System properties

A discrete-time system, as denoted in the figure, is a mathematical operator or mapping that transforms one signal, the input $x\left[n\right]$, into another signal, the output $y\left[n\right]$ by means of a fixed set of rules or operations. Discrete-time systems may be classified in terms of the properties that they possess.

The most important system properties are:

Memoryless

A system is memoryless if the output at any time $n=n_0$ depends only on the input at time $n=n_0$. In other words, a system is memoryless if, for any $n_0$, we are able to determine the output value $y\left[n_0\right]$ given only the input value $x\left[n_0\right]$.

Causality

A system property that is important for real-time applications is causality, which implies that for any index $n_0$, the response of the system at time index $n_0$ depends only on the input up to time index $n=n_0$. Thus, for a causal system, changes in the output cannot precede changes in the input.

Invertibility

A system property that is important in applications such as channel equalization and de-convolution is invertibility. A system is said to be invertible if the input of the system uniquely can be determined from the output. In order for a system to be invertible, it is necessary for distinct inputs to produce distinct outputs.

Additive

An additive system is one for which the response to a sum of inputs is equal to the sum of the outputs individually.

Homogeneous

A system is homogeneous if scaling the input by a constant $c$ results in a scaling of the output by the same amount of $c$.

Stability

In most practical cases it is important for a system to have a response, $y\left[n\right]$, that is bounded in amplitude whenever the input is bounded. A system with this property is said to be stable in the Bounded Input Bounded Output (BIBO) sense.

Linearity

A system that is both additive and homogeneous is said to be linear.

Referring to Fig. 2, assume that an input $x\left[n\right]$ which is applied to a discrete-time system, results in an output $y\left[n\right]$. Thus $x_1\left[n\right]$ results in $y_1\left[n\right]$ and $x_2\left[n\right]$ in $y_2\left[n\right]$. On the one hand, we can first apply separately $x_1\left[n\right]$ and $x_2\left[n\right]$ to the same system and then weight and combine both output into a new output: $w\left[n\right]=\alpha y_1\left[n\right]+\beta y_2\left[n\right]$. On the other hand, we can first weight, with the same parameters $\alpha$ and $\beta$, and combine both inputs into a new input $x\left[n\right]=\alpha x_1\left[n\right]+\beta x_2\left[n\right]$. When applying this new input $x\left[n\right]$ to the same system it results in an output $y\left[n\right]$. A system is linear if both outputs $w\left[n\right]$ and $y\left[n\right]$ are the same.

Time-Invariance

If a system has the property that a shift (a delay) of the input by $n_0$ samples results in a shift of the output by the same amount of $n_0$ samples, the system is said to by time-invariant.

Referring to Fig. 3, let $y\left[n\right]$ be the response of a discrete-time system to an arbitrary input signal $x\left[n\right]$. The system is said to be time-invariant if, for any delay of $n_0$ samples, the response to $x[n-n_0]$ is $y[n-n_0]$. In effect a system is time invariant if its properties or characteristics do not change with time.

Linear Time Invariance

A system that is both linear and time-invariant is referred to as a Linear Time Invariant system, abbreviated as LTI. Both properties, Linearity and Time Invariance, of an LTI system are important to understand how to simplify the mathematical analysis to obtain a greater insight and understanding of system behavior.