Directivity Enhancement of Movable Antenna Arrays with Mutual Coupling

Directi vity Enhancement of Mo v a ble Antenna Arrays with Mutual Coupling W ei Xu, Gr adua te Student Memb er , IEEE, L ip eng Zhu , Sen ior Member , IEE E , W enyan M a , Graduate S tudent Memb e r , IEEE, An Liu, Senior Memb er , IEEE, a n d Rui Zhan g , F e llow , IEEE Abstract —In con ventional antenna arrays, mutual coupling between antenna elements is often regarded as detrimen t al. Howe ver , under specific conditions, it can be harnessed to enhance the far -field directivity (i.e., beam forming gain). The- oretically , the directivity of a n N -antenna superdirecti ve array ov er the endfire direction can rea ch N 2 , significantly exceeding the directi vity of a traditional uncoupled array wh ich is N ov er all directions. Thi s paper in vestigates th e potential of mutual coupling effects in mov able antenna (MA) arrays for d irectiv ity enhancement. A low-complexity algorithm called Greedy Search and G radient Descent (GS-GD) is pr oposed to optimize the antenna positi on s for maximizing the array d irectiv ity over any giv en direction, where the antenna positions are fi rst selected sequentially from discrete grid poin ts and then contin uously refined through gradient descent ( GD) optimization. Numerical results demonstrate that the optimized MA array design by exploiting t h e antenna coupling achieves significant d irectiv ity gains compared to the con ventional uni form linear array (ULA) without an tenna coupli ng over all di rections. Additionally , the proposed GS-GD algorithm is shown to appr oach th e global optimum closely in most directions. Index T erms —movable antenna (MA ) array , mutual coupling, directivity , anten na position optimization. I . I N T R O D U C T I O N In r ecent years, multip le-input multiple - output (MIM O) technolog y has achieved remarkable success in wireless com - munication systems. By increasing the nu m ber of anten nas, MIMO systems can enh a n ce bo th beamf orming an d mu l- tiplexing g a ins, thereby sig n ificantly improving the achiev- able transmission r ate. Howe ver , the physical size constrain ts of mo dern devices a n d/or infr astructures necessitate p la c ing multiple anten nas in close pr o ximity , which inevitably leads to mutu al cou p ling between a n tenna elemen ts. Su c h mutual coupling distorts the signals and is usua lly regarded to be harmfu l to the system perform a n ce. Numerou s stud ies aim to elim inate or mitigate mutua l coupling b ased on phy sical isolation or signal pr ocessing technique s [ 1], [2]. Meanwhile, se veral recen t works indicate that coup led array s can im prove the far -field directivity (i.e., beamforming /array gain in a line- of-sight scenar io) by appr opriately adjusting the effecti ve radi- ation pattern [3], [ 4]. For example, for an N -an tenna u niform linear arr ay (ULA) with th e inter-antenna spacing app r oaching zero ( i.e., sup erdirective array [5]), the directi vity o ver the endfire direction ( i.e., the d irection aligned with the linear array) can reach N 2 [3], [4]. In com parison, the directivity for an un c oupled ULA with half-wavelength spacing (i.e., This work w as supported in par t by the Nationa l Key Resea rch and Dev elop- ment Program of China under Grant 2025ZD1301900; in part by the Zhejiang Provin cial K ey Laboratory of Information Pro cessing, Communication a nd Networ king (IPCAN), Hangzhou, China. (Correspondi ng authors: L ipeng Zhu; W enya n Ma.) W . Xu and A. Liu are with the College of Information Science and Electroni c E nginee ring, Zhejiang Uni versit y , Hangzhou 310027, Ch ina (e- mails: 12231 077@zju.edu.cn and anliu@zju.edu.c n). Lipeng Zhu is with the Stat e Ke y Laboratory of CNS/A TM and the Schoo l of Interdisc iplinary Scie nce, Beijing Institute of T ec hnology , Beijing 100081, China (E -mail: zhulp@bit.edu.cn). W . Ma, and R. Zhang are with the Department of Electric al and Computer Engineeri ng, National Uni versi ty of Singapo re, Singapore 117583 (e-mail s: wenya n@u.nus.edu, elezha ng@nus.edu.sg). ULAH) is al ways N over all dire c tions. This phen omeno n of c oupled arra y s is k nown as super d irectivity . Although superdirec ti ve arrays can achieve highe r b eamform ing gains over the endfir e direction , th ey suffer from sub stantially lo wer gain over oth er direction s co mpared to unco upled arrays. In other words, the d irectivity of an anten na array depen ds on the array geometr y (i.e., antennas’ positions) an d its maximum value v aries with the wave direction. Thus, th e maxim u m directivity over all wav e d irections c a nnot be simultaneously achieved b y any array with fixed-position antennas (FP As). Recently , movable antenn a (MA) array has been pr o posed to enhance wireless com munication p erform ance b y exploiting the degrees of f r eedom (DoFs) in antenna position op timiza- tion [ 6], [7], [8]. Joint ante n na position and ro tation design is also con sidered in six-d imensional MA ( 6DMA) systems [9], [10]. Howe ver , most existing studies assume th at no mutu al coupling exists b etween MA elements b y constrain ing the inter-antenna spacin g being greater than half-wav elength [11]. While prior works ha ve demo n strated through simulation s that mutual coupling can enhance the perfo rmance of dipole MA or reconfigu rable intelligent surface (RIS) systems [12], [13], to the best of our k nowledge, a theoretical fra m ew ork quantif y ing the fund amental beamf orming g a in of gen eral M A systems remains absent. T o this end , this paper takes an isotro p ic MA array as a case study to establish a systematic f ramework for harnessing mutu al cou p ling to en h ance MA arr ay directivity . In this pap er , we first provide the expression of th e directivity to illustrate the increased beamforming ga in ob tained by the coupled MA arr ay . Then, for any given d irection, we for m u- late an o ptimization prob lem to max imize the d irectivity by designing the antenna positions with in the mov able regio n . T o solve the optimiza tion problem, we pro pose a low-complexity Greedy Search an d Gradien t Descen t (GS-GD) al gorithm, where the antenna positions a re first selected sequ entially from discrete gr id points and then con tinuously refined thr ough gra- dient descent ( GD) op timization. Extensive simulation re su lts demonstra te the directivity gain of the coup led MA array and the efficiency of the pr oposed GS-GD algorithm . Notations : 0 N and I N refer to an all-zero column vector of dimension N and an N × N identity m atrix, respectively . ( · ) , ( · ) T , ( · ) H , ( · ) − 1 and ℜ ( · ) d enote conju gate, transpo se, conjuga te tran spose, inv erse and real p art, respectively . R an d C refer to the sets of r eal numb e rs and complex numbers, respectively . | a | and || a || 2 denote the amp litude o f scalar a and the 2-norm of vector a , respecti vely . a n and A mn refer to the n -th entry of vector a and the ( m, n ) -th entry of matrix A , respectiv ely . O ( · ) deno tes the or der o f co mplexity . o ( · ) denotes higher-order infin itesimals. I I . S Y S T E M M O D E L A N D P E R F O R M A N C E C H A R AC T E R I Z AT I O N A. Dir ectivity Model of the Cou pled MA Array As illustrated in Fig . 1, we co nsider a lin ear MA ar- ray with N isotropic antenn a elemen ts lo cated alon g the x -axis. T he anten na po sition vector (APV) is den oted by Fig. 1: Illustration of the considered linear MA array . x = [ x 1 , x 2 , · · · , x N ] T ∈ R N . W e assume that the an tennas can move on a line segment of length d max , and the minimu m spacing between antenn as is d min , i.e., d min ≤ | x m − x n | ≤ d max , 1 ≤ m 6 = n ≤ N . (1) The complex far-field pattern of the MA array over the direction θ ∈ [0 ◦ , 1 80 ◦ ] is expressed as a fun ction of u , cos θ ∈ [ − 1 , 1] for n otation simplicity , where u is the d irection cosine and is propo rtional to the discrete sp atial frequ ency [4]: y ( u, x ) = w H a ( u, x ) , where a ( u, x ) ∈ C N × 1 is the steering vector (SV) : a ( u, x ) = h e − j 2 π λ x 1 u , e − j 2 π λ x 2 u , · · · , e − j 2 π λ x N u i T , with λ denoting the wav elength . T he vector w ∈ C N × 1 represents the coupled excitation co efficients, whe re the n - th entry w n is pro portion al to the current excitation on the n -th antenn a. Then, the total radia ted power of the MA ar ray is given by the integral in the angu lar domain [5]: P r ad = 1 2 Z π 0 | a H (cos θ , x ) w | 2 sin ( θ ) dθ = w H R ( x ) w , where we defin e R ( x ) , 1 2 Z π 0 a ( u, x ) a H ( u, x ) du, which is also called impedance couplin g m atrix ca u sed by power radiation interactio n between antennas [5 ], [4], and the ( m, n ) -th entry of R ( x ) th at re presents the imped ance coupling coefficient between the m - th antenna and the n - th ante n nas is R mn ( x ) = 1 2 R 1 − 1 a m ( u, x m ) a n ( u, x n ) du = sinc  2 x n − x m λ  , with sinc ( x ) = sin( π x ) π x . According ly , the d irectivity o f the M A array over the direction θ , expressed in ter ms of u , takes the form of a Rayleigh q uotient ˚ G u ( x , w ) = | a H ( u, x ) w | 2 w H R ( x ) w . For a given x , th e optimal no rmalized w fo r maximizin g the directivity is given by w ∗ = R − 1 ( x ) a ( u, x ) || R − 1 ( x ) a ( u, x ) || 2 , yield ing the max imum directivity over u as a function of the APV : G u ( x ) = a H ( u, x ) R − 1 ( x ) a ( u, x ) . (2) In most existing works for MIMO system d esign, impedanc e coupling is neglected by co nstraining th e in te r-antenn a spacing being greater th a n half-wavelength, since the sinc functio n becomes n egligible for argumen ts larger th a n 1. Under this assumption, th e radiated p ower , optimal be a mformin g vector, and maximu m d ir ectivity simplify to P r ad = w H w , w ∗ = a ( u, x ) / √ N , and G u = N , respectively . B. Dir ectivity Gain Ana lysis Direct an alysis of th e rela tio nship betwe e n APV an d d ir ec- ti vity in (2) is ch allenging for a gen eral ar ray , prim arily due to the presence of the inverse co upling m atrix R − 1 ( x ) . T o facilitate analysis, we interpret the coupling matrix R ( x ) as the Gram matrix of th e pattern function s { a n ( u, x n ) } N n =1 with the inn er pr oduct d e fin ed as h a m ( u, x m ) , a n ( u, x n ) i , 1 2 Z 1 − 1 a m ( u, x m ) a n ( u, x n ) du. where a n ( u, x n ) = e − j 2 π λ x n u is the n - th entry of the SV a ( u, x ) . T o ortho gonalize these patter n functions, we a p ply Cholesky decomposition den oted by R ( x ) = L ( x ) L H ( x ) . The in verse of the re su lting lower -triang ular matrix, L − 1 ( x ) , actually serves as th e transfo rmation matrix that explicitly constructs a set of orth o norma l pattern fu nctions (or effectiv e SV) ˇ a ( u, x ) = ( L ( x )) − 1 a ( u, x ) v ia Gr am-Schmid t pro- cess. This process proceed s orth ogono rmalization recursively , with the n -th, n = 1 , 2 , · · · , N , pattern fun ction a n ( u, x n ) orthon ormalized as: ˜ a n ( u, x n ) = a n ( u, x n ) − n − 1 X m =1 h ˇ a m ( u, x m ) , a n ( u, x n ) i ˇ a m ( u, x m ) , (3a) ˇ a n ( u, x n ) = ˜ a n ( u, x n ) p h ˜ a n ( u, x n ) , ˜ a n ( u, x n ) i . (3b) By defining the effective beam formin g vector as ˇ w = L H ( x ) w , the dir e c ti vity simplifies to a can onical Rayleigh quotient form ˚ G u ( x , w ) = | ˇ a H ( u, x ) ˇ w | 2 ˇ w H ˇ w . The o ptimal ef fective beamfor ming vector is thus ˇ w ∗ = ˇ a ( u, x ) || ˇ a ( u, x ) || 2 , and the corre- sponding max imum dir ectivity is given by : G u ( x ) = || ˇ a ( u, x ) || 2 2 = N X n =1 | ˇ a n ( u, x n ) | 2 , (4) which are consistent with results in (2). This transfor mation incorpo rates the co upling matrix R ( x ) into the effectiv e SV ˇ a ( u, x ) , yielding a math ematical form iden tical to th at of an u n coupled arr ay . The key difference is that the elem ents in th e effective SV , i.e. , { ˇ a n ( u, x n ) } N n =1 , ar e g e nerally n ot unimod ular, owing to the transform ation by ( L ( x )) − 1 . This formu lation re veals a fundamental insight. For any giv en FP A array , the average d ir ectivity over the entire ang ular space is 1 2 R 1 − 1 P N n =1 | ˇ a n ( u, x n ) | 2 du , which is al ways eq u al to N because th e e ffective pattern functions are orthono rmal, as defined in (3). In particular, for arrays with elem ents located at in teger multiples of h alf-wav elength (includ ing ULAH), we have R ( x ) = I N , ˇ a ( u, x ) = a ( u, x ) , and thus the directivity remains a constant of P N n =1 | a n ( u, x n ) | 2 = N over all direction s. In contrast, for MA arrays, th e directivity over any direction can exceed N by le vera g ing the coupling effect v ia APV optimiza tio n, which is de m onstrated by the following three exam ples. 1) Example I: W e first con sid e r a simp le two-antenna case (i.e., N = 2 ). W e set x 1 = 0 with o ut loss of gen e rality . Based on ( 3), th e effectiv e pattern function s a r e given b y ˇ a 1 ( u, 0) = 1 , ˇ a 2 ( u, x 2 ) = a 2 ( u, x 2 ) − sinc  2 x 2 λ  q 1 − sinc 2  2 x 2 λ  . Then the directivity over any given u can be expressed as G u  [0 , x 2 ] T  = 2 1 − co s  2 π x 2 u λ  sinc  2 x 2 λ  1 − sinc 2  2 x 2 λ  . 0 0.5 1 1.5 2 0 1 2 3 4 Directivity (0.72,2.55) (1.19,2.24) (0,4) Fig. 2: Directivity versus x 2 λ for N = 2 and u = 0 , 0 . 5 , 1 . 0.5 1 1.5 2 0 0.5 1 1.5 2 (a) n = 2 . 0.5 1 1.5 2 0 0.5 1 1.5 2 (b) n = 3 . Fig. 3: For an ULA with d > λ 2 , approx imation of | ˇ a n ( u, x n ) | 2 via (5) against squared magnitude of (3b) versus d for n = 2 , 3 and u = 0 , 0 . 5 , 1 . Over the broad side direction (i.e., u = 0 ), this simplifies to G u = 2 1+sinc ( 2 x 2 λ ) . The o ptimal p osition x 2 ≈ 0 . 72 λ yields a maximum directivity of G ∗ u ≈ 2 . 55 . Over the endfire directio n (i.e., u = 1 ), x 2 → 0 yields superdirectivity G ∗ u → 4 . Both cases a chieve a d irectivity larger than the num ber of anten nas ( N = 2 ). Howe ver , over other d irections, the optimal solution for x 2 becomes analytically com plicated. T he relation ship between directivity of th e two-antenna MA array a n d x 2 is depicted in Fig. 2. 2) Example II: F or an array with inter-antenna spacing greater than half-wavelength, i.e., | x m − x n | > λ 2 , ∀ m 6 = n , the coupling ef f ects ar e weak. For conv enience, we define k mn = 2 π ( x m − x n ) λ , ∀ m 6 = n , and th e magnitude o f coupling coefficient   sinc  k mn π    < 0 . 22 since k mn π > 1 . By neglecting higher-order te r ms o  sinc  k mn π  , ∀ m 6 = n , the orthogonal- ization in (3) can be simplified as follows: ˇ a 1 ( u, 0) ≡ 1 , ˇ a 2 ( u, x 2 ) ≈ − sinc  k 12 π  + a 2 ( u, x 2 ) , · · · , ˇ a N ( u, x N ) ≈ − N − 1 X n =1 sinc  k nN π  e − j k nN u + a N ( u, x N ) . Under this fir st-order ap proxim ation, the squ ared magnitud e of eac h or thogon alized p attern simplifies to | ˇ a n ( u, x n ) | 2 ≈ 1 − 2 n − 1 X m =1 sinc  k mn π  cos ( k mn u ) . (5) T o validate this approximation , we con sider an ULA with element spacing d > λ 2 , i.e., x n = ( n − 1 ) d . T he behavior of th e approximated | ˇ a n ( u, ( n − 1) d ) | 2 in (5) as a functio n of d λ is illustrated in Fig. 3. Consequently , the overall directivity G u ( x ) is given by N X n =1 | ˇ a n ( u, x n ) | 2 ≈ N − 2 X 1 ≤ m G u ( x ( t ) ) ; on the other h and, it satisfies the position co nstraints in (7b). Otherwise, α is halved in a backtrack ing line search until the cond itions are satisfied or α is be low the threshold ǫ . C. Algorithm Summary The prop osed algorith m is summarize d in Algorithm 1. Lines 1-4 corre spond to the GS stage, while lines 5 -16 co rre- spond to the GD stag e. Th e co n vergence of GD is g uaranteed because the objectiv e function is u pper-bound ed, and non- decreasing in iteratio ns. Th e co m putationa l com plexity is analyzed as follows. The dir ectivity ev alu a tio n in (4) of M candidate grid points fo r M n +1 in line 3 can be reduced Algorithm 1 GS-GD for An tenna Position Optimiza tio n Input: N , θ , λ, d max , d min , d g , α 0 , ǫ . 1: %% T he first stage: GS to select the antenna positions from grid points. 2: for n = 1 · · · N − 1 do 3: Optimize M n +1 by solving (8). 4: end for 5: %% The second stage: GD to continuo usly refine the positions via backtracking line search. 6: Initialize x (0) = x GS . 7: for t = 0 · · · T − 1 do 8: Compute the gradients ∂ G u ( x ) ∂ x n , n = 2 , · · · , N as (9). 9: Initialize α = α 0 , r = − 1 , x test = x ( t ) . 10: while r < 0 or x test does not satisfy (7b ) do 11: Update x test as (10). 12: Compute r = G u  x test  − G u ( x ( t )) . 13: α ← 0 . 5 α . 14: if α < ǫ then 15: Return x ∗ = x ( t ) . 16: end if 17: end while 18: Update x ( t + 1) = x test . 19: end for 20: x ∗ = x ( T ) . Output: x ∗ . Parameter V alue Number o f antennas N 5 W av elength λ 0 . 3 m Minimum spa c in g d min λ/ 10 = 0 . 03 m Aperture size d max N − 1 2 λ , ( N − 1) λ , 2( N − 1) λ Grid spa c in g d g λ/ 20 = 0 . 015 m GS-GD iterations T 5 Initial step size α 0 1 T oleran c e ǫ 10 − 3 T ABLE II: Main simulation parameters. to computing { ˇ a n +1 ( u, x ) } x ∈ ˜ X with complexity O ( M n 2 ) , since  ˇ a m  u, x GS m  n m =1 are r ecursively computed in former steps. The comp lexity of calculating the gradien ts in line 8 is O  N 2  . Th e com plexity of calc u lating the objective f unction in line 12 is O  N 3  , and thus th e maximum complexity of the en tire wh ile loop is O  log 2  α 0 ǫ  N 3  . Hence, the overall computatio nal com plexity of th e p r oposed Algorith m 1 is O  M N 3 + T  N 2 + log 2  α 0 ǫ  N 3  . I V . N U M E R I C A L R E S U LT S This sectio n presen ts numeric a l re sults to validate the direc- ti vity e n hanceme nt of the MA array by takin g into the antenna coupling effect and the perfor mance of the pr oposed GS-GD algorithm . For c la r ity , the directivity is pr esented as a fu nction of the angle θ rath er than u in this section. Further more, the resu lts are confined to θ ∈ [0 ◦ , 9 0 ◦ ] , as the directivity is symmetric w .r .t. 90 ◦ . The main simulation parameters are summarized in T able II for ease of reference. First, we com pare the dir ectivity of th e optimized cou p led MA ar ray with the trad itional ULAH. In simulations, we employ an ES over all d iscretized g rid points to app roximate the glo bally optim al solution for prob lem (7). T he resu lts are shown in Fig. 4. The results dem onstrate that, over all directions, the opti- mized MA array achiev es a hig her directivity compared to the 0 10 20 30 40 50 60 70 80 90 (°) 0 5 10 15 20 25 Directivity (90,7.8) (60,6.1) (30,9.2) (0,24.2) Fig. 4: Directivity of the coupled MA array compared to the UL AH in different values of θ and d max . 0 10 20 30 40 50 60 70 80 90 (°) 0 5 10 15 20 25 Directivity Fig. 5: Comparison of the propo sed GS-GD algorithm with baselines in different values of θ . ULAH. Specifically , whe n θ → 0 ◦ , the supe r directive array is o ptimal, and th e direc tivity appr oaches N 2 . As θ increases, the optim al antenna positions exhib it irregular patterns, and there is at least 2 0% imp rovement in the directivity fo r the coupled MA ar ray compar ed to the trad itional ULAH without antenna cou pling wh en the movable region is sufficiently large. Wh e n θ → 90 ◦ , the o ptimal arr ay conver ges to a nearly uniform a r ray with spacing ar ound 0 . 8 λ , and there is around 50 % directivity g ain, which are consistent with the analysis in Section II- B2. Ad ditionally , over the vast major ity of d irections, the maxim um directivity for d max = ( N − 1 ) λ is the same as that for d max = 2 ( N − 1) λ , bo th o f which substantially exceed the directivity f or d max = 1 2 ( N − 1) λ , which implies that d max = ( N − 1) λ is sufficient to achiev e most of the directivity gains in practice. Next, we evaluate the perform ance of the propo sed GS-GD algorithm a gainst the following baseline m ethods: • ES: This method selects anten n a position s from grid points, and com pute the directivity of all feasible A PV cases to find the maximum directivity . • GS: This meth o d only perfor ms the first stage as men- tioned in Section III-A. • GD: This method in itializes the APV as a ULAH, and then perf orms the secon d stage as mentio ned in Section III-B, wher e we set T = 30 in s imulation s to e n sure conv ergence. • ULAH: For ULAH witho ut an tenna c oupling , the direc- ti vity is N over all directions. The perform ance of th e algorithm s is evaluated in Fig. 5. As shown in the figure, the p erform ance of the propo sed GS-GD approach e s ES over most directio ns. Moreover , GS- GD o utperfo rms the two baselines (GS and GD) over all directions. Th is su p eriority arises from its ab ility to mitigate the in herent limitations of both methods: G S is constrained by its sequential optimization process, which myop ically adjusts individual antenna p ositions, thus lacking the capability to co-optim ize th e entir e array , wh ile GD is pron e to local optima wh en initialized fro m a ULAH. By le veraging both approa c h es, GS-GD enab les a more robust search, c o nsistently conv erging to a so lu tion c lo ser to the global o p timum. V . C O N C L U S I O N In this p aper, we inv estigated the directivity of MA arr ays with antenna coup ling. In contrast to traditional uncoupled arrays, coupled MA ar rays can e nhance the d irectivity (i.e., beamfor ming gain) by co ncentratin g the rad iated p ower more effecti vely toward specific directions. W e aimed to optimize the positions of anten nas f or maximizing the dir ectivity of the MA ar ray over any given d irection. A low-complexity GS- GD a lgorithm was proposed , wh e re the antenna positions ar e first selected seque ntially from discrete g rid poin ts thr ough GS and th en contin uously refined throug h GD optimization. Numerical results d emonstrated the direc tivity ga in o f the optimized coup led MA array and th e efficiency of the proposed GS-GD algorithm. R E F E R E N C E S [1] J. G, I. Singh, and D. K . Choudha ry , “Gain and isola tion improv ement techni ques for MIMO antenna: A compendi ous survey , ” Resul ts in Engineerin g , vol. 25, p. 104482, Mar . 2025. [2] T . T . V o, L. Ouvry , A. Sibille, and S. Bories, “Mutual coupling modeling and calibra tion in antenna arrays for A OA estima tion, ” in 2018 2nd URSI Atl. R adio Sci. Meet. -RASC . Gran Cana ria: IEEE, May 2018, pp. 1–4. [3] T . L. Marzet ta, “Super-di recti ve antenna a rrays: Fun damentals and ne w perspecti ves, ” in 2019 53rd A silomar Conf . Signals Syst. Comput. Paci fic Grov e, CA, USA: IEEE, Nov . 2019, pp. 1–4. [4] A. Pizz o and A. Lozano, “Superdi recti vity in linear holographic MIMO, ” in 2024 IEEE 25th Int. W orkshop Signa l Pr ocess. Adv . W ir el. Commun. SP A WC . Lucca, Italy: IEEE, Sep. 2024, pp. 386–390 . [5] L. Han, H. Y in, M. Gao, and J. Xie, “ A superdire cti ve beamforming approac h wi th impedance coupli ng and fi eld coupling for compact antenna arrays, ” IEEE Open J. Commun. Soc. , vol. 5, pp. 7262–7277, 2024. [6] L. Zhu, W . Ma, and R. Zhang, “Modeling and perfo rmance analysis for mov able antenn a enabled wireless communications, ” IEEE T rans. W ir el. Commun. , vol. 23, no. 6, pp. 6234–6250, Jun. 2024. [7] — —, “Mov able antenna s for wire less communication : Opportunitie s and chall enges, ” IEEE Commun. Mag. , vol. 62, no. 6, pp. 114– 120, Jun. 2024. [8] W . Ma, L. Z hu, a nd R. Z hang, “Compressed sensing ba sed cha nnel estimati on for m ov able ante nna communicat ions, ” IEE E Commun. Lett. , vol. 27, no. 10, pp. 2747–2751, Oct. 2023. [9] X. Shao, Q. Jiang, and R. Zhang, “6D mov able antenn a based on user distrib ution: Modeling and optimization , ” IEEE T rans. W ir el. Commun. , vol. 24, no. 1, pp. 355–370, Jan. 2025. [10] X. Shao, R. Zhang, Q. Jiang, and R. Schober , “ 6D mova ble antenna enhanc ed wireless netw ork via discrete position and rotati on optimiza - tion, ” IE E E J. Sel. Areas Commun. , vol. 43, no. 3, pp. 674–687, Mar . 2025. [11] L. Zhu, W . Ma, and R. Z hang, “Mov able-anten na array enhanced beam- forming: A chievi ng full array gai n with null steeri ng, ” IEEE Commun. Lett. , vol. 27, no. 12, pp. 3340–3 344, Dec. 2023. [12] P . Mursia, F . Dev oti, M. Rossanese, V . Sci ancalep ore, G. Gradoni, M. D. Renzo, and X. Costa-Pére z, “T3dris: Adv ancing conformal ris design through in-depth analysis of mutual coupling ef fects, ” IEE E T rans. Commun. , vol. 73, no. 2, pp. 889–903, Feb . 2025. [13] J. Zhu, F . Han, Y . Guo, and Y . W ang, “Mutual coupling-en hanced mov able antenna arrays: Breaking minimum-spacing constraint s, ” IEEE Commun. Let t. , vo l. 30, pp. 672–676, 2026. [14] D. Y . Levin, S. Markovi ch-Golan, and S. Gannot, “Near- field superdi- recti vity: An analytical perspecti ve, ” IEEE/ACM T rans. Audio Speech Lang. Pro cess. , v ol. 29, pp. 1661–1 674, 2021.