Time sequence and spectral analysis - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 159

2.4.1 Convolution theorems
Consider a function of time, k ( t ), whose Fourier transform, K ( f ), is the product of
two Fourier transforms. Then,

K ( f ) = G ( f ) · H ( f ), (2.263)

where G ( f )and H ( f ) are the Fourier transforms of the functions of time, g( t )and
h ( t ), respectively. The Fourier integral representation of k ( t )is

∞

G ( f ) H ( f ) e i 2π ft df . k ( t ) = (2.264)

−∞

We can write

∞ ∞

g( t 1 ) e − i 2π ft 1 dt 1 , H ( f ) h ( t 2 ) e − i 2π ft 2 dt 2 .

= =

G ( f ) (2.265)

−∞ −∞

Thus,

∞
−∞ ·

∞ ∞

g( t 1 ) e − i 2π ft 1 dt 1 h ( t 2 ) e − i 2π ft 2 dt 2 e i 2π ft df k ( t ) = · ·

−∞ −∞

∞

g( t 1 ) ∞
−∞

∞

e i 2π f ( t − t 1 − t 2 ) df = h ( t 2 ) · · dt 2 dt 1 . (2.266)

−∞ −∞

From (2.262),

∞

e i 2π f ( t − t 1 − t 2 ) df = δ( t − t 1 − t 2 ). (2.267)

−∞

Hence,

∞

g( t 1 ) ∞
−∞

k ( t ) = h ( t 2 )δ( t − t 1 − t 2 ) dt 2 dt 1 , (2.268)

−∞

or

∞

k ( t ) = g( t 1 ) h ( t − t 1 ) dt 1 . (2.269)

−∞

k ( t ) is called the convolution of g( t )and h ( t ). For continuous signals this is the
analogue of the convolution of discrete sequences considered in Section 2.1.2. The
Fourier transform of the convolution of two functions of time is the product of their
Fourier transforms, just as the z-transform of the convolution of two discrete time
sequences is the product of their z-transforms, as shown in Section 2.1.2.