Determine whether the following systems are Time invariant or not

Determine whether the following systems are time invariant or not

i) Y(t) = tx(t)  

ii) Y(n) = x(2n)

i) Y(t) = tx(t)  

    Y(t) = T[x(t)] = tx(t)  

The output due to delayed input is,  

    Y(t,T) = T[x(t - T)] = tx(t - t)  

If the output is delayed by T, we get  

    Y(t -T) = (t - T) x( t - T)  

The system does not satisfy the condition, y(t,T) = y(t – T).  

    Then the system is time invariant.  

ii) Y(n) = x(2n)  

     Y(n) = x(2n)  

    Y(n) = T[x(n)] = x(2n)  

If the input is delayed by K units of time then the output is,  

     Y(n,k) = T[x(n-k)] = x(2n-k)  

The output delayed by k units of time is, 

    Y(n-k) = x[2(n-k)]  

Therefore, y(n,k) is not equal to y(n-k). Then the system is time variant.