2014 IEEE International Conference on Acoustic, Speech and Signal Processing (ICASSP)

978-1-4799-2893-4/14/$31.00 ©2014 IEEE 2982

x[n]

X[k] =

√2

N

k+N−1∑n=k

x[n]e−jγn, k = 0, . . . ,K − 1,

k

ω0 +Δ ω0

100π 120π Δ

Va, Vb, Vc ≥ 0 ϕa, ϕb, ϕc ∈ [0, 2π]

Va = Vb = Vc ϕa = ϕb +2π3

= ϕc − 2π3

N ω0

⎡⎣ va[n]

vb[n]vc[n]

⎤⎦ =

1

2ejγ

ω0+Δ

ω0nv +

1

2e−jγ

ω0+Δ

ω0nv∗ +wa,b,c[n],

∀n ∈ R γ�= 2π

Nv

�=

[Vae

jϕa , Vbejϕb , Vce

jϕc

]{wa,b,c[n]}n∈Z

σ2I3

I3 3 × 3Va = Vb = Vc

ϕa = ϕb +2π3

= ϕc − 2π3

{va[n]}, {vb[n]}{vc[n]}

k+N−1∑n=k

ejαn =sin(αN/2)

sin(α/2)ejα(k+

N−12 ), ∀α ∈ R,

⎡⎣ Va[k]

Vb[k]Vc[k]

⎤⎦ =

1√2Pe

jγ Δω0

kv +

1√2Qe

−jγ2ω0+Δ

ω0kv∗

+

√2

N

k+N−1∑n=k

wa,b,c[n]e−jγn,

k = 0, . . . ,K − 1

P =sin(γN Δ

2ω0)

N sin(γ Δ2ω0

)ejγ Δ

ω0

N−12

Q =sin(γN 2ω0+Δ

2ω0)

N sin(γ 2ω0+Δ2ω0

)e−jγ

2ω0+Δ

ω0

N−12 .

P QΔ k

PQ Δ

⎡⎣ V0[k]

V+[k]V−[K]

⎤⎦ =

1

3

⎡⎣ 1 1 1

1 α α2

1 α2 α

⎤⎦

︸ ︷︷ ︸�=H

⎡⎣ Va[k]

Vb[k]Vc[k]

⎤⎦ ,

∀k = 0, 1, . . . ,K − 1 V0[k], V+[k] V−[k]

α = ej2π/3

V0[k] = Pejγ Δ

ω0kC0 +Qe

−jγ2ω0+Δ

ω0kC∗

0 +W0[k]

V+[k] = Pejγ Δ

ω0kC+ +Qe

−jγ2ω0+Δ

ω0kC∗

− +W+[k]

V−[k] = Pejγ Δ

ω0kC− +Qe

−jγ2ω0+Δ

ω0kC∗

+ +W−[k]

k = 0, . . . ,K − 1

[C0, C+, C−]T =

√2

6Hv

⎡⎣ W0[k]

W+[k]W−[k]

⎤⎦ =

√2

3NH

k+N−1∑n=k

wa,b,c[n]e−jγn,

∀k = 0, . . . ,K − 1 HHH = 3I3

{W0[k]}K−1k=1 {W+[k]}K−1

k=1 {W−[k]}K−1k=1

2σ2

3N

C0 C+

C−

KK

ν0 = PC0e1 +QC∗0 e2 + w0

ν+ = PC+e1 +QC∗−e2 + w+

ν− = PC−e1 +QC∗+e2 + w−,

e1 =[1, e

jγ Δω0 , . . . , e

jγ Δω0

(K−1)]T

,

e2 =

[1, e

−jγ2ω0+Δ

ω0 , . . . , e−jγ

2ω0+Δ

ω0(K−1)

]T

,

ν0 = [V0[0], . . . , V0[K − 1]] ν+ = [V+[0], . . . , V+[K − 1]]ν− = [V−[0], . . . , V−[K − 1]]

e1 e2

w0 w+ w−

R R

K ×K k, l

[R]k,l =2σ2

3N2

{N − |k − l| −N ≤ k − l ≤ N0

.

2983

R− 1

2

ν0 = PC0e1 +QC∗0e2 +w0

ν+ = PC+e1 +QC∗−e2 +w+

ν− = PC−e1 +QC∗+e2 +w−,

em�= R

− 12 em m = 1, 2

ν0 = R− 1

2 ν0 ν+ = R− 1

2 ν+ ν− = R− 1

2 ν−

w0 = R− 1

2 w0 w+ = R− 1

2 w+ w− =

R− 1

2 w− IK

θ = [C0, C+, C−,Δ]

θ

θ = maxθ L(θ)

L(θ) = 3K log π − ||ν0 − PC0e1 −QC∗0e2||2

− ||ν+ − PC+e1 −QC∗−e2||2

− ||ν− − PC−e1 −QC∗+e2||2 .

ΔC0 C+ C−

C0 =z0 − κ2(C0)

∗

κ1

C+ =z+ − κ2(C−)

∗

κ1,

C− =z− − (C+)

∗κ2

κ1,

z0�= P ∗

eH1 ν0 + Qν

H0 e2, z+

�= P ∗

eH1 ν+ +

QνH−e2, z−

�= P ∗

eH1 ν−+Qν

H+e2, κ1

�= |P |2eH

1 e1 + |Q|2eH2 e2

κ2�= 2P ∗Qe

H1 e2

C0 =κ1z0 − κ2z

∗0

κ21 − |κ2|2 ,

C+ =κ1z+ − κ2z

∗−

κ21 − |κ2|2 ,

C− =κ1z− − κ2z

∗+

κ21 − |κ2|2 .

C0 C+ C−

L(θ) = 3K log π − ||ν0||2 − ||ν+||2 − ||ν−||2

+κ1

(|C0|2 + |C+|2 + |C−|2

)+

{κ∗2

(C2

0 + 2C+C−

)}.

Δ

Δ = argmaxΔ

[κ1

(|C0|2 + |C+|2 + |C−|2

)

+{κ∗2

(C2

0 + 2C+C−

)}].

Δ[−Nω0

2, Nω0

2

)

Vaejϕa = Vbe

j(ϕb+2π3

) = Vcej(ϕc−

2π3

)

C0 = 0 C− = 0

⎧⎨⎩

ν0 = w0

ν+ = PC+e1 +w+

ν− = QC∗+e2 +w−

.

C+

C(b)0 = 0, C

(b)+ =

z+κ1

, C(b)− = 0.

V+[k] = e−jγa(Δ)V+[k − 1]− e−j2γV+[k − 2]

+W+[k]− e−jγa(Δ)W+[k − 1] + e−j2γW+[k − 2],

k = 2, . . . ,K − 1

a(Δ) = 2 cos

(γω0 +Δ

ω0

).

ΔK

⎡⎣ E1ν0

E1ν+

E1ν−

⎤⎦ = a(Δ)

⎡⎣ E2ν0

E2ν+

E2ν−

⎤⎦+wAR[k],

2984

E1�=

⎡⎢⎢⎢⎣

e−j2γ 0 1 0 . . . . . . 00 e−j2γ 0 1 0 . . . 0

. . . . . . 00 . . . 0 e−j2γ 0 1

⎤⎥⎥⎥⎦ ,

E2�=

⎡⎢⎢⎢⎣

0 e−jγ 0 . . . 00 0 e−jγ 0 . . . 0

0 . . . . . . 00 . . . 0 e−jγ 0

⎤⎥⎥⎥⎦ .

wAR[k] k = 0, . . . ,K − 1

⎡⎣ T 0 0

0 T 0

0 0 T

⎤⎦ ,

T�= (E1 − a(Δ)E2)R(E1 − a(Δ)E2)

H .a(Δ)

a(Δ) =(νH0 E

H2 T

−1E2ν0

+νH+E

H2 T

−1E2ν+ + ν

H−E

H2 T

−1E2ν−

)−1

×(νH0 E

H2 T

−1E1ν0 + ν

H+E

H2 T

−1E1ν+

+νH−E

H2 T

−1E1ν−

).

a(Δ)T a(Δ)

a(Δ)T

Δ = 0

Δ = ω0/γ cos−1 (a/2)− ω0,

cos−1(·)

N = 48ω0 = 2π · 60

500 K = 8

Δ = 2.5 · 2π=

V 2a+V 2

b+V 2

c

σ2

Va = Vb = 1Vc = βVa ϕa = π

4ϕb = ϕa − 2π

3ϕc = ϕa + 2π

3+ ε

β = 0 ε = 0

C0 C+

C− C(b)0 C

(b)+ C

(b)−

C(b)0

C(b)+

C(b)−

C0

C+

C−

K = 8 N = 48Δ = 2.5× 2π

γΔω0

Δs =ω0

γ

1

K

K−2∑k=0

(V+[k + 1])− (V+[k]) .

3(K − 1)3K

K

γΔ

ω

γ Δω0

K = 8 N = 48 Δ = 2.5 × 2π

2985

2986