Linearity and Time Invariance

Given system:

Input Output

X0[n]=δ[n] -> Y0[n]=δ[n-1]

X1[n]=δ[n-1] -> Y1[n]=4δ[n-2]

X2[n]=δ[n-2] -> Y2[n]=9 δ[n-3]

X3[n]=δ[n-3] -> Y3[n]=16 δ[n-4]

... -> ...

Xk[n]=δ[n-k] -> Yk[n]=(k+1)2 δ[n-(k+1)] -> For any non-negative integer k

Time Invariant System?

Suppose the system is defined as the third line where input is $ X_2[n]= dirac[n-2] $ and output: $ Y_2[n]=9 dirac[n-3] $ with a time delay of 2 seconds.

Using the system then time delay method:

dirac[n-2] -> SYSTEM -> 9 dirac[n-3] -> Time Delay -> 9 dirac[n-5]

Using the time delay then system method:

dirac[n-2] -> Time Delay -> dirac[n-4] -> SYSTEM -> 25 dirac[n-6]

Because the two outputs are not the same, the system is not time-invariant.

Necessary Input to Yield u[n-1]

In order to obtain $ u[n-1] $ as the output instead of a delta function, the desired input would simply be $ X_0 = u[n] $.