262 SAMPLED-DATA CONTROL SYSTEMS 
the autocorrelation function and power spectrum given by 
oe) — ee 
204 
Pa na ag 
(10.43) 
The sampled power spectrum for this function was obtaincd in Sec. 10.3 as 
i en e7 2017 
1 
Sal?) = 7 GS eo) Ses) (10.38) 
which can be written in terms of the real frequency as 
se (ja See (10.44) 
T cosh wT’ — cos wT 
Certain characteristics of this spectrum are immediately evident from an 
inspection of (10.44). The spectrum is periodic in the variable w as 
expected and, since the maximum value of cos w7' is unity and minimum 
value of cosh wT is also unity, the magnitude of the fluctuations in the 
spectrum depend greatly on the value of w:7'. If this quantity #17’, which 
is proportional to the ratio between the signal bandwidth and the sam- 
pling frequency 1/7, is large, then cosh w1T is large and the spectrum is 
virtually constant. From another point of view, this means that if the 
sampling period is long compared to the signal fluctuations, the samples 
are nearly uncorrelated and the spectrum is ‘‘almost white,” that is, 
almost constant. On the other hand, as the sampling period becomes 
very short, (10.44) becomes indeterminate and a limiting process must be 
followed. Although the form of power spectrum given by (10.44) cor- 
rectly represents the impulse-modulated signal, it is clear from (10.29) 
that a multiplying factor of JT? must be included in S¥,(jw) to maintain a 
comparative amplitude as 7 approaches zero. For small 7, consider 
: 6 5 RR sinh wT 
2 Ox == Se 
ba MSra\J@) = ae cosh w17' — cos wT’ 
2 E 
= at aes ce ow? — va (Jw) (10.45) 
The limiting process as 7’ — 0 given in (10.45) shows that with the proper 
amplitude modifying factcr, the sampled power spectrum has the shape of 
the continuous power spectrum as the sampling operation is made faster 
and faster. 
Another approach to the same problem of fast sampling may be made 
through consideration of (10.42), which expresses the power spectrum of 
the signal at the output of a zero-order-hold circuit. Substituting (10.44) 
in (10.42) and manipulating the expression slightly, one obtains, in terms 
