Time sequence and spectral analysis - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 105

and

G ( z ) G ∗ (1/ z ∗ ). Φ ( z ) = (2.58)

The factors of G ( z ) can be arranged in only one minimum delay way. When in
minimum delay form, the factors of G ( z ) have the general term − ( z j − z ) with

| z j |≥ 1.

The term G ( z ) then becomes the z-transform of the only ( n + 1)-length min-

imum delay wavelet corresponding to Φ ( z ). Shorter wavelets can be minimum

delay but do not produce su ciently long autocorrelations. Longer wavelets with

z-transforms of the form

G ( z ) z p , (2.59)

where p is a positive integer, give the correct autocorrelation, for then

G ∗ (1/ z ∗ )1/ z p G ( z ) G ∗ (1/ z ∗ ), G ( z ) z p Φ ( z ) = · = (2.60)

but are not minimum delay since they contain p of the dipole factors (0 z )in

their z-transforms and, hence, are not minimum delay. Putting them in minimum
delay form would yield p of the factors (1

+ 1 ·

z )or1 p + 0 · = 1, and the wavelet would

be reduced to the minimum delay wavelet.
It also follows that maximum delay wavelets are not uniquely determined by the
autocorrelation. Longer than ( n + 1)-length maximum or mixed delay wavelets will

give the correct autocorrelation.

2.2 Linear optimum Wiener filters

In a linear filtering operation, a given member of a time sequence is replaced by a
linear combination of itself and neighbouring members of the sequence. The time
sequence being filtered may be regarded as the input to a linear system character-
ised by its impulse response. In turn, the filtered sequence may be regarded as the
output of the linear system. It is, therefore, the convolution of the input sequence
with the impulse response of the system. If f j is the input sequence, and g j is a
deterministic wavelet representing the impulse response of the linear system, the
output sequence h j is given by the convolution

k = 0 g k f j − k .

N

h j = (2.61)

In general, the objective of optimum filtering is to make the output sequence h j of
the filtering operation as close as possible to a desired sequence d j . The departure