Time sequence and spectral analysis - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 103

The reverse wavelet of the reverse wavelet is the original wavelet. The z-transform
of the autocorrelation of the reverse wavelet is then
R ( z ) F ( z )

z n =Φ

( z ), (2.48)

identical to that of the original wavelet.
Now, from (2.45), we have

f n − 1
z n − 1 +···+

f n
z n +

R ( z )
z n =

f 0

f 0 + f 1 1

+···+ f n 1

n ∗

=

z ∗ z ∗

F ∗ 1
z ∗

=

f n 1 z ∗ 1 1 z ∗ 2

1
z +

z ∗ n =

z + z +

··· . (2.49)

On substitution in expression (2.47) from (2.41) and (2.49), we find the z-transform
of the autocorrelation takes the form

z ) 1 z ∗ 1 z ) 1 z ∗ n f n f n ( z 1 Φ ( z ) = +

z +

··· ( z n +

z +

z ) 1 z ∗ 1 z z ) 1 z ∗ n z . = |

f n |

( z 1 + + ··· ( z n + + (2.50)

z n

We see that the z-transform of the autocorrelation depends on the product of
z-transforms of paired dipole wavelets . For example, the j th pair, z j z 1 z ∗ j z , + +

is the product of the z-transforms of the wavelets

↑ , z ∗ j ),

( z j
↑

,1) and (1

respectively. The first dipole wavelet is the reverse wavelet of the second and vice
versa. Thus, any number of the component dipole wavelets can be reversed without
changing the autocorrelation. Hence, there are 2 n possible ( n 1)-length wavelets

with the same autocorrelation. Wavelets have unique autocorrelations but there is
no unique wavelet corresponding to a given autocorrelation.
The wavelets corresponding to a given autocorrelation may be classified accord-
ing to their delay properties. If the wavelet has its dipole components arranged so
that in magnitude the leading coe

+

cient is greater than, or equal to, the second,
it is called minimum delay .Iftheleadingcoe cient in each dipole component

is less than, or equal to, the second, it is called maximum delay .Otherwise,itis
mixed delay .