Time sequence and spectral analysis - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 110

Augmenting the system of equations (2.79) with expression (2.78) for the predic-
tion error, the prediction error equations take the matrix form

⎝ ⎠ ⎝ ⎠

1
γ 1
. . .
γ N

P N + 1
0
. . .
0

⎝ ⎠

φ ff ( 0 )

··· φ ff ( − N )

. . . . . .

. . .

= . (2.80)

φ ff ( N ) ··· φ ff (0)

2.2.2 Predictive deconvolution
Frequently, in the study of Earth's dynamics, we have a recorded time sequence
y j that is known to be the response to an excitation x j . If the impulse response of
the particular Earth system is b j , then the recorded response is the convolution of
b j with x j , if the system can be modelled as linear. Often, we want to recover the
excitation by removing the e ff ect of the linear system. Thus, we want to undo the

convolution

y j = b k x j − k (2.81)

k

that produced the observed output sequence y j . This is the problem of de-
convolution .
In terms of z-transforms, equation (2.81) becomes

= B ( z ) X ( z ).

Y ( z ) (2.82)

The solution to the deconvolution problem then requires the construction of a
sequence a j with z-transform A ( z ) such that

A ( z ) B ( z ) = 1, (2.83)

or such that the convolution of a j with b j produces the unit impulse sequence.
On multiplying equation (2.82) through on the left-hand side by A ( z ), from
(2.83) we see that the z-transform of the sequence being sought is given by

X ( z ) = A ( z ) Y ( z ), (2.84)

while, from (2.83), A ( z )isgivenby

1
B ( z ) .

A ( z ) = (2.85)

The sequence a j is called the inverse to the impulse response b j .
In practice, the inverse is modelled as the ( m + 1)-length finite wavelet,

= a 0

↑ , a 1 ,..., a m

a