Single channel (univariate) signal - Practical Biomedical Signal Analysis Using MATLAB - page 39

sequences). Each window is used to compute the windowed periodogram of the sig-
nal. Subsequently the periodograms are averaged. This method is implemented in
MATLAB Signal Processing Toolbox as pmtm .

2.3.2.1.3 Relation of spectral density and the autocorrelation function Ac-

cording to the Wiener-Chinchyn formula spectral density function S ( f ) is a Fourier

transform of the autocorrelation function R ( τ ) :

Z ∞

e i 2π f τ d τ

S ( f )= R ( τ ) (2.38)

− ∞

Assuming that R ( τ ) exists and

Z ∞

∞ |

R ( τ ) | d τ < ∞ (2.39)

−

R ( τ ) is connected to S ( f ) by inverse Fourier transform.

Z ∞

e − i 2π f τ df R ( τ )= S ( f ) (2.40)

− ∞

From the above formula it follows that the integral of the spectral density is
equal to R ( 0 ) . Usually, one-sided spectral density estimator S ( f ) is calculated for

f ∈ ( 0 , ∞ ) , since for real signals S ( f ) is a symmetric function.

x ( t )

R x (

τ )

F F

F − 1 F − 1

X ( f )

S x (

f )

FIGURE 2.7: Illustration of relations between signal x ( t ) , its Fourier transform

X ( f ) , its autocorrelation function R x ( τ ) , and its spectrum S x ( f ) .

2.3.2.1.4 Bispectrum and bicoherence Polyspectra or higher order spectra pro-
vide supplementary information to the power spectrum. Third order polyspectrum
is called bispectrum. The prefix bi refers not to two time series, but to the two fre-
quencies of a single signal. Power spectrum according to equation 2.38 is a Fourier
transform of autocorrelation function, which is also called second order cumulant.