Time sequence and spectral analysis - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 158

1/ T

1

- T /2 T /2

1/ T

-1/ T

0

t

f

Figure 2.7 Fourier transform (right) of a boxcar enclosing unit area (left) for
small T .

denoted by δ( t − t 0 ). On integrating the product of the boxcar shown on the left of
Figure 2.7, but centred on time t = t 0 , with a function of time, f ( t ), we find that
t 0 + T /2

f ( t )
T ·

f ( t ) b ( t ) dt = T , (2.258)

t 0 − T /2

f ( t ) lies between the minimum and maximum of

with b ( t ) the boxcar, and where

f ( t ) on the closed interval [ t 0 − T /2, t 0 + T /2], by the mean value theorem for

integrals. Then,

t 0 + T /2 t 0 + T /2

lim
T → 0

f ( t ) b ( t ) dt = f ( t )δ( t − t 0 ) dt = f ( t 0 ). (2.259)

t 0 − T /2 t 0 − T /2

The Fourier transform of the Dirac delta function, δ( t − t 0 ), is

∞

t 0 ) e − i 2π ft dt e − i 2π ft 0 δ( t − = . (2.260)

−∞

−

The Fourier integral representation of δ( t t 0 )isthen

∞

e − i 2π ft 0 e i 2π ft df . δ( t − t 0 ) = · (2.261)

−∞

Thus,

∞

e i 2π f ( t − t 0 ) df . δ( t − t 0 ) = (2.262)

−∞