Super Nested Arrays: Linear Sparse Arrays. with Reduced Mutual Coupling Part II: High-Order Extensions

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1 Super Nested Arrays: Linear Sparse Arrays with Reduced Mutual Coupling Part II: High-Order Extensions Chun-Lin Liu, Student Member, IEEE, and P. P. Vaidyanathan, Fellow, IEEE Abstract In array processing, mutual coupling between sensors has an adverse effect on the estimation of parameters (e.g., DOA). Sparse arrays such as nested arrays, coprime arrays, and minimum redundancy arrays (MRA) have reduced mutual coupling compared to uniform linear arrays (ULAs). These arrays also have a difference coarray with O(N ) virtual elements, where N is the number of physical sensors, and can therefore resolve O(N ) uncorrelated source directions. But these well-known sparse arrays have disadvantages: MRAs do not have simple closed-form expressions for the array geometry; coprime arrays have holes in the coarray; and nested arrays contain a dense ULA in the physical array, resulting in significantly higher mutual coupling than coprime arrays and MRAs. In a companion paper, a sparse array configuration called the (second-order) super nested array was introduced, which has many of the advantages of these sparse arrays, while removing most of the disadvantages. Namely, the sensor locations are readily computed for any N (unlike MRAs), and the difference coarray is exactly that of a nested array, and therefore hole-free. At the same time, the mutual coupling is reduced significantly (unlike nested arrays). In this paper, a generalization of super nested arrays is introduced, called the th-order super nested array. This has all the properties of the second-order super nested array with the additional advantage that mutual coupling effects are further reduced for >. Many theoretical properties are proved and simulations are included to demonstrate the superior performance of these arrays. Index Terms Sparse arrays, nested arrays, coprime arrays, super nested arrays, mutual coupling, DOA estimation. I. INTRODUCTION In array processing, mutual coupling between sensors has an adverse effect on the estimation of parameters (e.g., DOA) [] [7]. Sparse arrays such as nested arrays [8], coprime arrays [9], and minimum redundancy arrays (MRA) This work was supported in parts by the ONR grant N , and the California Institute of Technology. Copyright (c) 05 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a reuest to pubs-permissions@ieee.org. The authors are with the California Institute of Technology, Pasadena, CA 95 USA ( cl.liu@caltech.edu; ppvnath@systems.caltech.edu.) April 4, 06

2 [0] have reduced mutual coupling compared to uniform linear arrays (ULAs). These arrays also have a difference coarray with O(N ) virtual elements, where N is the number of physical sensors, and can therefore resolve O(N ) uncorrelated source directions. But these sparse arrays have shortcomings: MRAs do not have simple closed-form expressions for the array geometry [0]; coprime arrays have holes in the coarray [9]; and nested arrays contain a dense ULA in the physical array [8], resulting in significantly higher mutual coupling than coprime arrays and MRAs. For details please see the companion paper [] and references therein. In the companion paper [], a sparse array configuration called the (second-order) super nested array was introduced, which has many of the advantages of these sparse arrays, while removing some of the disadvantages. Namely, the sensor locations are well-defined and readily computed for any N (unlike MRAs), and the difference coarray is exactly that of a nested array, and therefore hole-free. At the same time, the mutual coupling is reduced compared to nested arrays. Super nested arrays were designed by rearranging the dense ULA part of a nested array in such a way that the coarray remains unchanged, but mutual coupling is reduced by reducing the number of elements with small inter-element spacings. uantitatively, this is described in terms of the weight function w(m), which is eual to the number of sensor pairs whose inter-element spacing is mλ/. It was shown in [] that the first three weight functions of second-order super nested arrays are, if N is even, w() = (), if N is odd, N 3, if N is even, w() = () N, if N is odd, 3, if N = 4, 6, w(3) = 4, if N is even, N 8, (3), if N is odd, Contrast this with the nested array which has w() = N, w() = N and w(3) = N. While w() and w(3) are significantly better in () and (3), there is plenty of room for improving w(), and possibly w(m), m > 3. In this paper, a generalization of super nested arrays is introduced and called the th-order super nested array. It has all the good properties of the second-order super nested array with the additional advantage that mutual coupling effects are further reduced for >. For a given number of physical array elements N, th-order super nested arrays have the following properties: (a) the sensor locations can be defined using a simple algorithm, (b) the physical array has the same aperture as the nested array, (c) the difference coarray is exactly identical to that of the nested array (hence hole free), and (d) the weight functions are further improved, compared even to second-order super nested arrays. Like the parent nested array, the physical sensor locations of th-order super nested arrays are related to two integers N and N (Fig. 3(a) of []). The detailed description of th-order super nested arrays depends on whether N is even or odd. For odd N, there is a simple closed-form expression for the sensor locations, but for April 4, 06

3 3 even N, the locations have to be defined recursively as we shall elaborate. A MATLAB code to find the sensor locations of th-order super nested arrays is given in []. The proof that th-order super nested arrays have a coarray identical to the parent nested array is rather involved, and one of the main goals of the paper is to establish this very important result for both N odd and N even. We also analyze the weight functions w(m) in great depth (again, uite involved in its detail because of the intricate definition of the array geometry). The good news is that it is possible to improve the crucial weights w(), w(), and w(3), compared to nested arrays (see Theorem and Theorem 4). In particular, w() is only about half that of second-order super nested arrays. While the results of [] can in principle be regarded as special cases of this paper, it is very uneconomic to present them as special cases of this paper. The reason is that the proofs in this paper are very complicated they use induction, with the proofs in [] as the basis for induction. The clarity of [] would have been greatly compromised if it had been presented as a special case of this. A. Paper Outline Section II introduces th-order super nested arrays in terms of the parent nested array. The construction is based on some recursive rules to rearrange the sensors of nested arrays through successive systematic stages. In Section III, we formally define th-order super nested arrays for odd N. Many properties of these arrays are given, the highlights being (a) the result that the difference coarray is identical to that of the parent nested array (Theorem and Corollary ), and (b) that the weight functions (hence mutual coupling effects) are significantly reduced (Theorem ). Since the details are considerably different for even N, Section IV is dedicated to a presentation of this case. Detailed proofs of some of the claims of Section III and IV are relegated to Section V and VI, for ease of flow. Section VII presents simulation results and detailed comparison of performances, demonstrating clearly that th-order super nested arrays with > outperform other arrays in the presence of mutual coupling. II. GENERAL GUIDELINES FOR THE CONSTRUCTION OF SUPER NESTED ARRAYS Assume that sensors are defined over physical locations nd, where n belongs to some integer set S. d = λ/ is the minimum sensor separation among sensors, where λ is the wavelength of the incoming wave. For instance, nested arrays with N and N have the following integer set: S nested = {,,..., N, (N + ), (N + ),... N (N + )}, (4) where N and N are positive integers. Second-order super nested arrays [] define the sensor locations using an integer set S (), which can be partitioned into five ULA portions (X (), Y(), X(), Y() set (Z () )., Z() ) and an additional Fig. summarizes the hierarchy among nested arrays, second-order super nested arrays, and th-order super nested arrays. It has been mentioned in [] that the sets X (), Y(), X(), and Y() are obtained by rearranging April 4, 06

4 4 Parent nested array Dense S () X () X () ULA Y () Sparse ULA Second-order super nested array Y () Z () Z () Rule Rule Rule Rule Rule 3 S (3) S (4) X (3) 3 X (3) X (3) Y (3) Y (3) Y (3) 3 Z (3) Z (3) Rule Rule Rule Rule Rule Rule Rule 3 X (4) 4 X (4) 3 X (4) X (4) Y (4) Y (4) Y (4) 3 Y (4) 4 Z (4) Z (4) Fig.. Hierarchy of nested arrays, second-order super nested arrays S (), and th-order super nested arrays S (). Arrows indicate the origin of the given sets. For instance, X (4) originates from X (3) while Y (3) 3 is split into Y (4) 3 and Y (4) 4. It can be observed that the sets X() and Y () result from the dense ULA part of nested arrays. The sparse ULA portion of nested arrays is rearranged into the sets Z () and Z (). N = 6 N + = 4 N + = 4 N + = 4 N + = 4 N + = 4 N + = 4 (a) (b) N + = 4 N + = 4 N + = 4 N + = 4 N + = 4 N + = 4 N = 6 Fig.. D representations of (a) second-order super nested arrays, S (), and (b) third-order super nested arrays, S (3), where N = 3 and N = 6. Bullets denote sensor locations while crosses indicate empty locations. the dense ULA part of parent nested arrays, as in Lemma of []. The sparse ULA part of parent nested arrays is reorganized into Z () and Z () of second-order super nested arrays []. The formal definition of th-order nested arrays will be given in the next section. To develop some feeling for it, first consider = 3. Third-order super nested arrays, as specified by the integer set S (3), consist of eight sets as follows: X (3), Y(3), X(3), Y(3), X(3) 3, Y(3) 3, Z(3), and Z(3), which can be recursively generated from the sets X (), Y(), X(), Y(), Z(), Z() in second-order super nested arrays. For instance, X (3) is identical to X (). X() is split into two sets X (3) and X (3) 3. The same connections also apply to Y(), Y(), Y(3) the elements in Z () and Z () are rearranged into Z (3) and Z (3), Y(3), and Y(3) 3. Finally,. Hence, it can be interpreted that the sets X(3) Y (3) for =,, 3 originate from the dense ULA of parent nested arrays while Z (3) and Z (3) emanate from the and April 4, 06

5 5 (a) X () X () Z () (b) Z () X (3) 3 Y () X (3) Y () X (3) Z (3) Z (3) Y (3) Y (3) 3 Y (3) Fig. 3. D representations of (a) second-order super nested arrays, S (), and (b) third-order super nested arrays, S (3), where N = 3 and N = 6. Bullets denote sensor locations while crosses indicate empty locations. The dashed rectangles mark the sets X (), Y (), Z (), and Z () for. Thin arrows illustrate how sensors migrate from S ( ) to S (). sparse ULA of parent nested arrays. Fourth-order super nested arrays (or super nested arrays with = 4) generalize third-order super nested arrays further. It can be deduced from Fig. that X (3) 3 and Y (3) 3 are divided into X (4) 3, X(4) 4 and Y (4) 3, Y(4) 4, respectively. Similarly, Z (3) and Z (3) are rearranged into Z (4) and Z (4). The remaining sets of fourth-order super nested arrays are the same as their correspondences in third-order super nested arrays. To be more specific, the defining rules to go from ( )th-order super nested arrays to th-order super nested arrays are Rule : X () Rule : X ( ) and Y () replicate X ( ) and Y ( ), respectively, for. That is, we simply copy these portions from the ( )th-order super nested array to the th-order super nested array. and Y ( ) specified in Section III and IV. are split into X (), X() and Y(), Y() Rule 3: Z ( ) and Z ( ) are reorganized into Z () and Z (), using appropriate rules., respectively, according to rules to be Next, we give a concrete example of how th-order super nested arrays are obtained from ( )th-order super nested arrays. Fig. and 3 depict the D/D representations of the second-order super nested array (in parts (a)) and the third-order one (in parts (b)), respectively, where the details of D representations can be found in Fig. of []. In this example, it is obvious that X () = X (3) and Y () = Y (3), which satisfy Rule. To explain Rule, we consider the following sets in Fig. 3: X () = {6, 8, 0}, X (3) = {6, 0}, X (3) 3 = {3}. (5) The middle element of X (), which is the element 8 in this case, is selected and relocated to the third layer of D representations. It becomes the element 3 in X (3) 3. The remaining elements in X(), which correspond to sensor locations 6 and 0, constitute X (3). Finally, Rule 3 can also be clarified using Fig. 3. In the second-order super nested array, we consider the sensor located at (N + ) = 8, which is the leftmost element of Z (). However, this sensor is removed from S () and inserted to S (3) at location 67, as indicated by a thin arrow in Fig. 3(b). This new sensor location is included in Z (3) = {67, 83}, which explains Rule 3. Furthermore, after all these operations, the first layer in D representations does not change while only some elements (8, 4, and 8 in Fig. 3) in the second layer are rearranged to somewhere else. April 4, 06

6 6 Summarizing, th-order super nested arrays can be recursively generated from ( )th-order super nested arrays, as elaborated in Fig.. In the following two sections, based on the parameter N, we will give formal definitions for super nested arrays, which are consistent with Rule,, and 3. These definitions also enable us to determine the sensor locations explicitly. III. TH-ORDER SUPER NESTED ARRAYS, N IS ODD Here is the formal definition of th-order super nested arrays if N is an odd number: Definition (th-order super nested arrays, N is odd). Let N be an odd number, N, and. th-order super nested arrays are characterized by the integer set S (), defined by ( ) S () = X () Y () Z () Z (). = For a positive integer satisfying, X () { X () = { Y () = d () = L () = and Y () ( )(N + ) + + d () (N + ) d (), if =,,...,,, if =, N+ are defined as l 0 l L () l 0 l L () ( N+ ), if =,,...,,, if =, where is the floor function. Z () and Z () are given by Z () = {l(n + ) l N }, }, }, Z () = {(N + )(N + ) + }. For convenience of the reader, here is a MATLAB code for th-order super nested arrays []. In particular, super_nested.m returns the set S () given the array parameters N, N, and. If =, the corresponding array configuration degenerates to nested arrays with parameter N and N. Putting = in Definition gives us Definition 7 in []. For any pair of N, N, and satisfying the assumption of Definition, super nested arrays can be characterized in a closed-form and scalable fashion. It can be inferred from Definition that the inter-element spacing of X (), Y (), and Z () are d (), d (), N +, respectively. For instance, in Fig. 3(b), it can be seen that X (3) and Y (3) are ULA with sensor separation. The sensor separation for X (3) and Y (3) is d (3) = 4. Z (3) is a ULA of sensor separation N + = 4. This property is Here X () and Y () for =,, could be slightly different across Definition 7 of [] and Definition. In Definition 7 of [], these sets are disjoint while in Definition, they might not be disjoint. Even so, both definitions lead to the same S () but the latter one possesses the symmetric property: X () = Y (), which will be more useful in the following development. April 4, 06

7 7 very similar to second-order super nested arrays. Notice from Fig. 3(a) that S () consists of a set of ULAs X (), X (), Y(), and Y(), each with sensor separation, another ULA Z() with sensor separation N + = 4, and finally a singleton Z (). Now we show that if an array is constructed according to Definition, then it satisfies Rule,, and 3 in Section II. This statement is obviously true for Rule. For Rule and 3, the details can be clarified by the following two lemmas: Lemma. Let N be an odd number and S () be a super nested array with order, as defined in Definition. Then X () is composed of even terms (related to even l) of X( ) and X () (N + ) consists of odd terms (related to odd l) of X ( ). These properties also hold true for Y( ), Y(), and Y() Proof: According to Definition, any element in X ( ) can be written as ( )(N + ) + + l where 0 l L ( ). If l = k is an even number, we obtain ( )(N + ) + + k = ( )(N + ) + + d () k, where 0 k L( ). Since k is an integer and x x < x +, 0 k L( ) is euivalent to 0 k L (). That is, even terms of X( ) are exactly X (). If l = k + is an odd number, the elements are, ( )(N + ) + + (k + ) [ ] = ( )(N + ) + + d () k (N + ), where k is a non-negative integer with 0 k + L ( ). The range of k can be rearranged to be 0 k L() because x x < x +. It can be deduced that odd terms in X( ) proof for Y ( ), Y(), and Y() is similar.. are exactly X () Lemma. Let N be an odd number. Assume that Z () and Z () satisfy Definition. Then Z () = Z ( ) \{( )(N +)}, where A\B denotes the relative complement of B in A. Z () = Z ( ) {(N + ( ))(N +) +}, Proof: This proof follows from Definition directly. We now prove that the X () and the Y () eual to the dense part of the parent nested array: Lemma 3 (Relation to dense ULA of nested array). Let X () (N + ). The parts of the super nested array, reduced modulo N + are exactly and Y () for =,,..., be defined in Definition April 4, 06

8 8. Define the sets A () = X () ( )(N + ) and B () = A () = Y () B () = {,,..., N }. ( )(N + ). Then Proof: This lemma is proved using induction on since when =, Lemma in [] holds true. Assuming Lemma 3 holds for, the case of becomes A () B () = = = = ( = (X () (X() ( (Y () (Y() ( = A ( ) A () B () ) (N + ))) ( )(N + ) ) (N + ))) ( )(N + ) ) B ( ) (Y ( ) ( )(N + )) = A ( ) Here Rule and Lemma are utilized. (X ( ) ( )(N + )) B ( ) = {,,..., N }. (6) Lemma 4 (Total number of sensors). If N is an odd number, the number of elements in S (), as defined in Definition, is N + N. Proof: The proof is given by induction on. According to Lemma in [], the cardinality of S () is N +N. If S ( ) has cardinality N + N, we will show that S () also has cardinality N + N. According to Rule, the number of elements in X () and Y () for remains unchanged in S ( ) and S (). Lemma does not alter the total number of elements since X () and X() (N + ) correspond to the even and odd terms in X ( ), respectively. It is also evident that Lemma preserves the total number of sensors. By induction, S () = S ( ) = N + N for. One of the most striking properties of the th-order super nested array is that the coarray is exactly identical to that of the parent nested array. This is proved in the following theorem and the corollary: Theorem. If N 3 is an odd number, N 3 4, and 3, then th-order super nested arrays are restricted arrays, i.e., the difference coarray is hole-free. Proof: This proof is based on induction on. Beginning with Theorem in [], we know S () are restricted arrays. If ( )th-order super nested arrays are restricted arrays, it can be inferred that th-order super nested arrays are still restricted arrays. The details are uite involved, and can be found in Section V. To clarify these details, a numerical demonstration of the mechanics of the proof is also provided in Section I of the supplementary document [3]. April 4, 06

9 9 (w(m) > 0) (w(m) > 0) Coarray index m (a) N = 3, N = 7, = Coarray index m (b) N = 33, N = 7, = 5. Fig. 4. An example to show that N 3 is not necessary in order to make the coarray of S () hole free. Here we consider the indicator function of w(m) > 0 for the super nested array with (a) N = 3, N = 7, = 5 and (b) N = 33, N = 7, = 5. It can be inferred that (a) is a restricted array, because w(m) > 0 for 3 m 3. However, (b) is not a restricted array since w(78) = w( 78) = 0. Corollary. If N 3 is an odd number, N 3 4, and 3, then th-order super nested arrays have the same coarray as their parent nested array. Proof: Due to Theorem, applying the chain of arguments in Corollary of [] proves this corollary. The sufficient conditions on N, N, and in Theorem guarantee that such array configuration is a restricted array. However, these conditions are not necessary. For instance, Fig. 4 examines the coarray of th-order super nested arrays if (a) N = 3, N = 7, = 5 and (b) N = 33, N = 7, = 5, where the indicator function (P ) is if the statement P is true and 0 if P is false. Theorem reuires N to be at least 3 = 95. In Fig. 4(a) (N = 3 < 95), it can be inferred that w(m) > 0 for 3 m 3 so this array configuration is a restricted array. On the other hand, the array in Fig. 4(b) (N = 33 < 95) is not a restricted array since w(78) = w( 78) = 0. Recall that the weight function w() of the second-order super nested array was as in E. (). The next theorem shows that the super nested array for > has significantly improved weight function w(), which is crucial to reducing the mutual coupling effects. Theorem. Assume that N 3 is an odd number, N 3 4, and 3. The weight function w(m) April 4, 06

10 0 of th-order super nested arrays satisfies w() =, w() = N 4 +, w(3) =. Proof: For m =, the sensors located at N (N + ) and N (N + ) contribute to w(). We need to show other combinations do not result in w(). It is obvious that the self-differences among X (), Y (), Z (), and Z () have sensor separation at least. Since these sets are defined in the increasing order, it suffices to show that the difference between the maximum element in one set and the minimum element in the succeeding set, is strictly greater than. Assume X () This is a contradiction since L () hand, it is obvious that Y () and Y () satisfies min(y () ) max(x () ) =. We have L () = (N )/(d () ). is an integer but (N )/(d () ) is not, if N is an odd number. On the other and X () + do not cause w(). Next, w() results from the self difference in X () Y (). First, we check the difference between min(y () ) and max(x () ): min(y () ) max(x () ) = 0, if N = 4r +,, if N = 4r + 3. Besides, consider the sensor pair located at ( )(N + ) + L () X () for some. The exact value of can be uniuely solved from the definitions of X () difference becomes (7) and (N + ) L () Y () and Y (). Their ((N + ) L () ) (( )(N + ) + L (), if N = 4r +, = 4, if N = 4r + 3. ) (8) If N = 4r +, there are L () pairs in X () and L () in Y () with separation. One more pair can be found in (8). In this case, w() becomes w() = L () + = r + = N On the other hand, according to (7), if N = 4r + 3, w() can be written as w() = L () + = r + = When m = 3, w(3) results from the following two pairs: ) min(x () ) and max(y () ). The difference is ) Z () and Z (). We obtain which completes the proof. 4 N 4 +, +. [(N + ) + ] [(N + ) ] = 3. (N )(N + ) [(N )(N + ) 3] = 3, April 4, 06

11 (a) X () X () (b) X (3) 3 X (3) X (3) Z () Z () Y () Y () Z (3) Z (3) Y (3) 3 Y (3) Y (3) Fig. 5. D representations of (a) the second-order super nested array S () and (b) the third-order super nested array S (3), where N = 6 (even) and N = 5. Bullets represent physical sensors while crosses denote empty space. Thin arrows illustrate the recursive rules (Rule and Rule 3) in Fig.. IV. TH-ORDER SUPER-NESTED ARRAYS, N IS EVEN For odd N, we presented three recursive rules between S () and S ( ), as described in Fig. 3, Lemma, and Lemma. For even N, the framework in Fig. still holds true but the details in Rule are different from Lemma. As an example, Fig. 5 displays D representations of super nested arrays with N = 6 and N = 5. The recursive rules are depicted by thin arrows in Fig. 5(b). First, the following sets are considered: X () ={9,, 3, 5}, X(3) ={9, 3, 5}, X(3) 3 ={38}. (9) It is clear that (9) justifies Rule in Fig.. However, (9) does not satisfy Lemma since X (3) contains an odd term, which is the element 5 in this example. On the other hand, Fig. 5 gives Z () = {34, 5, 68, 85}, Z () = {84}, Z (3) = {5, 68, 85}, and Z (3) = {65, 84}. It can be readily shown that these sets satisfy Rule 3 and Lemma precisely. Hence, it can be inferred from Fig. 5 that for even N, S () can be still generated from S ( ) using three recursive rules. Rule and Rule 3 can be utilized directly but Rule needs further development. The formal definition of super nested arrays when N is even is now given in a recursive manner as follows: Definition (th-order super nested arrays, N is even). Let N be an even number, N, and 3. A th-order super nested array is specified by the integer set S (), ( ) S () = X () Y () Z () Z (). These nonempty subsets X () ) (Rule ) For, X () =, Y (), Z (), and Z () satisfy = X ( ). ) (Rule ) X () and X() can be obtained from X( ) by April 4, 06

12 a) If the cardinality of X ( ) is odd, then X () = {Even terms of X( ) }, X () = {(Odd terms of X( ) ) + (N + )}, where the definition of even/odd terms are consistent with Lemma. b) Otherwise, we call the last element in X ( ) X () X () as the extra term. Then = {Even terms of X( ) } {the extra term}, = {(Odd terms of X( ), except the extra term) + (N + )}. Y () share similar properties as X () in Rule and. 3) (Rule 3) The sets Z () and Z () are given by Z () = {l(n +) l N }, Z () = {(N + )(N +) + }, which is euivalent to the recursive formula in Lemma. A MATLAB code for Definition is included in super_nested.m [], where the input parameters are N, N, and and the sensor locations S () are delivered as output. This function first takes second-order super nested arrays S () as an initial condition, then applies Definition multiple times to obtain S (3), S (4), up to S (). Next, we will clarify Rule in Definition using Fig. 5. According to (9), the cardinality of X () is 4 so Rule b is applicable. For X (), the extra term is 5, the even terms are 9 and 3, and the odd terms are and 5. Using the expressions in Rule b of Definition, we obtain X (3) and X (3) 3, which are identical to (9). On the other hand, if we consider Y () = {8, 30, 3} in Fig. 5, then the cardinality of Y () becomes 3, implying Rule a is applicable. The even terms and odd terms of Y () are 8, 3 and 30, respectively. As a result, Y (3) = {8, 3} and Y (3) 3 = {47}, which are consistent with Fig. 5. In short, for even N, super nested arrays are defined in a recursive fashion (Definition ). The only dissimilarity from the odd N case is that, sometimes the extra terms need to be considered (Rule b of Definition ). Next we will prove some important properties which result from Definition of the super nested array. Lemma 5 (Relation to dense ULA of nested array). Let S () be a super nested array, as defined in Definition, when N is an even number. Let A () =,,...,. Then = X () ( )(N + ) and B () = Y () ( )(N + ) for = A () B () = {,,..., N }. April 4, 06

13 3 Proof: First by Lemma in [], we know this lemma is true for =. Then we use proof by induction. Based on Rule of Definition, X () (X() (N + )) = X ( ) Therefore, the argument in the proof of Lemma 3 can be applied. and Y () (Y() (N + )) = Y ( ) Lemma 6 (Total number of sensors). Let S () be a th-order super nested array defined by Definition. Then S () = N + N. Proof: The proof is the same as that of Lemma 4. The coarray of the th-order super nested array is identical to that of the parent nested array. This was proved earlier for odd N. The same is true for even N, as shown by the theorem and corollary below. Theorem 3. If N + is an even number, N 3 4, 3, then th-order super nested arrays are restricted arrays. That is, their coarray is hole-free. Proof: The proof is similar to that of Theorem. The details are uite involved, and are presented in Section VI. As a numerical example, Section II of the supplementary document [3] illustrates these details in Section VI. Corollary. If N + is an even number, N 3 4, 3, then th-order super nested arrays have the same coarray as the parent nested arrays. Proof: This proof is identical to that of Corollary. The next theorem shows that the super nested array for > has significantly improved weight function w(), which is crucial to reducing the mutual coupling effects. Theorem 4. Assume that N + is an even number, N 3 4, and 3. Then, the weight function w(m) of th-order super nested arrays satisfies where k is an integer. w() =, w() = w(3) = 5, N +, if N = 8k, N, if N = 8k +, N, otherwise, Proof: The proof is uite similar to that of Theorem in [] and Theorem in this paper. The parameters A, B, A, B follow the same definition as in the companion paper []. For w(), the two sensor pairs are identical to those in the second-order ones, which have been identified in the proof of Theorem in []. For the weight function w(), there are some cases: April 4, 06

14 4 ) The self-differences of X () and Y () contribute to (A ) + + (B ) + pairs, which is N /. ) If N = 4r, then A = r and B = r. If A + is even, there is an extra term in X (). Note that the maximum ULA element in X () is less than the extra term by, as indicated in Lemma 7-3. The similar conclusion applies to Y (). Hence, depending on the even/odd properties of A and B, there is exactly one pair of sensors with sensor separation, in X () Y () when N = 4r. 3) When N = 4r +, A = r, B = r. If r = k is an odd number, A + and B + are both even numbers. One extra term exists in X () and another one can be found in Y (). There are two pairs of sensor separation. If r = k is an even number, A + and B + are odd numbers. There is no extra term in X () and Y (). Hence, w() is given by w() = N, if N = 4r, N +, if N = 4(k ) +, N, if N = 4(k) +, which proves the w() part. w(3) can be found in these sensor pairs: ) Four sensor pairs have been identified in the proof of Theorem in []. It is applicable because X () = X (), Y () = Y (), min(x() ) = min(x () ), and max(y() ) = max(y () ). ) One more pair exists between Z () and Z (). They are (N )(N +) Z () and (N )(N +) 3 Z (). Then the proof is complete. Remarks based on Theorem and 4 It seems that the super nested arrays with odd N is superior to those with even N in terms of the weight functions w(), w(), and w(3). However, in some scenarios, the super nested arrays with even N is preferred. For instance, suppose that we want to design super nested arrays with N = 4 physical sensors such that the number of identifiable sources is maximized. It was proved in [8] that the optimal N is given by N = (N )/ = 0, which is an even number. Another remark is that, w(), w(), and w(3) remain unchanged for 3. However, this phenomenon does not imply that super nested arrays for 3 have the same performance in the presence of mutual coupling. Instead, super nested arrays for > 3 could reduce the mutual coupling further. It is because the overall performance depends on the mutual coupling models, which are functions of the array geometry, as mentioned in (9) and (0) of the companion paper []. Super nested arrays with > 3 tend to make array geometries more sparse, as discussed extensively in Section III and IV. It can be shown that the weight functions like w(4), w(5), and so on, decrease as increases. Hence, ualititatively, mutual coupling could be reduced for super nested arrays with > 3. Furthermore, the judgement of the estimation error based on on the weight functions w(), w(), and w(3), is ualitative and does not always lead to right conclusions []. For example, Fig. 6 shows that for source spacing April 4, 06

15 5 TABLE I 7 CASES IN THE PROOF OF THEOREM n \ n X () (N +) X ( ), Y () (N +) ( )(N +) Case Case 0 Case 9 X () Case Case Case 0 X () (N +) Case 3 Case Case Y ( ), Case 4 Case 3 Case Y () Case 5 Case 4 Case 3 Y () (N +) Case 6 Case 5 Case 4 Z () Case 7 Case 6 Case 5 ( )(N +) Case 8 Case 7 Case 6 Z ( ) Case 9 Case 8 Case 7 θ = 0.00, the super nested array with =, N = N = 7 (w() = 6) outperforms the super nested array with = 3, N = N = 7 (w() = 9). V. PROOF OF THEOREM According to Theorem in [], second-order super nested arrays are restricted arrays. To prove the same for th-order nested arrays with >, we use induction. Thus, assume that S ( ) are restricted arrays. We need to show that S () are also restricted arrays under certain sufficient conditions. In the following development, we use D () to denote the difference set of th-order super nested arrays, S (). The main concept of the proof works as follows. Let n S ( ) \S () and n S ( ). It is obvious that n n belongs to D ( ). We need to show that there exist some n, n S () such that n n = n n. If the above statement holds true for every n S ( ) \S () and n S ( ), it is euivalent saying that S () is a restricted array. Table I lists 7 combinations, where n S ( ) \S () is divided into 3 subsets in each column and n S ( ) is partitioned into 9 categories in each row. In every case, given n and n, we need to identify the associated n and n such that (a) n, n S (), which will be elaborated in detail, and (b) n n = n n, which is simple to check. (Case ) Any n and n in this case can be written as n = ( )(N + ) + + l, n = ( )(N + ) + + l, April 4, 06

16 6 where 0 l L () ) L () and 0 l L (). According to Definition, L () L(), and we have these cases: < L() : The corresponding n and n can be expressed into two ways. They are n = ( )(N +)+ + l, n = ( )(N +)+ + (l ), n = ( )(N +)+ + (l +), n = ( )(N +)+ + (l + ), The membership of n and n can be derived as follows. Since L () and (). Next, n in (0) belongs to X () in () belongs to X (). That is, if (0) () < L(), we have n X () in (0) if 0 l L (). If 0 l + L (), then n L (),, () then we can find n, n S () using either (0) or (). Solving () leads to another sufficient condition N 7 4. ) L () 3) L () = L() and 0 l L () : The argument is the same as Case -. = L() and l = L () : Depending on l, we obtain two more cases, a) l L () : Under this condition, (0) is still applicable. We obtain l = L () L(), implying n X (). In addition, the maximum value of l is L (), which is less than or eual to L (). This property proves n X (). b) 0 l : The associated n and n are n =( )(N +)+ + (l ), It can be seen that n n =(N +) ( l ). X (), since l L (). We need to show that n Y () (3) under some sufficient conditions. Since 0 l, we obtain l. If L () becomes N 5 4., it can be inferred that n belongs to Y (). Therefore, the associated sufficient condition (Case, 5,, 4) In Case, n, n, n, n are n = ( )(N + ) + + l, n = ( )(N + ) + + l, n = + ( + l ), n = + ( 3 + l ), (4) (5) April 4, 06

17 7 where 0 l L () and 0 l L (). It can be concluded that n, n X () since + l ( ) + N l ( ) 3 + N + L (), L (). Here we apply some properties of the floor function: x x and x + n = x + n for integer n. Note that in Case 5,, 4, we can relate n and n with either X () or Y (). (Case 3, 6, 8,, 5, 7,, 4, 6) For any n and n in this combination, the corresponding n = n +(N +) and n = n + (N + ). We have n, n X () Y() {(N + )} S (). (Case 4, 0) Let us consider Case 4 first. The associated n and n are given by n = ( )(N + ) + + l, n = (N + ) l, where 0 l L () ) L () and 0 l L (). According to l, we have < L() : n and n can be written as n = ( )(N + ) + + l, n = (N + ) (l + ), n = ( )(N +)+ + (l +), n = (N +) (l ), (6) (7) Following Case -, if N 7 4, then n X () and n Y () in either (6) or (7). ) L () 3) L () = L() and 0 l L () : Case 4- applies. = L() and l = L () : a) 0 l L () : (6) can be applied to this case. It can be shown that n X () and n Y (). b) L () + l L () : To identify n and n in this case, we first introduce the remainder R = ( )(N + ) n + n, which is rewritten as Corollary 3. R <. R=(N +) L () l. (8) April 4, 06

18 8 Proof: If x is a real number, we obtain x < x x. This property implies Combining (9) to () and L () N + < + L () +, (9) N + < + L () + +, (0) N + < + l N +. () = L() gives 0 R <. However, if R = 0, both (9) and () achieve their upper bound. The condition that (9) being eual is N + is a multiple of, which contradicts with the condition that () being the eual. Hence, R <. Next, according to R, we can identify n and n. Let us consider the binary expansion of R, which is Then we define P satisfying R = r=0 a r r, a r {0, }. () a 0 = a = = a P = 0, a P =. (3) It can be deduced that (a) P is uniue for a given R, and (b) 0 P. Here we have three more cases, where is consistent with Case 4 in Table I: i) = : In this case, the proof techniue in (6) is applicable. n X () and n is an odd number less than N + so n X () Y ). ii) P, : Since, R is an even number and P. n and n become n =(+P )(N +)+ +P, n =P (N +)+R+ +P. It will be shown that n X () P and n X () P + under some sufficient conditions. It is obvious that n is the minimum element in X () P. n X () P + is euivalent to (4) R + +P = P + P + l 3, (5) for some integer l 3 satisfying 0 l 3 L () P +. According to the definition of P, in (3), the left-hand side of (5) is a multiple of P and l 3 is an integer. Next, we need to show P < R + +P P + P + L () P + under some sufficient conditions. According to (3) and the range of P and, we have R P and +P, yielding the lower bound. A sufficient condition for the upper bound is given by + +P P + P + L () P +. (6) Using x < x, P, and, (6) becomes N 3. April 4, 06

19 9 iii) P +, : n and n in this case are n = ( + P )(N + ), n = P (N + ) + R. (7) A sufficient condition for n belonging to Z () is +P N, implying N 5. On the other hand, n lives in X () P + when there exists some l 3 satisfying 0 l 3 L () P + and R = P + P + l 3. It suffices to solve P + P + L () P +, which gives another sufficient condition N. The proof for Case 0 is similar to Case 4. (Case 7, 6) First we consider Case 7, where n and n are given by n = ( )(N + ) + + l, n = l (N + ). (8) Here 0 l L () and l N. According to l, we obtain ) l N : n and n can be written as n = ( )(N + ) + + l, It is trivial that n X () n = (l + )(N + ). and n Z (). ) l = N : We obtain n and n to be n = + l, n = (N + )(N + ) +. It can be seen from (30) that n X () and n Z (). The proof for Case 6 follows the same argument for Case 7. (Case 9, 8) For Case 9, n and n are given by n = ( )(N + ) + + l, n = (N + )(N + ) +. where 0 l L () and. Rewriting (3) gives n and n n = + + l, n = (N + 3 )(N + ). (9) (30) (3) (3) We need to show that n X () Y () and n Z (). If N + 3 N, then n Z (), which leads to a sufficient condition N 3 5. The membership of n can be verified as follows. It is trivial that n is an odd number. In addition, n + + L () 8 has the same proof as Case 9. N +. We obtain n X () Y (). Case April 4, 06

20 0 (Case 3) In this case, n, n are given by n = ( )(N + ) l, where 0 l L () ) L () n = (N + ) l, and 0 l L (). According to l, we have two sub-cases < L() : The pair n and n can be written as n = ( )(N +) l, n = (N +) (l ), n = ( )(N +) (l +), n = (N +) (l + ), (33) (34) (35) Following the same discussion as Case -, we obtain a sufficient condition N 7 4. ) L () 3) L () = L() and 0 l < L () : This case is the same as Case 3-. = L() and l = L () : a) l L () : It can be shown that n Y () and n Y () -3a. b) 0 l, =, and = 3: In this case n and n are given by n = N + 3, n = + 4(L (3) 3 + ), due to the same reason in Case We know that n X (3), which is trivial, and n X (3) Y (3), since n is an odd number less than N +. c) 0 l, =, and 4: n and n can be written as n = (N + ) L () l, n =. (36) (37) It can be inferred that n is an odd number less than N +. Therefore, n, n X () Y (). d) 0 l and 3: We found that n and n can be expressed as n =( )(N +) 3 L (), n =( )(N + ) + + ( + 3 l ), (38) which satisfies n Y () and n X () under the sufficient conditions L () L() and l L (). We obtain a sufficient condition N 7 8. April 4, 06

21 (Case 9, ) In Case 9, n, n, n, and n can be written as n = ( )(N + ), n = ( )(N + ) + + l, n = ( )(N +)+, n = ( )(N +)+ + (l + ), n = ( )(N +), n = ( )(N +)+ + (l ), (39) (40) (4) where 0 l L () X (). The next argument is similar to that in Case -. A sufficient condition for n X (), n or n Y (), n X () is that L (), which leads to another sufficient condition N 9 4. Case is the same as Case 9. (Case 0, 3) In Case 0, n, n, n, and n become n = ( )(N + ), n = ( )(N + ) + + l, n = (N + ), n = + ( l ), (4) (43) where 0 l L (). It is trivial that n Y (). Besides, n X () since l L () L(). (Case 5) We can write n and n as n = ( )(N + ), n = (N + ), (44) where N. Based on, n and n are given by the following. ) N : In this case n = (N + ) and n = ( + )(N + ). It is evident that n, n Z (). ) = N : We obtain n = (N + ) ( ), n = (N + )(N + ) +. It can be seen that n is contained in Z (). The sufficient condition for n in Y () is L (). We obtain N +. (45) April 4, 06

22 (Case 7) In this case, we have n = ( )(N + ), n = (N + )(N + ) +, n =, n = (N + )(N + ), (46) (47) where. If N and N + N, n and n belong to X () Y () and Z (), respectively. Solving these ineualities leads to the following sufficient conditions for th-order super nested arrays: N 4 and N 3 4. The last step in the proof is to take the intersection of all these sufficient conditions. We obtain N 3 and N 3 4. Then all the (n, n ) pairs in those 7 cases exist simultaneously, implying this array configuration is a restricted array. VI. PROOF OF THEOREM 3 This proof follows the same strategy as that of Theorem, where induction on is applied. First of all, it is essential to characterize some properties of X () Lemma 7 (Properties on X () defined as Definition. Then X () as well as Y () before the induction step. when N is even). Suppose that, N is an even number and S () is possess the following properties: ) X () is a ULA with inter-element spacing. The first element is ( )(N + ) +. ) For, X () of ULA in X () is ( )(N + ) +. has a ULA portion with inter-element spacing. The minimum (or leftmost) element 3) If there is an extra term in X (), it is the maximum (rightmost) element of X () and it is larger than the maximum element in the ULA section of X (). 4) If n X (), then n (N + ) ± X (). Proof: ) We can prove this property by induction. When =, the closed-form expression is given by Definition 7 in [], which satisfies Lemma 7-. Suppose X ( ) ( )(N + ) +. According to Rule of Definition, X () is an ULA with sensor separation and first element is derived from odd terms in X( ). Therefore, the first element of X () is ( )(N + ) + and the inter-element spacing is. ) According to Rule of Definition, we have X () = X (+). Following Rule, X (+) has at least all the the even terms of X (), which constitute the ULA portion. Based on Lemma 7-, X () is a ULA of separation and its minimum element is ( )(N + ) +. Therefore, the ULA part in X (+). The minimum element in X () is then given by ( )(N + ) +. 3) In this case, Rule b of Definition indicates that the extra term is the largest one of X () of the ULA section of X () owns sensor separation while the last term is the second largest one in X (). Based on Lemma 7-, their difference is. April 4, 06

23 3 4) If n X (), then n (N + ) is an odd term of X ( ). Based on Rule of Definition and Lemma 7-, n (N + ) ± are even terms of X ( ), which is contained in X() This completes the proof. Next, assuming S ( ) is a restricted array, we need to show that S () is also a restricted array. Similarly, there are 7 cases, as listed in Table I. (Case, 3) Given n and n in this case, we have ) n belongs to the ULA portion of X () : This case is the same as Case - in the proof of Theorem. n and n can be written in two ways:. n = n, n = n, (48) n = n +, n = n +, (49) where (48) and (49) resemble (0) and (), respectively. According to Lemma 7-4, n lives in X (). In addition, at least one of the n in (48) or (49) belongs to X (). If neither n + nor n belongs to X (), then the ULA part of X () has aperture less than. On the other hand, n in (48) and (49) implies X () has aperture at least. This is a contradiction since X () must have larger aperture than X (). ) n is an extra term in X () : In this case, we only need to consider because when =, there is no extra term, by definition. Based on (49) and Lemma 7-4, we know that n + belongs to X () and n + (N + ) + 3 is contained in X (). Applying these rules multiple times yields, n = n ( )(N + ) + n = n ( )(N + ) + p= p, p= p. It ensures that n lives in X (). We need to show that n Y (). According to Rule in Definition, n is an even number. Its minimum value is attained when all the sets X (), X (),..., X () ( ) n ( )(N + ) ( + A ) + p, where A is given by Definition 7 in []. Therefore, n is lower-bounded by ( ) ( + A ) + p + p > + A. Thus, n belongs to Y (). (Case 4, 0) p= p= p= Let us consider Case 4. According to n, we obtain two cases: ) If n belongs to the ULA part of Y (), it is the same as Case -. ) If n is an extra term in Y (), following the idea of Case -, we can write n = n ( )(N + ) n = n ( )(N + ) p= p, p= p. (50) own extra terms, implying (5) April 4, 06

24 4 Here n belongs to X () and n is an odd number. Next we will show that n belongs to X (). Similar to Case -, n ( )(N + ) is upper-bounded by Therefore, n has an upper bound n ( )(N + ) (N + ) ( + B ) + (N + ) ( + B ) + < (N + ) ( + B ), ( + ( + ) p. p= ) p p= p= p which proves that n belongs to X (). The proof for Case 0 is similar to Case 4. (Case 9, ) Let n = ( )(N + ) and n X (). Based on n, we have two cases: ) n belongs to the ULA portion of X () : Following the steps of Case 9, in Section V, we obtain n = ( )(N + ) +, Then, n can be either in X () n = n +, n = ( )(N + ), n = n. or Y(), according to (5) or (53). It is trivial that n belongs to X (). ) n is an extra term in X () : n and n are given by n = ( )(N + ) + +, n = n ( )(N + ) + +. (5) (53) (54) Y () It can be seen from (54) that n belongs to X () + and n is contained in Y (). It can be proved by checking the lower bound of n, which is for. n ( )(N + ) + + ( ) ( + A ) + p + + p= = ( + A ) + ( + ) + ( + 3) > + A. (Other Cases) Proofs are the same as those in Section V. Next we discuss the sufficient conditions of S () being a restricted array. According to Definition, X () are not empty. Suppose there is only one element in X (), Lemma 7-4 implies there are at least elements in and April 4, 06

25 5 TABLE II ARRAY PROFILES FOR THE EXAMPLE IN SECTION VII Array ULA MRA Nested array, N = N = 7 Coprime array, M = 9, N = 7 S (), N = N = 7 S (3), N = N = 7 S (3), N = 6, N = 8 Aperture DOF Uniform DOF Restricted arrays Yes Yes Yes No Yes Yes Yes Max. sources w() 33 7 w() w(3) X (). Applying this argument many times yields that X() has at least elements. The same property holds for Y (). In addition, if N = 4r +, the number of elements between X () and Y () differs by. Hence, to guarantee this proof is valid, we need N + 4 = +. = Besides, the sufficient condition for N is N 3 4, following the same argument in the proof of Theorem. VII. NUMERICAL EXAMPLES In this section, we make a comparison among ULA, MRA, nested arrays, coprime arrays, and super nested arrays when the mutual coupling effect is present. The total number of sensors is 34 for each array configuration. The sensor locations for MRA cannot be found in the literature, so instead we select the approximate MRA with n = 8 and p = 3 (Reference L, Table 6 of [4]). The nested array has parameter N = N = 7. We choose M = 9, N = 7 in coprime arrays. For super nested arrays, there are three different cases: ) the super nested array with =, N = N = 7, ) the super nested array with = 3, N = N = 7, and 3) the super nested array with = 3, N = 6, N = 8. The sensor locations for these arrays are given by (4) for the nested array, (7) in [] for the coprime array, Definition 7 in [] for the super nested array with =, Definition for the super nested The sensor locations for the approximate MRA are 0,, 4, 30, 46, 6, 78, 94, 0, 6, 4, 58, 74, 90, 06,, 38, 54, 70, 86, 30, 304, 306, 308, 30, 3, 34, 37, 39, 3, 33, 35, 37, and 39. April 4, 06

26 ULA MRA Nested Coprime Super Nested, =,N = N = 7 Super Nested, = 3,N = N = 7 Super Nested, = 3,N = 6,N = 8 RMSE θ Fig. 6. Estimation error as a function of source spacing θ between two sources. The parameters are SNR = 0dB, K = 500. The sources have eual power and their normalized DOA are θ = θ 0 + θ/ and θ = θ 0 θ/, where θ 0 = 0.. Each point is an average over 000 runs. array with = 3, N = N = 7, and Definition for the super nested array with = 3, N = 6, N = 8. More details on these arrays are listed in Table II. The experiments in this section are conducted as in the companion paper []. Sensor measurements are generated from the model with mutual coupling, as in (8) of []. Then, for ULA, the MUSIC algorithm [5] is applied while for sparse arrays, the spatially smoothed MUSIC algorithm [8], [6], [7] is utilized to estimate the source directions. Note that no decoupling algorithms are involved. The parameters to be estimated are the normalized DOA: θ i = (d/λ) sin θ i, where d = λ/ is the minimum sensor separation, λ is the wavelength, and π/ θ i π/ is the DOA for the ith source. To compare the result uantitatively, the root-mean-suared error (RMSE) is defined as E = ( D i= (ˆ θi θ i ) /D) /, where ˆ θi is the estimated normalized DOA of the ith source, calculated from the root MUSIC algorithm, and θ i is the true normalized DOA. April 4, 06

27 7 A. Two Closely-Spaced Sources In this example, two closely-spaced sources with eual power are presented. The parameters are 0 db SNR, and K = 500 snapshots. The mutual coupling matrix is based on linear dipole antennas, as in (9) of []. We choose the carrier freuency f =.4GHz so λ = 0.49m. The dipole length l = λ/. The impedance Z A = Z L = 50 ohms. Two sources are located at θ = θ 0 + θ/ and θ = θ 0 θ/, where θ 0 = 0.. This experiment is repeated for 000 runs, yielding 000 instances of RMSE. In Fig. 6(a), the relationship between the source separation θ and its RMSE, which is the sample mean of 000 RMSE instances, is plotted. Some observations can be made from Fig. 6. First, all sparse arrays show a significant error reduction in almost all θ, compared to ULA. It can also be deduced from Fig. 6(a) that, as θ increases, the coprime array becomes slightly better than the second-order super nested array and the third-order super nested array with even N. The third-order super nested arrays with odd N shows the best performance over 0.00 θ 0.0 in Fig. 6, among all these array configurations. B. Performance Evaluation under Various Parameters The next simulation considers the performance over various SNR, number of snapshots, number of sources, and the mutual coupling matrices. The default parameter setting is 0 db SNR, K = 500 snapshots, and D = 0 sources with eual power. The sources are located at θ i = (i )/(D ) for i D. It will be observed from the simulations that the coprime array outperforms the other array configurations if the number of sources is small and the mutual coupling is small. The super nested array with = 3 and odd N exhibits the best performance when there are many sources and mutual coupling is severe. In Fig. 7(a), the RMSE is plotted as a function of SNR. We see that the super nested arrays with = 3 are the best and ULA is the worst. Fig. 7(b) shows the RMSE versus the number of snapshots K, where the super nested arrays with = 3 demonstrate a significant reduction on RMSE. The coprime array becomes more accurate as the number of snapshots increases, and it works better than the second-order super nested array when K is above 00. The relationship between the RMSE and the number of sources D is plotted in Fig. 7(c). The coprime array works the best if the number of sources is small. As D increases, the super nested arrays with = 3 own the minimum RMSE. The reason is, coprime arrays might own the least mutual coupling effect while third-order super nested arrays possess larger uniform DOF. If D is small, mutual coupling might be more important than uniform DOF. On the other hand, as the number of sources gets closer to the theoretical limit, as shown in Table II, the performance worsens for any array. This phenomenon happens sooner in the coprime array (around D = 30) than in the third-order super nested arrays (around D = 50), since the coprime array detects at most 6 sources while the third-order super nested arrays can resolve up to 305 sources. Fig. 8 examines the RMSE of various arrays under various amount of mutual coupling effect and different number of sources D = 0, 0, and 40. Note that the total number of sensors is 34, so D = 40 exceeds the resolution limit of ULA, as listed in Table II. Here E. (0) of [] is selected to be our mutual coupling model with B = 3. Notice that the larger the magnitudes of the mutual coupling coefficients c, c,... c B are, the more severe mutual coupling is. In this simulation, we first parametrize c and then c,..., c B are obtained from the assumption April 4, 06

28 8 (a) RMSE ULA MRA Nested Coprime Super Nested, =,N = N = 7 Super Nested, = 3,N = N = 7 Super Nested, = 3,N = 6,N = SNR (db) (b) RMSE ULA MRA Nested Coprime Super Nested, =,N = N = 7 Super Nested, = 3,N = N = 7 Super Nested, = 3,N = 6,N = Snapshots K (c) RMSE ULA MRA Nested Coprime Super Nested, =,N = N = 7 Super Nested, = 3,N = N = 7 Super Nested, = 3,N = 6,N = Number of sources D Fig. 7. Estimation error as a function of (a) SNR, (b) the number of snapshots K, and (c) the number of sources D. The parameters are (a) K = 500, D = 0, (b) SNR = 0dB, D = 0, and (c) SNR = 0dB, K = 500. The sources have eual power and normalized DOA θ i = (i )/(D ) for i D. Each point is an average over 000 runs. April 4, 06

29 9 (a) RMSE ULA MRA Nested Coprime Super Nested, =,N = N = 7 Super Nested, = 3,N = N = 7 Super Nested, = 3,N = 6,N = c (b) RMSE ULA MRA Nested Coprime Super Nested, =,N = N = 7 Super Nested, = 3,N = N = 7 Super Nested, = 3,N = 6,N = c MRA Nested Coprime Super Nested, =,N = N = 7 Super Nested, = 3,N = N = 7 Super Nested, = 3,N = 6,N = 8 (c) RMSE c Fig. 8. Estimation error as a function of mutual coupling coefficient c (see E. (0) of []). The parameters are SNR = 0dB, K = 500, and the number of sources (a) D = 0, (b) D = 0 (c) D = 40. The sources have eual power and are located at θ i = (i )/(D ) for i D. The mutual coupling coefficients satisfy c l /c k = k/l while the phases are randomly chosen from their domain. Each point is an average over 000 runs. April 4, 06

30 30 that the magnitude of mutual coupling coefficients is inversely proportional to the sensor separation. In each run, the phases of c, c,..., c B are randomly drawn from [ π, π), the root MUSIC algorithm is used to estimate the DOA of D sources, located at θ i = (i )/(D ) for i D. Finally the RMSE is evaluated. Each data point in Fig. 8 is the sample mean of 000 runs. Some observations can be drawn from Fig. 8. First, for any array geometry, as c increases, the associated RMSE increases. This is reasonable since larger c introduces more severe mutual coupling effect. Second, array configurations seem to have a direct impact on the robustness under mutual coupling. Most curves have turning points, or thresholds, such that the performance starts to become much worse. Hence larger thresholds imply the associated arrays are more tolerant to severe mutual coupling. Note that this threshold depends on the number of sources D. For instance, in coprime arrays, the thresholds in c are 0.8, 0.3, and 0.5 for D = 0, D = 0, and D = 40, respectively. In the super nested array with =, the threshold moves from 0.7, to 0.45, to 0.35, as D goes from 0, to 0, to 40. An interesting observation is that, the super nested array with = 3 and odd N are uite robust in the case of severe mutual coupling and many sources. Another way to interpret Fig. 8 is to consider a fixed D and a fixed c. In most cases, the super nested array with = 3 and odd N give the minimum RMSE. The exception occurs in Fig. 8(a) when D = 0 and 0. < c < 0.8, where coprime arrays become the best. This result is consistent with that in Fig. 7(c), where coprime arrays work slightly better if the number of sources is small. VIII. CONCLUDING REMARKS In this paper, we presented an extension of super nested arrays, called the th-order super nested arrays. These arrays preserve all the properties of nested arrays, while significantly reducing the effects of mutual coupling between sensors, by decreasing the number of sensor pairs with small separation. In the future, it will be of interest to apply to these arrays the decoupling algorithms developed in earlier literature for mitigating mutual coupling effects [] [7]. This will further improve the detection and estimation performance of these arrays. Another future direction of interest would be the extension of linear super nested arrays to the case of planar arrays. These extensions are currently under investigation. REFERENCES [] B. Friedlander and A. Weiss, Direction finding in the presence of mutual coupling, IEEE Trans. Antennas Propag., vol. 39, no. 3, pp , Mar 99. [] T. Svantesson, Modeling and estimation of mutual coupling in a uniform linear array of dipoles, in Proc. IEEE Int. Conf. Acoust., Speech, and Sig. Proc., vol. 5, 999, pp [3] M. Lin and L. Yang, Blind calibration and DOA estimation with uniform circular arrays in the presence of mutual coupling, IEEE Antennas Wireless Propag. Lett., vol. 5, no., pp , Dec 006. [4] F. Sellone and A. Serra, A novel online mutual coupling compensation algorithm for uniform and linear arrays, IEEE Trans. Signal Proc., vol. 55, no., pp , Feb 007. [5] Z. Ye, J. Dai, X. Xu, and X. Wu, DOA estimation for uniform linear array with mutual coupling, IEEE Trans. Aerosp. Electron. Syst., vol. 45, no., pp , Jan 009. April 4, 06

31 3 [6] J. Dai, D. Zhao, and X. Ji, A sparse representation method for DOA estimation with unknown mutual coupling, IEEE Antennas Wireless Propag. Lett., vol., pp. 0 3, 0. [7] E. BouDaher, F. Ahmad, M. G. Amin, and A. Hoorfar, DOA estimation with co-prime arrays in the presence of mutual coupling, in Proc. European Signal Proc. Conf., Nice, France, 05, pp [8] P. Pal and P. P. Vaidyanathan, Nested arrays: A novel approach to array processing with enhanced degrees of freedom, IEEE Trans. Signal Proc., vol. 58, no. 8, pp , Aug 00. [9] P. P. Vaidyanathan and P. Pal, Sparse sensing with co-prime samplers and arrays, IEEE Trans. Signal Proc., vol. 59, no., pp , Feb 0. [0] A. T. Moffet, Minimum-redundancy linear arrays, IEEE Trans. Antennas Propag., vol. 6, no., pp. 7 75, 968. [] C.-L. Liu and P. P. Vaidyanathan, Super nested arrays: Linear sparse arrays with reduced mutual coupling Part I: Fundamentals, to appear in IEEE Trans. Signal Proc. [] [3] [4] M. Ishiguro, Minimum redundancy linear arrays for a large number of antennas, Radio Science, vol. 5, no. 6, pp , 980. [5] R. Schmidt, Multiple emitter location and signal parameter estimation, IEEE Trans. Antennas Propag., vol. 34, no. 3, pp , Mar 986. [6] P. Pal and P. P. Vaidyanathan, Coprime sampling and the MUSIC algorithm, in Proc. IEEE Dig. Signal Proc. Signal Proc. Educ. Workshop, Jan 0, pp [7] C.-L. Liu and P. P. Vaidyanathan, Remarks on the spatial smoothing step in coarray MUSIC, IEEE Signal Proc. Lett., vol., no. 9, pp , Sept 05. Chun-Lin Liu (S ) was born in Yunlin, Taiwan, on April 8, 988. He received the B.S. and M.S. degrees in electrical engineering and communication engineering from National Taiwan University (NTU), Taipei, Taiwan, in 00 and 0, respectively. He is currently pursuing the Ph.D. degree in electrical engineering at the California Institute of Technology (Caltech), Pasadena, CA. His research interests are in sparse array processing, sparse array design, tensor signal processing, and filter bank design. He was one of the recipients of the Best Student Paper Award at the 4st IEEE International Conference on Acoustics, Speech and Signal Processing, 06, in Shanghai, China. April 4, 06

32 3 P. P. Vaidyanathan (S 80 M 83 SM 88 F 9) was born in Calcutta, India on Oct. 6, 954. He received the B.Sc. (Hons.) degree in physics and the B.Tech. and M.Tech. degrees in radiophysics and electronics, all from the University of Calcutta, India, in 974, 977 and 979, respectively, and the Ph.D degree in electrical and computer engineering from the University of California at Santa Barbara in 98. He was a post doctoral fellow at the University of California, Santa Barbara from Sept. 98 to March 983. In March 983 he joined the electrical engineering department of the Calfornia Institute of Technology as an Assistant Professor, and since 993 has been Professor of electrical engineering there. His main research interests are in digital signal processing, multirate systems, wavelet transforms, signal processing for digital communications, genomic signal processing, radar signal processing, and sparse array signal processing. Dr. Vaidyanathan served as Vice-Chairman of the Technical Program committee for the 983 IEEE International symposium on Circuits and Systems, and as the Technical Program Chairman for the 99 IEEE International symposium on Circuits and Systems. He was an Associate editor for the IEEE Transactions on Circuits and Systems for the period , and is currently an associate editor for the journal IEEE Signal Processing letters, and a consulting editor for the journal Applied and computational harmonic analysis. He has been a guest editor in 998 for special issues of the IEEE Trans. on Signal Processing and the IEEE Trans. on Circuits and Systems II, on the topics of filter banks, wavelets and subband coders. Dr. Vaidyanathan has authored nearly 500 papers in journals and conferences, and is the author/coauthor of the four books Multirate systems and filter banks, Prentice Hall, 993, Linear Prediction Theory, Morgan and Claypool, 008, and (with Phoong and Lin) Signal Processing and Optimization for Transceiver Systems, Cambridge University Press, 00, and Filter Bank Transceivers for OFDM and DMT Systems, Cambridge University Press, 00. He has written several chapters for various signal processing handbooks. He was a recipient of the Award for excellence in teaching at the California Institute of Technology for the years , and He also received the NSF s Presidential Young Investigator award in 986. In 989 he received the IEEE ASSP Senior Award for his paper on multirate perfect-reconstruction filter banks. In 990 he was recepient of the S. K. Mitra Memorial Award from the Institute of Electronics and Telecommuncations Engineers, India, for his joint paper in the IETE journal. In 009 he was chosen to receive the IETE students journal award for his tutorial paper in the IETE Journal of Education. He was also the coauthor of a paper on linear-phase perfect reconstruction filter banks in the IEEE SP Transactions, for which the first author (Truong Nguyen) received the Young outstanding author award in 993. Dr. Vaidyanathan was elected Fellow of the IEEE in 99. He received the 995 F. E. Terman Award of the American Society for Engineering Education, sponsored by Hewlett Packard Co., for his contributions to engineering education. He has given several plenary talks including at the IEEE ISCAS-04, Sampta-0, Eusipco-98, SPCOM-95, and Asilomar-88 conferences on signal processing. He has been chosen a distinguished lecturer for the IEEE Signal Processing Society for the year In 999 he was chosen to receive the IEEE CAS Society s Golden Jubilee Medal. He is a recepient of the IEEE Signal Processing Society s Technical Achievement Award for the year 00, and the IEEE Signal Processing Society s Education Award for the year 0. He is a recipient of the IEEE Gustav Kirchhoff Award (an IEEE Technical Field Award) in 06, for Fundamental contributions to digital signal processing. April 4, 06