Non-Parametric Methods - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 128

2

is thus to the multiplicative factor M

N

⎛ ⎜ ⎝⎠

e

the power of the sine wave

I

xx

M

present in this decomposition, that is to say, to the factor M
N

, the average power

1
M

spectral density of the signal x p in the interval of a frequency magnitude

:

centered on M

M

⎛⎞ =

() 2

ˆ [5.51]

I M x

MN

⎜ ⎝⎠

xx p

It is to be noted that the zero-padding technique introduces a redundancy in the

1
N

periodogram. In fact, knowing only its value for multiple frequencies of , we can

recreate the estimator of the autocorrelation function from the inverse Fourier
transform and, from this estimator, calculate the periodogram at any frequency by

frequencies with

the direct Fourier transform. The value of the periodogram at

N

0 ≤ ≤ N - 1 is sufficient. As a result, it is natural to measure the resolution of the

1
N

periodogram by , which is the frequency sampling value of this minimum

representation.

The variance of the periodogram is approximately equal to the value of the
power spectral density, and does not decrease with the length N of the recording
(see section 3.3). The periodogram variants which will be developed in the next
paragraph will reduce this variance. As a result, this procedure leads to a
deterioration of the resolution power. This resolution/variance compromise,
considered as important before obtaining the spectral parametric estimation, is the
main disadvantage of standard methods of spectral analysis.

5.2.3. Periodogram variants

The average periodogram or Bartlett's periodogram consists of splitting the

N
K

signal of N points into K sections of length , to calculate the periodogram on

each segment, and then to average the periodograms obtained: