Fundamentals of Array Antennas

Fundamentals of Array Antennas Nobuyoshi Kikuma 466-8555 Dept. of Computer Science and Engineering, Nagoya Institute of Technology Gokiso-cho, Showa-ku, Nagoya, 466-8555, Japan Abstract Array antenna technologies have contributed to development and progress of recent wireless systems. This lecture summarizes fundamentals of those array antenna technologies and also explains the mechanism and basic performances of array antennas. In addition, relating technologies, such as adaptive array antennas and direction-of-arrival estimation methods, are presented. 1. [1] LAN MUSIC [2] [7] [2], [8], [9] 2. 2. 1 1 K θ 1 E 0 g(θ) k ( E k = E 0 g(θ)exp j 2π ) λ d k sin θ (1) (k =1, 2,...,K) 1

λ d k k 1 E sum E sum = E 0 g(θ)d(θ) (2) K { D(θ) = A k exp j ( 2πλ )} d k sin θ + δ k (3) k=1 A k δ k k D(θ) (2) g(θ) D(θ) (pattern multiplication) 2π λ d k sin θ gm + δ k =2mπ (m = ±1, ±2,...) (5) θ gm grating lobe 2 (2) E sum θ 2 null -30 0 30-30 30-60 } 0 60-60 60-90 90-30 -30-20 -20-10 -10-40 0 0-30 -30-20 -20 0-10 -10-40 0 [db] [db] -90 90 d k ( c ) sinθ = cτ k θ (a) 6 (b) 6 2: d K d k d 1 #K #k #1 A K A k A 1 K k 1 Σ 1: K θ 0 δ k δ k = 2π λ d k sin θ 0 (4) (3) A k Dolph-Chebyshev array [10] 2. 2 2

3 K #1 #2 θ #K x 1 (t) x 2 (t) w 1 w 2 + x K (t) w K y(t) x(t) w y(t) 3: K x k (t) k w k k y(t) x(t) = Δ [x 1 (t),x 2 (t),...,x K (t)] T (6) w = Δ [w 1,w 2,...,w K ] T (7) y(t) y(t) =w H x(t) =x T (t)w (8) T H w k x k (t) 1 A k δ k wk = A k exp(jδ k ) (9) () (k, l) k l R xx Δ = E[x(t)x H (t)] (10) E[ ] P out = 1 2 E[ y(t) 2 ]= 1 2 wh R xx w (11) [1] SINR(Signal-to-Interference-plus-Noise Ratio) SINR = Δ + (12) db θ 0 x(t) =s(t)a(θ 0 )+n(t) (13) [ a(θ 0 )= g 1 (θ 0 )exp ( j 2πλ ) d 1 sin θ 0 ( g K (θ 0 )exp j 2π )] T λ d K sin θ 0 (14) s(t) a(θ 0 ) n(t) 3

g k (θ 0 ) k θ 0 g k (θ) =1(k =1, 2,...,K) y(t) SINR y(t) =s(t)w H a(θ 0 )+w H n(t) (15) SINR = E[ s(t)wh a(θ 0 ) 2 ] E[ w H n(t) 2 ] = E[ s(t) 2 ] w H a(θ 0 ) 2 w H E[n(t)n H (t)]w = P s w H a(θ 0 ) 2 P n w H w (16) (17) (18) P s = E[ s(t) 2 ] P n E[n(t)n H (t)] = P n I I SINR w H w = (18) w = a(θ 0 ) (19) (phased array) [10] 3. 3. 1 adaptive beamforming adaptive null steering [1] 1) (Minimum Mean Square Error: MMSE) 2) SNR (Maximum Signal-to-Noise ratio: MSN) 3) (Directionally Constrained Minimization of Power: DCMP) 4) (Power Inversion: PI) 5) (Constant Modulus Algorithm: CMA) 1) 4) 3. 2 MMSE MMSE 1960 Widrow [11] Widrow LMS Compton [12], [13] LMS LMS MMSE MMSE LMS MMSE 4

e(t) r(t) y(t) e(t) =r(t) y(t) =r(t) w H x(t) (20) E [ e(t) 2] = E [ r(t) y(t) 2] (21) = E [ r(t) w H x(t) 2] (22) = E [ r(t) 2] w T rxr w H r xr + w H R xx w (23) r xr r xr = E[x(t)r (t)] (24) w (23) (23) w R xx w (23) w opt w opt = R 1 xx r xr (25) MMSE [14] MMSE [15] 3. 3 MSN SNR MSN 1950 Howells IF [1] Applebaum SNR Howells-Applebaum loop MSN [16] Howells-Applebaum(HA) MSN SNR x(t) s(t) u(t) n(t) x(t) =s(t)+u(t)+n(t) (26) y s (t) y u (t) y n (t) y s (t) =w H s(t) =s T (t)w (27) y u (t) =w H u(t) =u T (t)w (28) y n (t) =w H n(t) =n T (t)w (29) P Sout = 1 2 E [ y s (t) 2] = 1 2 wh R ss w (30) P Uout = 1 2 E [ y u (t) 2] = 1 2 wh R uu w (31) P Nout = 1 2 E [ y n (t) 2] = 1 2 P nw H w (32) R ss R uu P n s(t) s(t) =s(t)v s = s(t)a(θ s ) (33) s(t) θ s v s R ss R ss = E [ s(t)s H (t) ] = P s v s v H s (34) P s SNR SINR SNR = P Sout = wh R ss w P Uout + P Nout w H R nn w (35) 5

R nn R nn = R uu + P n I (36) SNR w (35) SNR SNR w (35) (34) R xx = P s v s v H s + R nn (37) w opt = R 1 xx v s (38) [1] MSN v s = a(θ s ) MSN 3. 4 DCMP Frost MMSE fidelity constraint CMP [17] Frost fidelity constraint ( w 1 + w 2 + + w K =1 ) Frost prefilter (DCMP) [18] [19] 3 K DCMP : Directionally Constrained Minimization of Power N C T w = h (39) C =[c 1 c 2 c N ] (40) h =[h 1 h 2 h N ] T (41) c n (n =1,..., N) C h n (n =1,..., N) c n h h w H a(θ s )=h (42) y s (t) =w H {s(t)a(θ s )} = s(t)w H a(θ s )=hs(t) (43) c = a(θ s ) (39) DCMP DCMP ( min P out = 1 ) w 2 wh R xx w subject to C T w = h Lagrange [1] w opt = Rxx 1 C(C H Rxx 1 C) 1 h (44) N =1: c T w = h (44) w opt = γr 1 xx c, γ Δ = h c H R 1 xx c 3. 5 (45) PIAA : Power Inversion Adaptive Array (K 1) 6

[20] MMSE MSN DCMP PIAA PIAA PIAA CMPPIAA PIAA [1] 4. w opt = Rxx 1 t (46) t =[1, 0,, 0] T (47) 4. 1 LAN ( ) [4] Capon (LP:Linear Prediction) [2] MUSIC ESPRIT [4], [5] [2], [8], [9] Capon DCMP LP [9] 4. 2 3 K L s l (t) θ l (l =1, 2,...,L) a(θ l ) L x(t) = s l (t)a(θ l )+n(t) (48) l=1 = As(t)+n(t) (49) A =[a(θ 1 ), a(θ 2 ),, a(θ L )] (50) s(t) =[s 1 (t),s 2 (t),,s L (t)] T (51) A n(t) 0 ( ) σ 2 (= P n ) R xx R xx = E[x(t)x H (t)] = AE[s(t)s H (t)]a H + E[n(t)n H (t)] = ASA H + σ 2 I (52) S = Δ E[s(t)s H (t)] (53) (53) S S = diag{p 1,P 2,,P L } (54) Δ [ P l = E sl (t) 2] (l =1, 2,...,L) (55) P l 7

4. 2. 1 (beamformer) (uniform) () θ 2.2 ( ) w = a(θ) (56) θ 90 90 θ a(θ) θ (mode vector) P out = 1 2 ah (θ)r xx a(θ) (57) ( ) P BF (θ) = P out a H (θ)a(θ)/2 = ah (θ)r xx a(θ) a H (θ)a(θ) (58) R xx a(θ) P BF (θ) θ P BF (θ) [1] 4. 2. 2 Capon Capon (DCMP) (45) c = a(θ) h =1 P out = 1 2 wh CPR xx w CP (60) = 1 2a H (θ)r 1 xx a(θ) (61) Capon P CP (θ) =2P out = 1 a H (θ)r 1 xx a(θ) (62) R xx a(θ) P CP (θ) θ 4. 2. 3 Capon (Linear Prediction) 2 K 1 ˆx 1 (t) = K wkx k (t) (63) k=2 ˆx 1 (t) 1 ε(t) ε(t) Δ = x 1 (t) ˆx 1 (t) = K wkx k (t) (64) k=1 = w H x(t) (w 1 1) (65) w 2 w CP = R 1 xx a(θ) a H (θ)r 1 xx a(θ) (59) E[ ε(t) 2 ]=w H R xx w =2P out (66) 8

w 1 =1 (PIAA) w LP (46), (47) w LP K 1 > = L P LP (θ) = 1 w H LP a(θ) 2 (67) Capon R xx a(θ) P LP Capon R xx = ASA H + σ 2 I R xx e i =(ASA H + σ 2 I)e i (71) = ASA H e i + σ 2 e i (72) = μ i e i + σ 2 e i (73) =(μ i + σ 2 )e i (i =1, 2,...,K) (74) Δ λ i = μi + σ 2 (i =1, 2,...,K) (75) R xx λ 1 > = λ 2 > = > = λ L >λ L+1 = = λ K = σ 2 (76) σ 2 L λ L+1,...,λ K R xx e i =(ASA H + σ 2 I)e i = ASA H e i + σ 2 e i 4. 2. 4 MUSIC MUSIC(MUltiple SIgnal Classification) S L A L R xx = ASA H L [1] μ i (i =1, 2,...,K) e i (i =1, 2,...,K) ASA H e i = μ i e i (i =1, 2,...,K) (68) μ 1 > = μ 2 > = μ L >μ L+1 = = μ K = 0 (69) e H i e k = δ ik (i, k =1, 2,...,K) (70) δ ik = λ i e i = σ 2 e i (77) (i = L +1,,K) ASA H e i = 0 (i = L +1,,K) (78) A S A H e i = 0 (i = L +1,,K) (79) a H (θ l )e i =0 (l =1, 2,...,L; i = L +1,...,K) (80) e L+1,...,e K L {e 1,, e L } (signal subspace) 9

{e L+1,, e K } (noise subspace) MUSIC (K L) P MU (θ) Δ = a H (θ)a(θ) (81) K e H i a(θ) 2 i=l+1 a H (θ)a(θ) = a H (θ)e N EN Ha(θ) (82) Δ E N = [el+1,, e K ] (83) MUSIC θ L {θ 1,,θ L } (76) K > = L +1 5. [1] :, (2003). [2] S.U.Pillai : Array Signal Processing, Springer- Verlag New York Inc. (1989). [3] H.Krim and M.Viberg : Two Decades of Array Signal Processing Research The Parametric Approach, IEEE Signal Processing Magazine, vol.13, No.4, pp.67 94 (July 1996). [4] R.O.Schmidt : Multiple Emitter Location and Signal Parameter Estimation, IEEE Trans., vol.ap-34, No.3, pp.276 280 (Mar. 1986). [5] R.Roy and T.Kailath : ESPRIT Estimation of Signal Parameters via Rotational Invariance Techniques, IEEE Trans., vol.assp-37, pp.984 995 (July 1989). [6], 33, pp.77 84 2006. [7] P.J.Chung and J.F.Bohme DOA estimation using fast EM and SAGE algorithms, Signal Processing, vol.82, pp.1753 1762, Nov. 2002. [8] Y.Ogawa and N.Kikuma : High-Resolution Techniques in Signal Processing Antennas, IEICE Trans. Commun., vol.e78-b, No.11, pp.1435 1442 (Nov. 1995). [9] W.F.Gabriel : Spectral Analysis and Adaptive Array Supperresolution Techniques, Proc. IEEE, vol.68, No.6, pp.654 666 (June 1980). [10] C.A.Balanis, Antenna Theory: Analysis and Design, Wiley-Interscience, 3rd Ed., 2005. [11] B.Widrow, et al. : Adaptive Antenna Systems, Proc. IEEE, vol.55, No.12, pp.2143 2159 (Dec. 1967). [12] R.L.Riegler and R.T.Compton,Jr. : An Adaptive Array for Interference Rejection, Proc. IEEE, vol.61, No.6, pp.748 758 (June 1973). [13] R.T.Compton, Jr. : An Adaptive Array in a Spread-Spectrum Communication System, Proc, IEEE, vol.66, No.3, pp.289 298 (Mar. 1978). [14] Y.Ogawa, et al. : Fading Equalization Using an Adaptive Antenna for High-Speed Digital Mobile Communications, Proc. ISAP, vol.4, 4A2-3, pp.857 860 (Aug. 1989). [15] Y.Ogawa, et al. : An LMS Adaptive Array for Multipath Fading Reduction, IEEE Trans. Aerosp. & Electron. Syst., vol.aes-23, No.1, pp.17 23 (Jan. 1987). [16] S.P.Applebaum : Adaptive Arrays, IEEE Trans. Antennas & Propag., vol.ap-24, No.5, pp.585 598 (Sept. 1976). [17] O.L.Frost,III : An algorithm for linearly constrained adaptive array processing, Proc. IEEE, 60, 8, pp.926 935 (Aug. 1972). [18] K.Takao, et al. : An Adaptive Antenna Array under Directional Constraint, IEEE Trans. Antennas & Propag. vol.ap-24, No.5, pp.662 669 (Sept. 1976). [19] K.Takao and N.Kikuma : Tamed Adaptive Antenna Array, IEEE Trans. Antennas & Propag. vol.ap-34, No.3, pp.388 394 (Mar. 1986). [20] R.T.Compton, Jr. : The Power Inversion Adaptive Array : Concept and Performance, IEEE Trans. Aerosp. & Electron. Syst., vol.aes-15, No.6, pp.803 814 (Nov. 1979). 10