Home / Signals and Systems / Z Transform Introduction | Z Transform Properties

Z Transform Introduction | Z Transform Properties


 For the sake of analyzing continuous-time linear time-invariant (LTI) system, Laplace transformation is utilized. And z-transform is applied for the analysis of discrete-time LTI system.  Z-transform is fundamentally a numerical tool applied for a transition of a time domain into frequency domain and is a mathematical function of the complex-valued variable named Z. The z-transform of any discrete time signal x (n) referred by X (z) is specified as

$X\left( z \right)=\sum\limits_{n=-\infty }^{\infty }{x\left[ n \right]{{z}^{-n}}}$

Z transform is a non-finite power series as summing index number n changes from -∞ to ∞. However, it is valuable for values of z for which aggregate is finite (bounded). The values of z for which function f (z) is finite and lie down inside the region named as “region of convergence (ROC)”.


  1. The discrete Fourier transform (DFT) can be computed by assessing z-transform.
  2. Z-transform is extensively applied for analysis and synthesis of several types of digital filters.
  3. Z-transform is utilized in many applications such as linear filtering, finding linear convolution, and cross-correlation various sequences.
  4. System can be characterized (like stable/unstable, causal/anti-causal) using z-transform.


  1. It, in fact, decides whether given system is stable or not.
  2. It determines the character of sequences whether causal or anti-causal.
  3. It likewise determines finite or infinite length sequences.


Above figure displays the diagram of z-transform with the region of convergence (ROC). The z-transform possesses both real and imaginary components. Therefore a diagram of the imaginary component against real component is titled complex z-plane. The radius of the above circle is 1 so named as unit circle. The complex z plane is utilized to demonstrate ROC, poles, and zeros of a function. Complex variable z is carried in terms of polar form as

\[Z=\text{ }r{{e}^{j\omega }}\]

Whereas r is the radius of a circle, and ω is the angular frequency of the given sequence.


1) Linearity

The linearity property describes that if

${{x}_{1}}[n]\overset{z}\leftrightarrows{{X}_{1}}(z)$       and



\[{{a}_{1}}{{x}_{1}}\left[ n \right]\text{ }+\text{ }{{a}_{2}}{{x}_{2}}\left[ n \right]~~\overset{z}\leftrightarrows~~{{a}_{1}}{{X}_{1}}\left( z \right)\text{ }+\text{ }{{a}_{2}}{{X}_{2}}\left( z \right)\]                                                          

From preceding relation, we can infer that Z-Transform of a linear combination of two signals is equal to the linear combination of z-transform of two separate signals.

2) Time shifting

The Time shifting property describes that if

\[x\left[ n \right]~~~\overset{z}\leftrightarrows~~~X\text{ }\left( z \right)\]     then

\[x\text{ }\left[ n-k \right]~~~~~~~\overset{z}\leftrightarrows~~~~~~X\text{ }\left( z \right)\text{ }{{z}^{-k}}\]

 From above, it’s obvious that transferring the sequence circularly by ‘k’ number of samples is equal to multiplying its z-transform by z-k element.

3) Scaling

This property describes that if

\[x\left[ n \right]~~~\overset{z}\leftrightarrows~~~X\text{ }\left( z \right)\]     then\[{{a}^{n}}~~x\left[ n \right]\overset{z}\leftrightarrows~X\text{ }\left( z/a \right)\]                    

 So, we can say that scaling the function in z-transform is equal to multiplying it by factor ain time domain.

 4) Time reversal Property

The Time reversal property describes that if                                         

            \[x\left[ n \right]~~~\overset{z}\leftrightarrows~~~X\text{ }\left( z \right)\]     then

\[~x\text{ }\left[ -n \right]~~~~~~~~~\overset{z}\leftrightarrows~~~~~~~~X\text{ }\left( {{z}^{-1}} \right)\]

It implies that if the certain sequence is folded then in z domain, it is just equal to substituting z by z-1.

5) Differentiation in z-domain

The Differentiation property describes that if                                           

\[x\left[ n \right]~~~\overset{z}\leftrightarrows~~~X\text{ }\left( z \right)\]     then

   \[~n\text{ }x\text{ }\left[ n \right]~\overset{z}\leftrightarrows~-z\text{ }\frac{d\left( X\text{ }\left( z \right) \right)}{dx}\]                                                                                                            

6) Convolution Theorem

The Circular property describes that if

${{x}_{1}}[n]\overset{z}\leftrightarrows{{X}_{1}}(z)$       and

${{x}_{2}}[n]\overset{z}\leftrightarrows{{X}_{2}}(z)$       then

\[{{x}_{1}}\left[ n \right]\text{ }*\text{ }{{x}_{2}}\left[ n \right]~\overset{z}\leftrightarrows~{{X}_{1}}\left( z \right)\text{ }{{X}_{2}}\left( z \right)\]      

Convolution of two sequences in time domain equates to a multiplication of  Z transform of both sequences.

7) Correlation Property

The Correlation of two sequences describes that if

                                 ${{x}_{1}}[n]\overset{z}\leftrightarrows{{X}_{1}}(z)$       and

${{x}_{2}}[n]\overset{z}\leftrightarrows{{X}_{2}}(z)$       then

                        \[\sum\limits_{n=-\infty }^{\infty }{{{x}_{1}}\left( n \right)~{{x}_{2}}\left( -n \right)~}~~~~\overset{z}\leftrightarrows~~~~~{{X}_{1}}\left( z \right)\text{ }{{X}_{2}}\left( {{z}^{-1}} \right)\]                                                       

8) Initial value Theorem

Initial value theorem describes that if

                                               \[x\left[ n \right]~\overset{z}\leftrightarrows~X\text{ }\left( z \right)\]     then        

\[x\text{ }\left[ 0 \right]~~=~{{\lim }_{z\to \infty }}\text{ }X\left( z \right)\]               

9) Final value Theorem

Final value theorem describes that if

                                                      \[x\left[ n \right]~~~\overset{z}\leftrightarrows~~~X\text{ }\left( z \right)\]     then

\[{{\lim }_{n\to \infty }}\text{ }x\left[ n \right]~=\text{ }{{\lim }_{z\to 1\text{ }}}\left( z-1 \right)\text{ }X\left( z \right)~\]

About Ahmad Faizan

Mr. Ahmed Faizan Sheikh, M.Sc. (USA), Research Fellow (USA), a member of IEEE & CIGRE, is a Fulbright Alumnus and earned his Master’s Degree in Electrical and Power Engineering from Kansas State University, USA.

Check Also

Trigonometric Fourier Series Solved Examples

Why Fourier series? There are many functions that are important in engineering which are not …

Leave a Reply