EECS20N: Signals and Systems

Output Response

∀ n 3 0

y(n) = can s0 + ∑(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)

∑(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 ∈ Reals,

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

This property is called superposition.