SAMPLED DATA SYSTEMS WITH RANDOM INPUTS 278 
The mean-square value of the error of prediction for the filter whose 
transfer function is given by (10.94) is 
(22(f)) = 1 — ea (10.97) 
The expression for the error of the sampled-data filter is written deliber- 
ately as the sum of three terms. The first two of these terms are inde- 
pendent of the sampling period 7 and, furthermore, are identical to the 
error of prediction of the filter whose input is not sampled. It is 
reasonable, then, to describe these error terms as the error of prediction, 
as contrasted to the error due to sampling, which is termed ripple. The 
ripple is the third term of (10.96) and is an always positive quantity 
which increases monotonically with the sampling period 7 and vanishes 
as T’ approaches zero. 
For negative values of a the ideal operation is one of delay, and the 
physical system is trying to reproduce the message continuously with a 
delay of a sec. In the absence 
of noise, a filter operating on the 
continuous signal can obviously 
perform this function exactly, and 
no error results. With sampled- 
data inputs, however, the ripple 
effect remains, and for a specified 
sampling rate 7 there is a specific 

as —(a+T) —a =a 
minimum value of mean-square Time (t) 
error in the reproduction despite Fic. 10.9. Sketch showing general shape 
the fact that no noise is present of y(t) function for example problem. 
with the signal. A calculation of this minimum error will permit the 
designer to select the lowest sampling rate possible for a given specifica- 
tion of error. 
From an inspection of the expression for the minimum mean-square 
value of the error given in (10.85), it is clear that the error will decrease 
as long as the area under [y(¢)]? for £ > 0 increases, which usually con- 
tinues as the delay time isincreased. The y(t) function for the example 
problem is sketched in Fig. 10.9, which shows clearly that if a = —T 
all of the nonzero portion of y(t) is included in positive time and no 
further reduction in error may be realized by longer delays. This 
result merely expresses the well-known fact that two samples from a 
first-order Markov process supply all the information that is useful for 
the extrapolation of the process. A delay of one sampling period 
permits these two samples to be collected, and a longer delay, which 
would introduce more samples, would not reduce the error of extrapola- 
tion. This situation only applies to the particular problem chosen for 
