Digital Signal Processing Reference
In-Depth Information
Chapter 3
Estimation in Spectral Analysis
In this chapter we provide the necessary material for the statistical analysis of
various estimators (either of spectrum or of frequencies) described in the body of
this topic. More precisely, we insist on the methodology of the analysis of
estimators, particularly those based on the moments of the signal. After some
reminders on estimation theory, we provide results related to the statistical
performance of estimators of 1
st
and 2
nd
order moments of a stationary random
process or a process with a line spectrum. We then show how we can analyze the
performance of estimators using these moment estimates.
3.1. Introduction to estimation
3.1.1.
Formalization of the problem
Before entering into details, we will briefly describe the framework in which we
are based and which includes a large part of the problems tackled in the body of this
topic. The first section is deliberately general. Most of the results will be presented
without any demonstration, and anyway the theory of estimation is documented at
length in many topics; we recommend particularly the following works [AND 71,
BRO 91, KAY 93, POR 94, SCH 91, SÔD 89, TRE 71]. We briefly present the
elements that help formalize the problem of estimation, define the performance
criteria of an estimator and find the optimum estimators. In what follows, we
consider using
N
samples
()
0
{
n
=
of a signal which we group together in a
xn
N
×
N
:
vector;
x
∈
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