What is z transform

The Z transform has a real and imaginary part like Fourier transform. A plot of imaginary part versus the real part is called a Z plane or complex Z plane.

The pole-zero plot is the main characteristic of discrete time LTI systems full form. We can also check the stability of the system using a pole-zero plot. Z transform used for many signal processing

There are two types of Z transform :

1. Single sided Z transform
2. Double-sided Z transform 

1. Single sided Z transform :

As we know that a single-sided Z transform of discrete time signal x (n) is defined as,

X  (Z) =  ∑     x (n)  Z -n


Here "Z" is a complex variable. In this equation limits of summation are from the 0 too ∞. So while expanding the summation we will put only positive values of n (from the range n = 0 ton =  ∞). So this is single-sided or one-sided Z- transform.

2. Double-sided Z transform 

As we know that a double-sided Z transform of discrete time signal x (n) is defined as,

X  (Z) =  ∑     x (n)  Z -n

              n= - ∞

While expanding the summation we will put both positive as well as negative values of "n". Thus this is called a double-sided Z transform.

How to denote Z transform?

The relationship between x(n) and X(Z) is indicated as given below :

x (n) ↔ X (Z)

Note that X (Z) is Z transform of x (n), always when Z transform of the sequence is obtained then it is denoted by capital letter, X (Z) Here arrow is bidirectional. This is because we can also obtain x (n) from X (Z) using the inverse Z transform.

The Z transform of x (n) is also denoted as in this form,

X (Z) = Z  { x(n) }

So here x (n) and  X(Z) are called as Z transform pairs.

In same way if we want to analyze a system, which is already represented in frequency domain, a discrete time signal then we go for inverse Z transformation.