Estimation in Spectral Analysis - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 83

2

( ) ( ) ( )

Σ m , = 2 σ ⎡ c + m + c − m ⎤

⎣ ⎦

ss ss

[3.20]

( { }

⎡ ⎤

4

( )

4 4

() ( )

+ σδ − mb + n − 2 σ δ δ m

E

⎢ ⎥

⎣ ⎦

Finally, we would like to mention that very general theoretical results, as they
concern signals that are a combination of deterministic and random components
(stationary or non-stationary), were published by Dandawaté and Giannakis [DAN
95]. This last article makes it possible to include a very large class of signals and
provides analytical expressions of asymptotic distributions of moment estimates.
Finally, the reader is asked to read [DAN 95, LAC 97] for the analysis of estimator
of higher order moments.

3.3. Periodogram analysis

The periodogram is a natural estimation tool for the power spectral density. Let
us recall that its expression is:

2

N − 1 N − 1

1

∑ ∑

() ()

− j 2 π f

()

− j 2 π mf

If = xne = γ ˆ me [3.21]

xx

N

( )

n = 0 m =− − N 1

where:

Nm −− 1

1

∑

() ()( )

ˆ

γ m = x n x n + m

xx

N

n = 0

designates the biased estimator of the correlation function. The objective of this
section is to derive the asymptotic properties of I ( f ) for a real signal. Firstly we
calculate the bias of this estimator. The periodogram mean can directly be written
as:

N − 1

⎛ m ⎞

∑

{ }

() ()

− j 2 π mf

If = 1 − γ me

E

⎜ ⎟

xx

N

⎝ ⎠

( )

mN =− − 1

1/ 2

∫

() ( )

= SvWfv v −

x

− 1/ 2

2

1

( )

⎡

sin
sin

π
π

fN
f

⎤

where () Wf =

⎦ is the Fourier transform of the triangular window.

⎢ ⎥

( )

N ⎣