**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