Non-Parametric Methods - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 117

Chapter 5

Non-Parametric Methods

5.1. Introduction

A random discrete time signal () kk Z x is stationary in the wide sense if its

mean m x and its autocorrelation function ()

xx r κ defined by:

( ) ()

⎧

m = E x k

⎪ ⎨

x

[5.1]

( )

( ) ( ( )

() () )

r κ = E x k − m * x k + κ − m ∀ ∈ κ Z

⎪ ⎩

xx x x

are independent of the index k , that is to say independent of the time origin.

()
2

r κ
σ

2 xx

()

σ = r 0 is the variance of the considered signal. is the correlation

x xx

x

coefficient between the signal at time k and the signal at time k + . Usually we
limit ourselves only to the average and the autocorrelation function to characterize a
random stationary signal, even though this second order characterization is very
incomplete [NIK 93].

Practically, we have only one realization () , x kk ∈ of a random signal ()

x k ,

for which we can define its time average () x k :

N

1

+ ∑

() ()

x k = lim x k [5.2]

21 N

N →∞

kN =−