← Lecture 19 | Lecture 21 →

# The Transfer Function and the Impulse Response

We will now look at an important means of understanding the behaviour
of circuits and other linear systems. Previously, we solved differential
equations by finding first the general solution to the homogeneous equation,
then finding a particular solution, and then matching the solution to the
initial conditions. Unfortunately, the particular solution is not unique
and therefore nothing physical can be interpreted from the particular
solution or its effect on the coefficients of the general homogeneous solution.

Instead, we will see that the Laplace transform allows us to divide the
solution into two components:

- The response of the system due to the initial conditions, and
- The response of the system due to the input (or forcing function).

Consider the differential equation for the circuit shown in Figure 1.

Figure 1. An RC circuit.

If the charge on the capacitor (the output or response) is given by
then we may relate the forcing function (or input)
with the response through
. Assume, also, that
the charge on the capacitor at time
is
. Let us define
and therefore the differential
equation simplifies to
.
Using the Laplace transform, we get
that the solution is

.

We may solve this for to get

.

Notice that the first time depends only on the forcing function (or input) and
the second term depends only on the initial conditions. Using the inverse Laplace
transform and the convolution theorem, we get that

where

- is the response to
the input, and
- is the response to
the initial conditions.

In a stable system (necessary), the initial conditions are transient, that is, they decay to zero, and
therefore the focus is on the response to the input. In this example, the input response in the
frequency domain is the Laplace transform of the input () multiplied
by the factor . This last term is called the **transfer function**
and is written as

.

Notice that the transfer function is defined as and
may be thought of as the ratio between Laplace transforms of the the output and the input under
the assumption that the initial conditions are zero, that is, the system is initially quiescent.

Once we have the transfer function, it becomes straight-forward to find the response to a system:

.

# Example 1

Find the transfer function of a system where the output/response
to an input is defined by the differential equation

.

The Laplace transform gives us the expression

Therefore, the transfer function, the ratio between the Laplace transforms of the output and the input, is

.

Notice that for a linear system with constant coefficients, the transfer function is
a *rational polynomial*, that is, it is the ratio of two polynomials.

The degree of the denominator is the order of the system, in this case, two.

The zeros of the numerator are the *zeros of the transfer function*, while the
zeros of the denominator are the *roots of the transfer function*. In this case:

- The characteristic equation of the transfer function is
,
- The poles of the transfer function are -1 and -3, and
- The zero of the transfer function is -2.

The zeros and the poles of the transfer function determine the stability of the system and this will
be a significant focus later in this class and in your signals and systems course.

Suppose now we have a forcing function which has
, that is, . In this case,
the response of the system is the transfer function:

In this case, if we denote the inverse Laplace transform of the transfer function to be
, that is, , it follows that

.

Thus, the transfer function is the response of the system to the unit impulse forcing function
and this *impulse response* is usually denoted
.

Put another way, the impulse response and the transfer function both contain all information necessary about the
dynamics of a time-invariant system.

Conclusion: to determine the response of any linear time-invariant system, all it is necessary to do is to start
in a quiescent state, excite it with an impulse, and measure the response.

Note: the adjective *time-invariant* simply indicates that it does not matter when you choose time
; that is, the the system behaves the same way yesterday, today, and tomorrow.

# Example 2

Find the impulse response of the system described by the differential equation

.

The transfer function is

and therefore, we can find the inverse Laplace transform by using partial fraction decomposition

and hence

# Two Important Transfer Functions

Consider the system

.

That is, the response is the derivative of the input function. Taking the
Laplace transform
and therefore the transfer function, the ratio
of the Laplace transforms of the output over the input, is

.

Consider the system

.

That is, the response is the integral of the input function. Taking the
Laplace transform
and therefore the transfer function, the ratio
of the Laplace transforms of the output over the input, is

.

In the next section, we will see how engineers will often refer to an integrator
or a differentiator of a signal by their
transfer function and
, respectively.