where *h*(*t*) is an impulse response, is called the system function or transfer function and it completely characterizes the input/output relationship of an LTI
system. We can use it to determine time responses of LTI systems.

We can use Laplace Transforms to solve differential equations for systems (assuming the system is initially at rest for one-sided systems) of the form:

Taking the Laplace Transform of both sides of this equation and using the Differentiation Property, we get:

From this, we can define the transfer function *H*(*s*) as

We write its Partial Fraction Expansion as:

where

is the *residue* of the pole at *p _{j}*.

Thus

because the Inverse Laplace Transform of

is

__Example 1__ Find *y*(*t*) where the transfer function *H*(*s*) and the input *x*(*t*) are given. Use Partial Fraction Expansion to find the output *y*(*t*):

__Example 2__ Find the transfer function *H*(*s*) for the
differential equation. Assume zero initial conditions.

*y*'(*t*) + 2*y*(*t*) = 3*x*'(*t*).

__Example 3__ Now let the input to the system be *x*(*t*) =
5*u*(*t*). Find *y*(*t*).

As we saw for the Fourier Transform

*x*(*t*)**h*(*t*) ↔ *X*(*s*)*H*(*s*)

This is useful for studying LTI systems. In fact, we can completely characterize an LTI system from:

- The system differential equation
- or the system transfer function
*H*(*s*) - or the system impulse response
*h*(*t*).

__Example 4__ Find the step response *s*(*t*) to

*h*(*t*) = *e ^{-t}u*(

Hint: |

__Example 5__ Find the output of an LTI system with impulse response *h*(*t*)
= *e*^{bt}*u*(*t*) to an input *x*(*t*) =
*e*^{at}*u*(*t*), where a ≠ b.

We saw that a condition for bounded-input bounded-output stability was:

Let's look at stability from a system function standpoint. Given a Laplace
Transform *H*(*s*), we expand
*H*(*s*) with Partial Fraction Expansion:

The corresponding impulse response is:

What happens to *h*(*t*) as *t* → ∞? For a system
to be stable, its impulse response
must not blow up as *t* → ∞.

If Re{p},_{i} |
, then h(t) decays to 0 as t → ∞ and
the system is stable |

Therefore, the system is BIBO stable
if and only if all poles of *H*(*s*)
are in the left half plane of the
*s*-plane.

If you study CONTROL THEORY, you will learn
more about this. Using * feedback *, you can
build systems to steer the poles into the left half plane and thus stabilize the system. Here is an example of such a system.

__Example 6__ FEEDBACK

- You are given a system with impulse response
*h*(*t*) =*e*(^{t}u*t*)

Is the system stable?

- You now hook up the system up
into a "Feedback" system as shown. Find the new
impulse response or transfer function. Find the range on the parameter
A such that the system is stable.

You can find the inverse of a system using Laplace Transforms. This is because:

Take the Laplace Transform of both sides of this equation:

Therefore, the Laplace Transform of the inverse system is simply

__Example 7__ Find the inverse
system of