Which one of the following relations is not correct?

Concept:

Unit Impulse function:

A continuous-time unit impulse function δ(t), also called a Dirac delta function is defined as:

δ (t) = ∞ , t = 0

= 0,  otherwise

The unit impulse function is represented by an arrow with the strength of  ‘1’ which represents its area.

\(\mathop \smallint \limits_{ - \infty }^\infty δ \left( t \right)dt = 1\)

Properties of Delta function:

Scaling Property:

\(δ \left( {at} \right) = \frac{1}{{\left| {at} \right|}}δ \left( t \right)\)

Multiplication Property:

X(t).δ(t – t₀) = x(t₀)δ(t – t₀)

Sampling Property:

\(\mathop \smallint \limits_{ - \infty }^\infty x\left( t \right)δ \left( {t - {t_0}} \right)dt = x\left( {{t_0}} \right)\)

Important Expressions:

 X(t).δ(t) = x(0)δ(t)

\(\mathop \smallint \limits_{ - \infty }^\infty x\left( t \right)δ \left( t \right)dt = x\left( 0 \right)\)

Explanation:

From the concept part we can find that:

option(1) is true

From option (2)

\(\displaystyle\int_{-\infty}^\infty f(t) \ δ(\tau) \ d\tau =1\)

This option is false

As

\(\displaystyle\int_{-\infty}^\infty \ δ(\tau) \ d\tau =1\)

Both options (3) and (4) follow delta function properties,

Hence they are true.

Important Points

Differentiation Property:

\(\mathop \smallint \limits_{{t_1}}^{{t_2}} x\left( t \right){δ ^n}\left( {t - {t^0}} \right)dt = {\left( { - 1} \right)^n}x\left( {{t_0}} \right)\)

Even Signal property:

δ(-t) = δ(t)

Convolution Property:

\(\mathop \smallint \limits_{ - \infty }^\infty x\left( \tau \right)\delta \left( {t - \tau } \right)d\tau = x\left( t \right)\)

Integration Property:

\(\mathop \smallint \limits_{{t_1}}^{{t_2}} x\left( t \right)\delta \left( t \right)dt = x\left( 0 \right)\) for  (t₁ < t₂)

0, otherwise.