Fundamentals of Array Antennas

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1 Fundamentals of Array Antennas Nobuyoshi Kikuma Dept. of Computer Science and Engineering, Nagoya Institute of Technology Gokiso-cho, Showa-ku, Nagoya, , 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] 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

2 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 } [db] [db] 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]

3 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

4 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] 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

5 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

6 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 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

7 [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] 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

8 (beamformer) (uniform) () θ 2.2 ( ) w = a(θ) (56) θ 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] 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 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

9 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 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

10 {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] :, (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 (July 1996). [4] R.O.Schmidt : Multiple Emitter Location and Signal Parameter Estimation, IEEE Trans., vol.ap-34, No.3, pp (Mar. 1986). [5] R.Roy and T.Kailath : ESPRIT Estimation of Signal Parameters via Rotational Invariance Techniques, IEEE Trans., vol.assp-37, pp (July 1989). [6], 33, pp [7] P.J.Chung and J.F.Bohme DOA estimation using fast EM and SAGE algorithms, Signal Processing, vol.82, pp , Nov [8] Y.Ogawa and N.Kikuma : High-Resolution Techniques in Signal Processing Antennas, IEICE Trans. Commun., vol.e78-b, No.11, pp (Nov. 1995). [9] W.F.Gabriel : Spectral Analysis and Adaptive Array Supperresolution Techniques, Proc. IEEE, vol.68, No.6, pp (June 1980). [10] C.A.Balanis, Antenna Theory: Analysis and Design, Wiley-Interscience, 3rd Ed., [11] B.Widrow, et al. : Adaptive Antenna Systems, Proc. IEEE, vol.55, No.12, pp (Dec. 1967). [12] R.L.Riegler and R.T.Compton,Jr. : An Adaptive Array for Interference Rejection, Proc. IEEE, vol.61, No.6, pp (June 1973). [13] R.T.Compton, Jr. : An Adaptive Array in a Spread-Spectrum Communication System, Proc, IEEE, vol.66, No.3, pp (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 (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 (Jan. 1987). [16] S.P.Applebaum : Adaptive Arrays, IEEE Trans. Antennas & Propag., vol.ap-24, No.5, pp (Sept. 1976). [17] O.L.Frost,III : An algorithm for linearly constrained adaptive array processing, Proc. IEEE, 60, 8, pp (Aug. 1972). [18] K.Takao, et al. : An Adaptive Antenna Array under Directional Constraint, IEEE Trans. Antennas & Propag. vol.ap-24, No.5, pp (Sept. 1976). [19] K.Takao and N.Kikuma : Tamed Adaptive Antenna Array, IEEE Trans. Antennas & Propag. vol.ap-34, No.3, pp (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 (Nov. 1979). 10

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