Time sequence and spectral analysis - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 167

2 N +1

-4 f N -2 f N 0 2 f N 4 f N

f

Figure 2.12 The Fourier transform of the finite Dirac comb with 21 sample points
and unit sample interval t . The transform is periodic in frequency with period

equal to twice the Nyquist frequency, 2 f N =

Δ

1/ Δ t .

where T t is the record length. A graph of the Fourier transform of the

finite Dirac comb sampler is shown in Figure 2.12 for 2 N

= (2 N + 1) Δ

21 sample points
with unit sample interval. It is periodic in frequency, with period equal to twice the
Nyquist frequency ,2 f N

+ 1 =

t .
The Fourier transform of the finite Dirac comb sampler (2.290), illustrated in
Figure 2.12, requires some interpretation. The numerator has zeros at fT

= 1/ Δ

= 0, ± 1,

± 2,....The denominator has zeros at f Δ t = 0, ± 1, ± 2,...,orat f = 0, ± 1/ Δ t ,

± 2/ Δ t ,....Consider the value of the transform at f = n / Δ t , a zero of the denom-

inator. Since sin[π( n / Δ t ) T ] = sin[(2 N + 1) n π] = 0, the numerator has a zero there

too. By l'Hopital's rule,

sin π f (2 N t + 1) Δ

li m

sin ( π f

Δ

t )

n
Δ t

f →

1) cos π f (2 N

t + 1) Δ

= li m (2 N +

cos ( π f

Δ

t )

n
Δ t

f →

1) n (2 N + 1)
(

1) (

−

= (2 N +

1) n

−

1) 2 Nn ( 2 N

= − +

1 )

= +

( 2 N 1. (2.291)

Both numerator and denominator have zeros and are changing sign at the points
f t . Their quotient is therefore not changing sign at these points. The denom-

inator is a much weaker function of frequency than the numerator and, thus, near

= n / Δ