K

K 1 2 1 K

M N M(2 N 1)

K K K K K

f f(x 1, x 2,..., x K ) = K f xk (x k ), x 1, x 2,..., x K K K K f Yk (y k x 1, x 2,..., x k ) k=1 M i, i = 1, 2 Xi n n Yi n Xn 1 Xn 2 ˆM i P (n) e = {( ˆM 1, ˆM2 ) (M 1, M 2 )}. (R 1, R 2 ) n 2 nr i P (n) e 0 n M i C IC (R 1, R 2 )

(R 1, R 2 ) R 1 I(X 1 ; Y 1 X 2, Q), R 2 I(X 2 ; Y 2 X 1, Q) R 1 + R 2 I(X 1, X 2 ; Y 1, Y 2 Q), Q I(, ) Z 1 Z 2 N 0 2 Z i N ( 0, N 0 2 ) X1 X 2 P 1 P 2 g ij j i

g 2 12 (1 + P 1 ) g 2 22 g 2 21 (1 + P 2 ) g 2 11. X 2 g 11 X 1 R 2 R 2 1 2 2 ( 1 + g2 12 P ) 2 1 + g11 2 P. 1 R 2 = 1 2 2 (1+g2 22 P 2) R 2 = 1 2 2 (1 + g2 22 P 2) X 2 X 2 Y 1 X 1 R 1 1 2 2 (1 + g2 11 P 1) R 1 I(X 1 ; Y 1 X 2, Q) R 2 I(X 2 ; Y 2 X 1, Q), p(q)p(x 1 q)p(x 2 q) f(x 1 q) = f(x 1 ) f(x 2 q) = f(x 2 )

R 1 I(X 1 ; Y 1 X 2, Q), R 2 I(X 2 ; Y 2 X 1, Q) R 1 + R 2 {I(X 1, X 2 ; Y 1 Q), I(X 1, X 2 ; Y 2 Q)}, p(q)p(x 1 q)p(x 2 q)

2 K

C DoF = [ ] C ( ), 2 (1 + ) 2 ( ) M N DoF = (M, N) C = DoF 2 ( ) + o ( 2 ( ) ), o ( 2 ( ) ) o ( 2 ( ) ) 0. 2 ( ) K

y 1 = h 11 x 1 + h 12 x 2 + + h 1K x K y 2 = h 21 x 1 + h 22 x 2 + + h 2K x K y B = h B1 x 1 + h B2 x 2 + + h BK x K. B y 1, y 2,..., y B K x 1, x 2,..., x K h ij B K B h ij j i x 1, x 2,..., x K B = K K K x 1 K K B = K K K 1 K 1 K B < K K

x 1 = 1 x 1 + 2 x 2 + + K x K, y 1 y B y 2 = k = h 1k h 2k h Bk x k 1, 2,..., K x 1 1 2,,..., K 1 / span( 2,,..., K ), span 2,,..., K (K 1) 2,,..., K (B, K 1) B < K B B x 2 x 1, x,..., x K

K (K 1) O(K 2 ) K O(K 2 ) (M 1 N 1, d 1 ), (M 2 N 2, d 2 )... (M K N K, d K ) K k M k d k k N k k K = {1, 2,..., K} M k d k k rank( k ) = d k N k d k k rank( k ) = d k

j, k K, j k, j [jk] k = dj d k rank( k [kk] k ) = d k. [jk] N k M k k j k k k k d k d k [kk] [jk] DoF = rank( [kk] ) = (M k, N k ) d k k k [kk]

M N d (M N, d) K (M N, d) K d M + N K + 1. X M > 1 4 M 4 M X M = 2 4 8 DoF = (M, N) M 1, M 2, N DoF = (M 1 +M 2, N) M, N 1, N 2 DoF = (M, N 1 + N 2 ) M 1, M 2, N 1, N 2 DoF = (M 1 + M 2, N 1 + N 2, (M 1, N 2), (M 2, N 1)).

K 2 K K K k [k] = [kk] [k] + K i [ki] [i] + [k], i K, i k [kk] i N = K(K 1) [k] 0 [k] N (, ) [k] K K K i j = j i, i, j {1, 2,..., N} I = 1 2 N, I.

I [kk] [kk] I S = [kk] + I = + I, [kk] S n = + I 1, n, 2 1 2 K 2 1 K K U V n 1 n 2 S m n 1 + n 2 < m U V U V = {0}

(d 1, d 2,..., d K ) d i + d j 1, i, j {1, 2,..., K}, i j. M M

M K M MK M+K 1 M K [i] j (t) j i t x [i] j j i x [1] 1 x [1] 2 x [2] 1 x [2] 2

x [1] 1 x [1] 2 x [2] 1 x [2] 2 2 4 K K K 1+ 1 2 + 1 + + 1 K M, N 1 N 2 N 2 N 1 M N 1 M N 1 < M N 2 M > N 2 N 1 + N 2

8 7 9 8 2 4 6 5

K 2 9 8

k [k] = [ ] T u [k] 1 u [k] 2 u [k], k [k] k [ij] j i [k] k 9 8

k k [k] = [k] 1 u[k] 1 + [k] 2 u[k] 2 + [k] u[k], [k] j 5 1 j [ ] [1] = [11] [1] 1 [1] 2 [1] u [1] 1 u [1] 2 u [1] [ + [12] [2] 1 [2] 2 [2] ] u [2] 1 u [2] 2 u [2] [ + [1] [] 1 [] 2 [] ] [ij] 5 5 j i [2] j, [] j, j = 1, 2, 5 6 [ ] [12] [2] 1 [12] [2] 2 [12] [2] [1] [] 1 [1] [] 2 [1] [], 1 [ ] α [1] = α [1] 1, α[1] 2, α[1], α[1] 4, α[1] 5, α[1] 6, [ ] [12] [2] 1 [12] [2] 2 [12] [2] [1] [] 1 [1] [] 2 [1] [] α [1] = 5 1. [ij] [k] j u [] 1 u [] 2 u [],

α [2] α [] [ ] [21] [1] 1 [21] [1] 2 [21] [1] [2] [] 1 [2] [] 2 [2] [] α [2] = 5 1 [ ] [1] [1] 1 [1] [1] 2 [1] [1] [2] [2] 1 [2] [2] 2 [2] [2] α [] = 5 1. [ij] [k] i n n = 6, 7, 8 [1] 1 (n)u[1] 1 + [1] 2 (n)u[1] 2 + [1] (n)u[1], [2] 1 (n)u[2] 1 + [2] 2 (n)u[2] 2 + [2] (n)u[2] [] 1 (n)u[] 1 + [] 2 (n)u[] 2 + [] (n)u[]. [k] i [ [12] (n) [2] 1 (n) [12] (n) [2] 2 (n) [12] (n) [2] (n) [1] (n) [] 1 (n) [1] (n) [] 2 (n) [1] (n) [] (n) ] α [1] = 5 1, [ [21] (n) [1] 1 (n) [21] (n) [1] 2 (n) [21] (n) [1] (n) [2] (n) [] 1 (n) [2] (n) [] 2 (n) ] [2] (n) [] α (n) [2] = 5 1 [ [1] (n) [1] 1 (n) [1] (n) [1] 2 (n) [1] (n) [1] (n) [2] (n) [2] 1 (n) [2] (n) [2] 2 (n) [2] (n) [2] (n) ] α [] = 5 1.

[ij] (n) [k] i [1] 1 (n)α[2] 1 + [1] 2 (n)α[2] 2 + [1] (n)α[2] = 0, [1] 1 (n)α[] 1 + [1] 2 (n)α[] 2 + [1] (n)α[] = 0, [2] 1 (n)α[1] 1 + [2] 2 (n)α[1] 2 + [2] (n)α[1] = 0, [2] 1 (n)α[] 4 + [2] 2 (n)α[] 5 + [2] (n)α[] 6 = 0, [] 1 (n)α[1] 4 + [] 2 (n)α[1] 5 + [] (n)α[1] 6 = 0 [] 1 (n)α[2] 4 + [] 2 (n)α[2] 5 + [] (n)α[2] 6 = 0. [1] i (n), i = 1, 2, [1] 2 (n) = [1] (n) = α [2] 1 α [2] α [] 1 α [] α [2] 2 α [2] α [] 2 α [] α [2] 2 α [2] 1 α [] 2 α [] 1 α [2] 2 α [2] α [] 2 α [] n [1] 1 (n) [1] 1 (n). [1] (n) = [1] 1 (n)u[1] 1 + [1] 2 (n)u[1] 2 + [1] (n)u[1] ( ) = c (α [2] 2 α[] α[2] α[] 2 )u[1] 1 + (α[2] α[] 1 α[] α[2] 1 )u[1] 2 + (α[2] 1 α[] 2 α[2] 2 α[] 1 )u[1] = cs [1], c s [1] = (α [2] 2 α[] α[2] α[] 2 )u[1] 1 + (α[2] α[] 1 α[] α[2] 1 )u[1] 2 + (α[2] 1 α[] 2 α[2] 2 α[] 1 )u[1]

s [1] s [2] s [] s [2] = (α [1] 2 α[] 6 α[1] α[] 5 )u[2] 1 + (α[] 4 α[1] α[] 6 α[1] 1 )u[2] 2 + (α[1] 1 α[] 5 α[1] 2 α[] 4 )u[2] s [] = (α [1] 5 α[2] 6 α[1] 6 α[2] 5 )u[] 1 + (α[1] 6 α[2] 4 α[2] 6 α[1] 4 )u[] 2 + (α[1] 4 α[2] 5 α[2] 4 α[1] 5 )u[]. [k] [1] (1) [1] (2) [1] () [1] (4) = [1] (5) [1] (6) [1] (7) [1] (8) [1] 1 (1) [1] 2 (1) [1] (1) [1] 1 (2) [1] 2 (2) [1] (2) [1] 1 () [1] 2 () [1] () [1] 1 (4) [1] 2 (4) [1] (4) [1] 1 (5) [1] 2 (5) [1] (5) α [2] 2 α[] α[2] α[] 2 α [2] α[] 1 α[] α[2] 1 α [2] 1 α[] 2 α[2] 2 α[] 1 α [2] 2 α[] α[2] α[] 2 α [2] α[] 1 α[] α[2] 1 α [2] 1 α[] 2 α[2] 2 α[] 1 α [2] 2 α[] α[2] α[] 2 α [2] α[] 1 α[] α[2] 1 α [2] 1 α[] 2 α[2] 2 α[] 1 u [1] 1 u [1] 2 u [1]. [1] j (n) s [1] [1] j (n) 8 8 [1] 1 = [ [11] [1] 1 [11] [1] 2 [11] [1] [12] [2] 1 [12] [2] 2 [12] [2] [1] [] 1 [1] [] 2 ].

[2] [] ( [1] ) ( [2] ) ( [] ) 0. ( [1] ) ( [2] ) ( [] ) [ ] [ ] ( [1] ) ( [2] ) ( [] ) [ ] [ ] k 8 1 [ ] [k] = [k] [k] 1 [k] 2 [k] u [k] 1 u [k] 2 u [k] + [k] [k] [1] k [k], 6 1 k [k] k [k] ( [k] ) [k] N [( [1]) ], N [ ] (N ) = 8 5 = [k] 8 [k] N [( [k]) ] ( [k]) ( [k] = 6 [k]) [k] = 6.

( [k]) [k] = ( [k]) [k] ( = [k]) [ ] [kk] [k] 1 [k] 2 [k] = [k] u [k] 1 u [k] 2 u [k]. u [k] 1 u [k] 2 u [k] [k] k k 8 9 8 5 DoF = 15 1 > 9 8 15 1

n m m (n 1)m [(n 1)m d] d N + d < (n 1)m [(n 1)m d] [(n 1)m d] un = nm eq = n(n 1)d un eq = n nm n(n 1)d = n d = m 1 n 1. m m m [1] = [11] [1] + [12] [2] + + [1n] [n], [k] m 1 k [1] m 1 k [1 ] m m k

[k] s [k] s [k] [k] s [k] =. s [1] s [2] s [1] s [2] s [n] s [n] [1] = [11] s [1] + [12] s [2] + + [1n] s [n] = = [11] s [1] + [12] s [2] + + [1n] s [n] = s [1] ] = [ [11] [12] [1n] s [2] s [1] s [n] = [1] s [2], [1k] m 1 [1k] [1] m n [1k] m [1] m m n. d 1 d d N + s [n] d = 1. n = m. m > 1 n > 1

[(n 1)m 1] ( ), (4 4) (5 5) (2 2) m = n, d = 1 n > 2 m = n DoF = n2 n 2 1. ( ), (4 4) (5 5) ( ) P k k = 1, 2, k

P P P k R k = 1 E [ ( 2 1 + P hk 2)] h k k k E [ h 1 2] = E [ h 2 2] = E [ h 2] R = R 1 + R 2 + R = R 1 = E [ 2 ( 1 + P h1 2)]. [ ( DoF = E 2 1 + P h1 2) ] P 2 (1 + P h 1 2 = 1. ) 1 1 P [ ( E 2 1 + P h1 2) ] P 0 2 (1 + P h 1 2 =. ) P y 1 = h 11 x 1 + h 12 x 2 + h 1 x 2 + z 1,

h 1j j x j j P z 1 h 12 x 2 + h 1 x 2 + z 1 [ P h 11 R 1 = E [ 2 ]] 2 1 + P ( h 12 2 + h 1 2. ) + 1 R 2 = R = R 1 R 1 + R 2 + R [ P h 11 R = E [ 2 ]] 2 1 + P ( h 12 2 + h 1 2. ) + 1 DoF = P E [ 2 1 + 2 (1 + P h 11 2 ) P h 11 2 P ( h 12 2 + h 1 2 )+1 ]. h 12 = h 1 = 0 h 12 h 1 E P 0 [ 2 1 + 2 (1 + P h 11 2 ) P h 11 2 P ( h 12 2 + h 1 2 )+1 ] =,

k [k] P [k] ( [k], [k]) = P k [ ( 1 R k = E 8 2 + P )] [k] [k].

R = R 1 + R 2 + R [ ( R = E 8 2 + P )] [k] [k]. ( 8 E 2 + P [k] ) [k] 2 (1 + P h 11 2 ) 9 8 8 9 8

9 8 [ ] [ ] k [ ] [k] = [kk] [k] 1 [k] 2 [k] u [k] 1 u [k] 2 u [k] + [k] [k] + [k], [k] 8 1 N (0, N 0 ) [k] 8 6 [k] 6 1 k k P u [k] j, j = 1, 2, P k 8P

[ ] [ ] CN (0, 1) = P = tr { (, )} = tr {E [ ]} = tr {E [ ]} = = tr { E [ ] } = tr { P } 8 = P tr { } = = P 8 = = 8P. [k] k k [k] = [k] u [k] 1 u [k] 2 u [k] + [k]. [k] = ( [k]) [k] 1 [k] = ( [k]) [ ] [kk] [k] 1 [k] 2 [k] [k] = ( [k]) [k] 1 [ ]

=. [k] (, ) = E[ ] = E[ ] = E[ ] CN (0, N 0 ) = (, ) = N 0 = N 0 = N 0. N k [k] = +. =

= ( ) 1, ˆ = = + ˆ. ˆ = ˆ (ˆ ) k [k] [( [k] ) ] 1 ( [k] ) = [k] û [k] 1 û [k] 2 û [k] = [k] [k] = [k] [k] u [k] 1 u [k] 2 u [k] + [k] [k] = u [k] 1 u [k] 2 u [k] + ˆ [k]. E [(ˆ ) (ˆ ) ] ( = + 1 ) 1 SNR, ( + 1 ) 1 SNR + ˆ, ˆ = ( + 1 SNR ) 1 h i i h i

8 1 û i i ( + 1 ) 1 SNR i. i k [ ( [k] ) [k] = [k] + 1 ] 1 ( [k] ) SNR i [k] i ( b [k] i = [k] i ) [ ( [k] ) [k] + 1 ] 1 SNR [k] i, i [k] [k],i [k] i k i û [k] i = [k],i b [k] [k]. i

P

9 8

9 8

K 2