Time sequence and spectral analysis - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 125

This is just the DFT expressed by (2.148) of the time sequence shifted back by
n units. Once again, changing the summation index to = j − n ,wehavethat

=− 2 n g e − i 2 N k .

0

e − i 2 N kn G = Δ t (2.156) k

Because of the periodicity of the input time sequence, the summation range need
only cover one complete cycle. This leads to the shifting theorem for DFTs,
G k = e − i 2 N kn G k , k = 0,1,..., n , (2.157)

the DFT of a time sequence shifted back by n units is found by multiplying the
DFT of the unshifted time sequence by the factor
e − i 2 N kn . (2.158)

The DFT, G k , can be found for negative indices using its periodicity, as

e − i 2 N kn G − N + k , k

G k = = n + 1, n + 2,...,2 n . (2.159)

Thus, the DFT expressed by (2.148) can be found from the generalised DFT expressed
by (2.150) as

e i 2 N kn G k = G k , k = 0,1,..., n ,

e i 2 N kn G k = G k + N , k =− n , − n + 1,..., − 1. (2.160)

2.3.2 The DFT and the z-transform
The discrete Fourier transform (DFT) can be directly related to the z-transform.
From the definition of the z-transform in Section 2.1.2, the z-transform of the
sequence, g 0 , g 1 ,..., g N − 2 , g N − 1 ,is

N − 1

g j z j

G ( z ) = . (2.161)

j = 0

Its DFT, given by (2.150), is

N − 1 N − 1

g j e − i 2 N kj

g j z j

G =Δ t =Δ t =Δ t G ( z ), (2.162) k

j = 0 j = 0

e − i 2 N k . If we have a second sequence, h 0 , h 1 ,..., h M − 2 , h M − 1 , of length M ,
its convolution with g j is

with z =

N − 1

e j

=

g k h j − k ,

(2.163)

k = 0