Unit impulse or delta function

A signal, which satisfies the condition, δ(t)=limϵ→∞x(t)$\delta (t)=\underset{ϵ\to \mathrm{\infty }}{lim}x(t)$ is known as unit impulse signal. This signal tends to infinity when t = 0 and tends to zero when t ≠ 0 such that the area under its curve is always equals to one. The delta function has zero amplitude everywhere excunit_impulse.jpgept at t = 0.

Properties of Unit Impulse Signal

• δ(t) is an even signal.
• δ(t) is an example of neither energy nor power (NENP) signal.
• Area of unit impulse signal can be written as;
A=∫∞−∞δ(t)dt=∫∞−∞limϵ→0x(t)dt=limϵ→0∫∞−∞[x(t)dt]=1$$A={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (t)dt={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\underset{ϵ\to 0}{lim}x(t)dt=\underset{ϵ\to 0}{lim}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}[x(t)dt]=1$$
• Weight or strength of the signal can be written as;
y(t)=Aδ(t)$$y(t)=A\delta (t)$$
• Area of the weighted impulse signal can be written as −
y(t)=∫∞−∞y(t)dt=∫∞−∞Aδ(t)=A[∫∞−∞δ(t)dt]=A=1=Wigthedimpulse