2 
R(s) = SZ 6a Isl’ (6.43) 
2a0 
Now suppose we have the sampled discrete data from a continuous process with 
sampling interval Ar, and the data show an autoregressive process as shown by Eq. 5.7’ 
X,- 4X1 =€]. 
2 
a 
The covariance function is, from Eq. 5.49, R(r) = 7 <a’. R(r) = @2/1-a*)a’. 
=e 
Relation of A(1) to AR(1) by Covariance Equivalence. When s = rAt, the autoco- 
variance function for a continuous process must be equivalent to that of the discrete 
process at t = rAr. Setting s = rAt in Eq. 6.42 gives 
2 2 
R(rAt)= ar = =. (eeenie (6.44) 
0 0 
: 4 ate F C 
For discrete computation this is equivalent to R(r) = mee a’. 
@ 
Therefore 
eA! = g (6.45) 
2) 2 
(oj 
Ej (6.46) 
2a0 l-a 
or 
In 
—aoAt=I1na or Gees (6.45’) 
At 
and 
2 
o% = a (6.46’) 
—a 
From these relations, we can convert the continuous process A(1) into a sampled process 
AR(1), or vice versa, by inverting the coefficients from Eqs. 6.45’ and 6.46’. From Eq. 
6.43, the variance is 
2 
OZ 
RO) =—. (6.47) 
) ae 
The spectrum is given by the Fourier transform of R(s) as 
212 
