Z-Transform of Unit Impulse, Unit Step, and Unit Ramp Functions

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Z-Transform

The Z-transform (ZT) is a mathematical tool which is used to convert the difference equations in time domain into the algebraic equations in z-domain.

Mathematically, if $\mathrm{x(n)}$ is a discrete-time signal or sequence, then its bilateral or two-sided Z-transform is defined as −

$$\mathrm{Z[x(n)]\:=\:X(z)\:=\:\sum_{n=-\infty}^{\infty}\:x(n)z^{-n}}$$

Where, z is a complex variable.

Also, the unilateral or one-sided z-transform is defined as −

$$\mathrm{Z[x(n)]\:=\:X(z)\:=\:\sum_{n=0}^{\infty}\:x(n)z^{-n}}$$

Z-Transform of Unit Impulse Function

The unit impulse sequence or unit sample sequence is defined as −

$$\mathrm{x(n)\:=\:\delta(n)\:=\: \begin{cases}1 & \text{for }\:n \:=\: 0\\\\0 & \text{for }\:n \:\neq\: 0\end{cases}}$$

Therefore, the Z-transform of the unit impulse function is given by,

$$\mathrm{Z[x(n)]\:=\:X(z)\:=\:Z[\delta(n)]}$$

$$\mathrm{\Rightarrow\:X(z)\:=\:\sum_{n=o}^{\infty}\:\delta(n)z^{-n}\:=\:1;\:\:\text{ROC }\:\rightarrow\:\text{ All z i.e. entire z - plane}}$$

Or it can also be represented as,

$$\mathrm{\delta(n)\:\overset{ZT}\longleftrightarrow\:1;\:\text{ for All } \:z}$$

Z-Transform of Unit Step Function

The unit step signal or unit step sequence is defined as −

$$\mathrm{x(n)\:=\:u(n)\:=\:\begin{cases}1 & \text{for }\:n \:\geq\: 0\\\\0 & \text{for }\:n \:\lt\: 0\end{cases}}$$

Therefore, the Z-transform of unit step function is given by,

$$\mathrm{Z[x(n)]\:=\:X(z)\:=\:Z[u(n)]}$$

$$\mathrm{\Rightarrow\:X(z)\:=\:\sum_{n=0}^{\infty}\:u(n)z^{-n}}$$

$$\mathrm{\Rightarrow\:X(z)\:=\:\sum_{n=0}^{\infty}\:(1)\:\cdot\:z^{-n}\:=\:1\:+\:z^{-1}\:+\:z^{-2}\:+\:\dotso}$$

$$\mathrm{\Rightarrow\:X(z)\:=\:\frac{1}{(1\:-\:z^{-1})}\:=\:\frac{z}{z\:-\:1}}$$

The above summation or series converges if |z| > 1. Therefore, the ROC of the Z-transform of the unit step sequence is |z| > 1. Hence, the ROC is the exterior of the unit circle in the z-plane.

The Z-transform of the unit step sequence can also be represented as,

$$\mathrm{u(n)\:\overset{ZT}\longleftrightarrow\:\left(\frac{z}{z\:-\:1}\right);\:\:\text{ROC }\:\rightarrow\:|z|\:\gt\:1}$$

Z-Transform of Unit Ramp Sequence

The unit ramp sequence is defined as −

$$\mathrm{x(n)\:=\:r(n)\:=\:\begin{cases}n & \text{for }\:n \:\geq\: 0\\\\0 & \text{for }\:n \:\lt\: 0\end{cases}}$$

Thus, the Z-transform of the unit ramp sequence is given by,

$$\mathrm{Z[x(n)]\:=\:X(z)\:=\:Z[r(n)]}$$

$$\mathrm{\Rightarrow\:X(z)\:=\:\sum_{n=0}^{\infty}\:r(n)z^{-n}}$$

$$\mathrm{\Rightarrow\:X(z)\:=\:\sum_{n=0}^{\infty}\:nz^{-n}\:=\:0\:+\:z^{-1}\:+\:2z^{-2}\:+\:3z^{-3}\:+\:4z^{-4}\:+\:\dotso}$$

$$\mathrm{\Rightarrow\: X(z) \:=\: z^{-1} \left( 1 \:+\: 2z^{-1} \:+\: 3z^{-2} \:+\: 4z^{-3} \:+\: \cdots \right)}$$

$$\mathrm{\Rightarrow\: X(z) \:=\: z^{-1} \left( 1 \:-\: z^{-1} \right)^{-2}}$$

$$\mathrm{\Rightarrow\:X(z)\:=\:\frac{z^{-1}}{(1\:-\:z^{-1})^2}\:=\:\frac{z}{(z\:-\:1)^2}}$$

This series converges for $\mathrm{|z^{−1}| \:\lt\: 1}$. Hence the ROC is $\mathrm{|z| \:\gt\: 1}$, i.e., the ROC of the Z-transform of the unit ramp function is the exterior of the unit circle in the z-plane.