Z Transform Introduction | Z Transform Properties

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 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

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

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