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Brunel neering, [email protected]) [email protected]; London, (e-mail: London, College Imperial Engineering, 3 Electronic July Estonia, Tallinn, (SAMPTA), Applications and Theory orm esn i hns eaneigover Remaindering Chinese via Sensing Coprime ( uGni ihteDprmn fEetoi n optrEngi Computer and Electronic of Department the with is Gan Lu Conferenc International at part in presented was paper This ogu iadCn igaewt h eateto Electrica of Department the with are Ling Cong and Li Conghui ne Terms Index Abstract rvossuishv netgtdsas raswith arrays sparse investigated have studies Previous N ) nen ra sasto nenspae na in placed antennas of set signals. receive a and/or transmit is to configuration certain array antenna N O hsclsnosfroedmninl(D source (1D) one-dimensional for sensors physical ( N Acpieatnaarycnit ftoo more or two of consists array antenna coprime —A 2 ) O yepotn h ocp ftevirtual the of concept the exploiting by DOF udai ils atI ra Designs Array I: Part Fields, Quadratic Arypoesn,CieeRmidrTheo- Remainder Chinese processing, —Array ogu i uGan, Lu Li, Conghui .I I. 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[9], vectors integer are or ( = | Z r eeao arcso h uary;and subarrays; the of matrices generator are and D e ( det G Z G n over and = 2 2 G N 2 npeiu tde [4]–[12]. studies previous in P G { 2 d = x 1 where G : 1 I ebr IEEE Member, d where , 2 ersn eevrsno locations; sensor receiver represent Z ) = | 2 ) G epciey een erelate we Herein, respectively. P with 1 x sa2b- nee matrix. integer 2-by-2 a is ig fagbacintegers algebraic of rings I 2 − | steiett matrix), identity the is det( G 2 G x M 1 1 } ) , | and + | det( N G such , 1 G ideal and G G 2 al ) T o d 1 1 a s 1 | 2 lattices arising from the prime decomposition. However, the of coprime arrays allocated on coprime lattices and derives conditions pertaining to the coprimality of algebraic integers closed-form expressions of CRT array geometries, which and ideal lattices are non-trivial. are inherent in any rings of algebraic integers. Section This paper shows the connection between coprime al- V extends CRT to hole-free symmetric CRT whereby the gebraic integers and their corresponding lattices that are parameter identifiability is enhanced, after which examples 2 obtained by embeddings and represented by integer matrices. are provided for Z and A2 as special cases of quadratic In general, the coprimality of integer matrices is defined integers. Section VI concludes the paper. in matrix rings [15], [16]. The class of integer matrices Notations: Bold font lowercase letters (e.g., x1), bold font obtained from algebraic integers in this work may be seen uppercase letters (e.g., G), fraktur font letters (e.g., p1) and as special matrix rings. Principal advantages of relating calligraphy font alphabets (e.g., ) denote vectors, matrices, algebraic integers with matrices include commutativity, sim- principal ideals and sets respectively.D Z and Q denote plified expressions and the potential to exploit the nice rational integers 1, 0, 1 and rational numbers a {··· − ···} F properties of algebraic integers (e.g., the convenience to b a,b Z, b = 0 respectively. R denotes the F - check coprimality). For instance, the coprimality issues of dimensional{ | ∈ Euclidean6 space.} Re(m) and Im(m) represent some classes of matrices such as adjugate pairs and skew the real and imaginary parts of a complex number m circulant pairs [17], [18] can be addressed as special cases of respectively. N(m)= mmˆ denotes the norm of m where mˆ algebraic conjugate integers. Examples of algebraic integers is the algebraic conjugate of m (Section II-A). For example in quadratic fields including ring of Gaussian integers and in the ring of Gaussian integers, with m =3+2i, it can be ring of Eisenstein integers are studied in this paper. readily verified that N(m)= mmˆ = 13. An important advantage of Eisenstein integers is that the difference coarray becomes a subset of the hexagonal lattice II. REVIEW OF QUADRATIC FIELDS AND ALGEBRAIC A2. It is well known that A2 is the optimum lattice for sphere LATTICES packing in two-dimensional space [19]. Numerical analysis Let us first briefly review some definitions and preliminary reveals that the optimum packing density results in a 15.5% results related to quadratic field along with its ring of gain in DOF for a fixed physical area of the array. Due integers; and based on algebraic integers, the construction of to this reason, the hexagonal geometry is currently used in algebraic lattices, on which sensor arrays can be allocated. the design of some phased-array antennas [20], [21]. This [7], [14] paper together with its accompanying paper puts forward the application of hexagonal lattices and hence provides A. Quadratic Field and Its Ring of Integers the potential to decrease the physical array aperture without A quadratic field K is a field extension of degree 2 over sacrificing DOF. rational numbers Q, i.e., it is a Q-vector space of dimension The main contribution of this paper along with its ac- two. Note that Q K. For instance, √ 1 is not an element companying paper is that they further develop the design in Q but it is an element⊆ in the field extension− of Q. In order methods of 2D coprime arrays, which brings a new class of to be a field, this new field extended from Q must contain Q 2D array configurations, namely CRT arrays. Such arrays and all the powers and multiples of √ 1. In other words, can provide enhanced DOF, sparse geometry, and hole-free Q is extended into a new vector space− over Q, which is coarrays. By lattice representations, the configurations of generated by the powers of √ 1. Let i , √ 1 and Q(i) the proposed arrays along with their virtual coarrays enjoy denote this field extension. Every− element m −Q(i) can be simple closed-form expressions. This paper addresses the expressed as m = m + m i, m ,m Q, i.e.,∈ 1,i is the issues relating to geometry and the generation of CRT arrays 1 2 1 2 basis of Q(i). In this case, an algebraic∈ integer{ takes} the including the mapping between number fields and lattices, form of m + m i where m ,m , Z. The ring of integers coprimality issues pertaining quadratic integers and embed- 1 2 1 2 of Q(i) is the set of all algebraic∈ integers in Q(i), which ded matrices, and lattice representations of CRT arrays along can be represented by Z[i]= m + m i,m ,m , Z . with their coarrays, whereas the accompanying part II em- 1 2 1 2 In general, a quadratic field{ is denoted by K = Q∈(√}D), ploys CRT arrays to propose practical algorithms for angle where D is a square-free rational integer. Note that if D is a estimations in both active and passive sensing scenarios. perfect square, K = Q. The ring of integers is often denoted Other potential applications of Chinese remaindering over as , which is a set that contains all algebraic integers in quadratic fields include sparse 2D discrete fourier transform K K.O In quadratic fields, algebraic integers are also known as [22], the radar measurements on multiple targets [23], [24], quadratic integers which are roots of quadratic polynomials filter banks [9], imaging [25], [26] and direction finding with coefficients in Z. The minimal polynomial denoted as problems using compressive sensing [27]. f(X) of can be expressed as: The rest of the paper is organized as follows. Before K O 2 constructing coprime lattices from prime ideals in Section X D, if D 1 (mod 4) ; f(X)= 2 − 1 D 6≡ (1) X X + − , if D 1 (mod 4), III, the concepts of quadratic fields and their rings of integers  − 4 ≡ are briefly reviewed in Section II along with algebraic or alternatively, lattices. Based on CRT, Section IV proposes a new class f(X)= X2 + BX + C, (2) 3 where B = 0 and C = D when D 1 (mod 4), and The Voronoi cell of Λ is defined by 1 D − 6≡ F B = 1 and C = −4 when D 1 (mod 4). The proof (Λ) = y R : y y λ , λ Λ , (5) of f(X−) can be found in [14]. Let≡q and qˆ denote the two V { ∈ k k≤k − k ∀ ∈ } where ties are broken in a systematic manner. roots of f(X) respectively. With the notations above, it can be easily calculated that There are various ways to construct lattices, for instance, 1 1 from codes and groups. In this paper, lattices and their ideals q = B + B2 4C, and (3) −2 2 − are obtained from rings of quadratic integers Z[q] via the 1 p1 canonical embedding. qˆ = B B2 4C (4) −2 − 2 − In general, the canonical embedding builds a bridge Here 1, q and 1, qˆ are calledp the integral bases of between lattices and rings of algebraic integers as it es- Q(√D{) [14],} i.e., every{ } element in Q(√D) can be written as tablishes a bijective mapping between the elements in an m1 +m2q corresponding to the former basis or as m1 +m2qˆ algebraic number field of degree F and the vectors of the F - corresponding to the latter with m1,m2 Q. Accordingly, dimensional Euclidean space. In other words, the canonical every element in its ring of integers can∈ be formed by embedding σ sends an algebraic integer m to a lattice point m = m + m q or mˆ = m + m qˆ with m ,m Z. Here m = σ(m) in Euclidean space where m is an F -by-1 vector. 1 2 1 2 1 2 ∈ m1 +m2q and m1 +m2qˆ are called algebraic conjugates of The canonical embedding of any algebraic number field of each other, which can be viewed as a generalization of the degree F is given in [28, Definition 5.15]. complex conjugation. From (3) and (4), it can be verified Herein we consider quadratic fields where F = 2. Then 2 that with B 4C < 0 (D < 0), the two conjugations are the embeddings of Q(√D) are simply given by identical to each− other. σ (√D)= √D, and σ (√D)= √D. With the knowledge of the integral basis, the ring of 1 2 − The canonical embedding σ is a geometrical representation integers K can be denoted as Z[q], and m = m1 + m2q is O of Q(√D) that maps m Q(√D) to a vector of 2D called a quadratic integer of Z[q] for any m1 and m2 in Z, Euclidean space, i.e., σ(m∈) = (σ (m), σ (m))T R2. which generalizes rational integers in Z to quadratic fields. 1 2 ∈ Note that q =q ˆ since D is square-free, whereas Z[q] and For example, the embeddings of an arbitrary element m = 6 Z[ˆq] represent the same ring of integers as qˆ = B q. m1 + √2m2 Q(√2) are given by σ1(m)= m1 + √2m2 − − and σ (m) =∈m √2m . Therefore, an algebraic lattice Henceforth, we will use Z[q] as the notation of the ring of 2 1 − 2 integers of Q(√D). can be constructed by embeddings as follows: When D = 1, for example, f(X) = X2 +1 whose Given 1, q as an integral basis of Q(√D), a 2D al- − { } roots are q = i and qˆ = i. The ring of integers of gebraic lattice Λ = Λ( K ) = σ( K ) is a lattice whose Q(i) denoted by = Z[i−] is also known as the ring generator matrix is explicitlyO givenO by OK of Gaussian integers. In this case, 1,i is an integral 1 σ (q) { } G = 1 for D> 0, (6) basis of Z[i], since 1 3 (mod 4). Therefore, every 1 σ2(q) element in Z[i] can be− uniquely≡ expressed as m + m i   1 2 whereas if D < 0, Q(√D) is also known as an imag- with m ,m Z. The algebraic conjugation of m is 1 2 inary quadratic field where the canonical embedding is mˆ = m + m ∈qˆ = m m i which is also the complex 1 2 1 2 further simplified and can be formulated by σ(m) = conjugation of m. Another− example that will be used in (Re(m), Im(m))T . Hence the corresponding generator ma- this paper for illustrative purposes is the ring of Eisenstein trix is computed by stacking the real and imaginary parts of integers with D = 3. In this case, = Z[ω] with 1,ω − OK { } 1 and q: as its integral basis where ω = eiπ/3 = 1 +i √3 since D 1 2 2 1 Re(q) (mod 4). An arbitrary element in Z[ω] can be expressed≡ as G = for D< 0. (7) 0 Im(q) n = n1 + n2ω with nˆ = n1 + n2ωˆ being its algebraic   Note that G is a non-singular matrix whose absolute deter- conjugation where ωˆ = 1 i √3 . 2 − 2 minant equals to the fundamental volume of its correspond- ing lattice, i.e., V (Λ) = det(G) [28, Theorem 5.8]. B. Construction of Algebraic Lattices | | For example, in the case of D = 3 (Eisenstein integers), Definition 1: Given F linearly independent column vec- since the integral basis is 1,ω , the− generator matrix of the tors g , g ,..., g RF , an F -dimensional lattice Λ is { } 1 2 F ∈ corresponding algebraic lattice is given by defined as the set of integer combinations of the basis 1 1 Re(ω) 1 vectors, i.e., G = = 2 . (8) E 0 Im(ω) 0 √3 F    2  G Λ= xkgk : xk Z . The lattice constructed from E is shown in Fig. 1, which ( ∈ ) is also known as the hexagonal lattice A2 with the densest kX=1 Accordingly, the generator matrix of the lattice Λ is obtained sphere packing in dimension two. The fundamental volume by of A2 is V (A2)= √3/2 with a minimum distance 1. G = [g g g ]. Analogously, the ring of Gaussian integers gives rise to 1| 2| · · · | F 4

and the issue of coprimality pertaining algebraic conjugates will be addressed. Examples are provided in Gaussian and Eisenstein integers, which will be exploited to design CRT arrays in the following sections.

Y A. Prime Elements in Quadratic Fields To distinguish from prime numbers in Z (e.g., 2, 3, 5, 7 ), a non-zero element m in Z[q] is a ±prime± element± ± if··· and only if it is not a unit of Z[q] and whenever m divides a product in Z[q], it also divides one of X the factors. Herein, the unit is defined as a quadratic integer u Z[q] with N(u)= 1. In the case of Gaussian integers, there∈ are four units: ±1, i, and in Z[ω], the six units are Figure 1. An illustration of A2 lattice. 1, ( 1 √3)/2. ± ± ± For± example,± 7 is a prime number in Z but not a prime element in Z[ω] since it is reducible, i.e., 7=(1+2ω)(3 2ω). Analogous to the norm of a complex number, with− the notations above, the norm of a quadratic integer m = m1 + m2q in Z[q] is defined as the product of m and its algebraic conjugate mˆ , i.e., N(m) = mmˆ . As q and qˆ are Y roots of Equation (2), q +ˆq = B and qqˆ = C. Thus N(m) can be derived as follows: − N(m)= mmˆ = (m1 + m2q)(m1 + m2qˆ) 2 2 2 2 = m1 + m1m2(q +ˆq)+ m2qqˆ = m1 Bm1m2 + Cm2. − (10) X Since m1,m2,B,C Z, the norm of a quadratic integer is always in Z. In general,∈ for all m Z[q], it can be verified Figure 2. An illustration of Z2 lattice. from [30, Theorem 1.8] that if N(m∈) is a prime number in Z, then m is a prime element. In the cases of Gaussian and the integer lattice Z2 whose generator matrix is Eisenstein primes, the sufficient and necessary conditions of prime elements can be derived [31]. 1 Re(i) 1 0 G = = . (9) A Gaussian prime is a prime element in the ring of G 0 Im(i) 0 1 Gaussian integers of the form that satisfies one   2  m1 + m2i Fig. 2 depicts the configuration of Z . of the following: In general, given a number field of degree F , its ring of Both m and m are nonzero and N(m) = m2 + m2 integers can always construct the algebraic lattice that is • 1 2 1 2 K is a prime number. expressedO by means of the generator matrix. This construc- m = 0 and m = 0 (or m = 0 and m = 0), m tion provides a general and straightforward way of finding • 1 2 1 2 2 is a prime number6 and m 6 3 (mod 4) (or m is a pairwise coprime matrices from pairwise coprime algebraic 2 1 prime number and m | 3|(mod ≡ 4)). integers, which significantly simplifies the method of matrix | 1|≡ factorization from Smith form [9], [29] and extends integer In Z[ω], an Eisenstein integer of the form n = n1 + n2ω is matrices in Smith form to any matrices that correspond to a Eisenstein prime if either: n + n ω equals to the product of a unit and a prime coprime elements in K . • 1 2 O number of the form 3a 1, a Z, or N(n)= n2 + n n + n2−is a prime∈ number. III. PRIME IDEALS IN QUADRATIC FIELDS AND • 1 1 2 2 2 2 CONSTRUCTION OF IDEAL LATTICES For example, 2+i is a Gaussian prime because 1 +2 =5 is a prime number. Likewise, 2+√3i =1+2ω is a Eisenstein In the previous section, the construction of 2D algebraic prime since 12 +2+22 =7 is a prime number. lattices from quadratic fields was briefly reviewed. Similar to 1-D arrays [7] where two coprime rational integers in Z were applied to determine sensor positions, in 2D array design, B. Prime Factorization into Ideals the quadratic integers in Z[q] shall be coprime as well, An ideal in a quadratic field is a subset of Z[q] such to which Chinese Remainder Theorem applies. Therefore, that wheneverI x and m Z[q], mx belongs to . If the this section studies prime quadratic integers along with ideal is generated∈ by I a single∈ element in Z[q], thisI ideal is its corresponding prime ideals, from which the algebraic called a principal ideal. For instance, 5Z of Z is a principal lattices will be constructed. The computation of prime ideals ideal whose elements are 5, 10 15 . For simplicity, ± ± ± ··· 5 we can write 5Z = 5 . Similar to the prime elements in The canonical embedding of a principal ideal maps the Z[q] mentioned in Sectionh i III-A, a prime ideal is an ideal elements in the ideal to lattice points, which are similar to such that mn implies m or n . In a principal that of K defined in (6) and (7) for D > 0 and D < 0 ideal domain ∈(PID) I where every∈ I ideal∈ is I principal, prime respectively.O Therefore, an ideal lattice denoted as σ(p) that ideals are simply generated by prime elements. All quadratic is generated by a principal ideal p = m has a generator fields with class number one are PIDs. It has been proved matrix given by h i that the PIDs in imaginary quadratic fields are Q √ , ( D) Gm = GBm, where (13) for D = 1, 2, 3, 7, 11, 19, 43, 67, 163, while − − − − − − − − − m1 Cm2 the full list of PIDs in real quadratic fields is not known Bm = − . (14) m2 m1 Bm2 yet. Examples include D =2, 3, 5, 6, 7, 11, 13, 14, 17, 19  −  ··· Here G is the generator matrix of that expressed in [32]. For simplicity, we only consider PIDs henceforth. For OK example, 2+ √3i is a prime ideal of the ring of integers (6) or (7) for real or imaginary quadratic field respectively. Q √ h i Z Q √ Bm is called the matrix representation of m Z[q] and of ( 3), namely [ω], since ( 3) is a PID and ∈ √ − − det(Bm) = N(m) [28, Theorem 5.11]. Note that Bm 2+ 3i is a prime element (Section III-A). The elements | | in 2+ √3i are in the set 2+ √3i = 2+ √3i Z[ω]= is always an integer matrix by definition, i.e., all entries h i h i h i (2 + √3i)m : m Z[ω] . in Bm are rational integers. The following two lemmas { ∈ } Similar to the fundamental theorem of arithmetic to ra- discuss the properties of Bm from the perspectives of tional integers [16, Theorem 5.3], every non-zero element eigenvectors/eigenvalues and commutativity respectively. in a unique factorization domain (UFD) can be written as a Lemma 1: (1, q) and (1, qˆ) are left row eigenvectors of product of prime elements. More generally, a nonzero ideal Bm with eigenvalues m and mˆ respectively. Proof: The row vector (1, q) is a left eigenvector of Bm p of K where K = Q(√D) can be uniquely factored has i O if (1, q)Bm = m(1, q) [33, Definition 4.2]. Substituting αk (14) to the left hand side of the equation results in p = pk (11) h i m1 Cm2 kY=1 (1, q) − m2 m1 Bm2 where pk’s are distinct prime ideals. Particularly, if p is a  −  rational prime greater than 2, then [14] = m1 + m2q, m1q + m2( Bq C) − − D As q is the root of f(X)=0 where f(X) is given in (2), it p1p2, if =1;
p satisfies q2 + Bq + C =0. Then the second element in the p = p K =  p, if  D  = 1; (12) 2 h i O  p row vector becomes m1q + m2q = (m1 + m2q)q = mq.  2 − 2  p , if p D. Likewise, substituting qˆ = Bqˆ C and mˆ = m1 + m2qˆ | to (1, qˆ)B yields (1, qˆ)B− =m ˆ−(1, qˆ).  where p and p are distinct prime ideals, and a is the m m 1 2  p Lemma 2: Any two matrix representations of quadratic Legendre symbol defined by   integers are commutative. a 1 if for some x Z: a x2 (mod p), = ∈ ≡ Proof: By the eigendecomposition discussed in Lemma 1, p 1 otherwise.    − any two matrix representations Bm and Bn of two quadratic In the first case where p splits, p1 and p2 are distinct integers m and n respectively can be factorized as prime ideals of norm , thus they are coprime by nature. 1 1 p Bm = Q− PmQ and Bn = Q− PnQ (15) Particularly in PIDs, the primality of these two ideals can where be tested using the criteria given in Section III-A. Based 1 q m 0 n 0 on p and p , two coprime ideal lattices can be constructed Q = , Pm = and Pn = (16) 1 2 1q ˆ 0m ˆ 0n ˆ whereby two subarrays can be allocated on respectively.       Therefore, by Lemma 1 we can write 1 1 1 BmBn = Q− PmQQ− PnQ = Q− PmPnQ 1 1 1 C. Coprime Ideal lattices and their matrix representations = Q− PnPmQ = (Q− PnQ)(Q− PmQ)= BnBm.  In general, an ideal lattice Λ = σ( ) is constructed by 1 In other words, the commutativity of algebraic integers canonical embedding from an ideal I , which is a K implies the commutativity of the corresponding matrices. sublattice of the algebraic lattice Λ constructedI ⊆ O from . K Next, we will relate the coprimality of quadratic integers For example, 7A is an ideal lattice constructed fromO the 2 with the coprimality of their corresponding matrix represen- ideal 7 Z[ω], thus it is a sublattice of A constructed 2 tations and provide an alternative way of generating coprime from Zh [ωi]. ⊂ lattices from algebraic conjugate pairs. In Z[q], an integral basis of the principal ideal m h i generated by the quadratic integer m = m1 + m2q can be calculated as D. Connection with coprime algebraic integers m 1, q = m + m q, (m + m q)q In general, the left coprimality of integers matrices is { } { 1 2 1 2 } = m + m q, Cm + q(m Bm ) . defined by Bezout’s identity [16]–[18]: { 1 2 − 2 1 − 2 } 6

Definition 2: Two integer matrices Bm and Bn are left and column j of Bm. Therefore, in the dimension of two, coprime if and only if there exist integer matrices C and D the adjugate of Bm is such that m1 Bm2 Cm2 Bmˆ = − . (18) BmC + BnD = I. (17) m m  − 2 1  Theorem 2: Using the notations above, two adjugate 2- Likewise, Bm and Bn are right coprime if and only if there by-2 matrices B and B are coprime if and only if exist C′ and D′ such that C′Bm + D′Bn = I. m mˆ Theorem 1: In PIDs, two algebraic integers are coprime (a) GCD(m1,m2)=1 and GCD(2m1 +m2, 4C 1)=1, if and only if their corresponding matrix representations for B = 1, − obtained from canonical embeddings are left coprime. − (b) GCD(m ,m )=1 and GCD(m , C)=1, for B =0 Proof: See Appendix A  1 2 1 and C is even, Corollary 1: Two integers matrices generated from em- (c) GCD(m + m ,m m )=1 and GCD(m , C)=1 beddings are right coprime if and only if they are left 1 2 1 2 1 for B =0 and C is− odd. coprime. Proof: See Appendix B  Proof : See Appendix C  From Theorem 1 and Proposition 1, the coprimality of It is worth to notice that according to Theorem 1, Theorem two quadratic integers indicates the right and left coprimality 2 also provides the coprime conditions of two algebraic of their corresponding matrix representations obtained from conjugates m = m1 + m2q and mˆ = m1 + m2qˆ where embeddings and vice versa. Henceforth, we say two lattices q and qˆ are given in (3) and (4) respectively. For illustration are coprime lattices if their matrix representations are (left purposes, two examples of algebraic conjugate integers are and right) coprime. Exploiting this theorem, we will provide given, providing two classes of coprime matrices. conditions on the coprimality of adjugate matrix pairs next. Corollary 2: A Gaussian integer m and its conjugate are relatively prime if and only if GCD(m +m ,m m )=1. 1 2 1− 2 E. Adjugate Matrix Pairs Proof : The minimum polynomial of the ring of Gaussian integers is X2 +1=0 with the basis 1,i . By Theorem 2, Since the notion of greatest common divisor (GCD) can { } be generalized to an arbitrary commutative ring, it can be B =0 and C =1 match the assumptions of case (c), thus the coprimality condition becomes GCD(m +m ,m m )= defined in the rings of integers of quadratic fields as well 1 2 1 − 2 [34], [35]. Herein, the concept of GCD is generalized to 1 and GCD(m1, 1) = 1. Note that GCD(m1, 1) = 1 holds for all m Z.  quadratic integers in PIDs, i.e., if d = GCD(m,n) is a GCD 1 ∈ of m and n, then all the common divisors of m and n divide Corollary 3: An Eisenstein integer and its conjugate are d [34, Definition 6.1.3]. Similarly, the Bezout’s identity can relatively prime if and only if GCD(m1,m2) = 1 and also be generalized as if two integers m,n Z[q] are not GCD(m1 m2, 3)=1. − both equal to 0 then there exist α, β Z[q] such∈ that Proof : Likewise, in the case of Z[ω], the coefficients ∈ GCD(m,n)= αm + βn. become C = 1 and B = 1, which can be addressed to case (a) in Theorem 2.− The coprime conditions are Given GCD(m,n) = u where u is the unit in Z[q] and GCD(m ,m ) = 1 and GCD(2m + m , 3) = 1. By N(u) = 1, m and n are defined as coprime quadratic 1 2 1 2 Fact (1), GCD(m ,m )=1 is equivalent to GCD(2m + integers [34, Definition 6.1.4]. In other words, two quadratic 1 2 1 m ,m ) = 1, which can be combined with the second integers m and n are coprime if and only if there exist α 2 1 ′ condition by Fact (3), i.e., GCD(2m + m , 3m )= 1. and β such that mα + nβ = u. Because u is a unit, u 1 1 2 1 ′ ′ ′ − Applying Fact (1) again results in GCD(3m (2m + always exists. Then Bezout’s identity becomes mα+nβ =1 1 1 m ), 3m ) = GCD(m m , 3) = 1, which holds− if and where α = α u 1 and β = β u 1. Recall that the following 2 1 1 2 ′ − ′ − only if both GCD(m ,m−)=1 and GCD(m m , 3)=1 facts of GCD hold for a,b,c Z[q]: 1 2 1 − 2 ∈ hold.  1) GCD GCD , Z ; (a,b)= (a + αb,b) α [q] Remark: By Theorem 2, the class of coprime matrix pairs 2) GCD if and only∀ if GCD∈ α β (a,b) = 1 (a ,b ) = 1, is enriched to any matrices obtained from quadratic integers. Z+. α, β By exploiting the bijective mappings between algebraic 3)∀ GCD ∈ if and only if GCD and (a,bc)=1 (a,b)=1 integers and integer matrices, [17, Theorem 2] can be viewed GCD . (a,c)=1 as the coprimality of two algebraic conjugates and proved These facts can be proved straightforward and will be by the generalized GCD, i.e., employed in the following proofs of coprimality. GCD(m1 + m2q, m1 + m2qˆ) As mentioned in Section II-A, the algebraic conjugate of = GCD (m1 + m2q)(m1 + m2qˆ), 2m1 + m2(q +ˆq) m denoted by mˆ is also in Z[q] and can be written as m1 + m qˆ. The matrix generated by mˆ is the same as the adjugate = GCD(N(m), 2m1 Bm2). 2 −
of Bm, which is the transpose of the cofactor matrix of According to the coprimality relation between quadratic k+j Bm, i.e., adj(Bm)kj = ( 1) Mjk where Mjk is the integers and matrices stated in Theorem 1, some useful determinant of the matrix that− results from deleting row k classes of coprime matrices such as skew-circulant adjugates 7

derived in [18] can be viewed as a case of Corollary 2 with This implies that for all ak R/ and bj R/ there canonical embedding, i.e., the following two integer matrices exists z R/ such that ∈ I ∈ J ∈ IJ m1 m2 m1 m2 z ak (mod ) and Bm = − and Bmˆ = ≡ I (21) m2 m1 m2 m1 z b (mod ),   −  ≡ j J are coprime if and only if GCD(m1 + m2, m1 m2)=1. which can also be proved as follows: from coprimality that Similarly, an alternative way of describing Corollary− 3 would + = R, there exist xk and yj such that be as follows: xI +Jy = 1. For all a R/∈ Iand b R/∈ J, it can be k j k ∈ I j ∈ J Two integer matrices readily verified that every pair (ak,bj) forms the solution m m B = 1 2 and z xkbj + yj ak (mod ). (22) m m m−+ m ≡ IJ  2 1 2 We may check that m1 + m2 m2 Bmˆ = z yjak xkak + yj ak = (xk + yj )ak ak (mod ) m2 m1 ≡ ≡ ≡ I  −  z xkbj xkbj + yjbj = (xk + yj)bj bj (mod ). are coprime if and only if GCD(m1, m2) = 1 and ≡ ≡ ≡ J The pair (xk,yj ) serves as a “CRT basis” which can be GCD(m1 m2, 3)=1, i.e., m1 and m2 are relatively prime with their− difference being not divisible by 3. chosen as the basis of prime ideals in Z[q]. With this basis, the mapping from R/ R/ to R/ is bijective, i.e., all I⊗ J IJ solutions of z are identical given the different pairs (ak,bj ), which leads to the definition of CRT arrays and its cross- IV. DESIGN OF CRT-BASED SPARSE ARRAYS difference and sum coarrays: Definition 5 (CRT arrays): Given two coprime ideals We have proved that the coprimality of quadratic integers and in ring R, a CRT-based array is defined as: I in PIDs is a necessary and sufficient condition of the J = σ( )/σ( ) σ( )/σ( ), (23) coprimality of the corresponding matrices obtained from Z I IJ ∪ J IJ canonical embeddings. In this section, we will briefly review where σ( ) denotes the canonical embedding of and same with . I I the generalized CRT based on [14, Appendix 1], and then J propose a design method for coprime arrays based on Definition 6 (Cross-difference coarrays of CRT arrays): The cross-difference coarray generated by an CRT array CRT over rings of quadratic integers where the sensors are D deployed by the use of coprime lattices generated by coprime is given by: ideals. For illustrative purposes, CRT is employed over the = z1 z2 z1 σ( )/σ( ), z2 σ( )/σ( ) . D { − | ∈ I IJ ∈ J IJ } ring of Gaussian integers and the ring of Eisenstein integers Definition 7 (Sum coarray of CRT arrays): The sum respectively as examples. coarray generated by an CRT array can be expressed as: S To begin with, we extend the modulo operation to ideals, = z + z z σ( )/σ( ), z σ( )/σ( ) . S { 1 2 | 1 ∈ I IJ 2 ∈ J IJ } and define the sum and the product of ideals as follows: Note that because of the symmetry of the ideal lattices, is Definition 3: Let be an ideal of a ring R. Given x, y identical to . From the point of view that regards latticesD R, x is congruent toIy modulo , i.e., x y mod if and∈ S I ≡ I as sets of points, σ( )/σ( ) corresponds to /( ) in only if number fields. I IJ I IJ x y . (19) − ∈ I According to [14], the ring isomorphism (20) holds over any commutative ring, therefore over all PIDs where the Definition 4: Let and be two ideals of the ring R. ideals can be obtained from the prime decomposition (12). The sum of and I is theJ ideal I J Given = p1 = m and = p2 = n where + = x + y, x ,y , m,n IZ[q], the canonicalh i embeddingJ σ(p ) of ph isi given in I J { ∈ I ∈J} ∈ 1 1 and their product is the ideal (13) and similar with σ(p2). The product of these two ideals

= xkyj , xk ,yj . forms a principal ideal as well, i.e., = mn = p . IJ { ∈ I ∈J} With the notations above, expressionsIJ for theh numberi h ofi The quotient ringX is defined as the set that contains all sensors and the achievable DOF can be derived as follows: the cosets of the ideal , i.e., R/ = r + , r R . Let Proposition 1: If and that are used to allocate sensors ideals and be relativelyI primeI in{ a commutativeI ∈ } ring I J I J are decomposed from p , the total number of physical R, then + = R. For example, given = 3 and sensors is 2p 1 and itsh maximumi DOF is p2. I J Z I h i = 5 as two coprime ideals of , + = 3 + Proof : By− assumption = p = pR, the number of J5 =h 3i 2+5 ( 1) = 1 = Z, I =J 15h ,i and IJ h i h i h · · − i h i IJ h i elements in /pR is p which is the same as /pR [28, R/ = Z/3Z = 0, 1, 2 which is an equivalence class Definition 3.12].I Since the only identical elementJ that they withI [x] = [y] if and{ only} if x y 3 . − ∈ h i share is 0, the total number of nonidentical elements in The Chinese Remaindering Theorem [14] asserts that ( /pR) ( /pR) is 2p 1. there is a ring isomorphism IDefine∪ theJ maximum DOF− as the maximum number of R/ R/ R/ . (20) degrees of freedom that the array can achieve, i.e., the total IJ ≃ I× J 8

number of identical elements in the coarray. According to 3 the ring isomorphism of the generalized CRT given in (20), all the difference/sum vectors generated by coprime lattices 2 are nonidentical, thus the total number of elements in ( + I )/pR can be written as: 1 J /pR /pR = p2, (24) |I | · |J | where is the cardinality of a set. Note that as canonical 0 embedding| · | σ( ) is bijective, the number of lattice points -1 in σ( )/σ( · ) is the same as the number of elements in / J IJ  -2 J TheIJ rest of this section will demonstrate the proposed design method by Chinese Remaindering over PIDs in -3 quadratic fields, namely over Z[i] and Z[ω]. -4 -2 0 2 4 (a) A. Chinese Remaindering over Z[i] In the ring of Gaussian integers, we look for the p such 6 that D = 1 is a quadratic residue: − 4 x2 1 (mod p) ≡− for some x Z. The first few solutions are 22 1 2 (mod 5), 52 ∈ 1 (mod 13), and so forth. By performing≡ − 0 the prime decomposition≡− (12), these rational primes can be decomposed into prime ideals as 5 = 2+ i 2 i , -2 and 13 = 3+2i 3 2i .h Herei allh theih quadratic− i integersh arei Gaussianh primesih − as statedi in the criteria given -4 in Section III-A. Alternatively, it can be checked that all -6 pairs are relatively prime according to Corollary 2. Let us take the example of p = 5 to demonstrate the design -5 0 5 procedure. p =5 yields two prime ideals p = 2+ i and (b) 1 h i p2 = 2 i , whose corresponding matrices are coprime as 2 well byh − Theoremi 1. As 1,i is the integral basis of Z[i], Figure 3. Z arrays in the Voronoi cells (black polygons) constructed from decomposition over Gaussian integers of p = 5 (a) and p = 13 (b) with the an integral basis of 2+{i can} be calculated as h i first subarray (G(2+i)g1) in red stars and the second subarray (G(2−i)g2) (2 + i) 1,i = 2+ i, 1+2i . in blue dots. { } { − } Since the minimum polynomial over Z[i] is X2 +1 (B =0 and C =1), by canonical embedding given as (13) and (14), Using the matrix representation, the cross-difference coar- the generator matrix of 2+ i is h i ray consisting of vectors can be defined by 2 1 G(2+i) = − . (25) G = dg : dg = G(2 i)g1 G(2+i)g2 , 1 2 D { − − }   where g Z2/σ 2+i and g Z2/σ 2 i . G Notice that because G = I, the generator matrix of 2+i is 1 2 (2+i) and G ∈ are givenh byi (25) and (26)∈ respectivelyh − i and they identical to its matrix representation, i.e., G =h B i . (2 i) (2+i) (2+i) are coprime− because 2+
i and 2 i are coprime
integers Analogously, an integral basis of 2 i is given by − h − i (Theorem 1). According to the ring isomorphism of the (2 i) 1,i = 2 i, 1+2i , − { } { − } generalized CRT, with R = Z, = 2+i and = 2 i , whose generator matrix is it can be readily calculated thatI h = i5 =J 5Z andh − thusi 2 2 IJ 2h i 2 1 dg Z /5Z , yielding an array of 5 = 25 degrees of G(2 i) = . (26) ∈ − 1 2 freedom according to Proposition 1. The locations of the  −  elements of the first subarray are given by The determinant of G(2+i) is equivalent to the norm of 2+ h 2 i and same with G(2 i) and 2 i [28]. By the definition G(2 i)g1 σ 2 i /5Z , ofi the norm (10), it can− be provedh − i straightforward that if − ∈ h − i while those of the second subarray are
given by p = 2 + i 2 i , N( 2 + i )N( 2 i ) = N(p) 2 h i h ih − i h i h − i G(2+i)g2 σ 2+ i /5Z . and since p is a prime number, N(p) = p2 has and only ∈ h i 2 Therefore, = σ 2 i /5Z2 σ 2+ i /5Z2 and has three divisors, namely 1, p, and p . According to the Z h − i ∪
h i decomposition shown in (12), both 2+ i and 2 i are only 9 elements are actually used in the sparse array by h i h − i Proposition 1.

not units. This implies det(G(2+i)) = det(G(2 i)) = N( 2+ i )= N( 2 i |)= p =5. | | − | Another example that comprises more sensors is p = 13, h i h − i 9

4 ideal decomposition (12), these rational primes can be de- composed into prime ideals as: 7 = 2+√3i 2 √3i and 13 = 1+2√3i 1 2h√3ii .h With p =ih 7,− the twoi h i h ih − i 2 prime ideals decomposed from 7 are p1 = 2+ √3i h i h i and p2 = 2 √3i , where 2 + √3i and 2 √3i are prime elementsh − byi the criteria given in Section− III-A. 0 Because p1 and p2 are algebraic conjugate of each other (2+ √3i = 1+2ω and 2 √3i = 1+2ˆω), it can also be checked that these two conjugate− Eisenstein integers are -2 coprime according to Corollary 3 with m1 =1 and m2 =2. Similar to Gaussian integers, 2 + √3i in Z[ω] has an integral basis represented by h i -4 1+3√3i -5 0 5 (2 + √3i) 1,ω = 2+ √3i, − { } 2 (a)   whose corresponding generator matrix is 2 1 6 G = − 2 (2+√3i) √3 3√3  2  4 1 1 1 2 = 2 − , 0 √3 2 3 2  2    B i 0 (2+√3 ) and the integral basis of 2 √3i is| given{z by } h − i -2 5+ √3i (2 √3i) 1,ω = 2 √3i, − { } − 2 -4   with the generator matrix: -6 1 1 3 2 G = 2 . -5 0 5 (2 √3i) 0 √3 2 1 −  2   −  (b) B (2 √3i) − A Figure 4. 2 arrays in the Voronoi cells (black polygons) constructed from Here the matrices B(2+√3i) and B|(2 √3{zi) are} the ma- = 7 = 13 − decomposition of p (a) and p (b) over Eisenstein integers. trix representations of ideals 2 + √3i and 2 √3i h i h − i where two coprime ideals 3+2i and 3 2i can be respectively and they are coprime according to Theorem obtained from (12). The generatorh i matricesh − of thei corre- 1. Similar to the Gaussian case, it can be verified that sponding ideal lattices are given by det(B(2+√3i)) = det(B(2 √3i)) = N( 2+ √3i ) = | | | − | h i 3 2 3 2 N( 2 √3i ) = p = 7. With the notations above, the G(3+2i) = − , and G(3 2i) = . h − i 2 3 − 2 3 locations of the elements of the first subarray are given by   −  G e √ This coprime array produces 132 = 169 DOFs from 13 (2 √3i) 1 σ 2 3i /7A2, while those positions of × − ∈ h − i physical sensors according to Proposition 1. the second subarray are G √ e2 σ 2+ √3i /7A2, 2 1=25 (2+
3i) ∈ h i −Since Z is isomorphic to polynomial ring Z 2 where e A /σ 2+√3i , and e A /σ 2 √3i . [i] [x]/(x +1) 1 2 2 2
that gives rise to skew-circulant matrices, the sensor arrays The elements∈ in theh cross-differencei coarray∈ areh defined− i in 2

obtained from skew-circulant matrices in [9] can be viewed the same form as the Z array, i.e., de = G(2 √3i)e1 − − as CRT arrays over Z[i]. Symmetric Voronoi regions (pZ2) G e . By substituting R = A , = 2+ √3i and V (2+√3i) 2 2 I h i defined in (5) are used to modulo these sensors correspond- = 2 √3i to (20), the generalized CRT guarantees ing to algebraic integers in Z[i], as depicted in Fig. 3(a) and J h − i de A2/7A2, yielding an array of 49 degrees of freedom Fig. 3(b) for p =5 and p = 13 respectively. with∈13 sensors according to Proposition 1.

B. Chinese Remaindering over Z[ω] To derive the prime ideals in the ring of Eisenstein integers, we shall aim for p such that D = 3 is a quadratic Similarly, with a larger array aperture such as , the residue: − p = 13 2 two prime ideals decomposed from 13 are 1+2√3i = x 3 (mod p), h i h i ≡− 1+4ω and 1 2√3i = 1+4ˆω , which are coprime 2 h− i h − i h− i for some x Z. For example, solutions can be 2 3 according to Corollary 3 with m1 = 1 and m2 = 4. The (mod 7), 62∈ 3 (mod 13), and so forth. By performing≡ − generator matrices of the corresponding− ideal lattices are ≡− 10

Table I HOLE-FREE SYMMETRIC CRT ARRAY DESIGN

Require: A PID Q(√D). Steps: 1: Calculate a rational integer p Z such that Legendre symbol D =1. ∈ p 1 1 √ 2 2: Compute the integral basis 1, q of Z[q] as q = 2 B + 2 B 4C where B =0 and C = D if D 1 (mod 4), 1 D{ } − −
− 6≡ and B = 1 and C = − if D 1 (mod 4). − 4 ≡ 3: Decompose p into two coprime ideals p = m + m q and p = n + n q of Q(√D). h i 1 h 1 2 i 2 h 1 2 i 4: Compute generator matrices as m Cm n Cn G = G 1 2 and G = G 1 2 , 1 m m− Bm 2 n n − Bn  2 1 − 2   2 1 − 2  where G is defined in (6) for D> 0 and (7) for D< 0. 5: The sensors are allocated on a 2D space where the positions are given by G1x2 and G2x1 with x1 σ(Z[q])/2σ(p1) and x σ(Z[q])/σ(p ). ∈ 2 ∈ 2

given by p2 respectively. A hole-free Symmetric CRT array is an 5 1 extension of CRT array where x1 σ(Z[q]) / 2σ(p1) and G = − 2 and ∈ (1+2√3i) 2√3 3√3 x2 σ(Z[q]) / σ(p2) and the two subarrays are G1x2 and  2  G x∈ respectively. 1 7 1 2 G = 2 . Note that the two subarrays can be rewritten by means (1 2√3i) √ √3 − 2 3 2 of Voronoi cells (Definition 1) as σ(p ) (σ(p p )) = − −  1 ∩ V 1 2 Here the set of the difference coarray is A2/13A2 Thus this σ(p1) (pΛ) and σ(p2) (2pΛ) where Λ= σ(Z[q]) is the array provides 169 DOF with 25 sensors. The subarrays are algebraic∩V lattice that corresponds∩V to Z[q] of a quadratic field. allocated on the following two ideal lattices: The following proposition exploits the concept of Voronoi

σ 1 2√3i /13A2, and σ 1+2√3i /13A2. cells to guarantee the ’hole-free’ property of HSCRT: h − i h i Proposition 2 (Generating All Lattice Points in Λ Similar to the Gaussian
cases, Voronoi regions
(pA2) are ∩ employed to allocate sensors, yielding more compactV arrays. (pΛ)): HSCRT can generate at least all lattice points in V by using the cross-difference coarray. The examples of A array configurations are shown in Fig. Λ (pΛ) 2 ∩VProof : For simplicity, let us denote the two ideal lattices 4(a) and Fig. 4(b) for p = 7 and p = 13 respectively. To by p and p respectively. The ideal is highlight the shape of hexagonal Voronoi cell for illustra- Λ1 = σ( 1) Λ2 = σ( 2) to find a new range for x such that the difference vectors tive purposes, all A based arrays are rotated 90 degrees 1 2 can overspread which corresponds to from counterclockwise henceforth. Λ (pΛ) Λ/pΛ quotient group point∩V of view. According to CRT, for all d When all antennas act like receivers, the array configura- Λ (pΛ), there exist x Λ (Λ ) and x Λ (Λ ∈) tion given in Definition 5 can be redefined equivalently as a 1′ 1 2 2 such∩V that ∈ ∩V ∈ ∩V set that consists of sensor locations given by vectors, i.e., = z = G x x σ(Z[q])/σ(p ) d G2x1′ G1x2 (mod pΛ) Z { 2 1 | 1 ∈ 1 } ≡ − = G2x′ G1x2 y, y pΛ (2pΛ) z = G1x2 x2 σ(Z[q])/σ(p2) , 1 ∪{ | ∈ } − − ∈ ∩V1 = G x G x , x = x′ G− y. where G1 and G2 defined as (13) are generator matrices of 2 1 − 1 2 1 1 − 2 p1 and p2 respectively. Considering p = p1p2 and their corresponding matrices h i 1 G1 and G2, G2− y is in Λ1 (2Λ1). Note that Λ1 is a 1∩ V 1 sublattice of Λ and x′ G− y is identical to x′ + G− y V. HOLE-FREE SYMMETRIC CRT-BASED ARRAYS 1 − 2 1 2 because of the symmetry of Λ1. The proof is completed by In general, the elements in the coarrays may not be noting that x Λ (2Λ ). In short, by selecting x 1 ∈ ∩ V 1 1 ∈ contiguous, i.e., (or equivalently ) may contain holes, Λ (2Λ1) and x2 Λ (Λ2) results in d =Λ (pΛ). which cause ambiguitiesD when subspace-basedS algorithms ∩V ∈ ∩V ∩V  are applied. In this section, we provide conditions for Generally, the contiguous coarray can be defined as ele- hole-free and contiguous cross-difference and sum coarrays ments within a convex polygon, whereas in this paper, we by modifying the CRT arrays (Definition 5) under certain only consider convex regular polygons such as square and restrictions on quotient rings. The definition of hole-free hexagon. One remarkable advantage of HSCRT arrays is symmetric CRT (HSCRT) arrays is given as follows: that because of the symmetry of the algebraic lattices, their Definition 8: [Hole-free Symmetric CRT arrays, HSCRT] cross-difference coarrays are identical to the corresponding Assume the prime decomposition p = p p in Z[q], sum coarrays. As a result, Proposition 2 also applies to h i 1 2 with G1 and G2 being the generator matrices of p1 and the sum coarrays, implying that both passive and active 11

sensing algorithms that require contiguous coarrays can 15 employ HSCRT arrays. The design procedure of HSCRT is summarized in Table I. Next, we study the properties of 10 HSCRT and formulate the contiguous coarrays of hole-free 2 Z and hole-free A2, which are in the class of HSCRT. 5

0 A. Properties of HSCRT Arrays -5 1) Number of Physical Sensors: According to Proposition 1, the number of sensors in Λ1/pΛ and in Λ2/pΛ both -10 equal to p. After doubling the range of x to Λ (2Λ ) 1 ∩ V 1 and removing the duplicated sensors at the origin, the total -15 -20 -10 0 10 20 sensor number in 2Λ1 becomes 4(p 1). Thus the number of physical sensors of HSCRT is − (a) 4(p 1) + p =5p 4. − − 2) Perimeters and Areas of Physical Arrays: Given a 15 2 prime p, the perimeters of hole-free Z denoted as CG and 10 of hole-free A2 denoted as CE can be calculated as π 1 5 C =8pd, C =6pd(sin )− 6.928pd, G E 3 ≈ where d is the minimum inter-element spacing and the areas 0 acquired by the two array configurations are -5 2 2 2 2 π 1 2 2 AG =4p d , AE =3p d (sin )− 3.464p d . 3 ≈ -10 Therefore the perimeter and the area of A2 array are about 2 86% of those Z array, which implies that hole-free A2 is -15 more compact regarding the geometry. -20 -10 0 10 20 3) Number of Virtual Sensors in Contiguous Coarrays: According to Proposition 2, the cross-difference/sum coarray (b) of HSCRT can generate all lattice points in Λ (pΛ), Figure 5. Given the decomposition of p = 13, hole-free Z2 array (a) and which corresponds to Λ/pΛ from the quotient group∩ of V view. its contiguous cross-difference/sum coarray is within 13Z2 (b). The first Since Λ is a lattice with generator matrix G, pΛ is also a subarray G2x1 is in red stars and the second subarray G1x2 is in blue dots. Voronoi cells of 13Z2 and 26Z2 are also shown. lattice whose generator matrix is GBp where Bp = pI. The cardinality of Λ (pΛ) equals to the cardinality of Λ/pΛ 2) Hole-free A2: Analogously, hole-free A2 is defined as [28, Definition 3.12.]:∩V a type of HSCRT over the ring of Eisenstein integers Z[ω], 2 whose contiguous part of the coarray is also hexagonal with Λ/pΛ = det(Bp) = p , | | | | basis given by (8). Let lr denote the inscribed radius of the i.e., the number of sensors in Λ (pΛ) is p2. ∩V contiguous hexagonal cell (pA2). Using Proposition 2 and the geometry property of hexagonalV lattices, it can be easily 1 verified that lr = 2 p. The contiguous part of the cross- B. Examples of contiguous coarrays of HSCRT difference/sum coarray of hole-free A2 can be described as A (pA ), or equivalently 1) Hole-free Z2: A hole-free Z2 array is an HSCRT over 2 ∩V 2 C,E = d = (xk,yj ) d , lr yj lr, the ring of Gaussian integers Z[i], i.e., p1, p2 Z[i] and D { | ∈ D − ≤ ≤ Λ = Z2. The consecutive set of hole-free Z2 is∈ a uniform 2l √3x + y 2l − r ≤± k j ≤ r} rectangular array (URA) which can be expressed as C,G = where there are 2 elements in . Fig. 6(a) depicts D p C,E Z2 (pZ2), or equivalently an example of hole-free with D over Eisenstein ∩V A2 p = 13 = d = (x ,y ) d , l x l , integers whose cross-difference/sum coarray is shown in Fig. DC,G { k j | ∈ D − G ≤ k ≤ G lG yj lG,k,j =1, 2, lG , 6(b). − ≤1 ≤ ··· } where lG = 2 p according to Proposition 2. It can be verified 2 VI. CONCLUSION that the cardinality of C,G is p . An example of hole-free Z2 array is depicted inD Fig. 5(a) corresponding to Fig. 3(b) In this paper, it has been demonstrated that the problem and the effect of filling the holes is illustrated in Fig. 5(b). of designing planar coprime arrays can be solved through From this point of view, [9, Theorem 2] can be interpreted Chinese remaindering over quadratic fields. Inspired by the as a particular case of Z2. bijective mappings between the rings of integers and lattices, 12

15 quadratic integers. The feasibility of the proposed arrays will be employed for both passive and active sensing, which puts 10 forward the algorithms of angle estimations to sparser and more compact hexagonal arrays. 5 APPENDIX A 0 PROOF OF THEOREM 1

-5 By the assumption on the coprimality of m and n, there must exist α and β such that mα+nβ =1 with m,n,α,β ∈ -10 Z[q]. Without loss of generality, let 1, q be an integral basis of Z[q] where q2 + Bq + C =0 according{ } to (2). Taking the -15 algebraic conjugation of both sides of mα + nβ =1 yields -20 -10 0 10 20 mˆ αˆ +ˆnβˆ =1 where mˆ is the algebraic conjugate of m and (a) same with other elements. qˆ is the other root of f(X)=0 expressed (4). By expending all quadratic integers using the 20 basis, it can be easily proved that mα + nβ =1 if and only if mˆ αˆ +ˆnβˆ = 1. Let us write these two equations by the 10 matrix form: m 0 α 0 n 0 β 0 + = I. 0m ˆ 0α ˆ 0n ˆ 0 βˆ         0 (27) Define Pα and Pβ as eigenvalue matrices as Pm given in (16). Then (27) can be rewritten as PmPα + PnPβ = I. -10 According to Lemma 1, Q consists the eigenvectors of matrix representations and is expressed in (16), then left 1 and right multiplying Q− and Q respectively yields

-20 BmBα + BnBβ = I, (28) -20 -10 0 10 20 i.e., B and B are left coprime. (b) m n Next, we prove the sufficiency of the theorem. If Bm and = 13 Figure 6. Given the decomposition of p , hole-free A2 array (a) with Bn are assumed to be coprime, there exist Bα′ and Bβ′ where contiguous cross-difference/sum coarray within 13A2 (b). Voronoi cells of 13A2 and 26A2 are also shown. α1′ α2′ β1′ β2′ Bα′ = and Bβ′ = α′ α′ β′ β′  3 4  3 4 a new class of array configurations based on coprime lattices such that BmBα′ + BnBβ′ = I, which results the following: constructed from quadratic integers in PIDs is proposed, m α′ Cm α′ + n β′ Cn β′ =1, and (29) 1 1 − 2 3 1 1 − 2 3 which provides enhanced DOF and sparse array geometries, m1α3′ +m2α1′ +n1β3′ +n2β1′ Bm2α3′ Bn2β3′ =0. (30) and thus alleviates the mutual coupling effect. By exploiting − − the properties of PID, a lattice can be represented by a Let α′ = α1′ + α3′ q and β′ = β1′ + β3′ q be two quadratic integers in Z[q]. Then replacing q2 with Bq C yields generator matrix calculated by its corresponding quadratic − − integer, which significantly simplifies the notations of the mα′ + nβ′ = (m1α′ Cm2α′ + n1β′ Cn2β′ ) 1 − 3 1 − 3 sensor locations and generalizes the discussions of copri- + (m1α3′ + m2α1′ + n1β3′ + n2β1′ Bm2α3′ Bn2β3′ )q mality issues. The correlation between coprime quadratic − − (31) integers and matrices have been investigated in great detail, Substituting (29) and (30) to (31), it can be verified that whereby the coprimality of skew-circulant adjugate pairs mα′ + nβ′ =1. can be interpreted as special cases of adjugate matrices in Theorem 2. A modified configuration of CRT array is also APPENDIX B introduced for an enlarged contiguous coarray. Examples PROOF OF COROLLARY 1 over Z[i] and Z[ω] are provided for illustrative purposes while in general all quadratic coprime integers can be Assume Bm and Bn are left coprime. From Theorem chosen for CRT array design since the generalized CRT only 1, this assumption is equivalent to that their corresponding requires the coprimality of ideals. quadratic integers m and n in Z[q] are coprime, i.e., there exist α and β such that mα + nβ =1, which is equivalent In the accompanying paper, a new approach of obtaining to (28). By Lemma 2, (28) is equivalent to coprime matrices will be demonstrated, after which the multi-sublattice CRT arrays will be introduced, where the BαBm + BβBn = I, subarrays are built from three or more pairwise coprime i.e., Bm and Bn are right coprime. 13

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