Time sequence and spectral analysis - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 101

Thus, this represents the superposition of deterministic energy signals, f j , starting
at random times with random amplitudes. The autocorrelation of the stationary
stochastic sequence, h j ,is

= E ⎩

⎭

= E h l h ∗ l − k

∞

f m n l − m ∞

f n n ∗ l − k − n

φ hh ( k )

m =−∞ n =−∞

∞ ∞

f m f n E n l − m n ∗ l − k − n

=

m =−∞ n =−∞

∞ ∞ ∞

f m f n δ

0
k − m + n

f m f m − k = φ ff ( k ), = = (2.34)

m =−∞ n =−∞ m =−∞

in accordance with the relaxed definition (2.22) of autocorrelation for energy sig-
nals. Our relaxed definition of autocorrelation for an energy signal is consistent
with the ensemble average for stationary stochastic processes, provided that we
use the equivalent stationary stochastic process to represent the energy signal.
Similarly, we can construct stationary stochastic processes that lead to (2.26) as
a consistent definition of crosscorrelation for two energy signals.
In the case of one energy signal f j and one stationary power signal g j , we write

∞

h j = f k n j − k (2.35)

k =−∞

and

∞

g j = a k n j − k , (2.36)

k =−∞

where, by the Wold decomposition theorem, a k is a deterministic energy signal.
Then,

E h l g ∗ l − k φ h g ( k ) =

E ⎩

⎭

∞

f m n l − m ∞

a ∗ n n ∗ l − k − n =

m =−∞ n =−∞

∞ ∞ ∞ ∞

f m a ∗ n E n l − m n ∗ l − k − n f m a ∗ n δ

0
k − m + n

= =

m =−∞ n =−∞ m =−∞ n =−∞

∞

f m a ∗ m − k . = (2.37)

m =−∞

Therefore, in the relaxed definition (2.26) of crosscorrelation, for one energy sig-
nal and one power signal, we need to replace the power signal by its equivalent
deterministic energy signal found by Wold decomposition.