Time sequence and spectral analysis - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 156

g ( t ) 1

− T /2 T /2

0
t

Figure 2.5 The boxcar functiong( t ) to be used in modelling finite length records.

We conclude this section with the Fourier transforms of time functions to be
used in subsequent analyses.
First, there is the Fourier transform of the boxcar function of time, defined by

⎩

0, | t | > T /2,

g( t ) = (2.253)

1, | t |≤ T /2.

As illustrated in Figure 2.5, it is of unit height, extending over the interval − T /2 <

t < T /2.
The Fourier transform of the boxcar is

∞ T /2

g( t ) e − i 2π ft dt e − i 2π ft dt G ( f ) = =

−∞ − T /2

T /2
− T /2 =

e − i 2π ft
−

e − i 2π fT /2 e i 2π fT /2 −

T sin( π fT ) = =

π fT .

(2.254)

i 2π f − i 2π f

Introducing the sinc function, defined as

sin( x )
x ,

sinc( x ) = (2.255)

the Fourier transform of the boxcar function of time can be written

G ( f ) = T sinc(π fT ). (2.256)