Time sequence and spectral analysis - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 169

G ( f )

- f N 0 f N

f

Figure 2.13 Illustration of a band limited function of frequency that vanishes out-
side the Nyquist band,

− f N < f < f N .

is then

∞

1
Δ

−

n / Δ

δ( f t ). (2.297)

t

n =−∞

Using the infinite Dirac comb scaled by Δ t ,

t ∞

Δ δ( t − j Δ t ), (2.298)

j =−∞

as the sampler, the Fourier transform of the sampled record is the transform of the
product of time functions,

t ∞ h ( t ) = g( t ) ·Δ δ( t − j Δ t ). (2.299)

j =−∞

The transform is then

∞

f − df .

f ) ∞

n
Δ t

H ( f ) = G ( f − δ (2.300)

−∞

n =−∞

It is the convolution of the true transform, G ( f ), with the infinite Dirac comb in the
frequency domain.
To see the e ects of discrete sampling, we first suppose G ( f ) vanishes out-

side the band of frequencies

ff

t the Nyquist frequency,
as illustrated in Figure 2.13. Such a function of frequency is said to be band
limited . If the sampling interval

− f N < f < f N , with f N = 1/2 Δ

Δ t is made su ciently small, any function of