256 SAMPLED-DATA CONTROL SYSTEMS 
over one sampling period and expressed in the form 
© N 
BelGjl= > Jim aot » x(nT)a(nT + kT) 
k 
=O nrn=- 
1 T/2 
7 Hs p(o)p(o ++ —kT) do (10.21) 
where p(c) is a pulse of width y and height 1/y. From (10.14) the sample 
average in (10.21) is recognized to be ®,,(kT), so that (10.21) reduces to 
o 
T/2 
aa) > B.:(kT) 7 Se p(c)p(o +7 —kT) do (10.22) 
k=—o© 
The integral in (10.22) is the convolution of pulse trains and results in a 
periodic train of triangles of height 1/y7' and base 2y centered at ‘‘sam- 
pling instants” kT. A sketch of the autocorrelation of a typical sampled 
signal is shown in Fig. 10.3. As mentioned before, the gain factor 1/y 
- 
y 
Y 
LY 
Ly 
A 

Fia. 10.2. Pulse sequences corresponding to normal and shifted sampled signals. 
was introduced in the description of the sampling process to make later 
limiting processes simpler. If the gain factor is removed for the moment, 
one is left with a signal consisting of pulses of width y and height x(nT), 
and the modified autocorrelation function would be given by the expres- 
sion of (10.22) multiplied by y?. From Fig. 10.3 it is obvious that the 
value of the modified autocorrelation at the origin which is the mean- 
square value of the conventional sampled signal with no extra gain is 
v/T({®,:(0)|. That is, the mean-square value of the pulse amplitude- 
modulated wave equals the mean-square value of the original signal 
multiplied by a pulse duty factor, y/T. This result checks with physical 
reasoning, as it should. 
