Introduction to signals - Continuous and Discrete Time Signals and Systems - page 33

area = 1

1 d ( t )

1/e

t

t

−0.5e 0.5e

(a) (b)

Fig. 1.18. Impulse function δ( t ).
(a) Generating the impulse
function δ( t ) from a rectangular
pulse. (b) Notation used to
represent an impulse function.

(iii) The scaled and time-shifted version δ ( at + b ) of the unit impulse function
is given by

t + b
a

δ ( at + b ) = 1

a δ

. (1.45)

(iv) When an arbitrary function φ ( t ) is multiplied by a shifted impulse function,
the product is given by

φ ( t ) δ ( t − t 0 ) = φ ( t 0 ) δ ( t − t 0 ) . (1.46)
In other words, multiplication of a CT function and an impulse function
produces an impulse function, which has an area equal to the value of the
CT function at the location of the impulse. Combining properties (ii) and
(iv), it is straightforward to show that
∞

φ ( t ) δ ( t − t 0 )d t = φ ( t 0 ) . (1.47)

−∞

(v) The unit impulse function can be obtained by taking the derivative of the
unit step function as follows:
δ ( t ) = d u

d t .

(1.48)

(vi) Conversely, the unit step function is obtained by integrating the unit
impulse function as follows:

t

u ( t ) = δ ( τ )d τ. (1.49)

−∞

Example 1.12
Simplify the following expressions:

5 − j t
7 + t 2 δ ( t );

(i)

∞

(ii) ( t + 5) δ ( t − 2)d t ;

−∞

∞

e j0 . 5 πω+ 2 δ ( ω − 5)d ω . (iii)

−∞