4 -- 1 7 MIMO 2009 4 MIMO Multiple Input Multiple Output MIMO = = MIMO LAN IEEE802.11n MIMO Alamouti STBC Space Time Block Code 1 7-1 MIMO 7-2 MIMO 7-3 MIMO MIMO 7-4 MIMO 8 8-5 c 2010 1/(18)
4 -- 1 -- 7 7--1 7--1--1 MIMO 2009 4 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 2010 2/(18)
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) 7--1--2 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 2010 3/(18)
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 2010 4/(18)
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 2 1 + 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) 6 2000 2) 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, 1996. 3) 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.311-335, 1998. c 2010 5/(18)
4 -- 1 -- 7 7--2 2009 8 STC: Space-Time Code STC STBC: Space-Time Block Code STTC: Space-Time Trellis Code 2 STC 2 STBC W-CDMA 7--2--1 STBC STBC N = 2 STBC 1) STBC 2 2 2 2 X 1 X 2 1, 2 X 1, X 2 1 1 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 2010 6/(18)
3) N > 2 STBC 1/2 R 3/4 STBC Jafarkhani 6) STBC STBC Jafarkhani STBC partial diversity ML 7--2--2 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 00,01,02,03 2 0 1 10,11,12,13 3 2 3 20,21,22,23 30,31,32,33 7 2 QPSK, 4 STTC 2) 7 2 QPSK 2) QPSK, 4 STTC QPSK, 4 STTC 2) G G = 0 2 0 1 2 0 1 0 (7 25) S k 0, 1, 2, 3 2 1 2 c 2010 7/(18)
1) S. Alamouti, Space block coding: A simple transmitter diversity technique for wireless communications, IEEE J. Select. Areas. Commun., vol.16, no.5, pp.1451-1458, Oct. 1998. 2) 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.744-765, March 1998. 3) V. Tarokh, H. Jafarkhani, and A.R. Calderbank, Space-time block codes from orthogonal designs, IEEE Trans. Inform. Theory, vol.45, no.4, pp.1456-1467, July 1999. 4) G. Ganesan and P. Stoica, Space-time diversity using orthogonal and amicable orthogonal designs, Wireless Personal Commun., vol.18, pp.165-178, Aug. 2001. 5) G. Ganesan and P. Stoica, Space-time block codes: A maximum SNR approach, IEEE Trans. Inform. Theory, vol.47, no.4, pp.1650-1656, May 2001. 6) H. Jafarkhani, A quasi-orthogonal space-time block code, IEEE Trans. Commun.,, vol.49, no.1, pp.1-4, Jan. 2001. c 2010 8/(18)
4 -- 1 -- 7 7--3 MIMO 2009 5 MIMO 7--3--1 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 2010 9/(18)
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 7--3--2 1 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 2010 10/(18)
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 2010 11/(18)
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 2010 12/(18)
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) 7--3--3 2 5) 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 2010 13/(18)
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 2010 14/(18)
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.150-152, Oct. 1994. 2) 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.195-202, Jan. 2005. 3) 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. 2003. 4) 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 2002. 5) 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.461-471, Feb. 2004. c 2010 15/(18)
4 -- 1 -- 7 7--4 SVD-MIMO 2009 8 N M MIMO 7--4--1 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 7--4--2 SVD MIMO L c 2010 16/(18)
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 2010 17/(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), 2009 2), 2006 c 2010 18/(18)