EECS20N: Signals and Systems

Evaluation

Given input sequence x(0), x(1), x(2), …, the state response and the output sequence are determined by

s(0) = initialState = s0 , say and for n ≥ 0

s(n + 1) = As(n) + Bx(n),

y(n) = Cs (n) + Dx(n).

In this notation

  • s is a N × 1 column vector,
  • x is a M × 1 column vector,
  • y is a K × 1 column vector, and
  • A, B, C, and D are all matrices with dimensions that make the equations valid.

Recursive solution of these equations leads to

s(n) = Ans0 + ∑ (m = 0 to n-1) An-1-m Bx(m)

y(n) = CAn s0 + ∑ (m = 0 to n-1) C An-1-m Bx(m) + Dx(n)

The first term is the zero-input response and the second-term is the zero-state response.