Signals and Systems - Electrical Engineering

1.4 Representation Using Basic Signals 89

Ignoring the value att=0 we define theunit-stepsignal as

u(t)=

{

1 t> 0

0 t< 0

You can think of theu(t)as the switching of a dc signal generator from off to on, whileδ(t)is a very
strong pulse of very short duration.

The impulsesignalδ(t)is:

n Zero everywhere except at the origin where its value is not well defined (i.e.,δ(t)= 0 ,t6= 0 , and undefined

att= 0 ).

n its area is unity, i.e.,

∫t

−∞

δ(τ)dτ=

{

1 t> 0

0 t<0.

(1.26)

The unit-step signal is

u(t)=

{

1 t> 0

0 t< 0

Theδ(t)andu(t)are related as follows:

u(t)=

∫t

−∞

δ(τ)dτ (1.27)

δ(t)=

du(t)

dt

(1.28)

According to calculus we have

u 1 (t)=

∫t

−∞

p 1 (τ)dτ

p 1 (t)=

du 1 (t)

dt

and so letting 1 →0, we obtain the relation betweenu(t)andδ(t).

Remarks

n Since u(t)is not a continuous function, it jumps from 0 to 1 instantaneously around t= 0 , from the
calculus point of view it should not have a derivative. Thatδ(t)is its derivative must be taken with
suspicion, which makes theδ(t)signal also suspicious. Such signals can, however, be formally defined
using the theory of distributions.