Properties of Discrete Time Unit Impulse Signal

---

---

What is a Discrete Time Impulse Sequence?

The discrete time unit impulse sequence δ[n], also called the unit sample sequence, is defined as,

$$\mathrm{\delta [n] \:=\: \left\{\begin{matrix} 1\: for\: n \:=\: 0\ 0\: for \: n \: = \: 0\ \end{matrix}\right.}$$

Properties of Discrete Time Unit Impulse Sequence

Scaling Property

According to the scaling property of discrete time unit impulse sequence,

$$\mathrm{\delta[kn] \:=\: \delta[n]}$$

Where, k is an integer.

Proof − By the definition of the discrete time unit impulse sequence,

$$\mathrm{\delta [n] \:=\: \left\{\begin{matrix} 1\: for\: n \:=\: 0\ 0\:\: for \: n \:= \: 0\ \end{matrix}\right.}$$

Similarly, for the scaled unit impulse sequence,

$$\mathrm{\delta [kn] \:=\: \left\{\begin{matrix} 1\: for\: kn\:=\:0\ 0\:\: for \: kn \: =\: 0\ \end{matrix}\right.}$$

$$\mathrm{\Rightarrow\: \delta [kn]\:=\:\left\{\begin{matrix} 1\:\: for\: n\:=\:\frac{0}{k}\:=\:0\ 0\: for \: n\: = \: \frac{0}{k}\:=\: 0\ \end{matrix}\right. \:=\: \left\{\begin{matrix} 1\: \: for\: n\:=\:0\ 0\: \: for\: n\:=\: 0\ \end{matrix}\right. \:=\: \delta [n]}$$

Product Property

$$\mathrm{x[n]\:\delta\:[n \:-\: n_0] \:=\: x[n_0]\:\delta\:[n \:-\: n_0]}$$

Proof − By the definition of the unit impulse signal, we know,

$$\mathrm{\delta \left[n\:-\:n_{0} \right]\:=\:\left\{\begin{matrix} 1\: \: for\: n\:=\:n_{0}\ 0\: \: for\: n\:=\: n_{0}\ \end{matrix}\right.}$$

As from the expression it is clear that the impulse sequence has a non-zero value only at n = n₀. Therefore,

$$\mathrm{x[n]\:\delta\:[n\:-\: n_0] \:=\: x[n_0]\:\delta\:[n\:-\: n_0]}$$

Shifting Property

$$\mathrm{x\:[n]\:=\:\sum_{k\:=\:-\infty}^{\infty}x \:[k]\:\delta\: \left [ n\:-\:k \right ]}$$

Proof − By using the product property of the discrete time unit impulse sequence, we have,

$$\mathrm{x[n]\:\delta\:[n\:-\: n_0] \:=\: x\:[n_0]\:\delta\:[n\:-\: n_0] \:\: \dotso\: (1)}$$

Putting k in place of n0 in equation (1), we get,

$$\mathrm{x[n]\:\delta\:[n\: - \:k] \:=\: x[k]\:\delta\:[n\: -\: k]}$$

$$\mathrm{\Rightarrow\: \sum_{k\:=\:-\infty}^{\infty}x \:[n]\:\delta\: \left[n\:-\:k \right]\:=\:\sum_{k\:=\:-\infty }^{\infty}x\:[k]\:\delta\:[ n\:-\:k]}$$

$$\mathrm{\Rightarrow\: x \:[n]\:\sum_{k\:=\:-\infty}^{\infty}\delta\: \left [ n\:-\:k \right ]\:=\:\sum_{k\:=\:-\infty }^{\infty }\:x\:[k]\:\delta\: \left [ n\:-\:k \right ]}$$

$$\mathrm{\because\: \sum_{k\:=\:-\infty }^{\infty }\:\delta\: \left [ n\:-\:k \right ]\:=\:1}$$

$$\mathrm{\therefore\: x\:\left [ n \right ]\:=\:\sum_{k\:=\:-\infty }^{\infty }x\:\left [ k \right ]\:\delta\: \left [ n\:-\:k \right ]}$$

The discrete time unit impulse sequence is the first difference of discrete time unit step sequence. That is,

$$\mathrm{\delta\:[n] \:=\: u[n]\: -\: u[n\: -\: 1]}$$

Proof − By the definition of the discrete time unit step sequence,

$$\mathrm{u\:\left [ n \right ]\:=\:\sum_{k\:=\:0}^{\infty }\:\delta \:\left [ n\:-\:k \right ] \:=\:\delta\: \left [ n \right ]\:+\:\sum_{k\:=\:1}^{\infty }\:\delta \:\left [ n\:-\:k \right ] }$$

$$\mathrm{\because\: u\:\left [ n\:-\:1 \right ]\:=\:\sum_{k\:=\:1}^{\infty }\:\delta\: \left [ n\:-\:k \right ]}$$

$$\mathrm{\therefore\: u[n] \:=\: \delta[n] \:+\: u[n \:-\: 1]}$$

$$\mathrm{\Rightarrow\: \delta[n] \:=\: u[n] \:-\: u[n \:-\: 1]}$$