Discrete-time systems:

Definition

A discrete-time system is a mapping from set of the acceptable discrete-time signals, called as input set, to a set of discrete-time signals called as output set.

Definition 

A discrete-time system is deterministic in nature if its output to a given input does not depend on some random phenomenon.  If it does, the system is known as random (stochastic) system.

Definition 

A digital system is a mapping which assigns a digital output signal to every acceptable digital input signal.

A discrete-time system can be thought of as a transformation or operator, T, which maps an input sequence x(n) to the output sequence y(n) shown therefore:

We focus on the presence or absence of the following properties in discrete-time systems: shift invariance, linearity, causality and stability.

Filter 

A linear time-invariant (LTI) system is also referred as a filter, which means that, a filter is a system T with a single input and a single output signal which is linear and time-invariant both.

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