DSP - Time-Variant Systems

For a time variant system, also, output and input should be delayed by some time constant but the delay at the input should not reflect at the output. All time scaling cases are examples of time variant system. Similarly, when coefficient in the system relationship is a function of time, then also, the system is time variant.

Examples

a) $y(t) = x[\cos T]$

If the above signal is first passed through the system and then through the time delay, the output will be $x\cos (T-t)$. If it is passed through the time delay first and then through the system, it will be $x(\cos T-t)$. As the outputs are not same, the system is time variant.

b) $y(T) = \cos T.x(T)$

If the above expression is first passed through the system and then through the time delay, then the output will be $\cos(T-t)x(T-t)$. However, if the expression is passed through the time delay first and then through the system, the output will be $\cos T.x(T-t)$. As the outputs are not same, clearly the system is time variant.