High-order extended coprime array design for direction of arrival estimation

SHI Junpeng ,WEN Fangqing ,LIU Yongxiang ,LIU Tianpeng ,and LIU Zhen

1.College of Electronic Science,National University of Defense Technology,Changsha 410073,China;2.College of Electronic Countermeasure,National University of Defense Technology,Hefei 230037,China;3.College of Computer and Information Technology,China Three Gorges University,Yichang 443002,China

Abstract:Nonuniform linear arrays,such as coprime array and nested array,have received great attentions because of the increased degrees of freedom (DOFs) and weakened mutual coupling.In this paper,inspired by the existing coprime array,we propose a high-order extended coprime array (HoECA) for improved direction of arrival (DOA) estimation.We first derive the closed-form expressions for the range of consecutive lags.Then,by changing the inter-element spacing of a uniform linear array(ULA),three cases are proposed and discussed.It is indicated that the HoECA can obtain the largest number of consecutive lags when the spacing takes the maximum value.Finally,by comparing it with the other sparse arrays,the optimized HoECA enjoys a larger number of consecutive lags with mitigating mutual coupling.Simulation results are shown to evaluate the superiority of HoECA over the others in terms of DOF,mutual coupling leakage and estimation accuracy.

Keywords:high-order extended coprime array (HoECA),direction of arrival (DOA),degree of freedom (DOF),mutual coupling.

1.Introduction

Direction of arrival (DOA) estimation has been widely used in radar,sonar,wireless communication,etc [1,2].By employing the subspace-based methods,such as multiple signal classification (MUSIC) [3] and estimation of signal parameters via rotational invariance technique (ESPRIT) [4],anN-sensors uniform linear array (ULA) can identify up toN−1 sources.Its inter-element spacing is often limited to equal to or less than half a wavelength to avoid spatial aliasing,thus resulting in severe mutual coupling effect.

Some sparse arrays have been designed for increased degrees of freedom (DOFs) and reduced mutual coupling from the difference coarray perspective.The minimum redundancy array (MRA) [5] was firstly presented by providing a large hole-free aperture,but it has no exact expressions for sensor positions and also needs a timeconsuming search to design the array structure. To address these issues,two sparse arrays,i.e.,nested array [6,7] and coprime array [8−10],were proposed with closed-form expressions for sensor locations and virtual coarrays. It is proven that these two structures can provideN2virtual DOFs withNphysical sensors.However,the nested array suffers from heavy mutual coupling due to the densely located subarray,while the coprime array can suppress the mutual coupling at a cost of consecutive lags.In order to further improve the number of DOFs and reduce mutual coupling,various redesigned sparse arrays have been developed.The improved nested arrays,such as super nested array (SNA)[11,12],augmented nested array (ANA) [13],and maximum inter-element spacing constraint (MISC) [14],have been given full discussion with more DOFs and less mutual coupling. Due to the involved hole-free coarray,however,there exists some sensor pairs with small separations,especially with the increase of sensor numbers.As a result,the generalized coprime array was built with two different operations,named as coprime array with compressed inter-element spacing (CACIS) and coprime array with displaced subarrays (CADiS) [15].A generalized nested array (GNA) [16] was then proposed to suppress mutual coupling.Nevertheless,the number of consecutive lags is reduced,which decreases the DOA estimation performance using the spatial smoothing-based MUSIC (SS-MUSIC) algorithm [17].The authors in [18]extended the conventional coprime array (CCA) to thektimes coprime array by filling the holes,but it still has some extra sensor pairs with small separations.Furthermore,the authors in [19] proposed a thinned co-prime array (TCA) by deleting the redundant physical sensors existed in the CCA,which enjoys the minimum number of sensor pairs with small separation.In [20],two extended coprime arrays,including sliding extended coprime array(SECA) and relocating extended coprime array (RECA),were designed to increase the number of consecutive lags and reduce the mutual coupling.Besides,by generating the hole filling strategy,the padded coprime array and the filled difference coarray-based coprime array (FDCCA)were developed for improved DOA estimation,both of which can greatly increase the number of consecutive lags with limited mutual coupling [21−23]. However,FDCCA suffers from higher mutual coupling than the padded coprime array. Besides,by employing the coprime or nested arrays as the transmitter and receiver of conventional multiple input multiple output (MIMO)radar [24−27],the sparse MIMO radar was developed for more DOFs and less mutual coupling [28−30].

In this paper,inspired by the array structures in [19,20],we propose a high-order extended coprime array(HoECA) by introducing another ULA.To be special,the ULA has a flexible inter-element spacing (i.e.,the integer γ),which is coprime with that of the first ULA.We derive the closed-form expressions for the range of consecutive lags,which can be seen as the function of sensor numbers and flexible spacing.Three cases with different values of γ are presented.It is indicated that HoECA can achieve the largest number of consecutive lags when γ takes the maximum value.By comparing it with the other sparse arrays,the advantages of HoECA are presented as below.

(i) HoECA has a flexible inter-element spacing for one ULA,which can be used for developing the hole-free coarray.

(ii) HoECA enjoys more consecutive lags and unique lags than RECA and TCA.Specifically,HoECA possesses about O(2T2/5) consecutive lags,while RECA and TCA have O(T2/3) ones,whereTdenotes the total number of sensors.

(iii) HoECA has similar mutual coupling as RECA and TCA,which is much better than CACIS.

Finally,simulations verify the proposed results.

The rest of the paper are given as follows.Section 2 presents the coarray signal model with the mutual coupling effect.In Section 3,we propose the HoECA structure using another ULA with flexible inter-element spacing,and then compare it with the other existing sparse arrays.Section 4 shows the simulations to verify the effectiveness of HoECA and Section 5 concludes the work.

Notation:In this paper,the boldface letters are reserved for vectors,matrices,and tensors.For a matrixA,AT,A∗and,AHdenote the transpose,complex conjugate and conjugate transpose,respectively.vec(A) denotes the vectorization ofA.The Khatri-Rao product ofAandBis given byA◦B.The symbol diag(·) represents a diagonal matrix.INdenotes anN×Nidentity matrix.The symbol E[·] represents the expectation operation. ⎿a」 denotes the nearest integer to the numbera,⎿a」≤a,and 〈b,c〉 stands for all the integers between the numbersbandc.

2.Preliminaries

2.1 Coarray signal model

Suppose that there areKfar-field uncorrelated sources impinging into theT-sensors sparse array.The sensor locations are set as {did0,i=1,2,···,T},whered0is the unit spacing and often assumed as λ/2. λ denotes the wavelength of the carrier wave.We rewrite the sensor locations as an integer set S={di,i=1,2,···,T} for brevity.Then the received signal model can be given [6,8] by

whereA=[a1,a2,···,aK] denotes the steering matrix,ak=stands for the directional vector of thekth source,t=1,2,···,L,Lis the total number of snapshots,ands(t)=[s1(t),s2(t),···,sK(t)]Tis the source vector.n(t) represents the white Gaussian noise vector with mean zero and covariance matrixanddenotes the noise variance.

The covariance matrix of the signalx(t) is calculated as

In order to explore the difference coarray of the sparse array,we perform vectorization operation on the matrixRand we have

2.2 Mutual coupling

As shown,the signal model in (1) is built without consideration of mutual coupling.In fact,the mutual coupling is greatly related to the separation between two sensors.Then,the signal model in the presence of mutual coupling can be given by

whereCstands for the mutual coupling matrix.Note thatCis often modelled by a B-banded symmetric Toeplitz matrix,the entries of which are written [32] as

wheredi,dj∈S,1=c>|c1|>···>|cB|>|cB+1|=0,c1=c0ejπ/3,cn=c1e−j(n−1)π/8/n,n=2,···,B.c0represents the mutual coupling constant.Bdenotes the maximum spacing of sensor pairs with mutual coupling.Generally,the mutual coupling is evaluated by the coupling leakage

where |·|Fis the Frobenius norm.

3.HoECA

In this section,we propose an HoECA configuration by introducing another subarray with flexible inter-element spacing γ.Some cases are discussed for different values of γ.Then,the comparisons are provided to show the superiority of HoECA in enlarging the DOFs.

3.1 Array design

In this part,by reinforcing the conventional coprime array,such as CADiS and RECA,we further introduce another subarray to build an HoECA,as depicted inFig.1,whereMandNare coprime integers,N>M.

Fig.1 HoECA where M and N (or γ) are coprime integers

To be specific,the third subarray hasMsensors with a flexible inter-element spacing γ,which is also coprime with the sensor numberN.The total number of sensors is thus given byT=2M+N+⎿M/2」.Then,the sensor positions can be presented as

where P1,P2,P3,P4denote the sensor positions of the corresponding four subarrays.As shown,when we ignore the third ULA,the remained subarrays can be regarded as an improved RECA. Therefore,the set P0=P1∪P2∪P3holds for the following property [20].

Theorem 1The difference coarray of P0

contains the consecutive lags within the range 〈−c,c〉 and the number of consecutive lags equals 2c+1,wherec=MN+2M+N−1.

Note that Theorem 1 shows the range of consecutive lags constructed by the sensor positions P1,P2,P3.In order to explore the property of HoECA,the coarrays between the fourth subarray and the others should be further studied for the increased consecutive lags and unique lags.Then,the difference coarray between the fourth and the first ULAs can be set as

which satisfies the following property.

Lemma 1The difference coarray D14has consecutive lags in the range 〈c0,c1〉 and its mirrored range,where γ ∈〈1,N〉,c0=(γ −1)(M−1)+γ+M(N+2) andc1=γM+M(2N+1)−(γ −1)(M−1).

Proof We consider the positive part of D14for better illustration,which can be presented as

We need to prove that,for the consecutive lags within the range [c0,c1],there must exist two integersp∈〈1,M〉,n∈〈N+2,2N+1〉 holding for=γp+Mn,where γ ∈〈1,N〉.

We restate the conditionp∈〈1,M〉 as

which followsn∈〈N+2,2N+1〉. This completes the proof. □

Similar to (9),the difference coarray for the second and the fourth ULAs can be given by

It can be seen that the maximum value in D24can be calculated as=MN+γM+M−N,which is less than the value=M(N−γ)+M+N+γ−1>0).That is,when the consecutive lags in the subsets D0and D14are connected,the lags within the set D24can be neglected.Therefore,we can just consider the union set D0∪D14to explore the consecutive lags and unique lags.Based on the values of γ,we present the following three cases for discussion.

Case 1γ≤(M+N−1)/M

In this case,we consider that the consecutive lags in the subsets P0and D14are connected,which results inc0≤c+1.We thus get γ≤(M+N−1)/M.The union set D0∪D14contains all the lags in the range 〈−c1,c1〉 and the total number of consecutive lags equals 2c1+1,wherec1=2MN+2M+γ −1.For example,when γ=1,we havec1=2MN+2Mand all the lags are consecutive;If γ=3,the conditionN=2M+1 should be satisfied and we havec1=2MN+2M+2.

Case 2(M+N−1)/M<γ

In this case,we address that the consecutive lags in the subsets D0and D14are disconnected,which meansc0>c+1.Therefore,we can have (M+N−1)/M<γ

Case 3γ=N

In case of γ=N,the coarray D14can be rewritten as

Likewise,the difference coarray between the first and the second ULAs can be calculated as

and the coarray between the first ULA and the positionN−Myields

Furthermore,the difference value between the fourth ULA and the positionN−Mcan be given by

Combining with the coarrayswe can form the following union set:

where the position 2MN+Mis extracted from the coarray between the third and the fourth subarrays,and AB denotes the difference set between the coarrays A and B.As a result,similar to Lemma 1,the set D1holds for the following lemma.

Lemma 2D1contains all lags in the rangeand its mirrored part,where=MN+1 and=2MN+3M+N−1.

Combining Lemma 2 and Theorem 1,we can find that,in case of γ=N,HoECA has the following properties as in Theorem 2.

Theorem 2The difference coarray D of HoECA can be rewritten as D=D0∪D1.Then,D providesconsecutive lags within the rangeand also produces 2f+1 unique lags,where=2MN+3M+N−1 andf=3MN+M−1−(M−1)(N−1)/2.

ProofForm Lemma 2 and Theorem 1,sincec0

Then,we prove that D has 2f+1 unique lags,wheref=3MN+M−1−(M−1)(N−1)/2.It is shown that the maximum value in D is 3MN+M,which implies that the discrete lags in the positive part fall in the range3MN+M].

To determine the number of discrete lags in3MN+M],we first introduce an augmented range3MN+2M]. Because 3MN+2M−=(M−1)(N−1),it can be proven in [28] that the range,3MN+2M]possesses (M−1)(N−1)/2 discrete lags.

We also consider that the range (3MN+M,3MN+2M] just has one value 3MN+2M.Therefore,the range3MN+M] contains (M−1)(N−1)/2−1 inconsecutive lags,resulting inf=3MN+M−1−(M−1)(N−1)/2.□

3.2 Comparison

In Section 3.1,we have finished the discussion for the three cases,the results of which are then concluded in(20).It is shown that,in case of γ=N(i.e.,Case 3),HoECA can achieve the maximum number of consecutive lags by constructing the augmented coarray D1.Compared with Case 1,Case 2 can possess more consecutive lags with a large value of γ.Besides,due to the enlarged inter-element spacingN,Case 3 also enjoys the least mutual coupling.

For illustrative purposes,we suppose the case ofM=4,N=7,T=17 as an example.The corresponding virtual lags are depicted inFig.2,where γ can be set as 1,3,7 for the three cases,respectively.It is clear that Case 3 can obtain more consecutive lags than the others by increasing the inter-element spacing of the fourth subarray.Besides,Case 2 enjoys a higher number of consecutive lags and discrete lags than Case 1.

Fig.2 An example of HoECA where M=4, N=7, T=17

Then,for a given sensor numberT=2M+N+⎿M/2」,the numberof Case 3 can be rewritten as

WhenMis even,we have

WhenMis odd,we can get

As a result,HoECA can obtain about O(2T2/5) consecutive lags.For comparison,we can see that both TCA and RECA has about O(T2/3) ones.

In fact,for the givenTsensors,TCA possesses 4MN+2M+2N−1 consecutive lags and RECA has 4MN+4M+2N−1 ones,while HoECA has 4MN+6M+2N−1 ones. As shown inTable 1,whenT=17,B=2,HoECA obtains 149 consecutive lags,which is much larger than others. CACIS has the least consecutive or unique lags with the largest mutual coupling.We can thus conclude that HoECA performs the best among these coprime arrays.Furthermore,due to the increased spacing between the second and the fourth ULAs,HoECA has slightly less mutual coupling than RECA whenBis large.Compared with CACIS,HoECA will perform better with mitigating mutual coupling.

Table 1 Comparison of relevant sparse arrays

RemarkCompared with RECA,HoECA is proposed by introducing another ULA with flexible inter-element spacing.It is verified that HoECA enjoys a higher number of consecutive lags,thus achieving better estimation performance.

4.Simulation results

In this section,the SS-MUSIC algorithm is employed for DOA estimation.The DOF ratio is defined as γ(T)=T2/L(T),whereL(T) denotes the maximum number of consecutive lags in the positive part of the difference coarray.Moreover,we also use the root mean square error (RMSE) to evaluate the estimation performance of relevant sparse arrays,which is given by

4.1 DOF and mutual coupling

Note that the smaller γ(T) is,the higher the DOF capacity can be acquired.Fig.3depicts the DOF ratio γ(T) in terms of the sensor numberT.It is clear that HoECA has a higher DOF capacity than the other coprime arrays by introducing another subarray,such as RECA,TCA,and CACIS,but performs weaker relative to ANAI2 (the second ANA geometry) due to the increased inter-element spacing.Fig.4shows the mutual coupling leakageL(M) versus the numberT.HoECA has similar mutual coupling as TCA and RECA,especially with the increase of the numberT,all of which possess a smallerL(M)than the others.Moreover,ANAI2 has much larger mutual coupling than HoECA and TCA.CombiningFig.3andFig.4,we can conclude that HoECA enjoys higher consecutive lags with relatively less mutual coupling,especially when the sensor number is large.

Fig.3 DOF ratio versus the sensor number T

4.2 RMSE

In this simulation,we address the estimation accuracy of relevant arrays by employing the SS-MUSIC algorithm.

We employ 21 sensors for DOA estimation,where HoECA,RECA,TCA,ANAI2,and CACIS can obtain 227,201,181,257,and 129 consecutive lags.The corresponding mutual coupling leakages are given as 0.121 3,0.126 7,0.111 3.0.202 5,and 0.325 4,respectively.We also assumeK=11 sources impinging into the sparse arrays from directions −50o:50owith a step 10o.Fig.5exhibits the RMSE curves versus the signal-to-noise ratio(SNR),andL=500,c0=0.2.Fig.6depicts the RMSE in terms of the number of snapshots,where the SNR is set as 5 dB,andc0=0.2.Fig.7shows the RMSE versus the coupling coefficientc0,whereL=500 and the SNR is 5 dB.

Fig.5 RMSE versus the SNR

Fig.6 RMSE versus the number of snapshots

Fig.7 RMSE versus the coupling coefficient c0

It indicates that the accuracy of the sparse arrays improves with the increase of the SNR and the number of snapshots,but decreases along with the coupling coefficient.Meanwhile,it is shown inFig.5andFig.6that HoECA outperforms the other sparse arrays even with fewer consecutive lags in the case of severe mutual coupling.However,although TCA enjoys the least mutual coupling,it suffers from weaker performance compared with ANAI2 due to the limited number of consecutive lags.Furthermore,Fig.7also shows that ANAI2 performs worse and worse with the increase of coefficientc0,even weaker than TCA whenc0≥0.36.By contrast,whenc0≤0.04,ANAI2 has the best performance among these arrays.Therefore,in the presence of sever mutual coupling,the proposed coprime array performs much better than ANAI2.

5.Conclusions

In this paper,we propose an HoECA configuration for improved DOA estimation by introducing another subarray to the RECA.Note that the subarray has a flexible inter-element spacing,resulting in some different closedform expressions for the range of consecutive lags.It has been proven that HoECA can achieve the largest number of consecutive lags when the spacing takes the maximum value.The advantages of HoECA are then provided by comparing with the recently developed sparse arrays.Finally,simulation results are shown to evaluate the superiority of HoECA in terms of DOF,mutual coupling leakage,and estimation accuracy.