Gapped-Data APES - Spectral Analysis of Signals: The Missing Data Case - page 15

3.2 GAPES
Assume that some segments of the 1-D data sequence

N − 1

{ y n }

0 are unavailable. Let n =

y N − 1 ] T

···

y [ y 1 y 2

y 1

y P

T

y 2 ··· (3.1)

be the complete data vector, where y 1 ,..., y P are subvectors of y , whose lengths are
N 1 , ..., N P ,respectively, with N 1 + N 2 +···+ N P = N .Agapped-data vector

γ is formed by assuming y p , for p = 1 , 3 , ..., P ( P is always an odd number),

are available:

y 1

y P

T

y 3

γ ··· . (3.2)

Similarly,

y 2

y P − 1

T

y 4

µ ··· (3.3)

denotes all the missing samples. Then γ and µ have dimensions g × 1 and

( N − g ) × 1, respectively, where g = N 1 + N 3 + ···+ N P is the total number of

available samples.

3.2.1 Initial Estimates via APES
We obtain the initial APES estimates of h ( ω ) and α ( ω )from the available data γ

as follows.
Choose an initial filter length M 0 such that an initial full-rank covariance ma-
trix R canbebuilt with the filter length M 0 using only the available data segments.
This indicates

max(0 , N p − M 0 + 1) > M 0 . (3.4)

p ∈{ 1 , 3 ,..., P }

Let L p = N p − M 0 + 1 and let J be the subset of { 1, 3, . . . , P } for which L p > 0.

Then the filter-bank h ( ω )iscalculated from (2.11) and (2.12) by using the