Laplace transformation 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:
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 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.
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:
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:
By applying the z transform definition, we get:
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:
Using z-transforms, we may observe the following:
Making |z-1| < 1 is the only way to get this geometric series to a convergent state. Its ROC is this.
Geometric sequence
The provided geometric sequence is:
I'll use the definition once more:
This guy converges if |az-1| < 1. So, ROC is |z| > |a|.
Z-transform Plot
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:
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
and
then
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
then
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
then
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
then
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).
- 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.
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