Then the frequency response function of x; to x2 is 
J {Coma} + {Ouy,2,(0)]? 
Gx,x,(@)| = 6.28 
Cx SG) (6.28) 
The phase relation is 
Qux,x,(@) 
x(@) = Arg[G,,x,(@)] = tan) 4 => — 6.29 
Px,x,(@) g1Gx,x,(@)] = tan Creal) (6.29) 
and the coherency function is 
ls.,2(@)? 
Vxj5 0) et (6.30) 
Sxix(@)Sxx,(@) 
6.3 AUTOREGRESSIVE CONTINUOUS PROCESS 
So far only discrete processes have been dealt with in which difference equations 
have been used to formulate the processes. When we study the response of some dynamic 
systems, however, the processes are in many cases substantially continuous. Usually, 
however, for computational analysis these processes are sampled at a certain time inter- 
val Ar, and then the readings are digitized and are treated as discrete processes. The 
analysis technique for a discrete process has been shown in detail in the preceding 
sections. 
Sometimes, however, it is helpful in understanding the response dynamics of a 
system to treat the process directly as the continuous process. 
Although difference equations have been used to formulate discrete processes, dif- 
ferential equations are used to formulate continuous processes. We are more accustomed 
to dealing with differential equations to express the physical characteristics of the re- 
sponse of the dynamic systems than with difference equations. 
In this section, the differential equations that formulate the continuous process will 
be introduced. Then, following the derivation of Pandit and Wu,* the relation of these 
differential equations to the difference equations which formulate the digitized processes 
will be summarized, since in practical applications, we might want to determine the char- 
acteristics of the differential equations from the digitized data of responses and inputs. 
63.1 The First Order Continuous Autoregressive Process A(1) 
For the first order continuous autoregressive process, referred to as A(1), the first 
order differential equation 
“x + @,X(t) = Z(t) (6.31) 
can be formulated. Here Z(t) is the forcing function or the white noise, expressed as 
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