What is Z-transform?

Z-transform is a complex frequency domain representation of discrete-time signals. We can say that Z-transform is a generalization of discrete-time Fourier transformEvaluated over the unit-circle of Z-domain, and equivalent to Laplace transform.

Basics of Z-transform

Z-transform provides a way to solve linear, constant-coefficient equations of the order $K$, where the equation contains the first $K$differences of the sequence or function it refers to.

The following equation is an effective way to define Z-transform:

In the equation above, the value $r$defines the magnitude of $z$, and $\omega$represents the angle. The $z$variable is complex and is multiplied by the summation.

Types of Z-transform

On a surface level, Z-transform has two different types:

1. Bi-directional Z-transform

2. Uni-directional Z-transform

Bi-directional Z-transform is what we saw above. The only difference between these two is the range of the summation on which their multiplication is defined. In bi-directional Z-transform, the summation ranges from $-\infin$ to $+\infin$.

In uni-directional Z-transform, whereas, the summation ranges from $0$to $+\infin$. We can write it as follows:

One other thing left to discuss is the inverse Z-transform formula. It is stated as follows:

The $C$ in the integral is the counterclockwise path that encapsulates the poles of $X(z)$.

The Region of Convergence (RoC)

The Region of Convergence (RoC) is the set of the points for which the Z-transform converges. This means that the summation tends to end before $\infin$. ROC is used mainly in questions that revolve around Z-transform.

It is a ring or disc in the $z$-plane centered on the origin. Also, it cannot contain any poles. Poles are values of $z$ for which $Q(z)$ is $0$ in the equation:

Let's take a look at a diagram to understand better:

Example

Let's find the response of the system:

Here, all initial conditions are set to $0$.

Then, the following has $S(z)$ as the response:

Applications

We use Z-transform in the following:

• Mathematical and signal processing

• Digital filters

• Linear discrete system

• System design and analysis

• Telecommunication automatic controls