Basic System Parameters

Basic system parameters:

System parameters are classified as:

• Static | Dynamic
• Causal | Non-Causal
• Time invariant | Varient
• Linear | Non-Linear
• Stable | Unstable
• Invertible | Non-Invertible

1.Static and Dynamic system:

Static system: If output of system depends on only present values of input then system is called static system. These systems are also known as system without memory.

Example: y(t)= sin[x(t)] , y(t)= x(t)

Dynamic system: If output of a system depends on past or future values of input at any instant of time is called dynamic system.

Example: y(t)= x(t+1) , y(t)= x(2t) , y(t)= x(t^2) , y(t)= x[sgn(t)] , y(t)= x[u(t)]

Note:

• If any operator (square, cube,..... addition,.......square root,...... sin, cos,....,u(t),..., sgn(t),..... hyperbolic, exponential, modulus etc.) operates on time either in x(t) or y(t) than it is the case of Time scalling and system become Dynamic System.
• For static system there should not be any shifting or scalling in time either in x(t) or y(t).
• All integral and derivatives system are Dynamic system.

2.Causal | Non-Causal System:

Causal System: If Output of System does not depends on future value of input at any instant of time the system is called causal system.

Example: y(t)= x(t), y(t)= x(t) + x(t-1) , y(t)= 2tx(t) etc.

Non- Casual System: If output of system depends on future values of input at any instant of time system is called Non-Causal.

Example: y(t)= x(t+1), y(t)= x(t) + x(t+1) etc.

Anti-Causal System: If output of system depends on only future value of input than system is called Anti-Causal System. Anti-Causal System is a subset of Non-Causal system .

Example: y(t)= x(t+1)

Note: 

• If any operator operates on time either in x(t) or y(t)  than system is non causal.

3.Linear | Non-Linear System:

Linear system follow law of of superposition. Law of superposition is necessary and sufficient condition to prove the linearity of the system.

Law of superposition is equal to the sum of Law of Additivity and Law of Homogeneity. The system which does not follow law of superposition is said to be nonlinear system.

Law of Homogeneity:

Let us consider a system  y(t)=  x(t)

Let us consider another example y(t)=  x²(t)

So it does not follow Homogeneity so the system is Non-Linear system.

Law of Additivity

Examples:

1. y(t)= sin[x(t)] ….................. Non-Linear
2. y(t)=|t|.x(t) ………………… Linear
3. y(t)= sint.x(t) ……………... Linear
4. y(t)= |x(t)| ………………….Non Linear
5. y(t) = x(t) + 10 ……………. Non-Linear

Note:

• If any operator operates on x(t) or y(t) the system will be non linear.
• Linearity is  independent of coefficient.
• For linear system output should be zero for zero input.
• Zero input and zero output condition is necessary for linear system but not the sufficient condition.
• Conjugate operator is a non linear operator.

4.Time invariant | Variant System:

A system is said to be time invariant if any amount of delay provided in input must be reflected in output. Otherwise system will be time variant.

Note:

• If any operator operates and time either in x(t) or y(t) then the system will be time variant.

Example: y(t)= x[cost), y(t)= x[tan(e^t)], y(t)= x[logt]

• If coefficient in system relationship is a function of time then system will be time variant system.

y(t)=t.x(t)

• All split systems are time variant system.

y(t)= { x(t)       , t<0

           x(t+1)   , t>0 }

5.Stable | Unstable System:

Bounded input Bounded output criteria:

For stable system output should be bounded and finite for or finite or bounded input at all instant of time.

Eg: y(t)= 2x(t) ……….…..Stable

      y(t)= t.x(t) …………… Unstable

6.Invertible and Non-Invertible System:

For an  invertible system there should be an one to one mapping between input and output for all the instant of time.

Note:

When an inverter power system is cascaded with its inverse system than the  output of its composite system will be equal to input.

Example:

y(t)= x²(t) .................... Non Invertible

y(t)= dx(t)/dt ............... Non Invertible