Laplace transforms are mathematical operations that transform functions of a single real variable into functions of a complex variable. They are extremely useful in the study of linear systems for many reasons. The most important reason is that by using Laplace transforms one can tranform a linear differential equation into an algebraic equation. Another important reason for Laplace transforms is that they supply a different representation for a linear system. Such a representaion will be the basis for the classical control techniques. Given a function of time, x(t) for t>0, we define the Laplace transform of x(t) as follows:
A natural question to ask is: what are the conditions that the signal x(t) must satisfy in order for the Laplace transform to exist? The Laplace transform exists for signals for which the above transformation integral converges. However, the following fact supplies an easy way to check sufficient condition for the existence of a Laplace transform.
Fact Suppose that
- x(t) is piecewise continuous
- |x(t)|<Kexp(at) when t>0 for some positive real numbers K and a
then the Laplace transform L(x(t)) = X(s) exists for Re(s)>a. Note that Re(s) is the real part of s.
Since any physically realizable signal is continuous and will not blow up so fast that no exponential can bound it, the previous fact assures us that the Laplace transform can be defined for any physically realizable signal. Let us now consider a few examples of how to calculate Laplace transforms. Note: For this course, we will use the letter 'j' to represent the complex constant sqaure root of -1.
Example x(t)=Step Function
Example Exponential Function
We will now start studying the properties of the Laplace transform.
Linearity of Laplace Transforms: If 
then for any constants 
.
This property can be derived in the following way:
Laplace Transform of a Derivative: If 
then
This property can be derived in the following way:
This property shows that the process of differentiation in the time-domain corresponds to multiplication by s in the Laplace domain plus the addition of the constant -x(0).
Laplace Transforms of Higher Derivatives: If 
then
This property can be derived in the following way:
In general we have
Laplace Transform of a Signal Multiplied by an Exponential: If 
then
.
This property can be derived as follows.
This property shows that multiplication by an exponential in time corresponds to shifting in s.
Example: Find the Laplace transform for the following functions
We can find the Laplace transform for t and sin2t using the Laplace transform table. In part (a) we need to shift s by 3 whereas we shift s by -1 in part (b). Therefore the Laplace transforms of the above two functions are given as
