- Trending Categories
- Data Structure
- Networking
- RDBMS
- Operating System
- Java
- MS Excel
- iOS
- HTML
- CSS
- Android
- Python
- C Programming
- C++
- C#
- MongoDB
- MySQL
- Javascript
- PHP
- Physics
- Chemistry
- Biology
- Mathematics
- English
- Economics
- Psychology
- Social Studies
- Fashion Studies
- Legal Studies

- Selected Reading
- UPSC IAS Exams Notes
- Developer's Best Practices
- Questions and Answers
- Effective Resume Writing
- HR Interview Questions
- Computer Glossary
- Who is Who

# Signals and Systems: Linear and Non-Linear Systems

## Linear System

A system is said to be linear if it obeys the principle of homogeneity and principle of superposition.

### Principle of Homogeneity

The principle of homogeneity says that a system which generates an output y(t) for an input x(t) must produce an output ay(t) for an input ax(t).

### Superposition Principle

According to the principle of superposition, a system which gives an output 𝑦_{1}(𝑡) for an input 𝑥_{1}(𝑡) and an output 𝑦_{2}(𝑡) for an input 𝑥_{2}(𝑡) must produce an output [𝑦_{1}(𝑡) + 𝑦_{2}(𝑡)] for an input [𝑥_{1}(𝑡) + 𝑥_{2}(𝑡)].

Therefore, for a continuous-time linear system,

[𝑎𝑦_{1}(𝑡) + 𝑏𝑦_{2}(𝑡)] = 𝑇[𝑎𝑥_{1}(𝑡) + 𝑏𝑥_{2}(𝑡)] = 𝑎𝑇[𝑥_{1}(𝑡)] + 𝑏𝑇[𝑥_{2}(𝑡)]

Also, for a discrete-time linear system,

[𝑎𝑦_{1}(𝑛) + 𝑏𝑦_{2}(𝑛)] = 𝑇[𝑎𝑥_{1}(𝑛) + 𝑏𝑥_{2}(𝑛)] = 𝑎𝑇[𝑥_{1}(𝑛)] + 𝑏𝑇[𝑥_{2}(𝑛)]

Hence, we can say that a system is linear if the output of the system due to weighted sum of inputs is equal to the weighted sum of outputs.

Filter circuits, communications channels, etc. are few examples of linear systems.

## Non-Linear System

A system is said to be a non-linear system if it does not obey the principle of homogeneity and principle of superposition.

Generally, if the equation describing the system contains square or higher order terms of input/output or product of input/output and its derivatives or a constant, the system will be a non-linear system. Triangulation of GPS signals is an example of non-linear system.

## Numerical Example

Check whether the given systems are linear or non-linear systems −

$\mathrm{\frac{\mathrm{d^{2}}y(t) }{\mathrm{d} t^{2}}+3ty(t)=t^{2}x(t)} $

$\mathrm{3\frac{\mathrm{d}y(t) }{\mathrm{d} t}+4y(t)=x^{2}(t)} $

### Solution

The given system is,

$$\mathrm{\frac{\mathrm{d^{2}}y(t) }{\mathrm{d} t^{2}}+3ty(t)=t^{2}x(t)} $$

Consider an input 𝑥

_{1}(𝑡) which produces an output 𝑦_{1}(𝑡), then,$$\mathrm{\mathrm{\frac{\mathrm{d^{2}}y_{1}(t) }{\mathrm{d} t^{2}}+3ty_{1}(t)=t^{2}x_{1}(t)}\: \: \cdot \cdot \cdot (1)} $$

And an input 𝑥

_{2}(𝑡) which produces an output 𝑦_{2}(𝑡), then,$$\mathrm{\mathrm{\frac{\mathrm{d^{2}}y_{2}(t) }{\mathrm{d} t^{2}}+3ty_{2}(t)=t^{2}x_{2}(t)}\: \: \cdot \cdot \cdot (2)} $$

Now, the linear combination of the equations (1) and (2) gives,

$$\mathrm{ \begin{Bmatrix} a\mathrm{\frac{\mathrm{d^{2}}y_{1}(t) }{\mathrm{d} t^{2}}+a3ty_{1}(t)}\ \end{Bmatrix}+\begin{Bmatrix} b\mathrm{\frac{\mathrm{d^{2}}y_{2}(t) }{\mathrm{d} t^{2}}+b3ty_{2}(t)}\ \end{Bmatrix}=at^{2}x_{1}(t)+bt^{2}x_{2}(t) } $$

$$\mathrm{\Rightarrow \frac{\mathrm{d} ^{2}}{\mathrm{d} t^{2}}\left [ ay_{1}(t)+by_{2}(t) \right ]+3t\left [ ay_{1}(t)+by_{2}(t) \right ]}$$

$$\mathrm{= t^{2}\left [ ax_{1}(t)+bx_{2}(t) \right ]\: \: \cdot \cdot \cdot (3)}$$

Where, [𝑎𝑦

_{1}(𝑡) + 𝑏𝑦_{2}(𝑡)] is the weighted sum of outputs and [𝑎𝑥_{1}(𝑡) + 𝑏𝑥_{2}(𝑡)] is the weighted sum of inputs.Hence, the equation (3) shows that the weighted sum of inputs to the given system generates an output that is equal to the weighted sum of outputs to each of the individual inputs. Therefore, the given system is a

**linear system**.The given system is,

$$\mathrm{3\frac{\mathrm{d} y(t)}{\mathrm{d} t}+4y(t)=x^{2}(t)}$$

Let the output 𝑦

_{1}(𝑡) is corresponding to the input 𝑥_{1}(𝑡), then,$$\mathrm{3\frac{\mathrm{d} y_{1}(t)}{\mathrm{d} t}+4y_{1}(t)=x_{1}^{2}(t)\: \: \cdot \cdot \cdot (1)}$$

Similarly, the output 𝑦

_{2}(𝑡) is corresponding to the input 𝑥_{2}(𝑡), then,$$\mathrm{3\frac{\mathrm{d} y_{2}(t)}{\mathrm{d} t}+4y_{2}(t)=x_{2}^{2}(t)\: \: \cdot \cdot \cdot (2)}$$

Then, the linear combination (i.e., homogeneity and superposition) of the equations (1) and (2) can be written as,

$$\mathrm{\begin{Bmatrix} a3\frac{\mathrm{d} y_{1}(t)}{\mathrm{d} t}+a4y_{1}(t)\ \end{Bmatrix}+\begin{Bmatrix} b3\frac{\mathrm{d} y_{2}(t)}{\mathrm{d} t}+b4y_{2}(t)\ \end{Bmatrix}=ax_{1}^{2}(t)+bx_{2}^{2}(t)}$$

$$\mathrm{\Rightarrow 3\frac{\mathrm{d} }{\mathrm{d} t}\left [ ay_{1}(t)+by_{2}(t) \right ]+4\left [ ay_{1}(t)+by_{2}(t) \right ]=ax_{1}^{2}(t)+bx_{2}^{2}(t)}$$

Where, $\mathrm{\left [ ay_{1}(t)+by_{2}(t) \right ]}$ is the weighted sum of outputs but $\mathrm{\left [ ax_{1}^{2}(t)+bx_{2}^{2}(t) \right ]} $ is not the weighted sum of inputs. Here, the principle of superposition is not satisfied. Therefore, the given system is a

**non-linear system**.

- Related Articles
- Signals and Systems: Linear Time-Invariant Systems
- Signals and Systems – Filter Characteristics of Linear Systems
- Signals and Systems – Properties of Linear Time-Invariant (LTI) Systems
- Signals and Systems – What is a Linear System?
- Signals and Systems: Invertible and Non-Invertible Systems
- Signals and Systems – Symmetric Impulse Response of Linear-Phase System
- Signals and Systems – Response of Linear Time Invariant (LTI) System
- Signals and Systems – Transfer Function of Linear Time Invariant (LTI) System
- Signals and Systems: Causal, Non-Causal, and Anti-Causal Signals
- Signals and Systems – Causal and Non-Causal System
- Signals and Systems: Classification of Systems
- Signals and Systems: Time Variant and Time-Invariant Systems
- Signals and Systems: Even and Odd Signals
- Signals and Systems: Periodic and Aperiodic Signals
- Signals and Systems: Energy and Power Signals