The impulse response of a system is an important response. The impulse response is the response to a unit impulse.

**Impulse Response of A First Order
System**

o
The impulse response of a system is an important
response. The impulse response is the response to a unit impulse.

o
The unit impulse has a Laplace transform of unity
(1).That gives the unit impulse a unique stature. If a system has a unit
impulse input, the output transform is G(s), where G(s) is the transfer
function of the system. The unit impulse response is therefore the inverse
transform of G(s), i.e. g(t), the time function you get by inverse transforming
G(s). If you haven't begun to study Laplace transforms yet, you can just file
these last statements away until you begin to learn about Laplace transforms.
Still there is an important fact buried in all of this.

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Knowing that the impulse response is the inverse
transform of the transfer function of a system can be useful in identifying
systems (getting system parameters from measured responses).

In this
section we will examine the shapes/forms of several impulse responses. We will
start with simple first order systems, and give you links to modules that
discuss other, higher order responses.

A general
first order system satisfies a differential equation with this general form

If the
input, u(t), is a unit impulse, then for a short instant around t = 0 the input
is infinite. Let us assume that the state, x(t), is initially zero, i.e. x(0) =
0. We will integrate both sides of the differential equation from a small time,
, before t = 0, to a small time, after t = 0. We are just taking advantage of
one of the properties of the unit impulse.

The right
hand side of the equation is just G**dc**
since the impulse is assumed to be a unit impulse - one with unit area. Thus,
we have:

We can
also note that x(0) = 0, so the second integral on the right hand side is zero.
In other words, what the impulse does is it produces a calculable change in the
state, x(t), and this change occurs in a negligibly short time (the duration of
the impulse) after t = 0 That leads us to a simple strategy for getting the
impulse response. Calculate the new initial condition after the impulse passes.
Solve the differential equation - with zero input - starting from the newly
calculated initial condition.

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