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]=δ[n-2] $ and output: $ Y_2[n]=9 δ[n-3] $ with a time delay of . Using the same method as in Part D, we can determine whether this system is time invariant or not.

δ[n] -> time delay -> δ[n-3] -> system -> 16δ[n-4]

δ[n] -> system -> δ[n-1] -> time delay -> δ[n-4]

Since both cascades produce different outputs, this system is NON-time invariant.