EECS20N: Signals and Systems

State Response

Given input sequence x(0), x(1), x(2), …, the state response is

s(0) = initialState = s0 (say)

s(1) = as0 + bx(0)

s(2) = as(1) + bx(1) = a2 s0 + abx(0) + bx(1)

s(3) = as(2) + bx(2) = a3 s0 + a2bx(0) + abx(1) + bx(2)

…

So ∀ n 3 0

s(n) = an s0 + ∑ (m = 0 to n-1) a n- 1 - m bx(m) .

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

an s0

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

∑ (m = 0 to n) a n - 1 - m bx(m)