Time sequence and spectral analysis - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 178

For discrete, equispaced, windowed data, the spectral density estimator (2.332)
becomes

Δ

j =− N w j g j e − i 2π f k j Δ t

2

N

t

S gg ( f k )

= . (2.350)

I

This is the squared magnitude of the discrete Fourier transform of the windowed
data, divided by I .
When the window being used is the Parzen window (2.284), w( t ) = p ( t )and

M /2

151
560 M ,

p 2 ( t ) dt I = = (2.351)

− M /2

on performing the elementary integrations of the square of the piecewise cubic.
As seen previously (2.343), the true spectrum is convolved, or averaged, with the
square of the Parzen frequency window, whose half-power points are separated
by a distance 1.82/ M along the frequency axis. The single segment estimate is
an average over a band of frequencies of the same order. Hence, the selection of
the segment length, M , controls the frequency resolution of the spectral density
estimate.
In order to assess the significance of a spectral density estimate, it is neces-
sary to model the stochastic process generating the time sequence. Usually, we are
dealing with time sequences in which the signal we wish to discover is highly cor-
rupted by noise. Thus, a purely random Gaussian process is normally assumed to
be generating the sequence, new results being discovered by significant departures
from such a process. The coe cients of the discrete Fourier transform, used in the

spectral density estimate, are found by linear combinations of independent real-
isations of the Gaussian process assumed to be generating the time sequence. If
the process has zero mean, so will the coe cients, since they are linear, homogen-

eous combinations of the realisations of the process. If a given Fourier coe cient

is A ( f k ), then

A 2 ( f k )
var A ( f k )

(2.352)

2 distributed with one degree of freedom, with var[ A ( f k )] being the variance of
the coe

is χ

cient A ( f k ). Since there are two real Fourier coe cients involved in each

2

raw spectral density estimate, each such estimate is χ

2 distributed, the subscript 2

indicating two degrees of freedom.
The usefulness of the single segment spectral density estimator (2.350) lies in the
fact that the variance of the spectral density estimate can be reduced by breaking
the full record, of total length T ,intoκ multiple discrete segments of length M ,