# Properties of Z-Transform

## Z-Transform

The Z-Transform is a mathematical tool which is used to convert the difference equations in time domain into the algebraic equations in the z-domain. Mathematically, the Z-transform of a discrete-time signal or a sequence $\mathit{x}\mathrm{\left(\mathit{n}\right)}$ is defined as −

$$\mathrm{\mathit{X}\mathrm{\left(\mathit{z}\right)}\:\mathrm{=}\:\sum_{\mathit{n=-\infty }}^{\infty }\mathit{x}\mathrm{\left(\mathit{n}\right)}\mathit{z}^{-\mathit{n}}}$$

## Properties of Z-Transform

The following table highlights some of the important properties of Z-Transform −

PropertyTime-Domainz-DomainRegion of Convergence (ROC)
Notation$\mathrm{\mathit{x}\mathrm{\left(\mathit{n}\right)}}$
$\mathrm{\mathit{X}\mathrm{\left(\mathit{z}\right)}}$
$\mathrm{\mathit{R}}$
$\mathrm{\mathit{x}_{\mathrm{1}}\mathrm{\left(\mathit{n}\right)}}$
$\mathrm{\mathit{X}_{\mathrm{1}}\mathrm{\left(\mathit{z}\right)}}$
$\mathrm{\mathit{R}_{\mathrm{1}}}$
$\mathrm{\mathit{x}_{\mathrm{2}}\mathrm{\left(\mathit{n}\right)}}$
$\mathrm{\mathit{X}_{\mathrm{2}}\mathrm{\left(\mathit{z}\right)}}$
$\mathrm{\mathit{R}_{\mathrm{2}}}$
Linearity and Superposition$\mathrm{\mathit{a}\mathit{x}_{\mathrm{1}}\mathrm{\left(\mathit{n} \right)}\:\mathrm{+}\:\mathit{b}\mathit{x}_{\mathrm{2}}\mathrm{\left(\mathit{n} \right)}}$
$\mathrm{\mathit{a}\mathit{X}_{\mathrm{1}}\mathrm{\left(\mathit{z} \right)}\:\mathrm{+}\:\mathit{b}\mathit{X}_{\mathrm{2}}\mathrm{\left(\mathit{z}\right)}}$
$\mathrm{\mathit{R}_{\mathrm{1}}\:\cap \mathit{R}_{\mathrm{2}}}$
Time-Shifting$\mathrm{\mathit{x}\mathrm{\left(\mathit{n-k}\right)}}$
$\mathrm{\mathit{z}^{-\mathit{k}}\mathit{X}\mathrm{\left(\mathit{z}\right)}}$
$\mathrm{\mathrm{same\:as\:}\mathit{X}\mathrm{\left(\mathit{z}\right)}\:\mathrm{except}\:\mathit{z}\:\mathrm{=}\:\mathrm{0}}$
$\mathrm{\mathit{x}\mathrm{\left(\mathit{n\mathrm{+}\mathit{k}}\right)}}$
$\mathrm{\mathit{z}^{\mathit{k}}\mathit{X}\mathrm{\left(\mathit{z}\right)}}$
$\mathrm{\mathrm{same\:as\:}\mathit{X}\mathrm{\left(\mathit{z}\right)}\:\mathrm{except}\:\mathit{z}\:\mathrm{=}\:\mathrm{\infty}}$
Scaling in zdomain$\mathrm{\mathit{a}^{\mathit{n}}\mathit{x}\mathrm{\left(\mathit{n}\right)}}$
$\mathrm{\mathit{X}\mathrm{\left( \frac{\mathit{z}}{\mathit{a}}\right )}}$
$\mathrm{\left|\mathit{a}\right|\mathit{R}_{\mathrm{1}}<\left|\mathit{z}\right|<\left|\mathit{a}\right|\mathit{R}_{\mathrm{2}}}$
Time Reversal$\mathrm{\mathit{x}\mathrm{\left(\mathit{-n}\right)}}$
$\mathrm{\mathit{X}\mathrm{\left(\mathit{z}^{-\mathrm{1}}\right)}}$
$\mathrm{\mathrm{\left(\frac{1}{\mathit{R}_{\mathrm{2}}}\right)}<\left| \mathit{z}\right|<\mathrm{\left(\frac{1}{\mathit{R}_{\mathrm{1}}}\right)}}$
Time Expansion$\mathrm{\mathit{x}\mathrm{\left(\frac{\mathit{n}}{\mathit{k}}\right )}}$
$\mathrm{\mathit{X}\mathrm{\left(\mathit{z}^{\mathit{k}}\right)}}$
$\mathrm{}$
Conjugation$\mathrm{\mathit{x}^{*}\mathrm{\left(\mathit{n}\right)}}$
$\mathrm{\mathit{X}^{*}\mathrm{\left(\mathit{z}^{*}\right)}}$
$\mathrm{\mathit{R}_{\mathrm{1}}<\left|\mathit{z} \right|<\mathit{R}_{\mathrm{2}}}$
Convolution$\mathrm{\mathit{x}_{\mathrm{1}}\mathrm{\left(\mathit{n}\right)} * \mathit{x}_{\mathrm{2}}\mathrm{\left(\mathit{n}\right)}}$
$\mathrm{\mathit{X}_{\mathrm{1}}\mathrm{\left(\mathit{z}\right)} \mathit{X}_{\mathrm{2}}\mathrm{\left(\mathit{z}\right)}}$
$\mathrm{\mathrm{At\:least}\:\mathit{R}_{\mathrm{1}}\cap \mathit{R}_{\mathrm{2}}}$
Correlation$\mathrm{\mathit{R}_{\mathit{x}_{\mathrm{1}}\mathit{x}_{\mathrm{2}}}\mathrm{\left(\mathit{n}\right)}\:\mathrm{=}\:\mathit{x}_{\mathrm{1}}\mathrm{\left(\mathit{n}\right)} \bigotimes \mathit{x}_{\mathrm{2}}\mathrm{\left(\mathit{n}\right)}}$
$\mathrm{\mathit{X}_{\mathrm{1}}\mathrm{\left(\mathit{z}\right)} \mathit{X}_{\mathrm{2}}\mathrm{\left(\mathit{z}^{-\mathrm{1}}\right)}}$
$\mathrm{\mathrm{At\:least}\:\mathit{R}_{\mathrm{1}}\cap \mathit{R}_{\mathrm{2}}}$
Multiplication$\mathrm{\mathit{x}_{\mathrm{1}}\mathrm{\left(\mathit{n}\right)} \mathit{x}_{\mathrm{2}}\mathrm{\left(\mathit{n}\right)}}$
$\mathrm{\frac{1}{2\mathit{\pi j}}\oint_{c}^{}\mathit{X}_{\mathrm{1}}\mathrm{\left( v\right )}\mathit{X}_{\mathrm{2}}\mathrm{\left(\frac{\mathit{z}}{\mathit{v}}\right)}\mathit{v}^{\mathrm{-1}}\:\mathit{dv}}$
$\mathrm{\mathrm{At\:least}\:\mathit{R}_{\mathrm{1}} \mathit{R}_{\mathrm{2}}<\left| \mathit{z}\right|<\mathit{R}_{\mathrm{1}\mathit{u}}\mathit{R}_{\mathrm{2}\mathit{u}}}$
Differentiation in z-domain$\mathrm{\mathit{n}\mathit{x}\mathrm{\left(\mathit{n}\right)}}$
$\mathrm{\mathit{-z}\frac{\mathit{d}}{\mathit{dz}}\mathit{X}\mathrm{\left(\mathit{z}\right)}}$
$\mathrm{\mathit{R}_{\mathrm{1}}<\left|\mathit{z}\right|<\mathit{R}_{\mathrm{2}}}$
Accumulation$\mathrm{\sum_{\mathit{k=-\infty}}^{\mathit{n}}\: \mathit{x}\mathrm{\left(\mathit{k}\right)}}$
$\mathrm{\frac{1}{\mathrm{\left( \mathrm{1-}\mathit{z}^{-\mathrm{1}}\right)}}\mathit{X}\mathrm{\left(\mathit{z}\right)}}$
$\mathrm{}$
Parseval’s Theorem$\mathrm{\sum_{\mathit{n=-\infty}}^{\mathit{\infty}}\: \mathit{x}_{\mathrm{1}}\mathrm{\left(\mathit{n}\right)}\mathit{x}_{\mathrm{2}}^{*}\mathrm{\left(\mathit{n}\right)}}$
$\mathrm{\frac{1}{2\mathit{\pi j}}\oint_{c}^{}\mathit{X}_{\mathrm{1}}\mathrm{\left( v\right )}\mathit{X}_{\mathrm{2}}^{*}\mathrm{\left(\frac{\mathrm{1}}{\mathit{v}^{*}}\right)}\mathit{v}^{\mathrm{-1}}\:\mathit{dv}}$
$\mathrm{}$
Initial Value Theorem$\mathrm{\mathit{x}\mathrm{\left(\mathrm{\mathrm{0}}\right)}\:\mathrm{=}\:\displaystyle \lim_{\mathit{n} \to 0}\mathit{x}\mathrm{\left(\mathit{n}\right)}}$
$\mathrm{\displaystyle \lim_{\mathit{z} \to \infty }\mathit{X}\mathrm{\left(\mathit{z}\right)}}$
$\mathrm{}$
Final Value Theorem$\mathrm{\mathit{x}\mathrm{\left(\mathrm{\mathit{\infty }}\right)}\:\mathrm{=}\:\displaystyle \lim_{\mathit{n} \to \infty }\mathit{x}\mathrm{\left(\mathit{n}\right)}}$
$\mathrm{\displaystyle \lim_{\mathit{z} \to 1}\mathrm{\left ( \mathit{z-\mathrm{1}} \right)}\mathit{X}\mathrm{\left(\mathit{z}\right)} \mathrm{If \mathrm{\left ( \mathit{z-\mathrm{1}} \right )}}\:\mathrm{has \:no\: pole \:on\:or\:outside\:the\:unit\:circle.} }$
$\mathrm{}$

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