## Understanding Z-transform

Z-transform is a DSP tool that is used to filter system analysis. Z-transform allows people to identify values that exponentially increase or decrease with time. It encompasses the fundamental and functional foundation of our current communication systems. It includes rudimentary concepts such as infinite, finite, and numerical integration as its foundation when boing used. Z-transform converts a discrete-time signal into a complex frequency domain representation. The idea in the Z-transform is called the mathematical literature as a method of generating functions that were started in 1730. Z-transform was called the Laplace when it was re-introduced by W. Hurewicz in 1947.

Z transform is divided into bilateral z-transform and the Unilateral Z-transform.

### The Bilateral Z-transform

- The bilateral z-transform is also called the two-sided Z-transform of a time signal. It is defined as

### Unilateral Z-transform

- The unilateral z-transform is also called a one-sided z transform and is defined as.

Z-transform allows us to analyze the phase of sinusoidal components and the frequency of a system to characterize the system's response. If the Z-transform shows increasing output values, then the system shows instability for the value of x{n} and Z^-n. Note that it is also possible to examine the frequency response of the system using tables.

## Explaining x{n} and Z

Remember that we use Z-transform to understand and describe a system. It’s important to also know that, x{n} is a discrete signal. Therefore in z-transform x{n} is used as the system’s impulse response. Variable Z is used as a complex number.

## Commonly used Z-transforms

- Unit impulse sequence – This is the first Z-transform. It is also the simplest, and it’s denoted by δn.
- Unit step sequence
- The geometric sequence