Time sequence and spectral analysis - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 122

H 0

H 1

1/ 2 Δ t 1/ 2 Δ t − −

0

f

1/ T

Figure 2.3 Frequency domain representation of the discrete Fourier transform.
Frequency samples are spaced at intervals of 1/ T on the band

− Δ Δ

1/2 t to 1/2 t .

at intervals of f /2 from

the ends of the frequency record. The width of the frequency axis covered is

Δ f = 1/ T . Again, the first and last samples are taken at Δ

(2 n + 1) 1

Δ

(2 n + 1) Δ f = =

t .

(2.142)

T

The frequency domain representation of the DFT in Figure 2.3 may be regarded as
the counterpart of the time domain representation shown in Figure 2.2.
The question then arises as to how well the representation (2.134), based on the
DFT, approximates the original time sequence g j . The DFT of g j is

j =− n g j e − i 2π T t j

n

T

G k = . (2.143)

2 n + 1

Then, the representation of g j , based on the DFT, is

n

1
T

G k e i 2π T t j

g j = . (2.144)

k =− n

With back substitution for the DFT from (2.143), the approximation is

⎝

j =− n g j e − i 2π T t j ⎠

n n

1
T

T

e i 2π T t . g ≈ (2.145)

2 n + 1

k =− n