Unit Impulse Signal – Definition, Waveform and Properties

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An ideal impulse signal is a signal that is zero everywhere but at the origin (t = 0), it is infinitely high. Although, the area of the impulse is finite. The unit impulse signal is the most widely used standard signal used in the analysis of signals and systems.

Continuous-Time Unit Impulse Signal

The continuous-time unit impulse signal is denoted by δ(t) and is defined as −

$$\mathrm{\delta (t) \:=\: \left\{\begin{matrix} 1\; \;\: for\:\: t\:=\:0\ 0\; \; \:for\:\:t\:=\: 0 \ \end{matrix} \right.}$$

Hence, by the definition, the unit impulse signal has zero amplitude everywhere except at t = 0. At the origin (t = 0) the amplitude of impulse signal is infinity so that the area under the curve is unity. The continuous-time impulse signal is also called Dirac Delta Signal.

The graphical representation of continuous-time unit impulse signal δ(t) is shown in Figure-1.

Also, if the unit impulse signal is assumed in the form of a pulse, then the following points may be observed about a unit impulse signal−

• The width of the pulse is zero which means the pulse exists only at origin (t = 0).
• The height of the pulse is infinity.
• The area under the curve is unity.
• The height of the arrow represents the total area under the pulse curve.

Properties of Continuous-Time Unit Impulse Signal

Properties of a continuous-time unit impulse signal are given below −

The continuous-time unit impulse signal is an even signal. That means, it is an even function of time (t), i.e., δ(t) = δ(-t).

Sampling Property

$$\mathrm{\int_{-\infty }^{\infty }x(t)\delta (t)dt \:=\: x(0)}$$

Shifting Property

$$\mathrm{\int_{-\infty }^{\infty }x(t)\delta (t \:-\: t_{0})dt \:=\: x(t_{0})}$$

Scaling Property

$$\mathrm{\delta(at) \:=\: \frac{1}{\left | a \right |}\delta (t) }$$

Product Property

$$\mathrm{x(t)\delta(t) \:=\: x(0)\delta(t) \:=\: x(0);\: x(t)\delta(t\: -\: t_0) \:=\: x(t_0) \delta(t \: -\: t_0)}$$

Discrete-Time Unit Impulse Signal

The discrete-time unit impulse signal is denoted by δ(n) and is defined as −

$$\mathrm{\delta(n) \:=\: \left\{\begin{matrix} 1\;\: for\: n\:=\:0\ 0\;\: for\: n\:=\: 0\ \end{matrix}\right. }$$

The discrete-time signal is also called unit sample sequence. The graphical representation of a discrete-time signal or unit sample sequence is shown in Figure-2.

Properties of Discrete-Time Unit Impulse Signal

Following are the properties of a discrete-time unit impulse signal −

$$\mathrm{\delta (n) \:=\:u(n)\:-\:u(n\:-\:1)}$$

$$\mathrm{\delta (n\:-\:k)\:=\:\left\{\begin{matrix} 1\; \;\: for\;\: n\:=\:k\ 0\; \;\: for\;\: n\:=\: k\ \end{matrix} \right.}$$

$$\mathrm{x(n) \:=\: \sum_{k \:=\: -\infty }^{\infty }x(k)\delta (n\:-\:k)}$$

$$\mathrm{\sum_{n \:=\: -\infty }^{\infty }x(n)\delta (n \:-\: n_{0}) \:=\: x(n_{0})}$$

Relationship between Unit Impulse Signal and Unit Step Signal

The time integral of unit impulse signal is a unit step signal. In other words, the time derivative of a unit step signal is a unit impulse signal, i.e.,

$$\mathrm{\int_{-\infty }^{\infty }\delta (t)\: dt \:=\:u(t)}$$

And

$$\mathrm{\delta (t) \:=\: \frac{\mathrm{d} }{\mathrm{d} t}u(t)}$$