Z transform Homework Help
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We can define transformations as the act of changing something. In mathematics, it is normally applied to a function. The goal of transformation tends to vary depending on the type of transformation applied. In its simplest form, it could be to normalize a dataset. There are more complex forms of transformations, such as the z-transform, Fourier transforms, and Laplace transform. In this article, we stick to Laplace transform and z-transform because they are interrelated. In fact, it is often reported that the z-transform was part of the Laplace transform.
This form of transformation was first introduced by Pierre-Simon Laplace in 1730 and is named after him. What the function does is that it transforms a real-valued function into a complex function. Therefore we can say that the Laplace transform maps f (t) to another function f(s) where s is a complex number. It is particularly useful in transforming a complex number. Its applications in the real world are vast.
The Laplace transforms bears a close relationship with the Fourier transform. Some even refer to the two to be the same. However, they still have some differences. The main difference is that the Fourier transform can only function in stable signals while the Laplace transform can function in both unstable and stable signals. As a result, its application is very wide.
It can be divided into unilateral Laplace transform, bilateral Laplace transforms, and inverse Laplace transform.
Unilateral Laplace transform
In a mathematical equation, we can define it as
This is the most common form of the Laplace transform. As can be seen that the integral extends over one side of the real axis. In addition, the function f must be integrable over the real axis.
Bilateral Laplace transform
This has a close resemblance to the unilateral. The difference which comes to the integral in bilateral transform is that it extends over the two sides of the real axis.
Most define an inverse as the opposite of something. As mentioned, the Laplace transform transforms the variables of a function into complex variables. The inverse transform does the opposite- transforms a complex variable of a function to a real value.
It has a close resemblance to the Laplace transform. They are both intended to transform the real variables of a function to complex. What is the difference then? The z-transform takes discrete variables while the Laplace function takes continuous variables.
History of z-transform.
The origin of the z-transform could be said to be in the 1700s. Its first application was when Pierre-Simon Laplace applied transformation probability theory. He applied generating functions in the transformation. He gave very little concern about the continuous variables, which is known as the Laplace transformation.
Hurewicz reintroduced the idea of the z-transform later during the Second World War. It acquired its name when Raggazini and Zadeh applied it in a sampled-data group at Columbia University.
Definition of the z-transform
Like the Laplace transform, it can be divided into, unilateral, bilateral, and inverse transform. Remember that its variables are in a discrete state.
Properties of the z transform
This property states that if two, discrete-time signals are multiplied by a constant, their z-transforms will also be multiplied by a constant of the same magnitude.
This property implies that since the rectangular function in time is a sinc function in time, in the same way, a sinc function in time will be a rectangular function in time.
This property is important in explaining how a change in time will affect the z-domain. It shows that there is an equivalence in the linear phase shift frequency and the shift in time.
Convolution is the reason why signals are transformed into frequencies. It depicts the symmetry that exists between time and frequency.
Modulation is a very important component when it comes to signals. The ability to shift frequency signals is very important if we are to take advantage of the electromagnetic spectrum. Modulation is very important in television signals and radio signals.
Z-transform in Matlab
Matlab is a very versatile tool that has been consistently utilized by scientists and engineers. It enables them to perform complicated mathematical computations with ease. It is somehow impossible for a student to grasp all the contents of Matlab; that is why they need help from experienced personnel. Experienced personnel can do any mathematical computation with Matlab. Our z-transform using Matlab assignment experts are well versed with the software and can do any complicated task in a short time.
Matlab is well equipped and can allow you to perform inverse transform unilateral transform and bilateral transform calculations. Iztrans is Matlab’s function that is used for inverse transform computations and ztrans for unilateral and bilateral transforms.
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