DSP | Z-Domain | Z-Transform

Calculators Theory Z-Domain Filters

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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.

The Z-Transform may be of two types; unilateral (or one-sided) and bilateral (or two-sided). For discrete signal processing, we typically focus on the unilateral, right-sided transform.

If \(x(n)\) is a discrete-time signal or sequence, its uni-lateral Z-Transform is defined as;

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

You can establish a connection between the discrete-time Fourier Transform and the Z-Transform through the polar form of \(z\).

\[X(z)|_{z=re^{j \hat{\omega}}} = X(re^{j \hat{\omega}}) = \sum_{n=0}^\infty x[n](re^{j \hat{\omega}})^{-n}\] \[= \sum_{n=0}^\infty(x[n]r^{-n})e^{-j \hat{\omega} n} = F\{x[n]r^{-n}\}\]

When \(r = 1\), this evaluates \(X(z)\) over the unit circle as \(\omega\) varies.

Region of Convergence (ROC)

The set of points in the z-plane, for which the Z-transform of a discrete-time sequence \(x(n)\), that is \(X(z)\) converges is called the region of convergence (ROC) of the Z-transform \(X(z)\).

For any given discrete-time sequence, the Z-transform may or may not converge. If there is no point in the z-plane for which the function \(X(z)\) converges, then the sequence \(x(n)\) is said to be having no z-transform.

Z-Transform Theorems

The following table provides common transform theorems which can help quickly convert from the time domain to the z-domain. Transform theorems can be applied generally to any signal.

Transform Theorems

Property	x[n]	X(z)	ROC
Linearity	||ax_{1}[n] + bx_{2}[n]||	||aX_{1}(z) + bX_{2}(z)||	at least ||R_{x_1} \cap R_{x_2}||
Delay (time shift)	||x[n-n_{0}]||	||z^{-n_{0}} X(z)||	||R_{x}||, maybe exclude 0 or ||\infty||
Differentiation	||nx[n]||	||-z\frac{dX(z)}{dz}||	||R_{x}||, maybe exclude 0 or ||\infty||
Conjugation	||x^{*}[n]||	||X^{*}(z^{*})||	||R_{x}||
Convolution	||x_{1}[n]*x_{2}[n]||	||X_{1}(z)X_{2}(z)||	at least ||R_{x_1} \cap R_{x_2}||
First difference	||x[n]-x[n-1]||	||(1-z^{-1})X(z)||	at least ||R_{x} \cap \{|z|>0\}||
Accumulation	||\sum_{k=-\infty}^{n} x[k]||	||\frac{1}{1-z^{-1}} X(z)||	at least ||R_{x} \cap \{|z|<\infty\}||
Initial value	||x[0] = \lim_{z\to\infty}X(z)||
||R_{x}|| is the ROC of ||x[n]||, and so on

Z-Transform Pairs

The following table provides common transform pairs which can help quickly convert from the time domain to the z-domain. Transform pairs involve a specific signal and corresponding z-transform. The transform pairs in this section emphasize right-sided signals and causal systems.

Transform Pairs

x[n], ||n \geq 0||	X(z)	ROC
||\delta[n]||	||1||	|||z| > 0||
||au[n]||	||\frac{az}{z-1}||	|||z|>1||
||nu[n]||	||\frac{z}{(z-1)^{2}}||	|||z|>1||
||n^{2}u[n]||	||\frac{z(z+1)}{(z-1)^{3}}||	|||z|>1||
||a^{n}u[n]||	||\frac{1}{1-az^{-1}} = \frac{z}{z-a}||	|||z|>|a|||
||-a^{n}-u[-n]||	||\frac{1}{1-az^{-1}} = \frac{z}{z-a}||	|||z|<|a|||
||(n+1)a^{n}u[n]||	||\frac{1}{(1-az^{-1})^{2}} = \frac{z}{(z-a)^{2}}||	|||z|>|a|||
||na^{n}u[n]||	||\frac{a}{(1-az^{-1})^{2}} = \frac{az}{(z-a)^{2}}||	|||z|>|a|||
||e^{-na}u[n]||	||\frac{z}{(z-e^{-a})}||	|||z|>e^{-a}||
||r^{n}cos(\hat{\omega}_{0}n)u[n]||	||\frac{1-(r \cdot cos(\hat{\omega}_{0})z^{-1}}{1-(2 \cdot r \cdot cos(\hat{\omega}_{0})z^{-1}+r^{2}z^{-2}}||	|||z|>r||
||r^{n}sin(\hat{\omega}_{0}n)u[n]||	||\frac{1-(r \cdot sin(\hat{\omega}_{0})z^{-1}}{1-(2 \cdot r \cdot cos(\hat{\omega}_{0})z^{-1}+r^{2}z^{-2}}||	|||z|>r||
||a^{n}(u[n] - u[n-N])||	||\frac{1-a^{N}z^{-N}}{1-az^{-1}}||	|||z|>0||

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