Two-Dimensional MAPES via Expectation Maximization and Cyclic Maximization - Spectral Analysis of Signals: The Missing Data Case - page 70

We let y denote the L 1 L 2 M 1 M 2 × 1vector obtained by concatenating all the

snapshots:

=

y 0 , 0
.
y L 1 − 1 , L 2 − 1

y

S g γ

+

S m µ

,

(6.31)

where S g (which has a size of L 1 L 2 M 1 M 2 × g ) and S m (which has a size of

g )) are the corresponding selection matrices for the avail-
able and missing data vectors, respectively. Because of the overlapping of the vec-
tors

L 1 L 2 M 1 M 2 × ( N 1 N 2 −

{ y l 1 , l 2 } , S g and S m are not unitary, but they are still orthogonal to each other:

S g S m

= 0 g × ( N 1 N 2 − g ) .Soinstead of (6.12) and (6.13), we have from (6.31):

γ = S g S g − 1 S g y

S g y

= (6.32)

and

µ = S m S m − 1 S m y

S m y

= , (6.33)

S g and S m introduced above are defined as S g

S g ( S g S g ) − 1

where the matrices ,

S m

S m ( S m S m ) − 1 , and they are also orthogonal to each other: S g

S m

0 g × ( N 1 N 2 − g ) .
Now the normalized surrogate log-likelihood function in (6.8) can be written

=

as

1
L 1 L 2

ln p ( y | α ( ω ,ω

2 )

, Q ( ω ,ω

2 )) 1 1

1
L 1 L 2

1
L 1 L 2 [ y

2 )] H =− M 1 M 2 ln π − ln | D ( ω ,ω 2 ) |− − α ( ω ,ω 2 ) ρ ( ω ,ω 1 1 1

D − 1 (

× ω ,ω

2 )[ y

− α ( ω ,ω

2 ) ρ (

ω ,ω

2 )]

, (6.34)

1 1 1

where ρ ( ω ,ω 2 ) and D ( ω ,ω 2 )are defined as 1 1

e j ( ω 1 0 + ω 2 0) a ( ω ,ω 2 ) 1

.
e j [ ω 1 ( L 1 − 1) + ω 2 ( L 2 − 1)] a (

ρ ( ω ,ω

2 )

(6.35)

1

ω ,ω

2 )

1