One-Dimensional Missing-Data APES via Expectation Maximization - Spectral Analysis of Signals: The Missing Data Case - page 44

Now the normalized surrogate log-likelihood function in (4.6) can be written

as

1
L ln p ( y

1
L ln

| α ( ω ) , Q ( ω )) =− M ln π − | D ( ω ) |

1
L [ y

)] H D − 1 ( − − α ( ω ) ρ ( ω ω )[ y − α ( ω ) ρ ( ω )] (5.37)

where ρ ( ω ) and D ( ω )are defined as

e j ω 0 a ( ω )

.
e j ω ( L − 1) a (

ρ ( ω ) (5.38)

ω )

and

Q ( ω ) 0

.

. . .

D ( ω ) (5.39)

0 Q ( ω )

Substituting (5.30) into (5.37), we obtain the joint surrogate log-likelihood of γ
and µ :
1
L ln p ( γ

L −

1

)] H , µ | α ( ω ) , Q ( ω )) = LM ln π − ln | D ( ω ) |− [ S g γ + S m µ − α ( ω ) ρ ( ω

)] + D − 1 (

× ω

)[ S g γ

+

S m µ

− α ω

) ρ (

ω ,

(5.40)

( C J

where C J is a constant that accounts for the Jacobian of the nonunitary transfor-
mation between y and γ and µ in (5.30).
To derive the EM algorithm for the current set of assumptions, we note that
for given ˆ

Q i − 1 (

i − 1 (

α ω ) and ω ), we have (as in (5.18)-(5.20))

θ i − 1

µ | γ , ∼ CN ( b , K ) , (5.41)

where

E µ

θ i − 1

b = | γ ,

) S g S g ) S g − 1

γ

)
(5.42)

S m ρ ( S m D i − 1 ( D i − 1 ( S g ρ (

i − 1 ( i − 1 (

= ω ) ˆ α ω ) + ω ω − ω ) ˆ α ω