Time sequence and spectral analysis - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 100

2.1.4 White noise and Wold decomposition
Time sequences may have specific statistical properties. Consider a time sequence,

..., n − 1 , n 0

↑ , n 1 ,..., n j ,...,

(2.28)

which is completely uncorrelated. Its autocorrelation at lag k , time index l , is then

E n l n ∗ l − k

k E n l n ∗ l ,

0

φ nn ( k , l ) = = δ (2.29)

0

where δ

k is the Kronecker delta, if it is completely uncorrelated. Thus, the autocor-
relation vanishes for all lags, except zero lag where it is the power at the particular
time index l . Such a time sequence is called a white noise sequence .
If it is stationary and of unit power, its autocorrelation is

0

φ nn ( k ) = δ

k .

(2.30)

The sequence

...,0,0,1

↑ ,0,0,...

(2.31)

0

is called the unit impulse sequence . Its general term, f j , is equal to δ

j . Hence, in
the lag domain, the autocorrelation of a white noise sequence of unit power is the
unit impulse sequence.
Wold (1938) proved a very important theorem called the Wold decomposition
theorem (see Box et al. (1994)). The theorem shows that a stationary stochastic
sequence may be decomposed into the convolution of a deterministic energy signal
with a stationary white noise sequence of unit power.
Suppose we now represent a deterministic energy signal by the stationary stocha-
stic process that results from its convolution with a white noise sequence of unit
power. If we convolved the deterministic energy signal f j with the unit impulse
sequence, we would get

∞

0
j − k

h j = f k δ = f j , (2.32)

k =−∞

just the deterministic energy signal itself.
If we convolve it with white noise of unit power, we get the stationary sequence

∞

h j = f k n j − k . (2.33)

k =−∞