Test whether the system described by the equation y(n) = n x(n) is Linear or Shift invariant.

(a) To check linearity 

y1(n) = T{x1(n) = n x1(n) 

y2(n) = T{x2(n) = n x2(n) 

y3(n) = T{a1x1(n) + a2 x2(n)} = n[a1 x1(n) + a2x2(n)] = n a1 x1(n) + n a2x2(n) 

and y’3(n) = a1y1(n) + a2y2(n) = a1nx1(n) + a2nx2(n) 

Here y3 (n) = y’3(n) Hence this system is linear. 

(b) To check shift invariance

y(n) = T{x(n) = n x(n) 

y(n,k) = T{x(n-k) = n x(n-k} 

y(n-k) = n(n-k) x(n-k) 

Since y(n,k) = y(n-k) the system is shift variant.