Output Response

" n ³ 0

y(n) = caⁿ s₀ + S_{(m = 0 to n-1)}  ca ^{n- 1 - m} bx(m) + dx(n).

The response can be decomposed into the sum of the zero input response (response if input = 0)

caⁿ s₀

and the zero state response (response if initial state s₀ = 0)

S_{(m = 0 to n-1)}  ca ^{n - 1 - m} bx(m) + dx(n)

From this, you can see that the input/output behavior of the system is linear if the initial state is zero. Specifically, if

• input sequence x₁ produces output sequence y₁, and
• input sequence x₂ produces output sequence y₂

then for all u, w Î Reals,

• input sequence wx₁ + ux₂ produces output sequence wy₁ + uy₂

This property is called superposition.