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0066_Frame_C23 Page 4 Wednesday, January 9, 2002 1:52 PM

FIGURE 23.3 Description of singularity function: (a) impulse, (b) step, (c) ramp, (d) parabolic.
The signum function is derived from the step function according to
sgn t() = 2ut() 1
–
The Ramp Function
Integrating (23.4) yields the ramp function shown in Fig. 23.3(c). The unit ramp function r(t) is expressed as

 t, t ≥ 0
rt() =  (23.5)
 0, t < 0

Furthermore, integrating r(t) yields a unit parabolic signal of the form
 t 2
at() =  ---, t ≥ 0 (23.6)

2

 0, t < 0
This is depicted in Fig. 23.3(d).
Basic Continuous-Time Signals
Figure 23.4 shows some of the elementary signals that are often encountered in signal analysis. Some of
these signals can be derived directly from the singular functions discussed above. For example, the unit
rectangular pulse signal that extends from −τ/2 to τ/2 can be expressed as



∏ t () = ut + τ  ut – -- τ  (23.7)
-- –

2 
2

and this is depicted in Fig. 23.4(a).
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