Linear and Non linear system | Causal and Non causal system | Stable and Unstable system

Linear and Non linear system

    • A system is said to be linear if it satisfies the superposition principle.  

    • Superposition principle states that the response to a weighted sum of input signal be  equal to the weighted sum of the output corresponding to each of the individual input signal  

    • The continuous system is linear if,  

            F[a1x1(t) + a2x2(t)] = a1y1(t) + a2y2(t)  

    • The discrete system is linear if, F[a1x1(n) + a2x2(n)] = a1y1(n) + a2y2(n)  

    • Otherwise the system is non linear.  

    • A system is called linear if its I/O behavior satisfies the additivity and homogeneity  properties

    • Equivalently, a system is called linear if its I/O behavior satisfies the superposition  property

Causal and Non causal system 

    • A causal system is one whose output depends upon the present and past input values.

    • If the system depends the future input values, the system is said to be non causal. 

Eg. for causal system.  

Y(t) = x(t) + x(t - 1)  

Y(n) = x(n) + x(n - 3)  

Eg. For non causal system,  

Y(t) = x(t+3) + x2(t)  

Y(n) = x(2n)  

    • A system is called causal or non-anticipative if the output at any time t (or n) depends  only on the input at times t or before t (or n or before n); in other words, independent of the input at times after t (or n). All memory less systems are causal. Physical systems where the time is the independent variable are causal.  

    • Non-causal systems may arise in applications where the independent variable is not  the time such as in the image processing applications. 

Stable and unstable system

• When every bounded input produces bounded output then the system is called as  

stable system or bounded input bounded output (BIBO stable).  

• Otherwise the system is unstable.  

• A system is called stable if it produces bounded outputs for all bounded inputs.  

• Stability in a physical system generally results from the presence of mechanisms that  dissipate energy, such as the resistors in a circuit, friction in a mechanical system, etc.