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1 RCS 5 5G

3 ZF MMSE MIMO

5 1 1 0 x A v y = Ax + v MIMO (multi-input multi-output)

6 2 1 ZF (zero-forcing) MMSE (minimum mean-square-error) MIMO

7 R C T H diag[a 1 a N ] a 1,, a N N N tr{a} A det{a} A I 0 a p p 1 a l p - a p = ( n i=1 a i p ) 1 p (2.1) a 0 a l 0 - j R{ }, I{ } 2.2 {x(n); n =..., 2, 1, 0, 1, 2,...} n n 0 x(n 0 ) x(n) ( ) x(n) m x (n) = E[x(n)] (2.2)

8 4 2 1 E[ ] n x(n) p(x(n)) E[x(n)] = x(n)p(x(n))dx(n) (2.3) P (x(n)) E[x(n)] = x(n)p (x(n)) (2.4) E[ ] n x(n) x(n) ( ) r x (n, k) = E[x(n)x (n k)] (2.5) 2 n x(n) x(n k) p(x(n), x(n k)) E[x(n)x (n k)] = x(n)x (n k)p(x(n), x(n k))dx(n)dx(n k) (2.6) c x (n, k) = E[(x(n) m x (n))(x(n k) m x (n k)) ] (2.7) x(n) ( ) σ 2 x(n) = c x (n, 0) = E[(x(n) m x (n))(x(n) m x (n)) ] (2.8) 3 (2.2) n n (2.5) (2.7) n n k n k m x (n) = m x k r x (n, k) = r x (k) n n + l, l x(n) n n + l, l p(x(n 1 ), x(n 2 ),, x(n p )) = p(x(n 1 + l), x(n 2 + l),, x(n p + l)) (2.9) 1 ( E[X(n)] = x(n)p(x(n))dx(n) ) 2 3 t-

9 x(n) m x (n) r x (n, k) x(n) m m x (n) = m x (n + mt ) (2.10) r x (n, k) = r x (n + mt, k + mt ) (2.11) T [20, 21] [22] [19] x(n) x(n) = [x(n) x(n 1) x(n N + 1)] T C N (2.12) x(n) x(n) R x = E[x(n)x H (n)] (2.13) R x E[x(n)x (n)] E[x(n)x (n 1)]... E[x(n)x (n N + 1)] E[x(n 1)x = (n)] E[x(n 1)x (n 1)]... E[x(n 1)x (n N + 1)] E[x(n N + 1)x (n)] E[x(n N + 1)x (n 1)]... E[x(n N + 1)x (n N + 1)] r x (0) r x (1)... r x (N 1) r = x ( 1) r x (0)... r x (N 2)..... (2.14). r x ( N + 1) r x ( N + 2)... r x (0)

10 6 2 E[x(n)] = 0 x(n) R H x = ( E[x(n)x H (n)] ) H = E[(x(n)x H (n)) H ] = E[x(n)x H (n)] = R x r k = E[x(n)x (n + k)] = (E[x (n)x(n + k)]) = (E[x (n k)x(n)]) = rk r 0 r 1... r N 1 R x = E[x(n)x H r (n)] = 1 r 0... r N rn 1 rn 2... r 0 u C N x(n) y = u H x(n) E[ y 2 ] = E[yy ] = E[u H x(n)x H (n)u] = u H E[x(n)x H (n)]u = u H R x u y 2 x(n) y 2 0 E[ y 2 ] u H R xu u

11 A u C N (u H Au) H = u H A H u = u H Au R x λ 1,..., λ N λ i q i = [q i,1 q i,n ] T ( 0) R x q i = λ i q i (2.15) q H i q H i R x q i = λ i q H i q i q H i q i N q H i q i = q i,j 2 j=1 λ i = qh i R xq i q H i q i (2.16) q H i R xq i λ i ( ) (2.16) q i Rayleigh quotient R x λ i q i R x λ 1,..., λ N λ i q i (2.15) q H j, j i q H j R x q i = λ i q H j q i (2.17)

12 8 2 R x q j = λ j q j (2.18) R x λ j q i (2.17) (2.20) i j λ i λ j q H j R x = λ j q H j (2.19) q H j R x q i = λ j q H j q i (2.20) (λ i λ j )q H j q i = 0 (2.21) q H j q i = 0 (i j) (2.22) R x R x λ 1,..., λ N q 1,..., q N q i 2 = 1 i (2.15) 1 R x Q = Q λ... λ N (2.23) ] Q = [q 1... q N (2.24) Q H Q = I (2.25)

13 Q (2.23) Q H Q H R x Q = λ 1... λ N (2.26) q i Q [18] R x λ max λ max = max q 2 =1 qh R x q (2.27) R x λ 1 > λ 2 > > λ N q 1,, q N p q 1,, q N N p = N α i q i (2.28) i=1 5 1 R x = Q λ... Q H = λ N N λ i q i q H i (2.29) i=1

14 10 2 ( N ) p H R x p = p H λ i q i q H i p = i=1 N λ i p H q i q H i p (2.30) i=1 p H q i = α i (2.31) q H i p = α i (2.32) p H R x p = N λ i α i 2 (2.33) i=1 λ 1 > λ 2 > > λ N p H R x p p H R x p λ 1 N i=1 α i 2 (2.34) i 1 i α i = 0 p α 1 = 1 max q 2 =1 qh R x q = λ 1 (2.35) [9],[10] 2.3 ( Wirtinger ) f(z)

15 : D C f : D C f(z + z) f(z) lim z 0 z z D f D f 5 : R{f} x R{f} y = I{f} y = I{f} x 2 f(z) = z 2 = zz f(z + z) f(z) z + z 2 z 2 lim = lim z 0 z z 0 z = lim z 0 = lim z 0 (z + z)(z + z) zz z zz + z( z) + z( z) z (2.36) z 0 z = x + j y ( x, y R) x 0 z z j y y 0 z z y 0 z + z + x x 0 z + z z z z + z f(z) = z 2 = zz z z f d x, y R f df = f f dx + dy (2.37) x y 5 z = x + jy f(z) x, y f(x, y) x, y

16 12 2 f x f x (2.37) z = x + jy (2.38) z = x jy (2.39) dz = dx + jdy (2.40) dz = dx jdy (2.41) dx = 1 2 (dz + dz ) (2.42) dy = 1 2j (dz dz ) (2.43) df = f dz + dz + f dz dz x 2 y 2j = 1 ( ) f 2 x j f dz + 1 y 2 ( f x + j f y f z, z ) dz (2.44) df = f f dz + z z dz (2.45) (2.44) (2.45), : f z = 1 ( ) f 2 x j f y f z = 1 ( ) f 2 x + j f y (2.46) (2.47) z = x + jy f(z) = z f(z) = z z z = 1 ( ) (x + jy) (x + jy) j = 1 (1 j j) = 1 (2.48) 2 x y 2 z z = 1 ( ) (x jy) (x jy) + j = 1 (1 + j ( j)) = 1 (2.49) 2 x y 2 z z = 1 ( ) (x + jy) (x + jy) + j = 1 (1 + j j) = 0 (2.50) 2 x y 2 z z = 1 ( ) (x jy) (x jy) j = 1 (1 j ( j)) = 0 (2.51) 2 x y 2

17 z z z z z z z z f(z) = z 2 z 2 z z 2 z f z = 1 ( 2 x (R{f} + ji{f}) + j y = 1 ( R{f} I{f} ) + j ( R{f} 2 x y 2 y = zz z = z (2.52) = zz z = z (2.53) ) (R{f} + ji{f}) ) + I{f} x (2.54) f z = 0 (2.55) z z z f z = [z 1 z M ] T C M z m = x m + jy m, (x m, y m R) f(z) df = M m=1 ( f dx m + f ) dy m x m y m (2.56) dz m = dx m + jdy m dz m = dx m jdy m df = M m=1 { ( 1 f j f ) dz m + 1 ( f + j f ) } dzm 2 x m y m 2 x m y m (2.57) f [ z = f z 1 dz = ] f z M [ dz 1 dz M ] T (2.58)

18 14 2 f [ z = f z1 dz = ] f zm [ T dz1 dzm] (2.59) df = f f dz + z z dz (2.60) f : [ ] f f z = f z 1 z [ ( M 1 f = j f ) ( 1 f j f )] 2 x 1 y 1 2 x M y M [ ] f f z = f z1 zm [ ( 1 f = + j f ) ( 1 f + j f )] 2 x 1 y 1 2 x M y M (2.61) (2.62) f [ ] f f f = z z (2.63) f z, f z ( ) f f T z H = z (2.64) f(z) f + j f x 1 y 1 f + j f e f = x 2 y 2. f + j f x M y M (2.65)

19 [9] e f = 2 f z H (2.66) f (2.60) f ( ) f z = (2.67) z ( ) f f z = (2.68) z f f = f ( ) f f z = (2.69) z f = [( ) f ] f z z (2.70) f f = 0 f f z = 0 z = 0 (2.65) f (2.69) (2.65) : a, A ( z H z H a ) = a (2.71) ( z H z H Az ) = Az (2.72) x ( x T a ) = a x (2.73) ( x T Ax ) = Ax + A T x x (2.74). [17]

20 16 2 Z : ( { tr Z H Z H A }) = A (2.75) ( { tr Z H Z H AZ }) = AZ (2.76) [8] [9, 11] [15, 16]

21 x = [x 1 x N ] T C N A = [a 1 a N ] C M N y = [y 1 y M ] T C M x y = Ax + v (3.1) y A x v = [v 1 v M ] T C M A y x v R y = E[yy H ] R x = E[xx H ] R v = E[vv H ] = σ 2 vi 3.2 ZF ZF x x ˆx zf = W H zf y = x + W H zf v (3.2) Wzf H ZF N M ZF A (M = N) W H zf A = I (3.3) W H zf = A 1 (3.4) A M > N (3.3) W zf (3.3) (3.2) (SNR: signal-to-noise power

22 18 3 ratio) W zf (3.2) E [ (Wzf H v)h Wzf H v] = tr { Wzf H } E[vvH ]W zf = σvtr 2 { Wzf H W } zf (3.5) SNR W zf W zf = arg min tr { W H W } s.t. W H A = I (3.6) W C M N tr{ab} = tr{ba} L zf (W) = tr { W H W } + = tr { W H W } + N ϕ H n (W H a n e n ) n=1 N n=1 tr { (W H a n e n )ϕ H } n (3.7) L zf (W) W H = W + N a n ϕ H n n=1 = W + AΦ H (3.8) ϕ n N e n n 1 0 N Φ = [ϕ 1 ϕ N ] L zf(w) W H = 0 W zf = AΦ H (3.9) (3.6) A Φ = (A H A) 1 (3.10) W H zf = (AH A) 1 A H (3.11) M = N W H zf = A 1 (3.4) M = N

23 3.2. ZF 19 ZF 2 ZF ˆx ls = arg min x C N Ax y 2 2 (3.12) Ax y 2 2 = (Ax y) H (Ax y) = x H A H Ax x H A H y y H Ax + y H y x H Ax y 2 2 = A H Ax A H y = 0 (3.13) ˆx ls = (A H A) 1 A H y (3.14) ZF (3.6) A ZF (noise enhancement) ZF (3.5) (3.11) σ 2 vtr { W H zf W zf} = σ 2 v tr { (A H A) 1} (3.15) A [ ] Ξ A = U V H (3.16) 0 (M N) N U C M M, V C N N Ξ A Ξ = diag[ξ 1 ξ N ] A H A = VΞ 2 V H (3.17) (A H A) 1 = VΞ 2 V H (3.18) σvtr 2 { Wzf H W } N zf = σ 2 1 v ξ n=1 n 2 (3.19) A ξ n 0

24 M < N (3.3) W zf ZF. M < N y = Ax x x y A y = Ax x l 2 - ˆx mn = arg min x C N x 2 2 s.t. y = Ax (3.20) L mn (x) = x (Ax y) H ϕ (3.21) Lmn(x) x H = 0 ˆx mn = A H ϕ (3.22) A ϕ = (AA H ) 1 y (3.23) ˆx mn = A H (AA H ) 1 y (3.24) 3.4 MMSE ZF MMSE MMSE MMSE MMSE MMSE MMSE f ˆx mmse = f(y) (3.25) f J mmse [f] = E [ f(y) x 2 2 y ] (3.26) y x p(x y) y x x(y) = E[x y] = xp(x y)dx (3.27)

25 3.4. MMSE 21 1, J mmse [f] =E [ f(y) x 2 2 y ] =E [ f(y) x(y) + x(y) x 2 2 y ] = f(y) x(y) E [ x(y) x 2 2 y ] + {f(y) x(y)} H E [{ x(y) x} y] + E [ { x(y) x} H y ] {f(y) x(y)} = f(y) x(y) E [ x(y) x 2 2 y ] E [ x(y) x 2 2 y ] (3.28) f(y) = x(y) MMSE ˆx mmse = x(y) (3.29) MMSE 0 MMSE W lmmse ˆx lmmse = Wlmmse H y (3.30) W lmmse W lmmse = arg min E [ W H (Ax + v) x 2 ] W C M N 2 (3.31) J lmmse (W) = E [ W H (Ax + v) x 2 ] 2 = E [ (W H Ax + W H v x) H (W H Ax + W H v x) ] = E [ tr{(w H Ax + W H v x)(w H Ax + W H v x) H } ] = tr { W H AE[xx H ]A H W } + tr { W H AE[xv H ]W } tr { W H AE[xx H ] } + tr { W H E[vx H ]A H W } + tr { W H E[vv H ]W } tr { W H E[vx H ] } tr { E[xx H ]A H W } tr { E[xv H ]W } + tr { E[xx H ] } = tr { W H AR x A H W } tr { W H AR x } + σ 2 v tr { W H W } tr { R x A H W } + tr {R x } (3.32) J lmmse (W) W H = AR x A H W AR x + σ 2 vw = 0 (3.33) 1 x

26 22 3 W H lmmse = R xa H ( AR x A H + σ 2 vi ) 1 (3.34) MMSE MMSE (x, y) [1] 3.5 y x x x (subtractive interference cancellation) [23, 24] (CDMA: code division multiple access) [25] (SIC: successive interference cancellation) (PIC: parallel interference cancellation) SIC x SNR x v A x y x SNR x 1, x 2,..., x N x 2,..., x N x 1 y (1) sic = y = Ax + v ( N ) = a 1 x 1 + a i x i + v i=2 (3.35) x 1 ˆx sic,1 ˆx sic,1 = w H 1 y (1) sic (3.36) w 1 C N ZF MMSE

27 y x 1 y (2) sic ˆx sic,1 y (2) sic = y a 1ˆx sic,1 ( N ) = a 2 x 2 + a i x i + v + a 1 (x 1 ˆx sic,1 ) i=3 ( N ) a 2 x 2 + a i x i + v i=3 (3.37) ˆx sic,1 = x 1 y (2) sic y x 2 y (2) sic x 1 x 2 w 2 C N ˆx sic,2 = w H 2 y (2) sic (3.38) x N SIC SNR SNR A [25] SIC [24] PIC x x y (1) pic = y x ˆx (1) pic y (2) pic = y A offdiagˆx (1) pic = Ax A offdiagˆx (1) pic + v A diag x + v (3.39) A diag A A offdiag = A A diag y (2) pic ˆx(2) pic PIC PIC SIC x SNR 3.6 x S x S N y x ˆx S N x

28 24 3 P (ˆx y) ˆx S N ˆx map = arg max P (x y) (3.40) x SN 2 P (x y) = p(y x)p (x) p(y) (3.41) x P (x) ˆx ml = arg max p(y x) (3.42) x SN p(y x) (3.1) v p(y x) = ( 1 π M det{r v } exp y ) Ax 2 2 σv 2 (3.43) ˆx ml = arg min x S N y Ax 2 2 (3.44) Ax S S S N N 3.7 ( ) (3.1) y = ax + v (3.45) 2 [26]

29 x C, a = [a 1 a M ] T C M y 1,, y M SNR ( ) y 1,, y M ( ) SNR MRC: maximal ratio combining w mrc ˆx mrc = w H mrcy = w H mrcax + w H mrcv (3.46) SNR γ mrc = E[ wh mrcax 2 ] E[ w H mrcv 2 ] = σ2 xw H mrcaa H w mrc σ 2 vw H mrcw mrc (3.47) E[ x 2 ] = σ 2 x (3.47) SNR aa H w mrc 2.2 aa H aa H 1 γ mrc = σ2 xa H aa H a σ 2 va H a = σ2 xa H a σ 2 v w mrc = a (3.48) = a 1 2 σ 2 x σ 2 v + a 2 2 σ 2 x σ 2 v + + a M 2 σ 2 x σ 2 v (3.49) SNR SNR (3.45) (3.1) w mrc ˆx mrc = wmrcy H = wmrcax H + wmrcv H (3.50) ˆx mrc x

30 26 3 SNR γ mrc = E[ wh mrcax 2 ] E[ w H mrcv 2 ] = wh mrcar x A H w mrc σ 2 vw H mrcw mrc (3.51) w mrc AR x A H SNR γ mrc = wh mrcar x A H w mrc w H mrcr v w mrc (3.52) w mrc AR x A H R v AR x A H w = λr v w (3.53) 3.8 x A A y( ) x v y R y = E[yy H ] = AR x A H + σ 2 vi (3.54) R y M λ 1 λ 2 λ M, AR x A H M ν 1 ν 2 ν M R y M λ m q m λ m q m = R y q m = (AR x A H + σvi)q 2 m = (ν m + σv)q 2 m λ m ν m λ m = ν m + σv 2, m = 1, 2,..., M (3.55).

31 M > N A R x AR x A H M N 0. (3.55) { ν m + σ 2 λ m = v, m = 1,..., N σv, 2 m = N + 1,..., M (3.56). rank A H = N A H N (A H ) M N q N (A H ) R y q = σ 2 vq q σ 2 v M N (3.56) M N q N+1,, q M N (A H ) q H ma = 0, m = N + 1,..., M (3.57) Q S = [q 1,, q N ] Q N = [q N+1,, q M ] R(Q S ) R(Q N ) [2] (R( ) ) (3.57) R(Q N ) = N (A H ) q 1,, q M R(Q S ) = R(Q N ) ( ) R(A) = N (A H ) R(Q S ) = R(A) R(Q N ) = R(A) (3.57) A A 3.9 x x N M < N y = Ax [3, 4, 5] M < N y = Ax x x ˆx l0 = arg min x x 0 s.t. y = Ax (3.58) x l 0 l 0 - NP l 0 - l 1 - ˆx l1 = arg min x x 1 s.t. y = Ax (3.59)

32 28 3 A M < N x ϵ > 0 ˆx cl1 = arg min x x 1 s.t. Ax y 2 2 ϵ (3.60) µ > 0 (3.60) ( ˆx l1 l 2 = arg min µ x ) x 2 Ax y 2 2 (3.61) (3.61) l 1 - l 2 - l 1 l 2 Lasso (least absolute shrinkage and selection operator)[6] (3.60), (3.61) ˆx lasso = arg min x Ax y 2 2 s.t. x 1 t (3.62) [7]

33 (3.1) s r H v r = Hs + v (4.1) H h h L... h 1.. h h L hl H = C M M h L... h 0 h 0,, h L (DFT: discrete Fourier transform) D = 1 2π 1 1 2π 1 (M 1) j j 1 e M... e M M... 2π(M 1) 1 2π(M 1) (M 1) j j 1 e M... e M. H {h 0, h 1,..., h L } H = D H ΛD (4.2)

34 30 4. Λ = diag[λ 1 λ M ] h 0 λ 1. = MD. λ M h L 0 (M L 1) 1 (4.3) 1.. ZF (3.4) r = D H ΛDs + v (4.4) ŝ = W H r (4.5) W H = (D H ΛD) 1 = D 1 Λ 1 D H MMSE (3.34) = D H Λ 1 D (4.6) W H = σ 2 s H H ( σ 2 s HH H + σ 2 vi ) 1 = σs 2 D H Λ H D ( σs 2 D H ΛΛ H D + σvi 2 ) 1 ( 1 = D H Λ H ΛΛ H + σ2 v I) D E[ss H ] = σ 2 s I D W H ZF MMSE IDFT D H DFT D (FFT: fast Fourier transform) ZF MMSE (MLSE: maximum likelihood sequence estimation) σ 2 s [λ 1,..., λ M ] T,

35 4.3. MIMO 31 r = Hp + v (4.7) p = [p 1 p M ] T H (4.7) r = Ph + v (4.8) P p 1 p M... p 2 p P = 2 p 1 p p M p M 1... p 1 (4.9) h H (4.8) h H (4.1) s E[hh H ] MMSE ZF P 2 ZF 2 (4.8) h 4.3 MIMO MIMO (3.1) N M s C N r C M H C M N v C M r = Hs + v (4.10) (4.10) 2 E[hh H ] MMSE

36 32 4 incoming plane wave d sinθ θ antenna 0 d (M-1)d 4.1: MIMO (4.10) MIMO H MIMO [12] MIMO MIMO MIMO more is different MIMO [27] 4.4 N M d 4.1 n θ n d sin θ n ϕ n = 2π d sin θ n η (4.11) η 1 {s 1, s 2,, s N } m N r m = s n e jϕn(m 1) + v m (4.12) n=1 v m 0, σv 2 N r = [r 1 r M ] T = s n a(θ n ) + v (4.13) n=1

37 v = [v 1 v M ] T a(θ) = [ 1, e d sin θ j2π η,, e (N 1)d sin θ j2π η ] T (4.14) A = [a(θ 1 ) a(θ N )] s = [s 1 s N ] T (3.1) r = As + v (4.15) s A N M M > N (3.57) q H ma = 0, m = N + 1,..., M (4.16) q m R = E[rr H ] M N S(θ) = 1 M m=n+1 ah (θ)q m 2 (4.17) θ, θ = θ n (n = 1,, N) 0 3 MUSIC (multiple signal classification) [13] [14] 3 0 R

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43 MIMO MIMO

45 G 1 1

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untitled 1 SS 2 2 (DS) 3 2.1 DS................................ 3 2.2 DS................................ 4 2.3.................................. 4 2.4 (channel papacity)............................ 6 2.5........................................