**INTRODUCTION TO Z-TRANSFORM**

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)”.

**Merits OF Z-TRANSFORM**

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

**Merits OF REGION OF CONVERGENCE (ROC)**

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

**Z-TRANSFORM PLOT**

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.

**Z-TRANSFORM PROPERTIES**

**1) Linearity**

The linearity property describes that if

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

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

then

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

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 a^{n }in 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

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