Define Time Variant And Time Invariant System?

A system is called Time Invariant if we delay an input before processing, output will be equal to output delayed after processing. And if we delay an input before processing, output will not be equal to input delayed after processing, the system is Time Variant.

A more formal proof of why systems A and B above differ is now presented. To perform this proof, the second definition will be used.

System A:

Start with a delay of the input {\\displaystyle x_{d}(t)=\\,\\!x(t+\\delta )}

{\\displaystyle y(t)=t\\,x(t)}

{\\displaystyle y_{1}(t)=t\\,x_{d}(t)=t\\,x(t+\\delta )}

Now delay the output by {\\displaystyle \\delta }

{\\displaystyle y(t)=t\\,x(t)}

{\\displaystyle y_{2}(t)=\\,\\!y(t+\\delta )=(t+\\delta )x(t+\\delta )}

Clearly {\\displaystyle y_{1}(t)\\,\\!\\neq y_{2}(t)}, therefore the system is not time-invariant.

System B:

Start with a delay of the input {\\displaystyle x_{d}(t)=\\,\\!x(t+\\delta )}

{\\displaystyle y(t)=10\\,x(t)}

{\\displaystyle y_{1}(t)=10\\,x_{d}(t)=10\\,x(t+\\delta )}

Now delay the output by {\\displaystyle \\,\\!\\delta }

{\\displaystyle y(t)=10\\,x(t)}

{\\displaystyle y_{2}(t)=y(t+\\delta )=10\\,x(t+\\delta )}

Clearly {\\displaystyle y_{1}(t)=\\,\\!y_{2}(t)}, therefore the system is time-invariant.

More generally, the relationship between the input and output is {\\displaystyle y(t)=f(x(t),t)}, and its variation with time is

{\\displaystyle {\\frac {\\mathrm {d} y}{\\mathrm {d} t}}={\\frac {\\partial f}{\\partial t}}+{\\frac {\\partial f}{\\partial x}}{\\frac {\\mathrm {d} x}{\\mathrm {d} t}}}.

For time-invariant systems, the system properties remain constant with time, {\\displaystyle \\partial f/\\partial t=0}. Applied to Systems A and B above:

{\\displaystyle f_{A}=tx(t)\\qquad \\Rightarrow \\qquad {\\frac {\\partial f_{A}}{\\partial t}}=x(t)\\neq 0} in general, so not time-invariant

{\\displaystyle f_{B}=10x(t)\\qquad \\Rightarrow \\qquad {\\frac {\\partial f_{B}}{\\partial t}}=0} so time-invariant.

A system is called time invariant if its output , input characteristics dos not change with time.

Introduce a known delay y(n,k) to a given system and determine the output as y(n,k). As a second scenario, substitute the discrete time variable n by n-k in the given system, say y(n-k). 

If both y(n,k) = y(n-k),-------> system is time invariant. 

Otherwise, time variant

A system is called time invariant if its output , input characteristics dos not change with time.

e.g.y(n)=x(n)+x(n-1)

A system is called time variant if its input, output characteristics changes with time.

e.g.y(n)=x(-n).