l 1

p r i = ρ ij α j + w i j=1 ρ ij λ α j j p w i p α j = 1, α j 0, j = 1,..., p j=1 R B B B m j [ρ 1j, ρ 2j,..., ρ Bj ] T = }{{} α +

[,,..., ] R B p p α [α 1,..., α p ] [w 1,..., w p ] M m 1 m 2, m 1 m 3,..., m 1 m p p S x { = α : α j = 1, α j 0, j = 1,..., p} j=1 M (p 1) R B C M

γ W = 0 = γα. p C p { R B : = γα : α j = 1, α j 0, j = 1,..., p} j=1

C S x S p { R B : = /( T ), r C p } T = 1 S p S p f 1 m a f 2 m a m b p << B

p R [r 1,..., r N ] M SNR th = 15 + log 10 (p)db SNR SNR th d := p := U d := mean( ) 1 d [ ] :,j := [ ] :,j /([ ] :,j ) d := p 1 [ ] :,j := ([ ] :,j r) U d c := j=1,...,n [ ] :,j c := [c c... c] c 1 N [ ] := := [ u 0... 0] u = [0,..., 0, 1] T p p i := 1 p := (0, p ) p := (( ) )/( ) ) [ ] :,1:i := T k := j=1,...,n [ ] :,j [ ] :,i := [ ] :,k [] i := k SNR SNR th M := d [ ] :, N p M := d [ ] :, + r N p SNR th SNR SNR th p SNR p 1

SNR SNR th SNR SNR th A A ˆM B N Y [y 1, y 2,..., y N ] R p n y i R p i = 1,..., n Y = MS S 0, 1 T p S = 1 T n

[,,..., ] R p p S R p n Y M, S M M = M M s.t QY 0, 1 T p QY = 1 T N Q M 1 Q = 1/ M Q = M Q s.t QY 0, 1 T p QY = 1 T N 1 T p QY = 1 T N p Y p = [y i1, y i2,..., y ip ] y = Y p β 1 T p β = 1 1 T p QY = 1 T N 1T p QY p = 1 T N 1 T p Q = 1 T p Y 1 q m = 1 T p Y 1 1 T p Q = q m Q = M Q s.t QY 0, 1 T p Q = q m S ij α j

M [m 1,..., m p ] p M g(q) :,j g(q) i,: j i g(q) f(q) p Y (f(q)) (Q) Q Q 0 :=Y p Q 0 := M Y := Y A Q b A = Y T I p b = 0 pn Aeq Q = beq Aeq = I p 1 T p beq = qm T g(q) := (Q 1 ) T [H(Q)] i,j := [g(q) :,j g(q) i,: ] Q := (f, Q 0, A, b, Aeq, beq, g, H) g(q) f H(Q) f

L 1 N 1 L 2 N 2 = [x 1, x 2,..., x N ] T N = L 1 N 1 L 2 N 2 L 1 L 2 N 1 N 2 = [y k,1, y k,2,..., y k,m ] T k k = 1, 2,..., p M = N 1 N 2 x x = k + 1 k p L 1 N 1 L 2 N 2 L 1 N 1 L 2 N 2 k (N 1 N 2 ) 2 L 1 N 1 L 2 N 2 Y 1 D 1 H 1 F 1 Y 2 D 2 H 2 F 2 = + V D k H k F k Y k

= M + V,,

x(t 1, t 2 ) X(w 1, w 2 ) k x k (t 1, t 2 ) = x(t 1 + δ k1,t 2 +δ k2 ) δ k1, δ k2 k = 1,..., p X k (w 1, w 2 ) X k (w 1, w 2 ) = {j 2 π(δ k1 w 1 + δ k2 w 2 )}X(w 1, w 2 ) x k (t 1, t 2 ) T 1, T 2 y k [n 1, n 2 ] = x k (n 1 T 1 + δ k1, n 2 T 2 + δ k2 ) Y k [r 1, r 2 ] k Y k [r 1, r 2 ] = 1 T 1 T 2 m 1 = m 2 = X k ( 2π T 1 ( r 1 N 1 m 1 ), 2π T 2 ( r 2 N 2 m 2 )) X(w 1, w 2 ) X(w 1, w 2 ) = 0 u 1 (N 1 π)/t 1 u 2 (N 2 π)/t 2 Y k [r 1, r 2 ] x(t 1, t 2 )

Y = ΦX Y p 1 k y k [r 1, r 2 ] X N 1 N 2 1 x(t 1, t 2 ) Φ p N 1 N 2 Y X H k B F k

F k H = + = 1 k p M(v, h) v h = P r ( ) = P r (, M(v, h) ) v v,h = = v,h v,h P r (, M(v, h))p r (, M(v, h)) v P r ( ) P r (, M(v, h))p r ( )P r (M(v, h)) v

M(v, h) P r (, M(v, h)) P r ( ) P r (M(v, h)) V P r (, M(v, h)) { 1 2σ 2 M(v, h) 2 } P r ( ) P r ( ) = 1 Z { αa( )} M(v, h) = P r (, M)P r ( ) = M 2 + λa( ) λ M P r ( )

{ i } n i=1 { i } n i=1 { i, i } i = DH i + { i } n i=1 { i } n i=1 k yk t t k N t { i } n i=1

ˆω s = yk t ω s y s 2, ω s y s N t ω s = 1 y s N t ˆx t k = ˆω s x s y s N t

Ψ Ψ Ψ x N 1 R N

x[n] n = 1,..., N {ψ i } N i=1 N 1 N N Ψ = [ψ 1 ψ 2... ψ N ] {ψ i } N i=1 x x = N s i ψ i x = ψs i=1 s N 1 s i =< x, ψ i >= ψi T x x s Ψ x K K K s i K N 97.5 N x s s = Ψ T x K (N K) K N

K K M N x {ϕ j } M j=1 y j =< x, ϕ j > y j M y ϕ T j Φ M N x y = Φx = ΦΨs = Θs Θ = ΦΨ M N Φ x Φ K x R N y R M x M K y Φ x N M < N M < N x K K M K v K s 1 ϵ Θv 2 v 2 1 + ϵ Θ K Θ K Θ

Φ, Ψ µ(φ, Ψ) = N < ϕ k, ψ j >, µ(φ, Ψ) [1, N] 1 k,j n Φ Φ ϕ j,i 1/N Φ Ψ = I δ M N Θ M ck (n/k) c K M ck (n/k) N Φ Θ Ψ N M y Φ Ψ l 0 s K M = K + 1 ( K N ) s l 1 K M ck (N/K) ŝ = ś 1 Θś = y l 1

R n K K K > n R n = α 0 α 0 R K n. = = α 0, R k n k < n = α α 0 h l l h

l h, = SH H S h h α α R K α 0 K

α l h h, l l α 0 F l α F 2 2 ϵ F α F α l 1 α 1 F l α F 2 2 ϵ min α F l α F 2 2 + λ α 1, λ l 1

h α α 1 F l α F 2 2 ϵ 1 P h α 2 2 ϵ 2 P min α α 2 2 + λ α 1, [ ] [ ] F l F = = β βp h β β = 1 α = h α l α 0 0 SH = = SH 2 2 + c 0 2 2 t+1 = t + ν[h T S T ( SH t ) + c(x X 0 )] t t ν

0 h l 3 3 m α α 2 2 + λ α 1 = h α + m 0 0 = SH 2 2 + c 0 2 2 F F 1 = [ 1, 0, 1], 2 = T 1, 3 = [1, 0, 2, 0, 1], 4 = T 3

λ 0.01 28.70 9.37 0.05 29.09 8.96 0.1 29.30 8.75 0.15 29.36 8.68 0.2 29.47 8.57 0.25 29.51 8.53 0.3 29.58 8.46 0.4 29.56 8.48 0.45 29.43 8.61 λ λ = 0.35

λ 0.1 28.50 9.59 0.15 28.64 9.43 0.2 28.72 9.34 0.3 28.75 9.31 0.35 28.71 9.35 λ

22.02 44.61 20.21 1.5 1.5 20.21 = 27.64 = 10.59

29 = 9.09 = 29.20 = 8.9