# SISO Systems** **

For a** single-input, single-output system**, *M *= 1, *K* = 1,
so *Inputs *= *Outputs *= *Reals*. (The state, however, is *N*-dimensional.)
So

- the
*N*×*B*can be written as the column vector*b*= [*b*_{1},**…**,*b*]_{N }^{T}, - the 1
*N*matrix*C*can be written as a 1 ×*N*row vector*c*^{T}= [*c*_{1 },**…**,*c*], and_{N } - the 1 × 1 matrix
*D*can be written as a real number*d*.

With this notation *s*, *b*, *c *are all *N *×* *1 column vectors.

The response of the system to an input sequence *x*(0), *x*(1), *x*(2),
**…** is

s(n) =A^{n }s_{0 }+ ∑_{(m = 0 to n-1)}_{ }A^{n-}^{1-m}b x(m)

y(n) =c^{T}A^{n }s_{0 }+ ∑_{(m = 0 to n-1)}_{ }c^{T}A^{n-}^{1-m}b x(m) +dx(n).