Z Transform Basics

Laplace transform is a generalization of the continuous-time Fourier transform. 

Z Transform is a generalization of the discrete-time Fourier transform.

Z transform is a representation of the discrete-time signal in the z domain.

the mathematical Equation can be represented as 

As we know that discrete-time signal x(n) can be plotted by amplitude v/s n. The z transform can be plotted in two axis where we plot the real part of the equation in the horizontal axis and the imaginary part of the equation to the vertical axis, which we call as Z domain.

For the conversion of z transform to Fourier transform the Value of z sill be substituted by 

And so the equation becomes 

The z-transform is the Fourier transform of the sequence x(n)r^-n. For r=1, this becomes the Fourier transform of x(n). The Fourier transform therefore corresponds to the z-transform evaluated on the unit circle as shown:

As per this representation, the inherent periodicity of Fourier transform can be seen easily.

the z-transform does not converge for all sequences or for all values of z. The set of values of z for which the z-transform converges is called the Region of Convergence (ROC).

we can define ROC for x(n) as the set of all values of z for which X(z) attains a finite value. Hence, we need to specify the ROC every time we write a Z transform.

So for the existence of the z-Transform, the following condition must be true:

The ROC, therefore, consists of a ring in the z-plane:

Lets take an example for determining the z transform and ROC for a given Fourier transform.

Consider the signal x[n] = a^n u[n] z^-n

The z transform becomes 

Convergence occurs only if 

Which is only the case if |az^-1|<1 or equivalently |z| > |a|. In the ROC the series converges to 

Since this is a Geometric series. The z-transform has a region of convergence

for any finite value of a.

The Fourier transform of x[n] only exists if the ROC includes the unit circle, which requires that |a| < 1. On the other hand, if |a| > 1 then the ROC does not include the unit circle, and the Fourier transform does not exist. This is consistent with the fact that for these values of a the sequence a^n u[n]  is

exponentially growing, and the sum, therefore, does not converge.

Now, let us take the left side of the sequence. For that consider the following sequence 

x[n] = -a^n u[-n-1] 

The z transform is 

For |a^-1 z| < 1, or |z| < |a|, the series converges to

The expression for the z-transform (and the pole-zero plot) is exactly the same as for the right-handed exponential sequence—only the region of convergence is different. Specifying the ROC is therefore critical when dealing with the z-transform.

Okay, now let's consider sum of two exponentials 

the z Transform 

the first term in this sum converges for |z| > 1/2, and the second for |z| > 1/2. The combined transform X(z) therefore converges in the intersection of these regions, namely when |z| > 1/2. In this case

The Pole-zero plot and region of convergence of the signal 

We also have Finite length sequence signal as 

the z transform of the above sequence

                           

Since there are only a finite number of nonzero terms the sum always converges when az^-1 is finite. There are no restrictions on a (|a|< ∞), and the ROC is the entire z-plane with the exception of the origin z = 0 (where the terms in the sum are infinite). The N roots of the numerator polynomial are at

since these values satisfy the equation z^N = a^N. The zero at k = 0 cancels the pole at z = a, so there are no poles except at the origin, and the zeros are at

Properties of Z transform

1. Multiplication by a constant: Z[ax(t)] = aX(z), where X(z) = Z[x(t)].
2. Linearity:
   
3. Multiplication by 
   
   
   
4. Real shifting:  
   
5. Complex shifting: 

   
6. Initial value theorem:

   
7. Final value theorem:

   

Properties of the region of convergence

The properties of the ROC depend on the nature of the signal. Assuming that the signal has a finite amplitude and that the z-transform is a rational function:

1. The ROC is a ring or disk in the z-plane, centered on the origin.
2. The Fourier transform of x[n] converges absolutely if and only if the ROC of the z-transform includes the unit circle.
3. The ROC cannot contain any poles.
4. If x[n] is finite duration then the ROC is the entire z-plane except perhaps at z = 0 or z = ∞.
5. If x[n] is a right-sided sequence then the ROC extends outward from the outermost finite pole to infinity.
6. If x[n] is left-sided then the ROC extends inward from the innermost nonzero pole to z = 0.
7. A two-sided sequence (neither left nor right-sided) has a ROC consisting of a ring in the z-plane, bounded on the interior and exterior by a pole (and not containing any poles).
8. The ROC is a connected region.

This article is the courtesy of 

Advanced Training Course on FPGA Design and VHDL for Hardware Simulation and Synthesis Massimiliano Nolich (26 October - 20 November, 2009)-Digital Signal Processing , The z-transform.

Thank you and see you in the next one...