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# 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.