Gapped-Data APES - Spectral Analysis of Signals: The Missing Data Case - page 19

interpolated data. (This step is equivalent to applying APES to the complete
data.)
Step 3: Repeat steps 1-2 until practical convergence.

The practical convergence can be decided when the relative change of the cost
function in (3.16) corresponding to the current and previous estimates is smaller
than a preassigned threshold (e.g., 10 − 3 ). After convergence, we have a final =

−

K 1

spectral estimate

0 .Ifdesired, we can use the final interpolated data
sequence to compute the APES spectrum on a grid even finer than the one used in
the aforementioned minimization procedure.
Note that usually the selected initial filter length satisfies M 0

{ α ˆ ( ω

k )

}

k =

M due to
the missing data samples, so there are many practical choices to increase the filter
length after initialization, which include, for example, increasing the filter length
after each iteration until it reaches M

<

. For simplicity, we choose to use filter length

M right after the initialization step.

3.3 TWO-DIMENSIONAL GAPES
In this section, we extend the GAPES algorithm developed previously to 2-D data
matrices.

3.3.1 Two-Dimensional APES Filter
Consider the problem of estimating the amplitude spectrum of a complex-valued
uniformly sampled 2-D discrete-time signal

N 1 − 1 , N 2 − 1

{ y n 1 , n 2 }

0 , where the data matrix n 1 = 0 , n 2 =

has dimension N 1 N 2 .

Fora2-D frequency (

×

ω ,ω 2 )ofinterest, the signal y n 1 , n 2 is described as 1

2 ) e j ( ω 1 n 1 + ω 2 n 2 )

y n 1 , n 2 = α ( ω ,ω + e n 1 , n 2 ( ω ,ω

2 )

, n 1 = 0 ,..., N 1 − 1 ,

1 1

n 2 = 0 ,..., N 2 − 1 ,ω ,ω ∈ [0 , 2 π ) , (3.17)

1 2

where 2 ) denotes the complex amplitude of the 2-D sinusoidal compo-

nent at frequency (

α ( ω ,ω 1

ω ,ω 2 ) and e n 1 , n 2 ( ω ,ω 2 ) denotes the residual matrix (assumed 1 1