Maximum Likelihood Fitting Interpretation of APES - Spectral Analysis of Signals: The Missing Data Case - page 32

CHAPTER 4

Maximum Likelihood
Fitting Interpretation
of APES

4.1 INTRODUCTION
In this chapter, we review the APES algorithm for complete-data spectral esti-
mation following the derivations in [13], which provide a “maximum likelihood
(ML) fitting” interpretation of the APES estimator. They pave the ground for the
missing-data algorithms we will present in later chapters.

4.2 ML FITTING BASED SPECTRAL ESTIMATOR
Recall the problem of estimating the amplitude spectrum of a complex-valued
uniformly sampled data sequence introduced in Section 2.2. The APES algorithm
derived below estimates

N − 1

α ( ω )from y n }

0 for any given frequency

ω .

{

n =

Partition the data vector

y N − 1 ] T y = [ y 0 y 1 ··· (4.1)

into L overlapping subvectors (data snapshots) of size M × 1 with the following

shifted structure:

y l + M − 1 ] T y l = [ y l y l + 1 ··· , l = 0 ,..., L − 1 , (4.2)