ULA Fitting for Sparse Array Design
UULA Fitting for Sparse Array Design
Wanlu Shi, Sergiy A. Vorobyov,
Fellow, IEEE , and Yingsong Li,
Senior Member, IEEE
Abstract —Sparse array (SA) geometries, such as coprime andnested arrays, can be regarded as a concatenation of two uniformlinear arrays (ULAs). Such arrays lead to a significant increase ofthe number of degrees of freedom (DOF) when the second-orderinformation is utilized, i.e., they provide long virtual differencecoarray (DCA). Thus, the idea of this paper is based on theobservation that SAs can be fitted through concatenation of sub-ULAs. A corresponding SA design principle, called ULA fitting,is then proposed. It aims to design SAs from sub-ULAs. Towardsthis goal, a polynomial model for arrays is used, and based onit, a DCA structure is analyzed if SA is composed of multiplesub-ULAs. SA design with low mutual coupling is considered.ULA fitting enables to transfer the SA design requirements, suchas hole free, low mutual coupling and other requirements, intopseudo polynomial equation, and hence, find particular solutions.We mainly focus on designing SAs with low mutual couplingand large uniform DOF. Two examples of SAs with closed-formexpressions are then developed based on ULA fitting. Numericalexperiments verify the superiority of the proposed SAs in thepresence of heavy mutual coupling.
Index Terms —Difference coarray (DCA), direction-of-arrival(DOA) estimation, mutual coupling, polynomials, sparse array(SA), weight function.
I. I
NTRODUCTION
Array signal processing has a pivotal role in communica-tions [1], radar [2], sonar [3], and other applications due tothe abilities of spatial selectivity, suppressing interferences andincreasing signal quality. Based on the conventional uniformlinear array (ULA) with maximum λ/ interspace betweenantenna elements (here λ is the signal wavelength), manyalgorithms have been developed to perform beamformingand direction-of-arrival (DOA) estimation [4]–[11]. However,the degrees of freedom (DOF) of ULA is proportional tothe number of sensors. Utilizing traditional subspace basedtechniques, such as MUSIC [5], a ULA with N sensors canresolve up to N − sources. Besides, although ULAs withmaximum λ/ inter-element spacing successfully avoid spatialaliasing, they may suffer from mutual coupling effect. This work was supported in parts by the China scholarship council underGrant 201906680047, the Ph.D. Student Research and Innovation Fund ofthe Fundamental Research Funds for the Central Universities under Grant3072019GIP0808, and Academy of Finland under Grant 319822. Somepreliminary results that leaded to this paper have been submitted to
IEEEInt. Conf. Acoust., Speech, Signal Process. 2021 , Toronto, Canada.Wanlu Shi (e-mail: [email protected]) is with the College of In-formation and Communication Engineering, Harbin Engineering University,Harbin 150001, China. He was also a visiting student at the Department ofSignal Processing and Acoustics, Aalto University, Espoo, Finland. SergiyA. Vorobyov (email: sergiy.vorobyov@aalto.fi) is with the Department ofSignal Processing and Acoustics, Aalto University, Espoo, Finland. YingsongLi (e-mail: [email protected]) is with the College of Information andCommunication Engineering, Harbin Engineering University, Harbin 150001,China. (Corresponding author: Yingsong Li)
The problem of detecting more sources than sensors haslong been a problem of great interest in a wide range offields. To further increase the DOF, the concept of differencecoarray (DCA) has been developed. Using DCA, an N -sensorsparse array (SA) can have up to O ( N ) virtual consecutivesensors [12]. The well known minimum redundancy array(MRA) and minimum hole array (MHA), which are in fact SAstructures with difference coarrays, provide larger DOF [13],[14]. Such SAs, however, have failed to lead to a simpleclosed-form expression for representing them, which is adisadvantage for practical applications.Recent progress in nested [15] and coprime arrays [16],[17] have indicated the need for SA structures with closed-form expression and long DCAs. Inspired by this, a numberof SAs have been introduced and analyzed [18]–[24]. Fur-ther, the development of super nested array (SNA) [25]–[28]has led to a renewed interest in designing SAs with bothlong DCA and low mutual coupling. In [27] and [28], theauthors have proposed the SNA geometry and emphasizedthe influence of weight function w ( n ) (where w ( n ) refers tothe number of sensor pairs with interval nλ/ ) on mutualcoupling. Qualitatively, the first three weights of DCA, i.e., w (1) , w (2) , and w (3) , dominate the mutual coupling. SNAachieves a reduced mutual coupling and equivalent DOFcompared to nested arrays. In [29], augmented nested array(ANA) has been developed by rearranging the sensors withsmall separations in nested array leading to an increased DOFand decreased mutual coupling. Essentially, SNA and ANA areoriginated from the conventional nested array and both canreach minimum w (1) = 2 . Further, maximum inter-elementspacing constraint (MISC) array has been proposed in [30] tofurther reduce w (1) to w (1) = 1 .Although many SA geometries with good properties havebeen proposed, only few papers discuss principles of design-ing SAs with desired properties. SNA and ANA are bothpresented with a detailed design process. However, they useset operations. As a consequence, it leads to a complicateddiscussion of many cases that need to be analyzed [27]–[29]. The design of MISC array is somewhat intuitive withthe position set given directly [30]. In [31], an SA designmethod through fractal geometries is proposed which providesa trade-off between uniform DOF (uDOF), mutual coupling,and robustness. However, the fractal geometry still relays onthe existing SAs to act as a generator. Besides, the number ofsensors is exponentially increasing with the array order whichmakes it complicated for the design process to be practicallyuseful.Motivations of this paper are as follows. First, ULAs arespecial types of arrays, which can be easily parameterized by a r X i v : . [ ee ss . SP ] F e b nly 3 parameters such as the initial position, inter-elementspacing, and number of sensors. Second, properties of ULAsare easier to investigate than SAs. Third, the well knownnested and coprime arrays are in fact SAs that consist oftwo sub-ULAs and can be regarded as special cases of amore generic SA design principle. Finally, there still existsno systematic SA design method.The main objective of this paper is to formulate an SAdesign principle, which we call as ULA fitting. The basic ideaof ULA fitting is to design SA using concatenation of a seriesof ULAs. For example, the known SAs, such as nested andcoprime arrays, are in fact combinations of two sub-ULAswith different spacing. However, the case when there are morethan two sub-ULAs within one SA structure has not beeninvestigated to the best of our knowledge. Although, ANAand MISC arrays contain more than two sub-ULAs, they arestill special cases.The problem of modeling an SA with arbitrary sub-ULAsand further analyzing such model is then of interest. In thisrespect, the polynomial model [6], [32]–[34] is convenientfor the purposes of this paper. Note that in [32] and [33],the authors mainly focus on designing effective apertureequivalent to a ULA using sparse receive and transmit arrays.In [32], based on the fact that the radiation pattern is theFourier transform of the aperture function, the convolutionof receive and transmit aperture functions is translated intothe multiplication of their corresponding one-way radiationpatterns. The work [33] further extended the method of [32]by utilizing polynomial factorization to design sparse periodiclinear transmit and receive arrays. In this paper, the poly-nomial model is employed as a powerful tool which linksthe physical sensor array, DCA, and weight function. Thepolynomial model is established to analyze the case whenan SA consists of arbitrary sub-ULAs. Based on investigationof the polynomial, the ULA fitting principle is formulated.Moreover, the main components of ULA fitting, new SAdivision, design criteria, and pseudo-polynomial equation areintroduced. The ULA fitting enables to design SAs with lowmutual coupling, closed-form expressions and large uDOFusing pseudo-polynomials.The main contributions are the following. (i) We intro-duce a polynomial model to analyze DCA and establishbasic relationship between physical sensor positions and thecorresponding DCA and weight function. (ii) We use thepolynomial to model SA with arbitrary sub-ULAs and furtheranalyze the constitute of corresponding DCA. Properties ofeach component in DCA are investigated, and based onthese properties a novel SA design principle (ULA fitting)is proposed. (iii) We develop a pseudo polynomial functionfor designing SAs which enables finding specific solutions forSA geometries with closed-form expressions. (iv) Two novelgeometries with reduced mutual coupling are also proposed.The paper is organized as follows. Section II explains basicmodels, DCA concept, and coupling leakage effect. Section IIIpresents the idea of ULA fitting and establishes the polynomialmodel of SAs. In Section IV, the polynomial model is utilized to investigate the DCA of SAs. Section V gives the basiccomponents of ULA fitting, criteria for parameter selection,and lower bound on uDOF. The design procedure along withtwo specific examples of the use of ULA fitting are given inSection VI. Numerical examples are presented in Section VII.Section VIII concludes the paper.II. P RELIMINARIES
A. Signal Model
Let us consider an array with N sensors located at thepositions S × λ/ , where λ is the signal wavelength. Positionset S , referred to as the normalized position set , possesses unitunderlying grid and is an integer set: S = { p l , l = 0 , , · · · N − } . (1)For the array, the steering vector for a given direc-tion θ is given as a ( θ ) = [ e j (2 π/λ ) p sin θ , e j (2 π/λ ) p sin θ , · · · e j (2 π/λ ) p N − sin θ ] T , where [ · ] T is the transpose.Let us assume that Q uncorrelated narrowband source sig-nals are impinging on the array from azimuth directions θ =[ θ , θ , . . . , θ Q ] T , and L snapshots are available. More specif-ically, there are source signals { s q ( l ) , q = 1 , , · · · , Q } withpowers (cid:8) σ q , q = 1 , , · · · , Q (cid:9) where { l, l = 1 , , · · · , L } .Then the received signal in snapshot l can be expressed as x l = As l + n l , (2)where A is the steering matrix with columns { a ( θ i ) , i = 1 , · · · , Q } , s l = [ s ( l ) , . . . , s Q ( l )] T , and n l is the white noise vector which is assumed to be independentfrom the sources. B. Difference Coarray
The main building block in this paper is DCA. The essenceof DCA is the utilization of the second order statistics, namely,the covariance matrix of the received signal. Using (2), thecovariance matrix of the received signal x l is computed as R xx ( l ) = E [ x l x Hl ] = AR ss ( l ) A H + σ n I , (3)where R ss ( l ) = diag (cid:2) σ , σ , · · · , σ (cid:3) denotes the sourcecovariance matrix, σ n is the noise power, ( · ) H is the Hermitiantranspose, and diag[ · ] forms a diagonal matrix from a vector.Vectorized version of (3) is expressed as f = vec( R xx ) = ( A ∗ (cid:12) A ) g + σ n n , (4)where g = (cid:2) σ , σ , · · · , σ Q (cid:3) T , n = vec( I N ) , (cid:12) is theKhatri-Rao product, and vec( · ) is the vectorization operator.Comparing (2) and (4), f can be regarded as a receivedsignal of a virtual sensor array whose manifold is expressedas ( A ∗ (cid:12) A ) . This virtual sensor array is the well known DCAwhose sensor positions are given by D = { p m − p n , m, n = 0 , , · · · N − } . (5) Definition 1: (DOF)
Given an SA S , the DOF is thecardinality of its DCA D . efinition 2: (uDOF) Given an SA S , the uDOF is thecardinality of the central ULA of its DCA D , denoted as U .We discuss DCA here in two aspects, namely the coarraystructure and the weight function, which are defined as fol-lows. Definition 3: (Coarray Structure)
The coarray of an array S is the array geometry with sensor position set D . Definition 4: (Weight Function)
The weight function w ( n ) of an array S is the number of sensor pairs with interval n .For a given SA S , its weight function w ( n ) can be computedas [15] w ( n ) = b ( n ) ⊕ b ( − n ) , (6)where b ( n ) is the binary expression of S given by b ( n ) = (cid:26) , n ∈ S , elsewhere . (7)It can be concluded from (6) that the length of w ( n ) is { S } + 1 .Using DCA, the DOF of an SA can be significantly in-creased. However, it is the uDOF that determins the abilityof identifying uncorrelated sources [15], [17], [29]. Therefore,we mainly focus on designing SAs with high uDOF and lowmutual coupling. C. Coupling Leakage
In practice, sensors with small separation space interferewith each other owing to energy radiation and absorption. Itis known as mutual coupling. Therefore, a coupling matrix C should be incorporated into (2), that is, x l = CAs l + n l . (8)Typically, mutual coupling is a result of many factors, e.g., hu-midity, operating frequency, adjacent objects, and so on, whichleads to a complicated expression for C [35]. In this paper,only linear array is considered and, thus, a simplified C can beutilized. Based on the inverse proportion relationship betweensensor interspace and coupling coefficients, an approximate C can be formulated using a B-banded mode as [27], [35]–[37] C i,j = (cid:26) c | p i − p j | , | p i − p j | ≤ B, , elsewhere , (9)where p i , p j ∈ S and c a , a ∈ [0 , B ] are coupling coefficients,which satisfy the following relationships (cid:26) c = 1 > | c | > | c | > · · · > | c B | , | c g /c h | = h/g. (10)To evaluate the mutual coupling effect, the coupling leakage L is introduced as L = || C − diag( C ) || F || C || F , (11)where (cid:107) · (cid:107) F stands for the Frobenius norm of a matrix.Qualitatively, the higher L is, the heavier mutual coupling is.Considering the definition of weight function (Definition 4)together with (9), (10), and (11), it can be concluded that themutual coupling is mainly dominated by the weight function for small coarray indexes, i.e., w (1) , w (2) , and w (3) [27],[28]. It is simply because the non-zero weights for smallindexes in D indicate small separation space between sensorsin S which are interfering with each other.III. ULA F ITTING : I
DEA AND P OLYNOMIAL M ODEL
A. General Idea of ULA Fitting
The general idea of ULA fitting is to design SAs from sub-ULAs. Indeed, the well known SA structures, e.g., coprimeand nested arrays can be regarded as a combination of two sub-ULAs [15], [17]. However, the case when an SA is composedof more sub-ULAs still remains unclear. Here we utilize andfurther develop the polynomial model for an array as ananalitic tool for ULA fitting analysis.
0 2 4
12 23
29 32 38 41 sub-ULA 1 sub-ULA 1 sub-ULA 2 sub-ULA 2 sub-ULA 3 sub-ULA 3 sub-ULA 4 sub-ULA 4 sub-ULA 5 sub-ULA 5 sub-ULA 6 sub-ULA 6 gap interspace
Fig. 1: ULA fitting using 6 sub-ULAs.To begin with, we shall introduce several concepts. Fig. 1shows one specific SA geometry explained later in the paper.It consists of 6 sub-ULAs. There are two concepts thatwill be used throughout the paper. The inter-element spacingwithin each sub-ULA is referred to as interspace , and theinterval between adjacent sub-ULAs is referred to as gap .More specifically, a gap between two sub-ULAs is the distancebetween the last sensor in the former sub-ULA and the firstsensor of the latter sub-ULA. The sequence of sub-ULAsis organized by the initial position of each sub-ULA. Theinterspace, number of sensors, aperture, and initial position ofsub-ULA i are denoted by S i , N i , AP i , and I i , respectively.For example, S = 5 , N = 3 , AP = 10 , and I = 12 indicate that the interspace of sub-ULA is , number ofsensors in sub-ULA 2 is 3, aperture of sub-ULA 3 is 10, andinitial position of sub-ULA 4 is 12. Gap between sub-ULA i and sub-ULA j is expressed as gap i,j . For example, thegap between sub-ULA and sub-ULA is gap , . We justconsider the gaps between adjacent sub-ULAs. Other gaps canbe then computed using adjacent gaps and apertures. Withinone SA, the following relationships can be easily established AP j = S j × ( N j − ,I = 0 ,I j = I j − + AP j − + gap j − , j , j ≥ , gap m,n = (cid:80) n − i = m gap i,i +1 + (cid:80) n − j = m +1 AP j , m Based on z -transform and discrete Fourier transform (DFT),polynomial model for an array has been utilized for designingesired radiation patterns of linear arrays [6], [32], [33]. TheDFT of b ( n ) can be expressed as B ( k ) = max { S } (cid:88) n =0 b ( n )e − j πkn max { S } +1 , k = 0 , · · · , max { S } . (13)Substituting x = e − j πk max { S } +1 in (13), the polynomial expressionfor SA S in (13) is given as P SA ( x ) = max { S } (cid:88) n =0 b ( n ) x n . (14)Using (6) and (14), the DCA corresponding to SA S can bethen expressed in terms of the following polynomial model P DCA ( x ) = P SA ( x ) × P SA ( x − )= max { S } (cid:88) n = − max { S } w ( n ) x n . (15)Equation (15) establishes an important relationship betweenthe original SA S , corresponding DCA D , and weight function w ( n ) . In P DCA ( x ) , there is a one-to-one correspondencebetween each exponent and coarray index D , while coefficientscorrespond to the weight function w ( n ) . Note that (15) in factcontains all the information about the DCA, namely, coarraystructure and weight function. However, as we have analyzed,to design an SA with low mutual coupling, it is important tofocus only on w (1) , w (2) , and w (3) . Therefore, we discuss thecoarray structure and the weight function separately. Hence,two simplified versions of the traditional operators × and + are utilized for convenience. The simplified operators, (cid:101) × and (cid:101) + inherit all the computation rules of × and + , butthey omit coefficients. For example, ( x + x ) × ( x + x ) = x + 2 x + x while ( x + x ) (cid:101) × ( x + x ) = x + x + x .Besides, ( x + x ) + ( x + x ) = x + 2 x + x while ( x + x ) (cid:101) +( x + x ) = x + x + x .Consider sub-ULA with parameters { I , S , N } that canbe expressed using polynomial model as P sub1 ( x ) = x I + x I + S + · · · + x I + S ( N − = (cid:16) x + x S + · · · + x S ( N − (cid:17) x I . (16)Equation (16) simply demonstrates the fact that any ULA canbe regarded as a shifted version of a special type of ULAswhose initial position is . Herein, we give these type of ULAsnew definition and notation. Definition 5: (Prototype Array) Prototype array is a ULAwith first sensor located at reference point.Particularly, we present a simplified notation for prototypearrays which only contains interspace and number of sensors.Considering the prototype array of the aforementioned sub-ULA which has N sensors and interspace S , we have thefollowing notation P Aproto { S , N } = x + x S + · · · + x S ( N − . (17) C. Properties of Polynomial Model1) Shifting Property: Consider ULA i with parameters { I i , S i , N i } and the corresponding polynomial expression P i ( x ) , and also ULA j with polynomial expression P j ( x ) = P i ( x ) × x n . Then ULA j is the n -shifted version of ULA i with parameters { I i + n, S i , N i } , where n can be any integer.The shifting property in fact clarifies the mapping relationshipbetween polynomial multiplication and array shifting. 2) Duality Property: For arbitrary SA, there always existsa dual array which shares the same DCA structure with it.Assuming an SA with arbitrary configuration and its cor-responding polynomial expression P SA ( x ) , the method forobtaining its dual array is P dualSA ( x ) = P SA ( x − ) × x max (cid:100) P SA ( x ) (cid:101) , (18)where max (cid:100) P SA ( x ) (cid:101) represents the maximum exponent valueof P SA ( x ) .The proof of this property is straightforward. The DCA forthis SA can be expressed as P SADCA ( x ) = P SA ( x ) × P SA ( x − ) , (19)while for the dual array, using (18), the DCA can be computedas P dualDCA ( x ) = P dualSA ( x ) × P dualSA ( x − )= (cid:110) P SA ( x − ) × x max (cid:100) P SA ( x ) (cid:101) (cid:111) × (cid:110) P SA ( x ) × x max (cid:100) P SA ( x − ) (cid:101) (cid:111) = P SA ( x ) × P SA ( x − ) = P SADCA ( x ) . (20)IV. A NALYZING ULA F ITTING USING P OLYNOMIAL Now that we have established the polynomial model for anarray, we use it for ULA fitting analysis. Suppose an SA iscomposed of n sub-ULAs, then this SA can be modeled usingpolynomial as P SA ( x ) = P sub1 ( x ) + P sub2 ( x ) + · · · + P sub( n ) ( x ) , (21)thus yielding the corresponding DCA expression P SADCA ( x ) = P SA ( x ) × P SA ( x − )= n (cid:88) i =1 P sub( i ) ( x ) (cid:8) P SA ( x − ) − P sub( i ) ( x − ) (cid:9)(cid:124) (cid:123)(cid:122) (cid:125) IDCAs + n (cid:88) i =1 P sub( i ) ( x ) P sub( i ) ( x − ) (cid:124) (cid:123)(cid:122) (cid:125) SDCAs . (22)Without loss of generality, as shown in (22), a DCA canbe divided into two parts, namely self-difference co-arrays(SDCAs) and inter-difference co-arrays (IDCAs). Besides, ifan SA is composed of n sub-ULAs, the corresponding DCAwill have n SDCAs and A n IDCAs.Further, we investigate SDCA structures, IDCA structures,and weight function successively by considering two sub-ULAs, namely, sub-ULA 1 and sub-ULA 2 with parameters I , S , N } , { I , S , N } , respectively, and gap , . Note thatfor structure analysis, we use the simplified operators forconvenience. A. Analysis of SDCA Structures Using polynomial model, we first investigate the SDCAstructures of sub-ULA 1 and sub-ULA 2, which yields P sub1SDCA ( x ) = P sub1 ( x ) (cid:101) × P sub1 ( x − )= (cid:16) x + · · · + x ( N − S (cid:17) (cid:101) × (cid:16) x + · · · + x − ( N − S (cid:17) = x − ( N − S + · · · + x + · · · + x ( N − S , (23)and P sub2SDCA ( x ) = P sub2 ( x ) (cid:101) × P sub2 ( x − )= (cid:16) x + · · · + x ( N − S (cid:17) (cid:101) × (cid:16) x + · · · + x − ( N − S (cid:17) = x − ( N − S + · · · + x + · · · + x ( N − S . (24)Expressions (23) and (24) imply several properties forSDCAs, which can be summarized as follows. (i) The SDCAof an arbitrary linear array geometry is symmetric. (ii) TheSDCA of an N -element ULA contains N − distinct sensors,while the positive set is same as its prototype array. (iii) Theinitial position of any ULA has no effect on its SDCA. (iv) Theperiod of an SDCA for any ULA is the same as its interspace.In addition to these properties of SDCAs, in designing process,we can just consider the prototype arrays of each sub-ULA forconvenience. B. Analysis of IDCA Structures There are two IDCA structures between sub-ULA 1 andsub-ULA 2, namely IDCA and IDCA with the followingexpressions P IDCA12 ( x ) = P sub2 ( x ) (cid:101) × P sub1 ( x − ) ,P IDCA21 ( x ) = P ULA1 ( x ) (cid:101) × P ULA2 ( x − )= P IDCA12 ( x − ) . (25)Clearly, IDCA and IDCA are symmetric to each other.Further computing IDCA , yields P IDCA12 ( x ) = P sub2 ( x ) (cid:101) × P sub1 ( x − )= P sub1proto { S , N } (cid:124) (cid:123)(cid:122) (cid:125) prototype of sub-ULA 1 (cid:101) × (cid:16) x + · · · + x ( N − S (cid:17)(cid:124) (cid:123)(cid:122) (cid:125) N terms, period S x gap . (26)Based on (26), it can be seen that the initial position ofIDCA is determined by gap . Besides, based on theaforementioned shifting property, (26) provides us with a newinsight about the IDCAs between two sub-ULAs which infact implies a transfer process. One sub-ULA provides theprototype array to be transferred, while the other sub-ULAdetermines transfer times and transfer period. By default, thesub-ULA with larger interspace provides the transfer times andperiod.Without loss of generality, let us set S >S , i.e., sub-ULA 2provides the transfer times and period. Then, based on (26),if S > ( N − S , i.e., the interspace of sub-ULA 2 is larger than the aperture of sub-ULA 1, the number of sensors withineach period of IDCA is N . This fact is summarized thenas the following proposition. Proposition 1: If the interspace of the transfer sub-ULA islarger than the aperture of the prototype sub-ULA, then eachperiod of the corresponding IDCAs possesses the same numberof sensors with the prototype sub-ULA.Additionally, we have established that the gap between twosub-ULAs can determine the initial position of their IDCAs.Thus, by properly designing the gap between two sub-ULAs,we can perfectly acquire two IDCAs that are symmetricallydistributed on both sides of the reference point. Thus, we canjust choose the IDCA that is located at the positive side forconvenience of the analysis.Based on the analysis above, we summarize the propertiesof IDCAs as follows. (i) Two IDCAs between two sub-ULAsare symmetric to the reference point. (ii) The initial position ofIDCAs are determined by the gap between the correspondingsub-ULAs. (iii) The IDCAs between two sub-ULAs can beregarded as a transferring process, while one sub-ULA providsthe prototype array to be transferred, and the other sub-ULAdetermines transfer times and period. C. Example To better comprehend the properties of SDCA’s and IDCA’sstructures, an example of a specific SA structure is presented.The SA consists of 2 sub-ULAs with parameters { , , } , { , , } and gap , = 4 . The SA structure versus the SDCAsand the SA versus IDCA are illustrated in Figs. 2 and 3,respectively. 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 0 2 6 13 sub ULA 1 sub ULA 1 sub ULA 2 sub ULA 2 SDCA 1SDCA 2 PrototypeSA Prototype Fig. 2: Relationship between SA and the corresponding SDCAstructures. ULAs in the same color indicate that they have thesame prototype array. D. Analysis of Weight Function As argued before, only w (1) , w (2) , and w (3) are ofsignificance. For other coarray indexes, it is more importantto investigate their existence rather than specific weights forconstructing a long consecutive coarray structures. Proposition 2: The SDCA of a ULA does not introduceweight function for coarray indices smaller than its interspace. 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 sub ULA 1 sub ULA 1 sub ULA 2 sub ULA 2 IDCA 12 gap Prototype SA SA gap 0 2 6 13 gap 0 2 6 13 gap 0 2 6 13 Transfer periodperiod Fig. 3: Relationship between SA and IDCA . Sub-ULA is the prototype sub-ULA and sub-ULA is the transfer sub-ULA, gap , determines the initial position of IDCA . ULAsin the same color indicate that they have the same prototypearray. Proof: Consider ULA i with parameters { I i , S i , N i } andcorresponding polynomial expression P i ( x ) . The weight func-tion contributions given by its SDCA are the coefficients of P i SDCA ( x ) , which satisfy w ( n )= (cid:26) N i − m, n = m × S i , m = 0 , , · · · , N i , , elsewhere . (27)Therefore, a ULA can only introduce non-zero weights forcoarray indexes that are multiples of its interspace and noweights for smaller indexes. Proposition 3: The IDCA between two sub-ULAs does notintroduce weights for coarray index smaller than their gap.Proposition 3 is straightforward because the initial positionof IDCA is determined by the gap between the correspondingsub-ULAs.The ULA fitting principle can now be introduced based onPropositions 2 and 3 along with properties of SDCAs andIDCAs. V. ULA F ITTING A. General Components of ULA Fitting The objective is to specify the ULA fitting principle basedon which one can construct SAs with closed-form expressions,large uDOF, and low mutual coupling. In ULA fitting, an SAgenerally consists of base layer, transfer layer and additionlayer. First, we introduce these layers. 1) The Base Layer:Definition 6: (The Base Layer) The base layer consistsof sub-ULA(s) with the same structure that pad each period(provided by the transfer layer) to realize a dense coarray.Number of sub-ULAs in base layer is N base , and each sub-ULA has parameters { N b , S b , AP b } . The following propertieshold for the base layer. (i) S b is fixed and N b is increasing.(ii) The base layer can consist of more than one sub-ULA, eachsub-ULA has the same prototype array. (iii) The base layer hasthe ability of constructing a dense ULA. (iv) Aperture of baselayer is equivalent to AP b . The base layer is named by the interspace of its sub-ULAs. For example, sub-ULAs within one base layer haveinterspace . In ULA fitting, the main function of the baselayer is to cover specific coarray space. Fig. 4 shows how toutilize , , and base layers to cover a target dense ULA.Generally speaking, as Fig. 4 indicates, the base layer withlarger interspace always has more sub-ULAs. Besides, thearrangement of the sub-ULAs pertaining to one base layeris realized by properly designed gaps.The great significance of developing base layer with largeinterspace is to reduce the mutual coupling. This is quitestraightforward due to the augmented interspace and Propo-sition 2. For instance, as illustrated in Table II, the mutualcoupling is reduced successively with , , and base layers.Note that we just consider here the mutual coupling within thebase layer. Influence of gaps is omitted. 0 2 4 6 8 10 12 Target dense ULA1 base layer2 base layer sub ULA 1sub ULA 1 sub ULA 1sub ULA 1 sub ULA 2sub ULA 2 sub ULA 1sub ULA 1sub ULA 2sub ULA 2 sub ULA 3sub ULA 3 Fig. 4: Illustration of utilizing , and base layers forrealizing the target dense coarray space.TABLE I: Weight contributions of the three base layer casesin Fig. 4 layer w(1) w(2) w(3) base layer 11 10 9 base layer 0 10 0 base layer 0 0 9 2) The Addition Layer: Propositions 2 and 3 in fact explain the purpose of havingthe addition layer. For example, if S b = 3 is selected, it is ofimportance to complement weights w (1) and w (2) in orderto get large uDOF. One possible approach is to set severalgaps as and . However, this approach leads to a stiff designprocess. Another method is to introduce sub-ULAs (which arein fact the addition layers) equiped only with two sensors andinterspaces and . We control the values of w (1) , w (2) ,and w (3) mainly by the second method. However, it shouldbe pointed out that the purpose of the addition layer is notconfined to complement the weight function. Definition 7: (The Addition Layer) The addition layer con-sists of sub-ULA(s) that pad each period (provided by thetransfer layer) together with the base layer to realize a tensecoarray.umber of sub-ULAs in addition layer is N addition .Each sub-ULA in addition layer is parameterized as { N a ( i ) , S a ( i ) , AP a ( i ) } where i = 1 , . . . , N addition . The fol-lowing properties must hold for the addition layer. (i) Theaddition layer is not indispensable. (ii) The addition layercan complement weight function in particular coarray indexes.(iii) The addition layer can complement some holes withineach period. (iv) N a ( i ) , ∀ i are fixed (in most cases N a ( i ) = 2 ),while there is no particular requirement for selecting S a ( i ) (inmost cases S a ( i ) are fixed as well). 3) The Transfer Layer:Definition 8: (The Transfer Layer) The transfer layer is thesub-ULA that provides the transfer times and period.Parameters of the sub-ULA in transfer layer are denotedby { N t , S t , AP t } . The following properties must hold at thetransfer layer. (i) S t is determined by N b and N a , and it islarger than AP a and AP b , while N t is increasing. (ii) An SAstructure contains only one transfer layer. (iii) The transferlayer contains only one sub-ULA.Properties (i) and (ii) follow from Proposition 1. Property (i)guarantees that AP t dominates the whole aperture of the SAand further leads to a new SA domain division principle latershown in this paper.We have introduced now all the components for SA designby ULA fitting. Note that, in the SAs examples designed inthis paper (see Section VI), we only consider the case ofusing one base layer. SA design with multiple base layershas special requirements for transfer layer, which we leaveas a future work. Reconsidering (21) and (22), suppose thatan SA consists of n sub-ULAs. There will be n SDCAsand C n = n ! / n − n − n/ IDCAs in DCAdomain (positive side). These coarrays in DCA domain aredisorganized and hard to analyze. Hence, we provide newdivisions in both SA and DCA domains for convenience ofthe analysis. B. New Array Division and Parameter Selection1) New SA and DCA Division: Suppose that an SA is composed of n sub-ULAs wherethe m th sub-ULA is selected as the transfer layer. Then theSA domain can be divided into three partitions, separatedby the transfer layer, as illustrated in Fig. 5, namely theleft group (LG), the transfer layer (TL), and the right group(RG). The LG, TL, and RG can be expressed by polynomials,respectively, as P LG ( x ) = (cid:80) m − i =1 P sub i ,P TL ( x ) = P sub m ( x ) ,P RG ( x ) = (cid:80) ni = m +1 P sub i . (28)All the SDCAs and IDCAs of the SA can be easily mappedinto three ranges in DCA domain, namely the near end range(NER), the transfer range (TR), and the far end range (FER).In Fig. 5, blue rectangles indicate that the IDCAs in FERare generated by the sub-ULAs choosen separately from LGand RG. The remaining SDCAs and IDCAs can be furtherdivided into two partitions. Obviously, based on the properties of SDCA and IDCA structures, all the coarrays related tothe transfer layer share the same period S t . Hence, these n coarrays, namely one SDCA and thus n − IDCAs, areassigned to the TR as illustrated by red rectangles in Fig. 5.Other coarrays, shown as yellow rectangles in Fig. 5, areassigned to the NER. It is easy to check that all the IDCAs inFER have initial positions larger than AP t , which is in fact thecriteria of the new division. The polynomial model of NER,TR, and FER are, respectively, given as P NER ( x ) = ( (cid:80) ni =1 P SDCA i − P SDCA m ) (cid:101) +( (cid:80) m − i =1 (cid:80) m − j = i +1 P IDCA i,j ) (cid:101) +( (cid:80) n − i = m +1 (cid:80) nj = i +1 P IDCA i,j ) ,P TR ( x ) = P SDCA m (cid:101) +( (cid:80) m − i =1 P IDCA i,m ) (cid:101) +( (cid:80) nj = m +1 P IDCA m,j ) ,P FER ( x ) = (cid:102)(cid:80) m − i =1 (cid:102)(cid:80) nj = m +1 P IDCA i,j . (29)By elaborately designing the interspaces and gaps withinone SA, it is possible to guarantee a hole-free DCA. Usingpolynomial model, the constraint/requirement of a hole freeDCA can be expressed as P NER ( x ) (cid:101) + P TR ( x ) (cid:101) + P FER ( x ) = P proto { , max (cid:100) P SA (cid:101) + 1 } . (30)Note that equation (30) has multiple solutions. However,the need to guarantee the consecutiveness of the whole DCAmay increase the coupling leakage. Thus, a relaxed criteriacan be used for achieving a compromise between uDOF andcoupling leakage. The relaxed criteria then should guaranteethat the consecutive range contains AP t . It is based on the factthat AP t dominates the aperture of the entire SA. Therefore,the relaxed criteria together with corresponding constraints canbe expressed using polynomial model as, for example cons . { P NER ( x ) (cid:101) + P TR ( x ) (cid:101) + P FER ( x ) } = P proto { , J +1 } , such that gap i,j > , < i, j ≤ n,J ∈ (cid:104) max (cid:100) P sub m (cid:101) , max (cid:100) P SA (cid:101) (cid:105) num { gap i,j = 1 } = 1 , num { gap i,j = 2 } = 1 , (31)where cons . { P ( x ) } stands for the consecutive polynomialof P ( x ) , max (cid:100) P ( x ) (cid:101) returns the maximum exponent value, num { x } returns the number of x , and J is the positive intervalthat includes the consecutive range in DCA.Clearly, the relationship between uDOF and J is uDOF = 2 J + 1 . (32)Constraints in (31) are much easier to satisfy than con-straint (30).Note that (31) is just an example. C. Lower Bound of uDOF and Selection of S t In ULA fitting, as we have analyzed, uDOF is dominatedby AP t . Hence, the lower bound on the available uDOF isanalyzed through AP t . We have mentioned that there are one ar end range(FER)Transfer range(TR)Near end range(NER) sub-ULA m (Transfer layer) sub-ULA m+ sub-ULA 1 ... ... sub-ULA m -1 sub-ULA n Left groupLeft group Right groupRight groupSDCAs SDCAsIDCAs IDCAsSDCAsIDCAs 0 Array Aperture FDCA domain (positive set) IDCAsIDCAsSA domain Fig. 5: Illustration of the new array division in both SA domain and FDCA domain when ULA fitting is applied. Rectanglesof the same color indicate the mapping relationships. Mixed colors indicate that the adjacent ranges are overlapped.SDCA and n − IDCAs within TR with period S t . Then,based on Proposition 1 and properties of the transfer layer,we can formulate a criteria for selecting S t .Generally speaking, S t is determined according to N base , N b , N addition , and N a . Ignoring the coarrays in NER and FER,the maximum number of sensors within each period in TR is N base · N b + (cid:80) N addition i =1 N a ( i ) + 1 . Hence, if the interspace ofthe transfer layer is selected based on S t ≤ N base · N b + N addition (cid:88) i =1 N a ( i ) + 1 , (33)a long ULA in TR can be guaranteed by properly selecting S b , S a , and gaps.Let us set S t = N b + N a + z , where z ≤ is an integer.Further, considering total number of sensors N , we have (cid:26) N = N base · N b + (cid:80) N addition i =1 N a ( i ) + N t ,AP t = ( N t − S t . (34)Substituting S t into (34), we obtain AP t = − N + ( N + z + 1) N t − N − z. (35)Maximizing (35), yields the selection of N t , which is N t = N + z + 12 . (36)Considering the odevity of N + z , AP maxt can be expressedas AP maxt = N + z + (2 N − z − N + α , (37)where α satisfies α = (cid:26) , N + z is odd , , N + z is even . (38) Omitting α , which does not influence the monotonicity of AP maxt , (37) can be regarded as a function of z which ismonotonically increasing for z ≤ . Hence, z = 1 can beselected, for example, to pursue a long uDOF. By selecting z = 1 , AP maxt can be written as AP maxt = N + β , (39)where β satisfies β = (cid:26) − , N is odd , , N is even . (40)Therefore, according to (39) for z = 1 , the selection criteriaof S t along with the corresponding lower bound on uDOF are (cid:40) S t = N base · N b + (cid:80) N addition i =1 N a ( i ) + 1 , uDOF lower = N + β + 1 . (41)In fact, both S t = N base · N b + (cid:80) N addition i =1 N a ( i ) + 1 and S t = N base · N b + (cid:80) N addition i =1 N a ( i ) are good options.For instance, the nested and MISC arrays [15], [30] both canbe explained based on ULA fitting. The nested array is builtbased on one base layer consisted of only 1 ULA and a transferlayer also consisted of only 1 ULA, where S t = N b + 1 isselected. In turns, the MISC array is built based on one baselayer consisted of sub-ULAs, one transfer layer and oneaddition layer consisted of one sub-ULA with 2 sensors andinterspace 1, S t = N base · N b + (cid:80) N addition i =1 N a ( i ) is selected.VI. SA S D ESIGN VIA ULA F ITTING Before explaining the design procedure, as we have a lot ofsub-ULAs to be arranged, more notations need to be declaredfirst.ub-ULAs are differentiated by the layer they pertain toand the interspace. For example, the sub-ULAs that pertainto the base layer with interspace and addition layer withinterspace can be expressed as B and A , respectively,where subscripts indicate the interspaces. The transfer layer isexpressed by T . The precedence of the sub-ULAs is shown byarrows. For example, the sequence corresponding to the SAshown in Fig. 1 can be expressed as A → B → T → A → B → B . (42) A. SA Design Procedure The procedure of designing SAs via ULA fitting can besummarized as follows.1) Select the parameters of each layer and list targetequation.2) Determine the sequence of all sub-ULAs.3) Find specific solution which satisfies the design criteria,and further analyze uDOF and parameter selection.4) If no solution is obtained, then redo step 2 and selectanother sequence.Now that all the necessary components have been declared,we give two examples of SA design to show how the ULAfitting works. Since the earlier mentioned MISC array alreadyexhibits a solution when the base layer consisted of 2 sub-ULAs, we start here by designing an SA with the base layerconsisted of 3 sub-ULAs to demonstrate the abilities of ULAfitting that are not matched in the existing literature of SAs.Note that in all the designs, the total number of sensors isdenoted as N . Because of the space limitation, the solutionsare presented directly. B. ULA Fitting for SAs Design Using Base Layer Consistedof 3 sub-ULAs (UF-3BL) In the first example, we present the explanation of the designprocess to show how the ULA fitting works. Consider anSA that is built based on one base layer consisted of 3 sub-ULAs, one transfer layer, and two addition layers. The twoaddition layers consisted of 2-sensors each are selected withinterspaces and , respectively, to control weights w (1) and w (2) . Therefore, in this case, the following relationships canbe established (cid:26) N = 3 N b + N t + 4 ,S t = 3 N b + 5 . (43)The design problem can be expressed as cons . { P NER ( x ) (cid:101) + P TR ( x ) (cid:101) + P FER ( x ) } = P proto { , J + 1 } , such that gap i,j > , < i, j ≤ n,J ∈ (cid:104) max (cid:100) P sub m (cid:101) , max (cid:100) P SA (cid:101) (cid:105) . (44)Moreover, we get sub-ULAs to be arranged, where thesequence is selected as B → A → T → B → A → B . (45) Note that, based on the dual property, sequence (45) leads tothe same solution as the following permuted sequence B → A → B → T → A → B . (46)Hence, the significance of the dual property is to reduce thepossible permutations greatly. Considering (29) and (44), aparticular solution for this case is obtained as gap , = 4 , gap , = 2 + 3 N b , gap , = 3 , gap , = 4 , gap , = 3 . (47)With (47), P NER ( x ) , P TR ( x ) and P FER ( x ) are expressed as P NER ( x ) = { x + x + x } (cid:101) + x × P proto { , N b − } (cid:101) + { x + x + x + x + x } × P proto { , N b } , (48) P TR ( x ) = x (cid:101) + x (cid:101) × P proto { , N b } (cid:101) + x N b +2 + · · · + x N b N t +5 N t − (cid:124) (cid:123)(cid:122) (cid:125) consecutive (cid:101) + x N b +7 × P proto { , N b } × P proto { N b + 5 , N t } (cid:101) + x N b +9 × P proto { , N b } × P proto { N b + 5 , N t } , (49)and P FER ( x ) = x N b N t +5 N t +5 × P proto { , N b − } (cid:101) + { x + x } × x N b N t +5 N t +3 N b +6 × P proto { , N b } (cid:101) + x N b N t +5 N t +3 N b +11 × P proto { , N b − } (cid:101) + { x + x } × x N b N t +5 N t × P proto { , N b } (cid:101) + { x + x + x + x } × x N b N t +5 N t +3 N b +1 (cid:101) + { x + x } × x N b N t +5 N t +3 N b +6 × P proto { , N b } , (50)which leads to the following expression cons . { P NER ( x ) (cid:101) + P TR ( x ) (cid:101) + P FER ( x ) } = P proto { , N b N t + 5 N t + 3 N b } . (51)Comparing (44) and (51), we have J UF − = 3 N b N t + 5 N t + 3 N b − . (52)Considering the fact that N b and N t are integers and the totalnumber of sensors N is fixed, the following optimal selectionof N b and N t is obtained by maximizing (52) (cid:26) N b = (cid:98) N − (cid:99) , N ≥ ,N t = N − N b − , (53)where (cid:98)·(cid:99) is the floor operation. The final uDOF can then beexpressed as uDOF = N + 2 N − , N %6 = 0 , N + 2 N − . , N %6 = 1 , N + 2 N − , N %6 = 2 , N + 2 N − . , N %6 = 5 , , N ≥ , (54)here % stands for the remainder. The closed-form expres-sions for the proposed UF-3BL can be summarized as sub-ULA1 : { , , N b } , sub-ULA2 : { N b + 1 , , } , sub-ULA3 : { N b + 4 , N b + 5 , N t } , sub-ULA4 : { N t N b + 5 N t + 3 N b + 2 , , N b } , sub-ULA5 : { N t N b + 5 N t + 6 N b + 3 , , } , sub-ULA6 : { N t N b + 5 N t + 6 N b + 8 , , N b } . (55)Note that, in (53) and (54), N ≥ is required to obtain theoptimal uDOF.Further, the weight function of the proposed UF-3BL forcoarray indexes , , and , i.e., the weights w (1) , w (2) , and w (3) , can be easily found to be w (1) = 1 , w (2) = 1 , w (3) = 3 N b − . (56)Here w (1) stems from A (1) and w (2) is generated from A (2) , while w (3) is contributed by the three sub-ULAs ofthe base layer ( × ( N b -1)) and 2 gaps ( gap , and gap , ). C. ULA Fitting for SAs Design Using Base Layer Consistedof 4 sub-ULAs (UF-4BL) We further consider the case of using the base layer con-sisted of 4 sub-ULAs to design SA. Additionally, we use 3addition layers each consisted of 2-sensors with interspaces , , and to control w (1) , w (2) , and w (3) . In this case, sub-ULAs are considered with the following sequence A → B → A → B → T → B → A → B , (57)and, the following relationships are satisfied (cid:26) N = 4 N b + N t + 6 ,S t = 4 N b + 7 . (58)The corresponding design problem inherits (44), and onepossible solution can be obtained as gap , = 4 , gap , = 5 , gap , = 6 , gap , = 8 , gap , = 7 , gap , = 3 , gap , = 5 , (59)with corresponding P NER ( x ) , P TR ( x ) , and P FER ( x ) given as P NER ( x ) = { x + x + x + x } (cid:101) + { x + x + x + x } × x N b +5 (cid:101) +( x + x ) × P proto { , N b − } (cid:101) + { x + x + x + x + x + x + x N b +12 + x N b +15 } × P proto { , N b } , (60) P TR ( x ) = x (cid:101) +( x + x + x + x + x ) × x N b (cid:101) +( x + x ) × P proto { , N b } (cid:101) + x N b +13 + · · · + x N t N b +7 N t − (cid:124) (cid:123)(cid:122) (cid:125) consecutive (cid:101) +( x + x ) × x N t N b +7 N t P proto { , N b } (cid:101) +( x + x + x + x + x ) × x N t N b +7 N t +4 N b , (61)and P FER ( x ) = ( x + x + x + x ) × x N b N t +7 N t +4 N b x N b N t +7 N t − N b +8 × P proto { , N b − } (cid:101) +( x + x ) × x N b N t +7 N t × P proto { , N b − } (cid:101) + x N b N t +7 N t +4 N b +22 × P proto { , N b − } (cid:101) +( x + x + x + x ) × x N b N t +7 N t × P proto { , N b } (cid:101) + (cid:2) ( x + x + x + x ) × x N b N t +7 N t +4 N b (cid:101) +( x + x + x + x ) × x N b N t +7 N t +8 N b (cid:3)(cid:101) × P proto { , N b } , (62)leading to cons . { P NER ( x ) (cid:101) + P TR ( x ) (cid:101) + P FER ( x ) } = P proto { , N b N t + 7 N t + 4 N b + 13 } , (63)and, thus, J UF − = 4 N b N t + 7 N t + 4 N b + 12 . (64)Similar to (53), the optimal solution for parameters can befurther obtained as (cid:26) N b = (cid:98) N − (cid:99) , N ≥ ,N t = N − N b − . (65)Following (58), (64) and (65), the final uDOF is summarizedas uDOF = N + 2 N + 5 , N %6 = 0 , N + 2 N + 13 . , N %6 = 1 , , N + 2 N + 11 , N %6 = 2 , , N + 2 N + 12 . , N %6 = 3 , , N + 2 N + 13 , N %6 = 4 . N ≥ . (66)The structure of the proposed UF-4BL can be summarized inclosed-form as sub-ULA1 : { , , } , sub-ULA2 : { , , N b } , sub-ULA3 : { N b + 8 , , } , sub-ULA4 : { N b + 15 , , N b } , sub-ULA5 : { N b + 19 , N b + 7 , N t } sub-ULA6 : { N t N b + 7 N t + 4 N b + 19 , , N b } , sub-ULA7 : { N t N b + 7 N t + 8 N b + 18 , , } , sub-ULA8 : { N t N b + 7 N t + 8 N b + 25 , , N b } , (67)with w (1) = 1 , w (2) = 1 , w (3) = 2 , w (4) = 4 N b − . (68)Note that, for pursuing the optimal uDOF, both UF-3BL andUF-4BL pose a requirement to the total number of sensors N .hen N is less than required, UF-3BL and UF-4BL are stillvalid designs, but no longer satisfy (54) and (66).VII. N UMERICAL E XPERIMENTS In this section, numerical experiments are used to verify thesuperiority of the proposed SA geometries in high mutual cou-pling environment. SA structures considered here are nestedarray, coprime array, SNA, ANA, MISC, and two structuresproposed in Section VI. The performance is evaluated interms of coupling leakage, spatial efficiency, uDOF, DOAsidentifiability, and root-mean-square error (RMSE). The spatialefficiency is defined as Spatial Efficiency = J SA AP SA . (69)The RMSE is given as RMSE = (cid:118)(cid:117)(cid:117)(cid:116) P Q P (cid:88) p =1 Q (cid:88) q =1 (ˆ θ pq − θ q ) , (70)where P is the number of trials and ˆ θ pq represents the esti-mation of θ q in p th trial. For all the SA geometries tested,DOAs are computed based on spatial smoothing MUSICalgorithm [15], [38]. A. uDOF, Spatial Efficiency and Coupling Leakage In the first example, we investigate the uDOF, spatialefficiency, and coupling leakage of the SA structures tested.Fig. 6 (a) shows the uDOFs of all the SA structures tested ver-sus the number of array elements. Among the SA structures,the proposed UF-3BL and UF-4BL possess lower uDOF thanMISC, but higher than the nested array and SNA. Fig. 6 (b)depicts the spatial efficiency of all the SA structures tested.For number of sensors larger than , the spatial efficiency islarger than . Moreover, as the number of sensors increases,the spatial efficiency also increases.The coupling leakage versus number of sensors is presentedin Fig. 6 (c). It can be seen that the proposed UF-3BLand UF-4BL provide significantly reduced mutual couplingin comparison with SNA, MISC, and ANAI-2. The weightfunctions for small coarray indexes of relevant structures arealso summarized in Table II.TABLE II: Weight functions of SA structures tested Geometries w(1) w(2) w(3)Nested N -1 N -2 N -3Coprime 2 2 2SNA ( Q = 2 ) 1 or 2 N − or N − Q ≥ ) 2 N or N − or N + 1 M M M ANAI-2 2 M (cid:98) N (cid:99) − N b − UF-4BL 1 1 2 20 25 30 35 40 45 50 Number of Sensors u DO F UF-3BLUF-4BLSNA (Q = 2)SNA (Q = 3)MISCNested ANAI-1 ANAI-2 (a) uDOF versus number of sensors. 20 25 30 35 40 45 50Number of Sensors0.750.80.850.90.951 S p a ti a l E ff i c i e n c y MISC UF-3BLUF-4BLNestedSNA (Q = 2)SNA (Q = 3)ANAI-1 ANAI-2 (b) Spatial efficiency versus number of sensors. 20 40 60 80 100 120 140 160Number of Sensors0.050.10.150.20.250.30.35 C oup li ng L ea k a g e MISCUF-3BLUF-4BLSNA (Q = 2)SNA (Q = 3)ANAI-1ANAI-2 Nested (c) Coupling leakage versus number of sensors. Fig. 6: Illustration of uDOF, spatial efficiency, and couplingleakage. . Target Identifiability and RMSE Performance In our second example, we analyze the target identifia-bility and the RMSE performance for different conditions.First, the target identifiability in severe mutual coupling en-vironment is shown in Fig. 7. Herein, c = 0 . e jπ/ and c i = c e − j ( i − π/ /i, i = 2 , . . . , , and SNR = 0 dB. Allthe SA structures possess sensors and targets which areuniformly distributed in azimuth in the interval from − ◦ to ◦ . It can be seen from Fig. 7 that the proposed UF-3BL, UF-4BL, and SNA ( Q = 3 ) can identify all the targetssuccessfully, while other SA structures tested have missed andspurious targets. -90 -60 60 90 Target Identifiability CoprimeANAI-2ANAI-1SNA (Q=3)SNA (Q=2)NestedUF-4BLUF-3BLMISC Angel (degree) Fig. 7: Target identification with 35 sensors, | c | =0.5Second, the RMSE performance versus SNR is presented.In this case, we consider targets within interval [ − ◦ , ◦ ] , c = 0 . e jπ/ , c i = c e − j ( i − π/ /i, i = 2 , . . . , , and allSAs are composed of sensors. One can see from Fig. 8 thatthe proposed UF-4BL performs the best when SNR is largerthan -10 dB, while the proposed UF-3BL performs slightlyworse than the proposed UF-4BL in high SNR environment.However, both the proposed UF-4BL and UF-3BL have aconsiderable improvement compared to the other SA structurestested.Third, the RMSE performance versus the number ofsources is studied. In this case, c = 0 . e jπ/ and c i = c e − j ( i − π/ /i, i = 2 , . . . , are considered. The numberof sources varies from 20 to 100 and all the SA structurestested are composed of sensors. As illustrated in Fig. 9,the proposed UF-4BL and UF-3BL both perform well whenthe number of sources is smaller than . However, theperformance of the proposed UF-3BL deteriorates when thenumber of sources is larger than 70. When the number ofsources is larger than 90, MISC provides the best performance.This is because MISC provides a larger uDOF than that of theproposed UF-3BL and UF-4BL.Finally, the RMSE performance versus | c | is presented.In this example, sources are uniformly distributed in the -20 -15 -10 -5 10 15 20 SNR(dB) -4 -3 -2 -1 R M S E 34 Sensors Detect 37 Sources MISCUF-3BLUF-4BLNested SNA (Q = 2)SNA (Q = 3)ANAI-1ANAI-2 Coprime Fig. 8: Illustration of RMSE versus SNR with 34 sensors. 20 30 40 50 60 70 80 90 100 Number of Sources -5 -4 -3 -2 -1 R M S E MISCUF-3BLUF-4BLNestedSNA (Q = 2)SNA (Q = 3)ANAI-1 ANAI-2 Coprime Fig. 9: RMSE versus number of sources. All SA structures arecomposed of 44 sensors.interval [ − ◦ , ◦ ] , and all the SA structures tested arecomposed of sensors. It is apparent from Fig. 10 thatwhen | c | ≤ . , i.e., in low mutual coupling environment,the RMSE is mainly determined by the uDOF. However,when | c | ≥ . , the two proposed SA structures show betterperformance than the other SA structures tested.VIII. C ONCLUSION In this paper, an SA design principle, ULA fitting, isestablished. The ULA fitting enables using pseudo polynomialequations corresponding to arrays to design SAs with closed-form expressions, large uDOF, and low mutual coupling. Twoexamples of designing SA structures based on the proposedULA fitting are given. Numerical examples show the effec-tiveness of these SA structures. Although the general principleof ULA fitting has been introduced, we limited our detailed |c | -4 -3 -2 -1 R M S E MISCUF-3BLUF-4BLNestedSNA (Q = 2)SNA (Q = 3)ANAI-1 ANAI-2 Coprime Fig. 10: Illustration of RMSE versus | c | , SNR = 0 dB. AllSA structures are composed of 44 sensors.study by considering currently only one base layer. Thus, theproposed specific examples of SA structures (UF-3BL and UF-4BL) are limited in terms of uDOF. 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