Time sequence and spectral analysis - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 124

± F /2 = ± 1/2 Δ t

± T /2 = ± 1/2 Δ f

Δ t

Δ f

0

0

f

t

Figure 2.4 The time and frequency records may be imagined to be wrapped in
cylindrical form to give the proper periodicity in the N points.

From (2.150),

2 n 2 n n

g j e − i 2 N jk g j e − i 2 N jk g j e − i 2 N jk

G =Δ t =Δ t +Δ t . (2.152) k

j = 0 j = n + 1 j = 0

Replacing the summation index by = j − N , the first term on the right-hand side

becomes

t − 1 t − 1

g + N e − i 2 N k ( + N ) g + N e − i 2 N k ,

Δ =Δ (2.153)

=− n =− n

using the period, N , in the indices of the exponential. Using the relation (2.151),
we then have that

=− n g + n + 1 e − i 2 N k j = 0 g j − n e − i 2 N jk

G k =Δ t − 1

n

+Δ t

=− n g − n e − i 2 N k j = 0 g j − n e − i 2 N jk

t − 1

n

=Δ +Δ t , (2.154)

on using the periodicity of g j expressed by (2.147). On replacing the summation
index by j ,wefindthat

j =− n g j − n e − i 2 N jk

n

G k =Δ t . (2.155)