Page 108 - Advanced engineering mathematics

88 CHAPTER 3 The Laplace Transform

1

0.5

0 5 10 15 20
t

–0.5

–1
FIGURE 3.10 H(t − π)cos(t).

1
y

0.5

y = 1
0
0 2 4 6 8
t
t
a b
–0.5

–1
FIGURE 3.11 A pulse H(t − a) − H(t − b).
FIGURE 3.12 (H(t − π/2) − H(t − 2π))
sin(t).

Figure 3.11 shows the pulse H(t − a) − H(t − b) with a < b. Pulses are used to turn a signal
off until time t = a and then to turn it on until time t = b, after which it is switched off again.
Figure 3.12 shows this effect for [H(t − π/2) − H(t − 2π)]sin(t), which is zero before time
π/2 and after time 2π and equals sin(t) between these times.
It is important to understand the difference between g(t), H(t − a)g(t) and H(t − a)
g(t − a). Figures 3.13, 3.14 and 3.15, show, respectively, graphs of t sin(t), H(t − 3/2)t sin(t),
and H(t − 4)(t − 4)sin(t − 4). H(t − 3/2)t sin(t) is zero until time 3/2 and then equals t sin(t),
while H(t − 4)(t − 4)sin(t − 4) is zero until time 4, then is the graph of t sin(t) shifted 4 units to
the right.
Using the Heaviside function, we can state the second shifting theorem, which is also called
shifting in the t variable.

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October 14, 2010 14:14 THM/NEIL Page-88 27410_03_ch03_p77-120