Two-Dimensional MAPES via Expectation Maximization and Cyclic Maximization - Spectral Analysis of Signals: The Missing Data Case - page 67

γ l 1 , l 2 , µ l 1 , l 2 }

the joint normalized surrogate log-likelihood function of { is obtained

by substituting (6.17) into (6.9)

1
L 1 L 2

γ l 1 , l 2 , µ l 1 , l 2 }| α

ln p ( { ( ω ,ω 2 ) , Q ( ω ,ω 2 )) 1 1

tr Q − 1 (

L 1 − 1 L 2 − 1

1
L 1 L 2

=− M 1 M 2 ln π − ln | Q ( ω ,ω 2 ) |− ω ,ω 2 ) 1 1

l 1 = 0 l 2 = 0

S g ( l 1

2 ) e j ( ω 1 l 1 + ω 2 l 2 )

S m ( l 1

, l 2 ) γ l 1 , l 2 + , l 2 ) µ l 1 , l 2 − α ( ω ,ω 2 ) a ( ω ,ω 1 1

H

S g ( l 1

2 ) e j ( ω 1 l 1 + ω 2 l 2 )

S m ( l 1

, l 2 ) γ l 1 , l 2 + , l 2 ) µ l 1 , l 2 − α ( ω ,ω 2 ) a ( ω ,ω . 1 1

(6.18)

Just as in the 1-D case, the probability density function of µ l 1 , l 2 conditioned
on γ l 1 , l 2 (for given θ =

i − 1

θ b l 1 , l 2 and covariance

)iscomplex Gaussian with mean

K l 1 , l 2 :

matrix

θ i − 1

( b l 1 , l 2

K l 1 , l 2 )

µ l 1 , l 2 | γ l 1 , l 2 , ∼ CN , , (6.19)

where

E µ l 1 , l 2 γ l 1 , l 2 ,

i − 1

b l 1 , l 2 = θ

S m ( l 1

i − 1 (

2 ) e j ( ω 1 l 1 + ω 2 l 2 )

= , l 2 ) a ( ω ,ω

2 ) ˆ

α ω ,ω

1 1

l 2 ) S g ( l 1 l 2 ) − 1

S m ( l 1

l 2 ) Q i − 1 (

2 ) S g ( l 1

l 2 ) Q i − 1 (

2 ) S g ( l 1

+ , ω ,ω , , ω ,ω ,

1 1

× γ l 1 , l 2 −

2 ) e j ( ω 1 l 1 + ω 2 l 2 )

S g ( l 1

i − 1 (

, l 2 ) a ( ω ,ω

2 ) ˆ

α ω ,ω (6.20)

1 1

and

cov µ l 1 , l 2 γ l 1 , l 2 ,

θ i − 1

K l 1 , l 2

=

S m ( l 1

l 2 ) Q i − 1 (

2 ) S m ( l 1

S m ( l 1

l 2 ) Q i − 1 (

2 ) S g ( l 1

= , ω ,ω , l 2 ) − , ω ,ω , l 2 )

1 1

× S g ( l 1 l 2 ) − 1 l 2 ) Q i − 1 ( 2 ) S g ( l 1

S g ( l 1

l 2 ) Q i − 1 ( 2 ) S m ( l 1 , ω ,ω , , ω ,ω , l 2 ) . 1 1

(6.21)