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Signals and Systems – What is a Linear System?
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(to) 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.
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