Non-Parametric Methods - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 134

The summation occurring for all values of k between 0 and N - 1 and such that
for all values of I , k + k i is between 0 and N - 1. This method (which is indirect
because it passes from the estimation of the moment) is similar to the correlogram of
the 2 nd order. In practice, to reduce the variance of this estimator, we will consider
the estimated moment as a multidimensional truncation window w n-1 , defined by the
product of a standard one-dimensional window w:

n − 1

()

= ∏

( )

w κ w κ [5.65]

n − 1 i

i = 1

The other method (which is direct because it operates directly on the data),
called periodogram of higher order, is defined in the following manner:

n − 1 n − 1

⎛ ⎞

1

()

∑ ∏

()

S ν = X * ν X ν [5.66]

⎜ ⎜ ⎝ ⎠

n i i

N

i = 1 i = 1

N − 1

where () = ∑ . As it is of the 2 nd order, it is useful to carry

out an average of the periodograms from different sections of the original signal,
and to weight the different sections by a truncation window - Welsh's procedure
(see [NIK 93]).

()

− j 2 πν k

X ν x k e

i

k = 0

5.4. Bibliography

[KAY 81] KAY S., MARPLE S., “Spectrum analysis. A modern perspective”, IEEE
Proceedings, vol. 69, no. 11, pp. 1380-1418, 1981.
[MAR 87] MARPLE JR. S., Digital Spectral Analysis with Applications, Prentice Hall, 1987.
[MAX 96] MAX J., LACOUME J., Méthodes et techniques de traitement du signal et
applications aux mesures physiques. Volume 1: principes généraux et méthodes
classiques, Masson, 1996.
[NIK 93] NlKIAS C., PETROPULU A., Higher-order Spectra Analysis, Prentice Hall, 1993.
[OPP 75] OPPENHEIM A., SCHAEFER R., Digital Signal Processing, Prentice Hall, 1975.
[POR 94] PORAT B., Digital Processing of Random Signals, Prentice Hall, 1994.