Non-Parametric Methods - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 122

the autocorrelation function r xx and spectrum S xx , the autocorrelation function r vv and
the spectrum S vv of the filter signal yhx

=⊗ can be expressed by:

=⊗⊗ *

r h hr

[5.23]

yy xx

2

ˆ

S = h S

[5.24]

yy xx

5.2. Estimation of the power spectral density

Under the hypothesis of a continuous or discrete time random signal x ergodic
for the autocorrelation verifying the hypothesis [5.10], and using a recording of
finite length ()

()

≤≤ ≤ ≤ − of a realization x , we attempt to
obtain an estimator of power spectral density S xx , where we can obtain the static
characteristics (bias, variance, correlation) as well as the resolution, that is to say the
capacity of the estimator to separate the spectral contributions of two narrow band
components, for example two cisoids.

xt ,0 t T or xk ,0 k N 1

The filter bank method (see section 5.2.1), used in analog spectrum analyzers, is
more naturally used in the analysis of continuous time signals. The periodogram
method (see section 5.2.2) and its variants (see section 5.2.3) are used for the
analysis of discrete time signals, and are generally calculated using the fast Fourier
transform algorithms (FFT).

5.2.1. Filter bank method

The power spectral density describes the distribution of power along the
frequency axis. Thus, to estimate this density, it is natural to filter the signal
observed x ( t ) by a band pass filter bank of impulse response h whose center
frequencies f are distributed along the frequency axis. Thus, the power P of

e

filter signal yhx

=⊗

equals:

+∞

= ∫

()

P S f df

[5.25]

yy

−∞

+∞

2

ˆ

= ∫

() ()

hf S f f

[5.26]

xx

−∞