Signals and Systems – What is a Linear System?

Signals and Systems Electronics & Electrical Digital Electronics

What is a Linear System?

System − An entity which acts on an input signal and transforms it into an output signal is called the system.

Linear System − A linear system is defined as a system for which the principle of superposition and the principle of homogeneity are valid.

Superposition Principle

The principle of superposition states that the response of the system to a weighted sum of input signals is equal to the corresponding weighted sum of the outputs of the system to each of the input signals.

Therefore, if an input signal x₁(t) produces an output signal y₁(t) and another input signal x₂(t) produces an output y₂(t), then the system is said to be linear if,

$$\mathrm{T[ax_{1}(t)+bx_{2}(t)]=ay_{1}(t)+by_{2}(t)}$$

Where,a and b are constants.

Types of Linear Systems

Linear systems can be of the following two types −

Linear Time-Invariant [LTI] System
Linear Time-Variant [LTV] System

Linear Time Invariant (LTI) System

A system which is both linear and time-invariant is called the linear timeinvariant system. In other words, a system for which both the superposition principle and the homogeneity principle are valid and the input-output characteristics of the system do not change with time is called linear time invariant (LTI) system.

Therefore, for an LTI system, the output for a delayed input [x(t -t₀)] must equal to the delayed output [y(t -t₀)], i.e.,

$$\mathrm{y(t,t_{0})=y(t-t_{0})}$$

It means that if the input to the system is delayed by (t₀) units, then the corresponding output will also be delayed by(t_o) units. Also, for a linear timeinvariant system, all the coefficients of the differential equation describing the system are constants.

Linear Time-Variant (LTV) System

A system which is linear but time-variant is called the linear time-variant system. In other words, a system for which the principle of superposition and homogeneity are valid but the input-output characteristics change with time is called the linear time-variant (LTV) system.

Therefore, for a linear time variant (LTV) system, the output of the system corresponding to the delayed input signal [i.e.,x(t -t₀)] is not equal to the delayed output [y(t -t₀)], i.e.,

$$\mathrm{y(t,t_{0})
eq y(t-t_{0})}$$

It means that, if the input to an LTV system is delayed by $(t_{0})$ units, then the corresponding output will not be delayed by exactly $(t_{0})$ units. Also, for a linear time variant (LTV) system at least one of the coefficients of the differential equation describing the system vary with time.