RCS 5 5G 28 10 19
1 1 2 3 2.1........................................ 3 2.2............................... 3 2.3.............................. 10 3 17 3.1...................................... 17 3.2 ZF...................................... 17 3.3................................... 20 3.4 MMSE.................................... 20 3.5.................................. 22 3.6........................... 23 3.7.................................... 24 3.8.................................... 26 3.9.................................. 27 4 29 4.1........................................ 29 4.2.................................... 30 4.3 MIMO.................................... 31 4.4................................... 32 5 35 5.1...................................... 35 5.2................................... 37 5.3............................... 38 6 41
1 1 0 x A v y = Ax + v MIMO (multi-input multi-output)
2 1 ZF (zero-forcing) MMSE (minimum mean-square-error) MIMO
3 2 2.1 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)
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-
2.2. 5 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)
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 2...... 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 ] 0 4 4 u H R xu u
2.2. 7 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)
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)
2.2. 9 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
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)
2.3. 11 : 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
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
2.3. 13 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)
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)
2.3. 15 [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]
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]
17 3 3.1 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
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
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
20 3 3.3 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 2 2 + (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)
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) 2 2 + 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) 2 2 + 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
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
3.6. 23 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
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]
3.7. 25 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
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).
3.9. 27 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)
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 1 + 1 ) 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]
29 4 4.1 (3.1) s r H v r = Hs + v (4.1) H h 0 0... 0 h L... h 1.. h........ 0.. h........ L hl H =. 0........ 0 C M M........................ 0 0...... 0 h L... h 0 h 0,, h L (DFT: discrete Fourier transform) 1 1... 1 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)
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 4.2 1 [λ 1,..., λ M ] T,
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 3..... 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
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
4.4. 33 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
35 5 1 5.1 2009. 2 1 2
36 5 1974. Jordan 2005. 2008. σ- 9 IEEE Trans. Information Theory IEEE Trans. Signal Processing 1968. 2007. 30
5.2. 37 2009. 5.2 S. Haykin, Adaptive Filter Theory (5th edition), Pearson, 2013. 3 B. F.-Boroujeny Adaptive Filters, Theory and Applications, John Wiley & Sons, 1998. 2000. 2011 2015 C. Bishop, Pattern Recognition and Machine Learning, Springer, 2006.
38 5 5.3 T. M. Cover, J. A. Thomas, Elements of Information Theory, John Wiley & Sons, 2006. R. G. Gallager, Information Theory and Reliable Communication, John Wiley & Sons, 1968. Cover D. J. C. Mackay, Information Theory, Inference, and Learning Algorithms, Cambridge University Press, 2003. T. Richardson, Modern Coding Theory, Cambridge University Press, 2008. 1999. 2010. 2011 J. G. Proakis, M. Salehi, Digital Communications, McGraw-Hill, 2008. 1000 D. Tse, P. Viswanath, Fundamentals of Wireless Communication, Cambridge University Press, 2005. R. G. Gallager, Principles of Digital Communication, Cambridge University Press, 2008.
5.3. 39 2003. MIMO 2009. MIMO
41 6 5 5G 1 1
43 [1],,, 2000. [2],,, vol. 43, no. 4, pp. 188-195, 1999. [3] D.L. Donoho, Compressed sensing, IEEE Trans. Inf. Theory, vol.52, no.4, pp.1289-1306, April 2006. [4] E.J. Candes and T. Tao, Decoding by linear programming, IEEE Trans. Inf. Theory, vol.51, no.12, pp.4203-4215, Dec. 2005. [5] E.J. Candes, J. Romberg, and T. Tao, Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information, IEEE Trans. Inf. Theory, vol.52, no.2, pp.489-509, Feb. 2006. [6] R. Tibshirani, Regression shrinkage and selection via the lasso, J. R. Statist. Soc. B, vol.58, no.1, pp.267-288, 1996. [7] K. Hayashi, M. Nagahara, T. Tanaka, A User s Guide to Compressed Sensing for Communications Systems, IEICE Trans. Commun., Vol. E96-B, No. 03, pp.685-712, Mar. 2013 [8] L. Schwartz, 6,, 1971. [9] S. Haykin, Adaptive Filter Theory, 3rd Edition, Prentice Hall, 1996. [10] B. F.-Boroujeny Adaptive Filters, Theory and Applications, John Wiley & Sons, 1998. [11] T. Kailath, A. Sayed, B. Hassibi, Linear Estimation, Prentice Hall, 2000. [12] MIMO 2009. [13] R. O. Schmidt, Multiple Emitter Location and Signal Parameter Estimation, IEEE Trans. Antennas and Propag., Vol. AP-34, Vo. 3, pp. 276-280, 1986. [14],, Fundamentals Review, Vol. 8, No. 3, pp. 143-150, Jan. 2015.
44 6 [15] P. J. Schreier and L. L. Scharf, Statistical Signal Processing of Complex-Valued Data, Cambridge University Press, 2010. [16] A. Hjørungnes, Complex-Valued Matrix Derivatives, Cambridge University Press, 2011. [17] 1990. [18] 1990. [19] W. A. Gardner, Cyclostationarity in Communications and Signal Processing, IEEE Press, 1994. [20] L. Tong, G. Xu, and T. Kailath, Blind Channel Identification Based on Second- Order Statistics : A Time Domain Approach, IEEE Trans. Inform. Theory, vol.41, pp. 340-349, Mar. 1994. [21] L. Tong, G. Xu, B. Hassibi,and T. Kailath, Blind Channel Identification Based on Second-Order Statistics : A Frequency Domain Approach, IEEE Trans. Inform. Theory, vol.40, pp. 340-349, Mar. 1994. [22] G. Xu, T. Kailath, Direction-of-arrival estimation via exploitation of cyclostationarity - A combination of temporal and spatial processing, IEEE Trans. Signal Processing, vol. 40, no. 7, pp. 1775-1786, July 1992. [23] Thomas M. Cover and Joy A. Thomas, Elements of Information Theory, 2nd Edition, Wiley-Interscience, 2006. [24] D. Tse, P. Viswanath, Fundamentals of Wireless Communication, Cambridge University Press, 2005. [25] S. Moshan, Multi-user detection for DS-CDMA communications, IEEE Communications Magazine,pp. 124-136, Oct. 1996. [26] D. P. Bertsekas and J. N. Tsitsiklis, Introduction to Probability, Athena Scientific, 2008. [27] A. Chockalingam, B. S. Rajan, Large MIMO Systems, Cambridge University Press, 2014.