Verify the linearity, causality and time invariance of the system, y(n+2) = a x(n+1) + b x(n+3)

     Putting n-2 for n in given difference equation, 

y(n-2+2) = a x(n-2+1) + b x(n-2+3) 

y(n) = a x(n-1) = b x(n+1) 

i) Linearity 

y1(n) = T {x1(n) = a x1(n-1) + b x1(n+1) 

y2(n) = T{x2(n) = a x2(n-1) + b x2(n+1) 

y3(n) = T {a1 x1(n) + a2 x2(n)} 

    = a[a1 x1(n-1) + a2 x2(n-1)] + b[a1 x1(n+1) + a2x2(n+1)] 

    = a a1x1(n-1) + a a2x2(n-1) + ba1x1(n+1)+ b a2 x2(n+1) 

y’3 (n) = a1y1(n) + a2y2(n) 

        = a1[a x1(n-1) + b x1(n+1)] + a2[a x2(n-1) + b x2(n+1)] 

        = a a1 x1(n-1) + b a1 x1(n+1) + a a2x2(n-1) + b a2 

x2(n+1) Since y3(n) = y’3 (n), the system is linear.

ii) Causality 

        y(n) depends upon x(n+1), which is future input. Hence the system is non causal.

iii) Time invariance 

y(n,k) = T{x(n-k)} = a x(n-1-k) + b x(n+1-k) 

y(n,k) = a x(n-k-1) + b x(n-k+1) 

Since y(n,k) = y(n-k), the system is time invariant