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What is Z transform and what is their importance ?
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asked Nov 25, 2015 | 66 views

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fourier transform is for frequency domain analysis and LTI systems, while Z transform provides analysis and design of discrete time signals and discrete time LTI systems.

mathematical calculations are reduced using Z transform.

there are two types of Z transform:

single sided and double sided.

z domain of X(Z) is when real part of Z is plotted on x-axis and imaginary part on y-axis.

ROC is set for all the values of Z for which X(Z) attains a finite value.
answer dont have "what z transform is !"

►For a given discrete time sequence x[n], we can define the z-transform X(z) as :

Z(x[n]) = X(z) = x[n]z​-n   ​-infinite< n <infinite

where z ⇒ continuous complex variable

z = Re(z) + j(Im(z)).

►Z-transform is a useful tool in the analysis of discrete time signals and systems and is the discrete time counterpart of the Laplace Transform for continuous time signals and systems.

►By putting z = e​jw, we have

X(z) = x[n]e-jwn​ , -infinite< n < infinite;

which forms the DTFT(Discrete Time Fourier Transform).

Z transform can be simply considered as a mathematic tool which switches or transforms the signal from time domain to a complex frequency domain called z domain.the computation will be easier by taking the transform of a signal,the stability and casualty of a system can be determined from the transform.
edited Jan 13, 2016 by saiuday
–1 vote

z-Transform

Introduction

In this Article of Jagrutkumar Vaghela,we see a section, which is absolutely fundamental, we define what is meant by the z-transform of a sequence.

We then obtain the z-transform of some important sequences and discuss useful properties of the transform. Most of the results obtained are tabulated at the end of the Section. The z-transform is the major mathematical tool for analysis in such areas as digital control and digital signal processing.

Prerequisites

Before we learn Z-transform we have to familiar with a Function Of Sigma and other mathematical symbols.

• understand sigma (Σ) notation for summations

• be familiar with geometric series and the binomial theorem

• have studied basic complex number theory including complex exponentials.

The z-transform

DEFINITION- due to some mathematical expression involved in the definition of z-transform i could not able to write in this answer page but if learner want a full definition i can send just send your Queries to my mail id vaghel.jagrut@engineer.com. i send you full definition.

Now,

If you have studied the Laplace transform either in a Mathematics course for Engineers and Scientists or have applied it in, for example, an analog control course you may recall that

1. the Laplace transform definition involves an integral

2. applying the Laplace transform to certain ordinary differential equations turns them into simpler (algebraic) equations

3. use of the Laplace transform gives rise to the basic concept of the transfer function of a continuous (or analog) system.

The z-transform plays a similar role for discrete systems, i.e. ones where sequences are involved, to that played by the Laplace transform for systems where the basic variable t is continuous. Specifically:

1. the z-transform definition involves a summation

2. the z-transform converts certain difference equations to algebraic equations

3. use of the z-transform gives rise to the concept of the transfer function of discrete (or digital) systems.

Notes

1. The z-transform only involves the terms yn, n = 0, 1, 2, . . . of the sequence. Terms y−1, y−2, . . . whether zero or non-zero, are not involved.

2. The infinite series in (1) must converge for Y (z) to be defined as a precise function of z. We shall discuss this point further with specific examples shortly.

3. The precise significance of the quantity (strictly the ‘variable’) z need not concern us except to note that it is complex and, unlike n, is continuous.

There is more and more Knowledge of z-transform , this is just an introduction and overview of z-transform

Further in detail z-transform has Its unique properties and Relationships with the Unit-step function,ramp,impulse and the delta function.

Thank you

Best wishes

Jagrutkumar Vaghela