Non-Parametric Methods - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 126

The average value of this estimator may be written as:

( )

() () ()

Er κ = B κ r κ

[5.41]

xx − N N , xx

where

B −

is the even Bartlett window, that is:

N N

⎧ − κ

1 if κ ≤ − N 1

⎪
= ⎨
⎪ ⎩

()

B κ

[5.42]

N

− NN ,

0 otherwise

This estimator, though having a bias, guarantees that the estimation of the
Toeplitz hermitian autocorrelation matrix

( ) 1

⎡ r − k ⎤

is semi-definite

⎣ ⎦

≤≤
≤≤

N

xx

kN

positive.

It is easy to show that if the biased estimator of the autocorrelation function is
used, then the periodogram and the estimator I' xx are equal.

I ′

I =

[5.43]

xx xx

From formulae [5.38] and [5.41], we can immediately deduce that the average
value of the periodogram is equal to the circular convolution of the exact spectral
density and the Fourier transform of the Bartlett window

B − .

NN

ˆ

( )

EI = S B −

[5.44]

xx xx N N ,

where

B −

can be expressed by (this result is easily obtained by considering

N N

B −

as the convolution integral of (1 0, N- 1 ( k )) k ∈Ζ and (1 0, N- 1 ( -k )) k ∈Ζ divided by N ):

NN

2

( )

()

⎛ ⎞

sin πν N

ˆ

()

B ν = ⎜ N

[5.45] ⎟

− NN ,

⎜ ⎟

N sin πν

⎝ ⎠

This function has a unit integral over a period (by applying Parseval's theorem),
and tends towards infinity when N tends towards infinity at integer abscissa. Thus, it
tends towards the Dirac comb Ξ 1 of period 1 which is the identity element for the
circular convolution integral. Thus, the periodogram is asymptotically unbiased:

( )

lim EI = S

[5.46]

xx xx

N →∞