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 𝑦₁(𝑡) for an input 𝑥₁(𝑡) and an output 𝑦₂(𝑡) for an input 𝑥₂(𝑡) must produce an output [𝑦₁(𝑡) + 𝑦₂(𝑡)] for an input [𝑥₁(𝑡) + 𝑥₂(𝑡)].

Therefore, for a continuous-time linear system,

[𝑎𝑦₁(𝑡) + 𝑏𝑦₂(𝑡)] = 𝑇[𝑎𝑥₁(𝑡) + 𝑏𝑥₂(𝑡)] = 𝑎𝑇[𝑥₁(𝑡)] + 𝑏𝑇[𝑥₂(𝑡)]

Also, for a discrete-time linear system,

[𝑎𝑦₁(𝑛) + 𝑏𝑦₂(𝑛)] = 𝑇[𝑎𝑥₁(𝑛) + 𝑏𝑥₂(𝑛)] = 𝑎𝑇[𝑥₁(𝑛)] + 𝑏𝑇[𝑥₂(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 𝑥₁(𝑡) which produces an output 𝑦₁(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 𝑥₂(𝑡) which produces an output 𝑦₂(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, [𝑎𝑦₁(𝑡) + 𝑏𝑦₂(𝑡)] is the weighted sum of outputs and [𝑎𝑥₁(𝑡) + 𝑏𝑥₂(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 𝑦₁(𝑡) is corresponding to the input 𝑥₁(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 𝑦₂(𝑡) is corresponding to the input 𝑥₂(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.