These systems are descibed by difference equations, i.e.,
General formula:
If 
is constant, i.e. 
, then 
, so the solution formula is
The Laplace Transform technique and transfer function assume 
does not
grow faster than an exponential function, i.e.,
The Laplace Transform is given by
The integral converges for 
whose real part 
i.e. 
.
We use the Laplace Transform to find the solution of the state equation,
Then if
If we have the general linear system
we can use the Laplace Transform to solve the system, so
then
For SISO (single input, single output) systems, we can divide
by 
, so
The last expression is called the transfer function of a linear system,
usually denoted by 
. Since
where adj
is the matrix of cofactors. Then
is a polynomial of degree 
, and
is a matrix of polynomials of
degree 
. So
Then 
is a matrix of polynomials of the form
Example 
.
Let us compare this result with the one obtained by the time-domain
calculations. If 
if 
, with 
if 
,
then
As before, let 
, then
Using the Laplace Transform tables for each of the three terms above, we obtain
which coincides with the time-domain result.
