One-Dimensional Missing-Data APES via Expectation Maximization - Spectral Analysis of Signals: The Missing Data Case - page 45

and

cov µ | γ ,

θ i − 1

K =

) S g S g ) S g − 1

S m D i − 1 (

) S m

S m D i − 1 ( D i − 1 ( S g D i − 1 (

) S m = ω − ω ω ω . (5.43)

Expectation: Following the same steps as in (5.21) and (5.22), we obtain the
conditional expectation of the surrogate log-likelihood function in (5.40):
E 1 )

Q i − 1 (

i − 1 (

L ln p ( γ , µ | α

( ω ) , Q ( ω )) | γ , α ˆ ω ) , ω

tr 1 ) S m KS m

1
L ln L D − 1 (

=− M ln π − | D ( ω ) |− ω

)] H

+ [ S g γ + S m b − α ( ω ) ρ ( ω )][ S g γ + S m b − α ( ω ) ρ ( ω + C J . (5.44)

Maximization: To maximize the expected surrogate log-likelihood function
in (5.44), we need to exploit the known structure of D ( ω ) and ρ ( ω ). Let

S g γ +

z 0
.
z L − 1

S m b (5.45)

denote the data snapshots made up of the available and estimated data samples,
where each z l , l = 0 ,..., L − 1, is an M × 1vector. Also let Γ 0 ,..., Γ L − 1 be the

M blocks on the block diagonal of S m KS m . Then the expected surrogate
log-likelihood function we need to maximize with respect to

M ×

α ( ω ) and Q ( ω )

becomes (to within an additive constant)

tr Q − 1 (

H

Γ l

L − 1

+ z l ) e j ω l z l ) e j ω l

) 1
L

− ln | Q ( ω ) |− ω − α ( ω ) a ( ω − α ( ω ) a ( ω .

l = 0

(5.46)

The solution can be readily obtained by a derivation similar to that in Section 5.3:

) S − 1 ( a H (

ω ω ) Z ( ω )

α ˆ 2 ( ω ) = (5.47)

) S − 1 ( a H ( ω ω ) a ( ω )