Signals and Systems - Section 1 (5)

For linearity

y₁[n] = x₁[n] + x₂[n - 10] ...(1)

y₂ = x₂[n] + x₂[n - 10] ...(2)

y₁[n] = x₁[n] + x₂[n] + x₂[n - 10] + x₂[n - 10] ...(3)

Now find y₁[n] + y₂[n]

Corresponding to x₁[n] + x₂[n]

It is same as equation (3) hence linear.

But in part (c) y[n] = x²[n]

⇒ y₁[n] = x²₁[n], y₂[n] = x²₂[n]⇒ y₁[n] + y₂[n] = x²₂[n]...3

But y₁[n] + y₂[n]

Corresponing x₁[n] + x₂[n] is y₁[n] + y₂[n] = {x₁[n] + x₂[n]}²

= x₁²[n] + x₂²[n] + 2x₁[n] x₂[n]....4

Equations (3) and (4) are not same hence not linear.

By applying time shifting and scaling property.

H(z) = 6 + z^-1 - z^-2, solve it by considering H(z) = 0 z = 1/3, -1/2 in H(z) only numerator.

In transfer function, no of zeros = no. of poles, hence it is all pass fliter.

Because no.of zero is more than Pole, therefore when you find transfer function.

Numerator power will higher than denominator, and such transfer functions is not realizable.