4 1 7 Ver.1/ MIMO MIMO Multiple Input Multiple Output MIMO = = MIMO LAN IEEE802.11n MIMO Alamouti STBC Space Time Block Code
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1 MIMO MIMO Multiple Input Multiple Output MIMO = = MIMO LAN IEEE802.11n MIMO Alamouti STBC Space Time Block Code MIMO 7-2 MIMO 7-3 MIMO MIMO 7-4 MIMO c /(18)
2 MIMO s 1 (i) #1 #1 y 1 (i) #Nt #Nr s Nt (i) y Nt (i) 7 1 MIMO 7 1 N t N r MIMO T it k (1 k N t ) s k (i) l (1 l N r ) k h lk l y l (i) N t y l (i) = h lk s k (i) + n l (i) k=1 (7 1) n l (i) l N r Y(i) Y H (i) = [ y 1 (i) y 2 (i) y N r (i) ] H Y(i) (7 1) (7 2) Y(i) = HS(i) + n(i) (7 3) H h lk N r N t H = [ h 1 h 2 h Nt ] h H k = [ h 1k h 2k h N rk ] (7 4) (7 5) S(i) n(i) N t N r S H (i) = [ s 1 (i) s 2 (i) s N t (i) ] n H (i) = [ n 1 (i) n 2 (i) n N r (i) ] (7 6) (7 7) c /(18)
3 S(i) N t 0 Nt R s = S(i)S H (i) n(i) 0 Nr R n = n(i)n H (i) Y(i) 0 Nr R y (7 3) R y = Y(i)Y H (i) = H S(i)S H (i) H H + n(i)n H (i) = HR s H H + R n (7 8) S(i) n(i) MIMO I[S(i), Y(i)] I[S(i), Y(i)] = dy { } pys [Y(i), S(i)] ds p ys [Y(i), S(i)] log 2 p y [Y(i)]p s [S(i)] 1) p ys [Y(i), S(i)] Y(i) S(i) p y [Y(i)] p s [S(i)] Y(i) S(i) I[S(i), Y(i)] (7 9) S(i) p s [S(i)] MIMO C (7 9) C = max I[S(i), Y(i)] p s[s(i)] (7 10) 1) W Hz WC (7 10) C (7 9) I[S(i), Y(i)] = dy p y [Y(i)] log 2 p y [Y(i)] + ds p s [S(i)] dy p ys [Y(i) S(i)] log 2 p ys [Y(i) S(i)] (7 11) p ys [Y(i) S(i)] S(i) Y(i) p ys [Y(i) S(i)] n (i) (7 3) p ys [Y(i) S(i)] = 1 π Nr detr n exp { [Y(i) HS(i)] H R 1 n [Y(i) HS(i)] } (7 12) c /(18)
4 det( ) (7 11) ds p s [S(i)] dy p ys [Y(i) S(i)] log 2 p ys [Y(i) S(i)] = { ( ) ds p s [S(i)] log 2 π N r 1 detr n log 2 tr } [Y(i) HS(i)][Y(i) HS(i)] H R 1 n = log 2 ( π N r detr n ) N r log 2 (7 13) p s [S(i)] tr( ) (7 11) Y(i) Y(i) p y [Y(i)] p y [Y(i)] = 1 π Nr detr y exp [ Y(i) H R 1 y Y(i) ] (7 14) S(i) Y(i) (7 11) (7 13) ( ) dy p y [Y(i)] log 2 p y [Y(i)] = log 2 π N r N r detr y + (7 15) log 2 (7 13) (7 10) MIMO C C = log 2 det ( R 1 n R y ) = log2 det ( I Nr + R 1 n HR s H H) (7 16) (7 8) I N r N r s k (i) S(i) R s N r R s = P t N t I Nt (7 17) P t n(i) R n = σ 2 ni Nr (7 18) σ 2 n (7 16) C ( C = log 2 det I Nr + P ) t HH H N t σ 2 n (7 19) HH H N r N r Q Q min(n t, N r ) HH H N r N r U c /(18)
5 HH H = UDU H D = diag[λ 1, λ 2,..., λ Q, 0,..., 0] (7 20) (7 21) D N r N r diag[ ] λ q (1 q Q) HH H (7 20) (7 19) [ ( C = log 2 det U I Nr + ( = log 2 det I Nr + = log 2 Q = Q q=1 q=1 ( 1 + P tλ q N t σ 2 n P t N t σ 2 n P ) t D N t σ 2 n ) ( log P ) tλ q N t σ 2 n ) ] D U H (7 22) MIMO Q q SNR P tλ q N t σ 2 n N t = N r = N H 0 1 H C P t 2, N 3) W W C 1) ) G.J. Foschini, Layered space-time architecture for wireless communication in a fading environment when using multi-element antennas, Bell Labs Technical Journal, vol.1, no.2, pp.41-59, ) G.J. Foschini and M.J. Gans, On limits of wireless communications in a fading environment when using multi-element antennas, Wireless Personal Communications, vol.6, no.3, pp , c /(18)
6 STC: Space-Time Code STC STBC: Space-Time Block Code STTC: Space-Time Trellis Code 2 STC 2 STBC W-CDMA STBC STBC N = 2 STBC 1) STBC X 1 X 2 1, 2 X 1, X X 2 2 X 1 STBC 1 1 STBC M 2M M = 1 1, 2 H 1, H 2 STBC 2 2 Y 1, Y 2 STBC X 1 = H 1 Y 1 + H 2 Y 2 (7 23) X 2 = H 2 Y 1 H 1 Y 2. (7 24) STBC N = 1 M = 2 MRC 3 db STBC STBC N > 2 Tarokh 3) Ganesan 4, 5) PAM STBC 3) STBC N = 2, 4, 8 c /(18)
7 3) N > 2 STBC 1/2 R 3/4 STBC Jafarkhani 6) STBC STBC Jafarkhani STBC partial diversity ML STTC STTC N M STTC QPSK, 4 STTC t U t = (U t,1, U t,2 ) G modulo 4 (X t,1,., X t,n ) N QPSK 1 S k 0 00,01,02, ,11,12, ,21,22,23 30,31,32, QPSK, 4 STTC 2) 7 2 QPSK 2) QPSK, 4 STTC QPSK, 4 STTC 2) G G = (7 25) S k 0, 1, 2, c /(18)
8 1) S. Alamouti, Space block coding: A simple transmitter diversity technique for wireless communications, IEEE J. Select. Areas. Commun., vol.16, no.5, pp , Oct ) V. Tarokh, N. Seshadri, and A.R. Calderbank, Space-time codes for high data rate wireless communication: Performance criterion and code construction, IEEE Trans. Inform. Theory, vol.44, pp , March ) V. Tarokh, H. Jafarkhani, and A.R. Calderbank, Space-time block codes from orthogonal designs, IEEE Trans. Inform. Theory, vol.45, no.4, pp , July ) G. Ganesan and P. Stoica, Space-time diversity using orthogonal and amicable orthogonal designs, Wireless Personal Commun., vol.18, pp , Aug ) G. Ganesan and P. Stoica, Space-time block codes: A maximum SNR approach, IEEE Trans. Inform. Theory, vol.47, no.4, pp , May ) H. Jafarkhani, A quasi-orthogonal space-time block code, IEEE Trans. Commun.,, vol.49, no.1, pp.1-4, Jan c /(18)
9 MIMO MIMO M 1 m (1 m M) N rm N t N r N r = M m=1 N rm m it (7 2) N rm Y m (i) Y m (i) = H m S(i) + n m (i) (7 26) H m n m (i) N rm N t N rm S(i) (7 6) N t M S(i) = F m d m (i) m=1 (7 27) d m (i) m p m p m (1 p m N rm ) m F m d m (i) N t p m (7 27) S(i) S(i) = Fd(i) F = [F 1 F 2 F M ] d H (i) = [d H 1 (i) dh 2 (i) dh M (i)] (7 28) (7 29) (7 30) F d(i) (7 29) (7 30) N t P P P = M m=1 p m (7 26) Y m (i) N r Y(i) Y H (i) = [Y H 1 (i) YH 2 (i) YH M (i)] (7 31) Y(i) = HS(i) + n(i) (7 32) c /(18)
10 H n(i) N r N t N r H H = [H H 1 HH 2 HH M ] n H (i) = [n H 1 (i) nh 2 (i) nh M (i)] (7 33) (7 34) P N t (7 29) F m N rm = 1 p m = 1 P = N r N r N t 1) 1 Channel Inversion HH H N r N r 0 F H H H (HH H ) 1 F = 1 ξ H H (HH H ) 1 (7 35) S(i) = 1 ξ H H (HH H ) 1 d(i) (7 36) ξ S H (i)s(i) = P t ξ = 1 P t d H (i)(hh H ) 1 d(i) (7 37) (7 32) (7 36) Y(i) = 1 ξ d(i) + n(i) (7 38) 0 HH H ill condition (7 37) ξ SNR F F = 1 ξ H H (HH H + ζi Nr ) 1 (7 39) ζ ξ F 0 ζ = M/P t SINR 2) c /(18)
11 2 Sphere Decoding d(i) τd H -1 S(i) H n(i) y 1 (i) mod τ Encoding Fading channel y M (i) mod τ (a) Modulo vector precoding d(i) d 1 d 2 d K -r 21 /r 22 Q H mod τ mod τ S(i) H Fading channel n(i) y 1 (i) mod τ y M (i) mod τ (b) QR-based, successive precoding 7 3 Sphere decoding (7 35) F HH H SNR modulo vector precoding 3) 7 3(a) d(i) d d(i) d(i) + τ d d = a + jb (7 40) (7 41) j τ a b N r (7 41) d ξ d = arg min d [d(i) + τ d] H (HH H ) 1 [d(i) + τ d] (7 42) (7 32) Y(i) = 1 ξ [d(i) + τ d] + n(i) (7 43) Y(i) modulo τ/ ξ τ d/ ξ f τ [Y(i)] = 1 ξ d(i) + n(i) (7 44) f τ [ ] modulo c /(18)
12 τ τ = 2(d max + /2) (7 45) d max (7 42) d N r N r = N t QR-based successive precoding 4) 7 3(b) H QR H = RQ (7 46) R N r N r Q N r N r N r S(i) S(i) = Q H S(i) r 11 s 1 HS(i) = R S(i) = r 21 s 1 + r 22 s 2 = Dd(i) (7 47). r pq R (p, q) s p S(i) p D N r N r D = diag[r 11 r 22 r NrN r ] (7 48) (7 47) S(i) s 1 = d 1 (i) s 2 = d 2 (i) r 21 s 1 r 22 s 3 = d 3 (i) r 31 s 1 r 32 s 2 r 33 r 33. d p (i) d(i) p S(i) modulo s 1 = d 1 (i) [ s 2 = f τ d 2 (i) r ] 21 s 1 = d 2 (i) r 21 r 22 ] s 3 = f τ [ d 3 (i) r 31 r 33 s 1 r 32 r 33 s 2. r 22 s 1 + τ d 2 = d 3 (i) r 31 r 33 s 1 r 32 r 33 s 2 + τ d 3 (7 49) c /(18)
13 d p modulo DPC Dirty Paper Coding (7 47) S(i) S(i) = R 1 D[d(i) + τ d] (7 50) d d p N r S(i) S(i) = Q H R 1 D[d(i) + τ d] (7 51) τ d (7 49) modulo (7 42) ) 1 Channel Block Diagonalization channel block diagonalization HF H 1 0 HF =... (7 52) 0 H M H m N rm p m 0 0 F (N r N rm ) N t H m H m = [H H 1 HH m 1 HH m+1 HH M ]H (7 53) H m Q m H m = Ũ m Σ m [Ṽ 1 m Ṽ0 m] H (7 54) Ũ m (N r N rm ) (N r N rm ) Σ m Q m 0 0 (N r N rm ) N t Ṽ 1 m Q m N t N t Q m Ṽ 0 m 0 N t N t (N t Q m ) Ṽ 0 m H m H m (m m) c /(18)
14 m N rm (N t Q m ) H m Ṽ 0 m Q m H m Ṽ 0 m = U m Σ m [V 1 m V 0 m] H (7 55) U m N rm N rm Σ m Q m 0 0 N rm (N t Q m ) V 1 m Q m (N t Q m ) (N t Q m ) Q m V 0 m 0 (N t Q m ) (N t Q m ) (N t Q m Q m ) p m Q m p m (N t Q m ) V 1 m (N t Q m ) p m V 1 m p m > Q m Q m m p m = Q m V 1 m V1 m V 1 m Ṽ0 m F F = [Ṽ 0 1V 1 1 Ṽ 0 MV 1 M ]Λ1/2 (7 56) Λ P P P t 5) 2 1 m p m = 1 m N rm w m 7 4 w 1 HH 1 w M HH M -1 Channel 1 H 1 n 1 y 1 Receiver 1 w 1 x 1 d S(i) Channel M Receiver M Precoder H M n M y M w K x M 7 4 m x m (i) (7 26) x m (i) = w H my m (i) = ( w H mh m ) S(i) + w H m n m (i) (7 57) c /(18)
15 x m (i) M X(i) X(i) = [x 1 (i) x 2 (i) x M (i)]h (7 58) Y(i) X(i) H m w H mh m F channel inversion F w m MMSE w m w m = Y m (i)y H m(i) 1 Y m (i)d m(i) = [ H m F d(i)d H (i) F H H H m + n m (i)n H m(i) ] 1 [ Hm F d(i)d m(i) ] = ( H m FF H H H m + σ 2 ni Nr ) 1 Hm F m (7 59) σ 2 n n m (i)n H m(i) = σ 2 ni Nr (7 60) d(i)d H (i) = I Nr (7 61) {w m 1 m M} F {w m 1 m M} F 1) D. Gerlach and A. Paulraj, Adaptive transmitting antenna arrays with feedback, IEEE Signal Processing Letters, vol.1, no.10, pp , Oct ) C.B. Peel, B.M. Hochwald, and A.L. Swindlehurst, A vector-perturbation technique for near-capacity multi-antenna multiuser communication - Part I: channel inversion and regularization, IEEE trans. Commun., vol.53, no.1, pp , Jan ) B.M. Hochwald and C.B. Peel, Vector precoding for the multi-antenna, multi-user channel, in Proceedings Allerton Conference on Communication, Control, and Computing, Monticello, LI, Oct ) R. Fischer, C. Windpassinger, A. Lampre, and J. Huber, MIMO precoding for decentralized receivers, in Proceedings IEEE International Symposium on Information Theory, Lausanne, Switzerland, p.496, June/July ) Q.H. Spencer, A.L. Swindlehurst, and M. Haardt, Zero-forcing methods for downlink spatial multiplexing in multi-user MIMO channels, IEEE Trans. Signal Processing, vol.52, no.2, pp , Feb c /(18)
16 SVD-MIMO N M MIMO MIMO H L MIMO L H SVD: Singular Value Decomposition H = V L ΣU H L (7 62) U L H H H N U 1 L N L H H H = UΛU H (7 63) Λ diag(λ 1,, λ L, 0,, 0) N λ j HH H j λ j λ 1 λ 2... λ L > λ L+1 =... = λ N = 0 (7 64) V L HH H M V 1 L M L HH H = VΛ V H (7 65) Λ diag(λ 1,, L λ, 0,, 0) M H H H Σ diag( λ 1,, λl ) H L = min(m, N) (7 5) SVD MIMO 7 5 L SVD λi λ i MIMO L SM multiplexing gain SVD MIMO L c /(18)
17 7 5 MIMO H H H L U L = (u 1 u L ) L HH H L V L = (v 1 v L ) VL H H e V H L HU L V H L V LΣU H L U L Σ (7 66) L E-SDM: Eigenbeam Eigenmode -SDM Water Filling i P i P total P N P total P i J = N ( Pi λ ) i N log 2σ + 1 a 2 P i P total i=1 i=1 (7 67) J/ P i = 0 P i 0 P i ( ) 1 P i = max a 2σ2, 0 (7 68) λ i c /(18)
18 a N P i = P total i=1 (7 69) 7 6 SN 2σ 2 /λ i 2σ 2 /λ i 1/a i 7 6 1), ), 2006 c /(18)
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