, we propose a novel sparse array using two ULAs (NSA-U2). We first derive the analytical expression of displacement between two subarrays, which can decrease the complexity of array design, and then discuss the selection of interelement spacing of each subarray to suppress the strong mutual coupling. Moreover, the effectiveness of conclusion is demonstrated by simulation experiments. The main contribution of this paper is summarized as follows: (1)Under the fixed number of sensors and same interelement spacing, the paper gives the design of NSA-U2, which can obtain the biggest aperture of the virtual ULA compared with other arrays using two ULAs. Moreover, the paper derives the analytical expression of displacement and the aperture of the consecutive virtual ULA and gives a low-complexity array design method(2)The paper analyzes the selection of interelement spacing of each subarray, which ensures that the weight of small interelement spacing is low. When mutual coupling is strong, robust DOA estimation can be obtained
The rest of this paper is organized as follows: Section II presents the model of sparse array. Section III proposes the design method of NSA-U2 in detail. Section IV analyzes the performance facing mutual coupling in NSA-U2. Section V is the simulation experiments. Section VI summarizes the paper. Throughout the paper, we make use of the following notations shown in Table 1.
2. Problem Formulation
Assume that there are sensors. Based on GCLA [15], NSA-U2 is made of two ULAs, having and sensors, respectively, where and and are coprime integers (generally assuming ). We set the unit interelement spacing to , where is the wavelength of impinging signals. So, the sensors’ location set is all integers, defined as where and and are integers. Here, is the compressed factor, are still coprime integers, and is the displacement between two subarrays. Suppose that there are far-field narrowband signals impinging on this array from with power . Then, the received data is where the manifold matrix is denoted as and the steering vector can be given by where . The signal data vector is where , and is the number of snapshots. And the noise vector is usually Gaussian random variables with zero means and variance . From the references mentioned above in Section I, the covariance matrix is use to construct the virtual array. Then, we can have the covariance matrix of the received data denoted as where . Vectorize the covariance matrix and obtain the vector where , , and are the diagonal elements of . Hence, the vector can be seen as the received data of a virtual linear array [4, 15], whose location is defined as the difference coarray, given by where the difference coarray is the union of self-difference coarray , where crossdifference coarray , where and the corresponding mirrored coarray and . And the aperture of virtual ULAs in coarray is equivalent to the maximum number of consecutive lags in .
CACIS and CADiS [14, 15], respectively, set and . Comparing CACIS with CADiS, we notice that CACIS has the bigger consecutive integer range while it can still be improved, and CADiS has the bigger number of unique coarray lags while its consecutive integer range is much smaller. Hence, the value of is a main factor to the number of consecutive lags. SCA in [17] gives a solution that when , setting can obtain the maximum number of consecutive lags. Then, we need to find a more general solution of under different . Because is a special case, where nested CADiS in [?] reveals that can have the maximum number of consecutive lags, we will take the into consideration in the following section.
3. Proposed Array Design Method
In order to have the maximum number of consecutive lags, defined as , the key of the NSA-U2 design method is to solve the value of and the selection of with the fixed . In this section, we will, respectively, give the setting of and with the detailed proof.
3.1. The Analytical Expression of
Based on (8), we solve the analytical expression of in two steps. The first step is solving to obtain the maximum number of consecutive lags in . The second step is adjusting when considering the effect of to .
In first step, in order to obtain the maximum number of consecutive lags of , the positive and negative lags must be connected. Before solving , we introduce the proposition proposed in [15], which tells the analytical expression of lags and holes in :
Proposition 1. contains the continuous integer , where contains the holes located at , where are integersThen, we further propose the proposition 2 based on Proposition 1:
Proposition 2. meets the following properties: (a)The lags in can be divided into three parts, which are given by , , and (b)The holes in can be divided into two parts, which are given by and
Proof. (a)Due to the property (a) of Proposition 1, we can have that contains all consecutive integers of . contains the values, which are not consecutive and less than . The minimum value of is , where . The value no less than can be denoted by , where . So, any value in is given by . contains the values, which are not consecutive and more than . Similarly, The maximum value of is , where . The value no more than it can be denoted by , where . So, any value in is given by (b)The holes can be divided into two parts, where corresponds to the holes in , and corresponds to the holes in . The maximum value of is . Considering the property (b) of Proposition 1, any hole no more than is given by , where . Similarly, the minimum value of is . So, any hole no less than is given by , where The maximum number of consecutive lags of means that positive and negative lags are connected. Thus, we should let the number of holes in be aligned by as many as possible. Through the analysis in Appendix A, we give the theorem about the expression of :☐
Theorem 3. can achieve the maximum number of consecutive lags defined as , when where the integers and cannot equal to 0 simultaneously. Moreover, the values of are depended on the odevity of .
Proof. See Appendix A.
In second step, with the theorem 3, we use to fill the holes in and obtain the detailed expression of , which is the function of . We propose the theorem 4, which states that☐
Theorem 4. The analytical expression of and are showed in Table 2.
Proof. See Appendix B.
We show an example in Figure 1, where , , and . Thus, from Table 2. The physical sensors of two subarrays have one common sensor located at the th sensor of subarray 1 and th sensor of subarray 2, which proves the Table 3. And ; so, positive holes in can be aligned by , which proves the Theorem 3, and the minimum value in can be aligned by , which proves the Theorem 4. Hence, the figure demonstrates that the proposed theorems are right.☐
3.2. The Selection of Sensor Number of Each Subarray
From Table 2, we have the analytical expression of and , which is determined by and . With a fixed number of sensors , the values of meet the following theorem to maximize the value of .
Theorem 5. With the fixed values of and , when satisfies the value of achieves the maximum. Moreover, under the selection of , with the increase of , increases.
Proof. See Appendix C.
In conclusion, with a fixed , we take the values of into (1) and can have the sensors location of NSA-U2.☐
4. The Proposed Sparse Array Facing Mutual Coupling
4.1. The Mutual Coupling and DOA Estimation
When there exists mutual coupling between physical sensors, (2) can be changed to where is the mutual coupling matrix with size . Generally, in linear array, can be a -band symmetric Toeplitz matrix [22–27]. The element in th row and th column is defined as where are coupling coefficients satisfying . (14) reveals that the sparse array that has a much larger interelement spacing can have weaker mutual coupling. To quantify the effect of mutual coupling, we introduce the following definitions [18–21]:
Definition 6. The weight function of the virtual array is defined as the number of coarray lags index , which can be expressed as When , the value of is smaller, and the mutual coupling is less significant.
Definition 7. The coupling leakage is defined as where the smaller value of implies the weaker mutual coupling.
Thus, we can have the corresponding proposition about the mutual coupling of NSA-U2:
Proposition 8. The weight function of is as follows: (a)The weight function of has (b)The weight function of haswhere and cannot be 0 at the same time. (c)The weight function of has(d)The weight function of is the sum of that of and
Proof. See Appendix D.
Due to Proposition 8, in order to have the weak mutual coupling, the value of should be small. So, the value of , which decides the value of , is important to the values of and . The detailed analysis will show in the next part.
Applying DAA to the received data (7) of virtual sensors, we can have the MUSIC spectrum and obtain the DOA of signals.☐
4.2. The Relationship between and
With a fixed value of , although a big can have a big , a small can cause a big . In order to show the relationship, we set , and based on Proposition 8, . Thus, can be . Firstly, we resolve the weight function of difference coarray corresponding to different values of , which is shown in Figure 2. The figure presents that if is small, when is small, can be . In order to quantify the influence of mutual coupling, we assume that , , and . Moreover, we compare NSA-U2 with the existing sparse arrays, such as NLA [6], MISC [20], ANAI-1, and ANAII-1 [20], under the same number of sensors. Figure 3(a) shows the relationship between and in NSA-U2. With the increase of , increases, but also increases. When mutual coupling is severe, a big array aperture may not obtain the high accuracy. Figure 3(b) compares of different kinds of sparse arrays with same . When and , NSA-U2 can have a small but also a small . ANAII-1 and MISC have the larger and also the bigger .
(a)
(b)
In conclusion, the selection of needs to balance the values of and to obtain a favorable performance, where the influence to DOA estimation will be presented by simulation experiments.
5. Simulation Results
The definition of root mean square error (RMSE) is given as follows,
is the number of Monte Carlo, is the number of target signals, and is the DOA of the estimated source by the Monte Carlo experiment.
Simulation 9. The MUSIC spectrum in NSA-U2 under different .
When we use DAA [?] to estimate DOAs, the maximum number of detectable signals is in theory, which is much more than . In first simulation, we present the MUSIC spectrum with mutual coupling, when there are multiple signals more than . Assume located at and , . Set , , and and can be . Moreover, set , , and . Thus, we apply DAA and obtain the spectrum under different , which is shown in Figure 4. The figure shows that we can estimate underdetermined signals in NSA-U2. However, with the increase of , the influence of mutual coupling to DOA estimation is more obvious, where the peaks (blue lines) gradually deviate from the true DOAs (red lines).
Moreover, we set and vary the from . The spectrum is shown in Figure 5. When the target angle difference increases, if the DOAs of signals are more close to , the accuracy will decrease or even deteriorate. The RMSE of is , while the RMSE of is . And when , the DOA estimation of signal from the direction higher than fails. When the target angle difference decreases, it can have the higher accuracy, such as , whose RMSE is . But if the difference is lower than the resolution of NSA-U2, such as , the DOA estimation fails.
Simulation 10. RMSE performance comparison of different SNRs.
In this simulation, we show the RMSE performance with mutual coupling under different SNRs. The mutual coupling coefficients and the structure of NSA-U2 are the same as simulation 9, and SNR is from to with intervals. Set , and results are shown in Figure 6(a). When , the RMSEs decrease with SNR increasing, but when , the RMSEs go to be flat due to the mutual coupling. When , is the main factor to affect the RMSEs, but when , is the main factor. So, when , NSA-U2 can obtain the highest accuracy. Moreover, we compare the NSA-U2 under , with NLA, MISC, ANAI-1, and ANAII-1, where the RMSEs are shown in Figure 6(b). Considering Figure 3, this figure shows that the RMSE of NSA-U2 is lower than NLA, ANAI-1, and MISC due to the weaker mutual coupling. ANAII-1 has the lowest RMSE because it has the largest , but the complexity to design ANAII-1 is much higher than NSA-U2, especially when is big.
Simulation 11. RMSE performance comparison of different snapshots.
Similar to the simulation 10, we set and vary from . The RMSEs are shown in Figure 7. The simulation results show that the performance of RMSEs is improved with the increase of snapshot numbers, but when , RMSEs are gradually flatted. Other conclusions are the same as that in simulation 10.
Simulation 12. RMSE performance comparison of different .
Based on the setting of simulation 10, we further consider the RMSEs of NSA-U2 with different numbers of sensors defined as under . We vary from . Through the conclusion in Table 2, we can have the corresponding values of as . The results are shown in Figure 8. The figure shows that the bigger can cause the bigger , which means the higher accuracy. But due to the mutual coupling, the RMSEs goes to be flat when , and the gap between the RMSEs corresponding to the bigger is not obvious. Although the difference of between and and that between and is big, the RMSEs between and and that between and are very close.
Simulation 13. RMSE performance comparison of different .
In this simulation, we do the experiments about the effect of number of signals to the RMSEs. Based on the setting of simulation 10, we set , , and vary from . The results in Figure 9 show that when , with the increase of , the RMSEs increase. If is small, such as , the threshold, defined as the value of when , is even smaller than . The threshold becomes bigger with increasing. Moreover, the RMSEs of and also prove that the feasibility of estimating underdetermined signals under enough SNR, where it needs shown in figure.
Simulation 14. RMSE performance comparison of different .
determines the strength of mutual coupling. So, in this simulation, we compare the RMSEs of different sparse arrays under different . The sparse arrays are the same as simulation 10. We set and . varies from to with intervals. The RMSEs are shown in Figure 10. For all array geometries, as increases, which causes more severe mutual coupling effect, the associated RMSEs increase. And NSA-U2 under and has the similar robust DOA estimation as MISC and ANAII-1 facing severe mutual coupling.
(a)
(b)
(a)
(b)
6. Conclusion
In this paper, based on GCLA, the design of sparse array using two ULAs is proposed, named NSA-U2. The paper analyzes the relationship between the displacement between two subarrays and the number of consecutive lags in difference coarray. Then, the selection of the number of subarray sensors and the analytical expressions of are derived, which can obtain the maximum number of consecutive lags. At last, facing mutual coupling, the paper analyzes the balance between and in NSA-U2. Through the simulations, NSA-U2 both has a low-complexity design method and obtains the robust DOA estimation facing severe mutual coupling under proper . Considering the advantages of NSA-U2, the design of sparse array using ULAs can be conducted in the future research.
Appendix
A. Proof of Theorem 3
Lemma 15. If there exists a positive value , where both and, then any positive hole, where and , can have , where .
Proof. Let . Due to the property (b) of Proposition 2, other positive holes are located at , where . Then, considering the property (a) of Proposition 2, other negative lags of , which are no less than , are located at , where . When , we have known that and have the same range of values; so, . Due to , the lemma is true.☐
lemma 15 reveals that all positive holes in can be aligned by , when . Therefore, let , where , and which can be simplified to
Then, can be . (A.2) requires that should be divisible by 2 and as big as possible. The former requirement is depended on the odevity of and values of . The latter requirement means that and . When , can never be divided by ; so, cannot be simultaneously.
Moreover, we find that some values of let two subarrays have one common sensor, where meets that
The possible values of some parameters are listed in Table 3. Thus, with a fixed value of , the number of sensors of subarray 2 can be ; so, can be bigger as .
B. Proof of Theorem 4
Situation 16. We first consider the situation that reaches the maximum values, and then all values in can be aligned. Based on theorem 3, the number of consecutive lags in is . Next, we expect that the minimum value in can be aligned by ; so, we have (B.1) and (B.2), respectively, tell that should be divisible by and .
Case 17. If meets (B.1), the next hole is located at , which cannot be divisible by . So, is given by .
Case 18. In order to let meet (B.2), should be no less than . When , for any value of . Thus, if and meets (A.3), the hole located at cannot be divisible by ; so, .
If , which can be divisible by , . If , which can be divisible by , . And If , which can be divisible by both and , both and . Hence, is the biggest, and that which one is bigger is depended on are positive or not, and are the smallest. In order to maximize value of , we need to let be as large as possible and also meet (B.1) or (B.2). Thus, we need to discuss odevity of parameters to make sure the values of .
We first consider . If is odd and no matter is even or odd, can be divisible by 2, and . If is even and is odd, and can be divisible by 2, then if , ; otherwise, . And we then consider . If is odd and no matter is even or odd, can be divisible by 2, and . If is even and is odd, can be divisible by 2, and . At last, for the sake of clarity, we list the value of and the analytical expression of in Table 2 under different situations.
Situation 19. In this part, we consider the situation that the hole cannot be aligned by ; so, it must be aligned by . Assume that , where . If , then ; otherwise, if , then . Considering that should be as big as possible, we set ; so, In order to ensure whether there is a common sensor in this situation, take (B.3) into (A.3), and each side is divided by , which can be expressed as So, and , but ; thus, (B.4) is false, and two subarrays have no common sensors. Then, we make sure whether the minimum value of given by can be aligned by , so we have Take (B.3) into (B.5), and each side of each equation is divided by and , respectively; then, (B.6) indicates that there are no integer values and to satisfy (B.5); so, the values in cannot be aligned by . As a result, the maximum number of consecutive lags is .
Hence, we compare that we can obtain in two situations. In situation 19, the maximum values of can be , when . Thus, when , is smaller than any case in situation 16. And when , is also smaller than any case in situation 16. So, we can obtain the maximum number of consecutive lags in situation 16, and the theorem is proved.
C. Proof of Theorem 5
Considering that is decided by , we define a function where ,, and are integers chose from . Then, take the derivative of the function and have
Let and have . Thus, define a new function about as . Take the derivative of it and have , where for every . Thus, we only discuss the situations that and . Based on the conclusions in Table 2, we give the detailed analysis.
Case 19. When , . If is even and is odd, is even, and and . If is odd and is even, due to, is even, and and . Those two situations are same. Because we assume , when is even, . If is odd and is odd, is odd, and and . Because are coprime integers, when is odd, and
Case 20. When , and . If is even and is odd, is even, , and . So, and . If is odd and is even, is even, , and . So, and . If is odd and is odd, is odd, and and . Because are coprime integers, when is odd, .
In conclusion, when is even, and when is odd,
When are the fixed integers, we consider the function of , where
Take the derivative of it and have
Obviously, for every , which means with the increase of , increases.
D. Proof of Proposition 8
(a)If we assume that there exists another pair meeting that which can be rewritten as
Because are coprime integers and , then and . Thus, the assumption is false. And each element in only corresponds to one unique ; so, . (b)Due to , it also satisfies the property (a). Based on theorem 3, and have no common part. However, and have one common part , where . And due to , cannot be at the same time. Because the maximum value of is , every in the common part has . Hence, considering property (a), the value of weight function of the common part is . Moreover, the two subarrays have one common sensor; so, . is the self-coarray lags; so, the weight function of each uniform linear subarray is , and .(d)Because , the value of weight function of is the sum of that of the weight function of each subset.
Data Availability
The authors claim that the data used in this article are provided by their simulations, and this is developed without using any data in a published article to support their results. 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 they have no conflicts of interest.
Acknowledgments
The authors claim that the data used in this article are provided by our simulations.