20 SAMPLED-DATA CONTROL SYSTEMS 
function f(¢) at that instant. In other words, the analogue of the signal 
datum is the area of the impulse, and if properly used and interpreted, 
this should cause no difficulties. In some applications where the area 
of the actual sampling pulse is finite, say y, the only modification made 
in the impulse approximation is to replace the unit-area impulse sequence 
with one whose areas are y._ These cases will be taken up when the par- 
ticular situation applies. 
|F*(jw)| 
2a 
T 0 
ip 
Frequency 
Fia. 2.7. Spectrum of impulse-sampled function. 
The implications of the impulse approximation in the frequency domain 
will now be considered. This can be studied by using the form given in 
(2.3) by computing the various Fourier coefficients C;, by evaluating 
these coefficients in the integral 
Sy ae jet 
Ce. =F if dr(t)e ? dt (2.14) 
T J-T2 
Since the area of the impulse at the origin is unity, the integral has a 
value of unity, so that the Fourier coefficients are all equal regardless of 
the value of k and are given by 
C. = (2.15) 
Sle 
Thus, from (2.5), the Fourier transform of the impulse-modulated func- 
tion f*(t) is given by 
+o 
G2) = - » Flj(@ — kwo)] (2.16) 
k=—o 
where wo is 27/T’. 
Thus, when impulse sampling is used, the repeated spectra which 
result from the sampling process are all equal in relative magnitude and 
are separated by a frequency wo. The main difference between this 
spectrum and that obtained with a finite-width sampling function is that 
the repeated spectra are all equal, instead of diminishing, as is the case 
with finite sampling function. These repeated spectra are plotted in 
Fig, 2.7: 
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