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Z-Transform of Unit Impulse, Unit Step, and Unit 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 |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,
$$\mathrm{\mathit{u\left ( n \right )\overset{ZT}{\leftrightarrow}\left ( \frac{z}{z-\mathrm{1}} \right );\: \: }\; \; ROC\to \left|z \right|> 1}$$
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 |𝑧−1| < 1. Hence the ROC is |𝑧| > 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.
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