Z-Transform of Unit Impulse, Step and Ramp Functions

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

$$\mathrm{\mathit{Z\left [ x\left ( n \right ) \right ]\mathrm{\, =\,}X\left ( z \right )\mathrm{\, =\,}\sum_{n\mathrm{\, =\,}-\infty }^{\infty }x\left ( n \right )z^{-n}}}$$

Where, z is a complex variable.

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

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

Z-Transform of Unit Impulse Function

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

$$\mathrm{\mathit{x\left ( n \right )\mathrm{\, =\,}\delta \left ( n \right )\mathrm{\, =\,}\begin{Bmatrix} \mathrm{1}& \mathrm{for}\: n\mathrm{\, =\,}\mathrm{0}\ \mathrm{0}& \mathrm{for}\: n
eq \mathrm{0}\ \end{Bmatrix} }}$$

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

$$\mathrm{\mathit{Z\left [ x\left ( n \right ) \right ]\mathrm{\, =\,}X\left ( z \right )\mathrm{\, =\,}Z\left [ \delta \left ( n \right ) \right ]}}$$

$$\mathrm{\mathit{\Rightarrow X\left ( z \right )\mathrm{\, =\,}\sum_{n\mathrm{\, =\,}\mathrm{0}}^{\infty }\delta \left ( n \right )z^{-n}\mathrm{\, =\,}\mathrm{1};\; \; }ROC\rightarrow All\: z\: \mathit{i.e.}entire\: z-plane}$$

Or it can also be represented as,

$$\mathrm{\mathit{\delta \left ( n \right )\overset{ZT}{\leftrightarrow}\mathrm{1;}\: \: }for\: all\: z}$$

Z-Transform of Unit Step Function

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

$$\mathrm{\mathit{x\left ( n \right )\mathrm{\, =\,}u \left ( n \right )\mathrm{\, =\,}\begin{Bmatrix} \mathrm{1}& \mathrm{for}\: n\geq \mathrm{0}\ \mathrm{0}& \mathrm{for}\: n< \mathrm{0}\ \end{Bmatrix} }}$$

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

$$\mathrm{\mathit{Z\left [ x\left ( n \right ) \right ]\mathrm{\, =\,}X\left ( z \right )\mathrm{\, =\,}Z\left [ u \left ( n \right ) \right ]}}$$

$$\mathrm{\mathit{\Rightarrow X\left ( z \right )\mathrm{\, =\,}\sum_{n\mathrm{\, =\,}\mathrm{0}}^{\infty }u \left ( n \right )z^{-n}}}$$

$$\mathrm{\mathit{\Rightarrow X\left ( z \right )\mathrm{\, =\,}\sum_{n\mathrm{\, =\,}\mathrm{0}}^{\infty }\left ( \mathrm{1} \right )\cdot z^{^{-n}}\mathrm{\, =\,}\mathrm{1}\mathrm{\, +\,}z^{\mathrm{-1}}\mathrm{\, +\,}z^{\mathrm{-2}}\mathrm{\, +\,}\cdot \cdot \cdot }}$$

$$\mathrm{\mathit{\Rightarrow X\left ( z \right )\mathrm{\, =\,}\frac{\mathrm{1}}{\left ( \mathrm{1}-z^{-\mathrm{1}} \right )}\mathrm{\, =\,}\frac{z}{z-\mathrm{1}}}}$$

The above summation or series converges if |?| > 1. Therefore, the ROC of the Z-transform of the unit step sequence is |?| > 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,

$$\mathrm{\mathit{u\left ( n \right )\overset{ZT}{\leftrightarrow}\left ( \frac{z}{z-\mathrm{1}} \right );\: \: }\; \; ROC\to \left|z \right|> 1}$$

Z-Transform of Unit Ramp Sequence

The unit ramp sequence is defined as −

$$\mathrm{\mathit{x\left ( n \right )\mathrm{\, =\,}r \left ( n \right )\mathrm{\, =\,}\begin{Bmatrix} n& \mathrm{for}\: n\geq \mathrm{0}\ \mathrm{0}& \mathrm{for}\: n< \mathrm{0}\ \end{Bmatrix} }}$$

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

$$\mathrm{\mathit{Z\left [ x\left ( n \right ) \right ]\mathrm{\, =\,}X\left ( z \right )\mathrm{\, =\,}Z\left [ r \left ( n \right ) \right ]}}$$

$$\mathrm{\mathit{\Rightarrow X\left ( z \right )\mathrm{\, =\,}\sum_{n\mathrm{\, =\,}\mathrm{0}}^{\infty }r \left ( n \right )z^{-n}}}$$

$$\mathrm{\mathit{\Rightarrow X\left ( z \right )\mathrm{\, =\,}\sum_{n\mathrm{\, =\,}\mathrm{0}}^{\infty }n z^{^{-n}}\mathrm{\, =\,}\mathrm{0}\mathrm{\, +\,}z^{\mathrm{-1}}\mathrm{\, +\,}\mathrm{2}z^{\mathrm{-2}}\mathrm{\, +\,}\mathrm{3}z^{\mathrm{-3}}\mathrm{\, +\,}\mathrm{4}z^{\mathrm{-4}}\mathrm{\, +\,}\cdot \cdot \cdot }}$$

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

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

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

This series converges for |?⁻¹| < 1. Hence the ROC is |?| > 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.