Page 64 - Mechatronics with Experiments

50 MECHATRONICS
2.2.3 Mathematical Idealization of the Sampling Circuit
Let us consider the limiting case of the sampling circuit as a mathematical idealization for
further analysis. Let us consider that the RC value goes to zero.
1
RC → 0; → ∞
RC
This means that as soon as the switch is turned ON, y(t) will reach the value of the y(t).
Therefore, there is no need to keep the switch ON any more than an infinitesimally small
+
period of time. The ON time of the switch can go to zero, T → 0 (Figure 2.4).
0
y(t) ≃ y(kT) (2.4)
With this idealization in mind, the sampling operation can be viewed as a sequence of
periodic impulse functions.
+∞
∑
(t − kT)
k=−∞
This also says that the sampling operation acts as a “comb” function (Figure 2.5). If we
represent the sequence of samples of the signal with {y(kT)}, the following relationship
holds,
∞
∑
{y(kT)} = (t − kT)]y(t) (2.5)
k=−∞
Now, we will consider the following three questions concerning a continuous time
signal, y(t), and its samples, y(kT) that is sampled at a sampling frequency w = 2 ∕T by
s
an A/D converter (Figure 2.6).
Question 1: what is the relationship between the Laplace transform of the samples

and the Laplace transform of the original continuous signal?
L{y(kT)} ? L{y(t)}

Question 2: what is the relationship between the Fourier transform of the samples and
the Fourier transform of the original continuous signal? Shannon’s sampling theorem
provides the answer to this question.
F{y(kT)} ? F{y(t)}

Switch(t)

1.0

–3T –2T –T 0 T 2T 3T Time

FIGURE 2.5: Idealized mathematical model of the sampling operation via a “comb” function.