Time sequence and spectral analysis - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 97

The subroutine CONV finds the time sequence h j that is the convolution of the
sequence f j with the sequence g j :

SUBROUTINE CONV(H,F,G,L,M,N)

C
C CONV finds the convolution of the M-length sequence F with the N-length
C sequence G as the L-length sequence H. L=M+N-1.
C

IMPLICIT DOUBLE COMPLEX(A-H,O-Z)
DIMENSION H(L),F(M),G(N)
L=M+N-1
DO 10 I=1,L
H(I)=(0.D0,0.D0)

10 CONTINUE

DO 20 I=1,M
DO 30 J=1,N
K=I+J-1
H(K)=H(K)+F(I)*G(J)

30 CONTINUE

20 CONTINUE

RETURN
END

2.1.3 Expected value, auto- and crosscorrelation
Time sequences may be deterministic or stochastic (random). In the latter case, a
particular time sequence may be regarded as a single realisation of the stochastic
process. In both cases, for their treatment, we introduce the expected value oper-
ator E { ··· }

.
Of course, the expected value of a deterministic time sequence is just the sequence
itself. The expected value of a stochastic variable is found by multiplying a given
value of the variable by the probability that it can take on this value, and averaging
the result over all possible values of the variable.
We can then regard a stochastic time sequence as the realisation of a sequence
of random variables generated by a stochastic process. Each trial of the process
will result in a new time sequence. The expected value of any quantity at a specific
time, say t 0 , is found by averaging across an infinite ensemble of realisations of the
stochastic process, as illustrated in Figure 2.1.
Because the expected value operator is an averaging operator, it is linear. Thus,
the expected value of a constant times a stochastic variable is the constant times
the expected value of the stochastic variable. Also, the expected value of a sum
of stochastic variables is equal to the sum of the expected values of the stochastic
variables.
The autocorrelation of the sequence f j at lag k , time index l ,isdefinedas

E f l f l − k . φ ff ( k , l ) = (2.17)

At zero lag, the autocorrelation gives the mean squared amplitude, or power, of the
sequence at time index l .