One-Dimensional Missing-Data APES via Expectation Maximization - Spectral Analysis of Signals: The Missing Data Case - page 43

obtained by concatenating all the snapshots

y 0
.
y L − 1

=

y S g γ + S m µ , (5.30)

where S g ( LM g )) are the corresponding selection ma-

trices for the available and missing data vectors, respectively. Because of the over-
lapping of

× g ) and S m ( LM × ( N −

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

other:

S g S m = 0 g × ( N − g ) . (5.31)

Instead of (5.6) and (5.7), we have from (5.30)

γ = S g S g − 1 S g y

S g y

= (5.32)

and

= S m S m − 1 S m y

S m y

µ = . (5.33)

S g and S m introduced above are defined as

The matrices

S g S g S g − 1

S g

(5.34)

and

S m S m S m S m − 1

, (5.35)

and they are also orthogonal to each other:

S g S m

= 0 g × ( N − g ) . (5.36)

Note that S g S g and S m S m are diagonal matrices where each diagonal element
indicates how many times the corresponding sample appears in y owing to the
overlapping of . Hence both S g S g and S m S m canbeeasily inverted.

{ y l }