Basics of Z-transform With Graphical Representation
Tags: ztransform, graphicalrepresentation, basics, signalsandsystems
Published: 2022-11-17 09:58:55

[Laplace transformation](https://byjus.com/maths/laplace-transform/) analyzes linear time-invariant (LTI) systems that operate continuously. Additionally, the z-transform is used to analyze the discrete-time LTI system. A mathematical expression of a complex-valued variable called Z, the Z-transform is mostly used as a numerical tool to convert from the time domain to the frequency domain. Any discrete temporal signal x (n), which is referred to by X (z), has the following z-transform:
![Image description](https://dev-to-uploads.s3.amazonaws.com/uploads/articles/tmcimf59r0accawnehja.png)
As the summation index number n shifts from - to +, the Z transform is a non-finite power series. But it is helpful for z-values where the aggregate is finite (bounded). In this context, "region of convergence" (ROC) refers to the set of z-values for which the function f (z) has a finite upper bound.

## What is Z-transform?
There are many applications for the [z-transform in MATLAB](https://www.theengineeringprojects.com/2022/09/introduction-to-z-transform-in-signal-and-systems-with-matlab.html) in studying discrete signals and systems. We are familiar with continuous or analog signals in the temporal domain. However, digital processing is the foundation for contemporary communication and systems. As a result, we are compelled to convert our analog impulses to digital signals. 

The first step is to convert the analog signal into a digital representation by taking samples at a rate more significant than the Nyquist sampling rate. The passage of time between them is discrete. Each sample happens at t=nTs, where Ts stands for sampling time. Following sampling, we must quantize the data to be stored, analyzed, or sent, assigning each sample to one of M possible levels.
![Image description](https://dev-to-uploads.s3.amazonaws.com/uploads/articles/uu8czirbdwpva1tmzsks.jpg)

## Definition
Let's say that the sequence is as follows:

y[n] = y0, y1, y2,.....

The sequence, in this case, contains samples of analog signals at each location.

This sequence's z transform is described as follows:
![Image description](https://dev-to-uploads.s3.amazonaws.com/uploads/articles/0m9lm92fguuloo85evxs.png)

A correct definition of Y(z) as a function of z requires converging the infinite series.

In the same way, as s is just a complex variable in the Laplace transform, z is also a complex variable. Still, unlike n, it is continuous, making the two transformations equivalent. On the other hand, not all sequences or z-values result in the z-transform converging. The zone of convergence is a collection of z values at which the z-transform converges (ROC).

We shall now witness various transformations of well-known signals.

**Unit impulse**
This short yet crucial sequence can be stated as follows:
![Image description](https://dev-to-uploads.s3.amazonaws.com/uploads/articles/dnpug28q05kxo0co6kk6.png)

By applying the z transform definition, we get:
![Image description](https://dev-to-uploads.s3.amazonaws.com/uploads/articles/4ndon1ubdalzstw4htum.png)

In this instance, the entire z-plane serves as the ROC.

Based on the definition, we know that Z(n-k) = z-k if and only if n-k is negative.

**Unit step**
Another typical sequence is this one. The definition of a unit step is:
![Image description](https://dev-to-uploads.s3.amazonaws.com/uploads/articles/jw2y3qp4wufdpwkpeiiz.png)

Using z-transforms, we may observe the following:
![Image description](https://dev-to-uploads.s3.amazonaws.com/uploads/articles/p3chcgeodyimekx3r4us.png)

Making |z-1| < 1 is the only way to get this geometric series to a convergent state. Its ROC is this.
![Image description](https://dev-to-uploads.s3.amazonaws.com/uploads/articles/o9o1d4odyz0ecmis9175.png)

**Geometric sequence**
The provided geometric sequence is:
![Image description](https://dev-to-uploads.s3.amazonaws.com/uploads/articles/1pmf56y06mgurcz1h49q.png)

I'll use the definition once more:
![Image description](https://dev-to-uploads.s3.amazonaws.com/uploads/articles/f377sef4mx8n27tzgq28.png)

This guy converges if |az-1| < 1. So, ROC is |z| > |a|.
![Image description](https://dev-to-uploads.s3.amazonaws.com/uploads/articles/yl2p8r4n2b9zogts8ox6.png)

## Z-transform Plot
![Image description](https://dev-to-uploads.s3.amazonaws.com/uploads/articles/syd6ryd8yl9il0zakaum.jpg)
In the preceding illustration, we can see the z-transform diagram together with the region of convergence (ROC). The z-transform is composed of actual and fictitious parts. A complex z-plane is a figure that contrasts an imaginary component with a real one. The circle above has a radius of 1, hence the term "unit circle." A function's ROC and its poles and zeros can be shown on the complex z-plane. Z is a complex variable that is represented by the polar form as:
![Image description](https://dev-to-uploads.s3.amazonaws.com/uploads/articles/5lp7xsd9u6ghr6jg8ar0.png)

Where;
r = the circle's radius
ω = a given sequence's angular frequency

## Z-transform Properties
**1. Linearity**
As defined by the linearity property, if
![Image description](https://dev-to-uploads.s3.amazonaws.com/uploads/articles/t48cmkz5mtp4yctii09b.png)
and
![Image description](https://dev-to-uploads.s3.amazonaws.com/uploads/articles/hb299w6g283r95hi7jww.png)
then
![Image description](https://dev-to-uploads.s3.amazonaws.com/uploads/articles/1jl5cgne4tqyorfu95ka.png)

Given the above, it follows that the Z-Transform of a linear mix of two signals is equivalent to the linear mixture of the Z-Transforms of the two individual signals.

**2. Time shifting**
According to the time-shifting feature, if
![Image description](https://dev-to-uploads.s3.amazonaws.com/uploads/articles/50fpusadbrslhug5v61v.png)
then
![Image description](https://dev-to-uploads.s3.amazonaws.com/uploads/articles/ckdp3hlzp7z3l6u8wxfi.png)
To reiterate what was said previously, a z-transform multiplied by a z-k element is equivalent to a circular transfer of the pattern by k samples.

**3. Scaling**
This characteristic states that if
![Image description](https://dev-to-uploads.s3.amazonaws.com/uploads/articles/v0j9qqsle95ym0uvvtsr.png)
then
![Image description](https://dev-to-uploads.s3.amazonaws.com/uploads/articles/ceflvztmtl3ku6h83ihp.png)
The z-scaling transform of a function is equivalent to the time domain's multiplication by a factor a.

**4. Time reversal Property**
According to the Time reversal feature, if
![Image description](https://dev-to-uploads.s3.amazonaws.com/uploads/articles/pgnngtwqcymjb3txvs9x.png)
then
![Image description](https://dev-to-uploads.s3.amazonaws.com/uploads/articles/2p22iozx9glvgvwrhyg1.png)
It suggests that folding a specific sequence into the z domain is equivalent to replacing z with z-1.

## Merits of Z-transform
- The z-transform helps calculate the [discrete fourier transform (DFT)](https://www.allaboutcircuits.com/technical-articles/an-introduction-to-the-discrete-fourier-transform/).
- Numerous digital filters are analyzed and created using Z-transform in this way.
- The Z-transform is used for various tasks, including linear filtering, locating linear convolution, and cross-correlating different sequences.
- You can use the z-transform to categorize systems as stable, casual, unstable
, or anti-causal.

## Conclusion
Z-transform is beneficial for studying signals that have been discretized in time. Consequently, you get a series of numbers in the temporal domain. By applying the z transform, we may examine these sequences' stability, frequency response, and other properties in the frequency domain (also known as the z domain). As a result, applying z transforms on continuous signals is equivalent to applying Laplace transforms.