Improved Uniform Linear Array Fitting Scheme With Increased Lower Bound on Uniform Degrees of Freedom for DOA Estimation

Recently, a uniform linear array (ULA) fitting (UF) principle is proposed for sparse array (SA) design using pseudo-polynomial equations. Typically, it is verified that the designed SAs via UF enjoy a lower bound on uniform degrees of freedom (uDOFs), which is <inline-formula> <tex-math notation="LaTeX">$\approx 0.5N^{2}$ </tex-math></inline-formula> with <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> sensors. Herein, an improved UF (IUF) scheme is proposed to significantly increase the lower bound on uDOF, which is realized by introducing two sub-ULAs, with the name of adjoint transfer arrays (ATAs), on both sides of a transfer sub-ULA. The ATAs together with the transfer sub-ULA serve as a new layer to construct an adjoint transfer layer (ATL) that has improved aperture in comparison with the traditional transfer layer to improve the lower bound on uDOF, which is derived in detail. Furthermore, two novel SAs are developed based on the ATL to verify the effectiveness of the proposed IUF scheme to increase the uDOF. Numerical simulations verify the superiority of devised SAs for direction-of-arrival (DOA) estimations.

sub-ULA [11], while CAs consist of two sparse ULAs whose 38 interelement spacing are Co-prime (CP) integers [12], [13]. 39 Motivated by NAs and CAs, a great deal of SAs has been 40 developed for various applications [14], [15], [16], [17], [18], 41 [19], [20], [21]. For example, using the parallel arrangements, 42 the linear SAs can be utilized to implement 2-D DOA estima-43 tion [19]. 44 However, the NA-based SAs, such as improved NA [17] and 45 multiple-level NA [21], are sensitive to mutual coupling (MC) 46 that happened between elements in the arrays. Meanwhile, the 47 CA-based SAs have low MC, but they are short of degrees 48 of freedom (DOFs) [12], [13]. From the previous reports, 49 we know that the MC is a primary problem for active sensing 50 to achieve high performance in electromagnetic engineering 51 applications [22], [23]. To solve this problem, super NAs 52 (SNAs) [24], [25], [26], [27] have been developed, which 53 inherit the DOF of NAs with decreased MC, which renew 54 the study interest in designing SA geometries with high DOF 55 and low MC. 56 In [26] and [27], it is clarified that the MC can be analyzed 57 using weight function. Specifically, the first three values of 58 weight function, namely, w(1), w(2), and w(3), have primary 59 effects on the MC. Following this property, many SAs are 60 proposed and discussed [28], [29], [30], [31] to analyze the 61 SAs. For instance, the augmented NAs (ANAs) [28], the 62 maximum interelement spacing constraint (MISC) arrays [30], 63 the two-side extended NAs (TSENAs) [29], and the extended 64 padded CAs (ePCAs) [31] have been designed to achieve 65 reduced MC or improved DOF. In [32] and [33], a sys-66 tematically SA design procedure, namely, the ULA fitting 67 (UF) principle, is proposed, analyzed, and used for DOA 68 estimations. UF allows to design SAs using a combination of 69 several ULAs, by which the MC can be regulated. Using UF, 70 several SAs with good properties are developed, such as UF 71 using three-base layer (UF-3BL) and UF using four-base layer 72 (UF-4BL) [33]. 73 Typically, the achievable lower bound on uniform DOF 74 (uDOF) is O(N 2 /2) for the UF method with N sensors. In this 75 article, an improved UF (IUF) scheme is proposed by intro-76 ducing two sub-ULAs, termed adjoint transfer arrays (ATAs), 77 into both sides of the transfer sub-ULA in traditional UF. Then, 78 a new adjoint transfer layer (ATL) is formed using the ATAs 79 and the transfer sub-ULA to provide an enlarged aperture, and 80 hence, the lower bound on uDOF is improved. The proposed 81 1557-9662 © 2022 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
See https://www.ieee.org/publications/rights/index.html for more information.   where is the Khatri-Rao product, 1 n = vec(I N ), and h = 135 [σ 2 1 , . . . , σ 2 Q ] T . The famous DCA is then originated from (4), 136 where h is referred to as the collected signal for a virtual array 137 with manifold (A * A). This virtual array is the so-called DCA 138 of array S (whose manifold is A). Therefore, the DCA of SA 139 with position set (1) is given as where ⊕ is convolution, and c(n) represents the binary expres-150 sion of S with value 1 denoting a physical sensor, and 0 for 151 otherwise. Fig. 1 gives an example of the binary expression 152 for an SA.

154
MC is nonnegligible in practical applications. Considering 155 the MC, the received data given in (2) are rewritten as 156 x q = CAs q + n q (7) 157 where C represents the MC matrix. For linear arrays, the MC 158 matrix can be approximately expressed as [26], [28], [30] 159 where p k , p l ∈ S and c d , d ∈ [0, D] are elements of C, which 161 satisfy The parameter for evaluating the MC effect is the coupling 164 leakage, which is given by [26], [28], [30] 165 177 179 The relationships among array position set S, array binary for the SA shown in Fig. 1 can be expressed as P(x) = 183 In summary, the polynomial model of SA S, whose physical 185 sensor position set is given in (1), is established as

187
Then, based on (6), (12), and (13), the DCA of an SA with 188 position set S can be formulated as where × is multiplication.

205
The i th sub-ULA is expressed as {α i , β i , γ i } with polynomial 206 expression P sub i (x), where i = 1, . . . , n. Besides, the distance 207 between adjacent sub-ULAs is referred to as gap i,i+1 .

208
In UF, sub-ULAs are divided into three layers, namely, the 209 base layer, the addition layer, and the transfer layer, where 210 each layer has specific functions.

211
In the base layer, the sub-ULAs have completely the same 212 structure with N base denotes the number of sub-ULAs, while 213 S b and N b indicate interspace and number of sensors within 214 each sub-ULA, respectively. Based on the properties of base 215 layer, we have N base = S b . Besides, the base layer is named 216 by S b . For instance, the base layer with S b = 3 is referred to 217 as the three-base layer.

218
In the addition layer, N addition denotes the number of sub-219 ULAs, while S a (i ) and N a (i ) (i = 1, . . . , N addition ) are the 220 interspace and number of sensors within each sub-ULA. Based 221 on the properties of addition layer, one can get The transfer layer only contains one sub-ULA, which is also 224 known as transfer sub-ULA, with S t represents the interspace, 225 and N t is the number of sensors. Basically, S t can be selected 226 based on the following equations: In [33], it has been verified that in the UF scheme, the 232 DCA of an SA is symmetric and can be divided into two 233 partitions, namely, the self-DCAs (SDCAs) of each sub-ULA, 234 and the inter-DCAs (IDCAs) between each sub-ULA pair. 235 In this article, the SDCA of sub-ULA i is denoted as SDCA i , 236 and the IDCA between sub-ULA i and sub-ULA j is denoted 237 as IDCA i, j . Based on [33], the SDCAs and IDCAs can be 238 calculated using where P sub-i (x) and P subj (x) are polynomial expressions for 243 sub-ULA i and sub-ULA j , respectively.

244
Based on the properties of SDCAs and IDCAs, one can 245 get the following conclusions. The SDCA (positive set) of a 246 ULA has the same structure as its prototype array. The IDCAs 247 can be regarded as a duplication and transfer procedure, where 248 one ULA (referred to as the prototype sub-ULA) is duplicated 249 and transferred, and the other ULA (referred to as the transfer 250 sub-ULA) gives the number of duplication and the period of 251 transfer. In UF, the sub-ULA within transfer layer is the only 252 transfer sub-ULA that determine the number of duplication 253 and the transfer period. Besides, the SDCA of the transfer 254 sub-ULA provides a periodic frame in DCA domain, and the 255

(20) 268
By dividing the sub-ULAs into three groups, the SDCAs of 269 each sub-ULA and the IDCAs between sub-ULA pairs can 270 be conveniently mapped into three ranges in DCA domain, 271 namely, the near end range (NER), the transfer range (TR), 272 and the far end range (FER). For a better understanding, the 273 array division and the mapping relationships are demonstrated 274 in Fig. 4, and polynomials for NER, TR, and FER are as 275 follows: where + is the simplified plus operation, which did not 278 consider the coefficients when computing, i.e., x +x = x. 279 Besides, in this article, the operator × has the same feature 280 with +.

282
The design procedure of the UF principle can be summa-283 rized as follows.   The objective function in UF has the following form: To improve the lower bound on uDOF, two sub-ULAs on both The number of sensors and the interspace of LATA and 326 RATA are denoted as N LATA , S LATA , N RATA , and S RATA , respec-327 tively. The LATA, transfer sub-ULA, and the RATA will be 328 regarded as a new layer, termed the ATL, which has enlarged 329 aperture in comparison with A P t and, hence, increases the 330 lower bound on uDOF. The following properties hold for the 331 ATL.  2) S LATA and S RATA are determined by S t using the follow-334 ing equation: where · and · are the floor and ceil operations, 337 respectively.

338
3) The gap between LATA and the transfer sub-ULA is 339 S LATA . Similarly, the gap between RATA and the transfer 340 sub-ULA is S RATA .

341
As all the necessary components and parameters for ATL 342 have been declared, we present an example of ATL and its 343 influence in TR in Fig. 5.

344
The polynomial model of the ATL can be written as where One can see from (29)  in this article, the sequence for the sub-ULAs is assigned as 368 follows: When the ATL is used to design SAs, the selection function 371 for S t will change from (16) and (17) to Besides, the following relationship is established: To date, the basic structure of ATL is presented. The IUF 380 scheme is composed of the base layer, the ATL, and the addi-381 tion layer. One should note that, in IUF, the array division in 382 both physical domain and DCA domain follows (20) and (21). 383 Moreover, the rest design procedure for the IUF scheme is the 384 same as the traditional UF principle provided in Section III-C. 385

386
In ATL, the number of sensors in LATA and RATA has 387 great influence on TR. Based on (29), it can be seen that with 388 the increasing of N LATA and N RATA , the regularity within the 389 periods in periodic frame will be destroyed.

429
The aperture of ATL can be written as  439 Equation (42) shows significant improvement for the lower 440 bound on the uDOF, which is larger than the uDOF of most 441 existing SAs. One should note that (42) is derived based 442 on (30). When (31) is utilized, the achievable lower bound 443 on uDOF will become smaller; meanwhile, the design process 444 will become easier.

446
The IUF scheme provides a quite flexible SA design pro-447 cedure shown in Section III-C, and the parameters for each 448 layer are selected based on their properties. After determining 449 the sequence of all sub-ULAs, the following problem is to list 450 and solve the objective function.

451
For solving objective functions similar to (34), the strat-452 egy provided in [33] still works. After completing the first 453 two steps in the design procedure, the following parameters, 454 namely, S a (i ), N a (i ), N addition , i = 1, 2, . . . , N addition , S b , and 455 N base , are obtained, and S t is obtained from (30) or (31). The 456 optimal selection for N b and N t is obtained via maximizing J . 457 Therefore, the solution of the objective function is the gaps of 458 adjacent sub-ULAs. For a specific sequence, the strategy for 459 solving the objective function obeys the following steps.

472
It is worthy to point out that the strategy provided earlier is 473 similar to the Mathematical Induction process. Nevertheless, 474 some particular solutions fail to generalize for N b > 2. From 475 the abovementioned discussions, we can see that the IUF 476 scheme enables the designer to determine the parameters and 477 sequence of all sub-ULAs flexibly based on the properties of 478 each layer. Though the desired solution may not exist for some 479 setups, the value of S t can be reduced to get an easier design 480 process.

482
In this section, two SAs are designed using IUF as exam-483 ples. Both are designed based on ATL with increasing number 484 of sensors.

A. Design Example 1: S t Is Odd 486
The first SA is designed via ATL with increasing number of 487 sensors in ATAs and two one-base layers (ATLI-1BL). In this 488 regard, the parameters satisfy (43) 490 Here, we set N b = N 1 b = N 2 b and follow (36)- (38), which 491 yields: In ATLI-1BL, there are five sub-ULAs, where the sequence is 494 selected as To pursue a high uDOF, the target function of ATLI-1BL can 497 be written as Based on the properties of ATL, one can simply get 501 Furthermore, we begin to solve the target function (46).

503
According to the strategy provided in Section IV-D and [33], 504 first, we do the initial step and set N b = 2. In this case, the 505 expression of the TR is Using (52)-(54), one can get Following the steps for solving the objective function, we now 541 need to check if the obtained J ATLI-1BL satisfies the constraints 542 in (46). In this case, the aperture of the ATL is A P ATL = 543 Obviously, we have J ATLI-1BL > 544 A P ATL , and therefore, (51) is the correct solution for (46).

B. Case 2: S t Is Even 560
The second SA is designed via ATL with increasing number 561 of sensors in ATAs, one two-base layer and one addition layer 562 with S a = 2 (termed ATLI-2BL). In this case, one can get

564
In ATLI-2BL, there are six sub-ULAs, where the sequence is 565 selected as Based on (67)-(69), one can get Maximizing (71) under the condition of (62), one can get 602 the optimal parameter selection and the uDOF expression for 603 ATLI-2BL as , N ≥ 10.

639
In terms of the coupling leakage, as shown in Fig. 7(b),   The probability of angle resolution versus SNR for all the 666 SAs tested is given in Fig. 9, where two targets are closely 667 located at θ 1 = 5.0 • and θ 2 = 5.1 • . The estimated θ 1 and θ 2 668 are denoted asθ 1 andθ 2 , respectively. The two targets are 669 resolved when the following equation is satisfied [6]: (76) 671 Fig. 9(a) shows the probability of angle resolution in non-672 coupling environment, where all SAs tested are composed of 673 30 sensors. It can be seen that the proposed ATLI-1BL and 674 ATLI-2BL have much higher angle resolution than other com-675 petitors. The probability of angle resolution in high-coupling 676 scenario is demonstrated in Fig. 9(b), where the number of 677 sensors is 35 and |c 1 | = 0.3. Clearly, the proposed ATLI-2BL 678 has the highest probability of angle resolution. Note that since 679 other competitors fail to reach such high angle resolution in 680 this case, hence, their results are omitted to keep the figure 681 concise and clear.

683
In the third example, the root-mean-square error (RMSE) 684 performance in different conditions is analyzed.   [−60 • , 60 • ]. From Fig. 10(a), it is shown that the ATLI-689 1BL and ATLI-2BL have better performance due to their high 690 uDOF. In non-coupling environment, the RMSE performance 691 is basically based on uDOF of all the SAs tested. In the 692 second case, we consider the RMSE performance versus SNR 693 in high coupling environment. Here, the number of sensors is 694 chosen as 35, the number of targets is set as 45 with azimuth 695 [−50 • , 50 • ], and c 1 = 0.3e j π/3 . In Fig. 10(b), it is shown 696 that the ATLI-2BL has considerable performance improvement 697 compared with other SAs tested.

730
To better evaluate the proposed SAs, an experiment is 731 carried out in the South China Sea. The receiving array has 732 80 uniformly located hydrophones, where the interspace is 733 6.25 m, and the deployment depth is 120 m. The observed 734 space is [−90 • , 90 • ], and the direction of the receiving ship 735 (fixed above the receiving array) is 65 • , while some other 736 ships are working around. The analyzed frequency ranges 737 from 50 to 120 Hz, and the total observe duration is 500 s.

738
First, the conventional beamforming (CBF) algorithm is 739 utilized to track the targets within the observed space [38]. 740 Here, all the data collected by the receiving array are used, 741 and the observed result is shown in Fig. 13(a) to serve as the 742 reference. Based on the observed results shown in Fig. 13(a) Fig. 13(b)-(g), where the SS-MUSIC algorithm is utilized 749 to obtain the tracking results. Array configurations selected 750 in this experiment are the MISC, nested, ANA12, ULA, and the proposed ATLI-1BL and ATLI-2BL. Overall, the tracking 752 results of the ATLI-1BL and ATLI-2BL have fewer missing 753 and spurious targets. As shown in Fig. 13(b), one can tell 754 that the ATLI-1BL has the best tracking performance, which 755 is consistent with the tracking result shown in Fig. 13(a).