Time sequence and spectral analysis - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 175

The definition (2.320) of h ( t ) is equivalently the product h ( t ) b ( t )g( t ), where

b ( t ) is the familiar boxcar function (2.253). The sample power spectral density
estimator can then be written as

=

∞

b ( t )g( t ) e − i 2π ft dt

2

1
T

S gg ( f )

=

−∞

∞

f 1 ) df 1

2

1
T

= T sinc(π f 1 T ) G ( f − (2.331)

−∞

by the convolution theorem, (2.274), for the Fourier transform of the product of
two functions of time. Thus, finite record length plays the same role here as in
the calculation of Fourier transforms, considered in Section 2.4.2. The true Fourier
transform, G ( f ), is convolved with the sinc function, with attendant serious fre-
quency mixing. Again, the same procedure is used to suppress frequency mixing;
namely, multiplication of the time sequence by a good window, such as the Parzen
window, before the transform is calculated.
For discrete, equispaced data sequences, the finite sampled record is given as
in (2.286), but with the finite Dirac comb scaled by Δ t , and the sample spectral
density estimator (2.328) takes the form

j =− N g j e − i 2π fj Δ t

2

N

1
T

S gg ( f )

= Δ t . (2.332)

This is just the squared magnitude of the discrete Fourier transform of the sampled
record, divided by the record length. In the discrete case, we have only the samples

E g( k Δ τ) t )g ∗ ( k φ gg ( j Δ τ) = Δ Δ t − j (2.333)

of the continuous autocorrelation for stationary sequences. Modelling the sampling
process, as before, by multiplication with the infinite Dirac comb in lag,

t ∞ Δ δ(τ − j Δ t ), (2.334)

j =−∞

the sampled autocorrelation is

t φ gg (τ) ∞

Δ δ(τ − j Δ t ). (2.335)

j =−∞

Its Fourier transform, the power spectral density for equispaced, discrete data, is

=Δ t ∞ φ gg ( j ) e − i 2π fj Δ t S gg ( f ) . (2.336)

j =−∞