Page 107 - Advanced engineering mathematics

3.3 Shifting and the Heaviside Function 87

y

y = 1

t

FIGURE 3.8 The Heaviside function.

y

y = 1

t
t = a

FIGURE 3.9 Shifted Heaviside function.

H(t − a) can be used to turn a signal (function) off until time t = a and then to turn it on. In
particular,

0 for t < a
H(t − a)g(t) =
g(t) for t ≥ a.
To illustrate, Figure 3.10 shows H(t − π)cos(t). This is the familiar cosine function for
t ≥ π, but is turned off (equals 0) for t <π. Multiplying a function f (t) by H(t − a) leaves the
graph of f (t) unchanged for t ≥ a, but replaces it by 0 for t < a.

We can also use the Heaviside function to define a pulse.If a < b, then
⎧
⎪0for t < a
⎨
H(t − a) − H(t − b) = 1for a ≤ t < b
⎪
0for t ≥ b.
⎩

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October 14, 2010 14:14 THM/NEIL Page-87 27410_03_ch03_p77-120