4.1 THE LAPLACE TRANSFORM

For a signal x( t), its Laplace transform X( s) is defined by

The signal x( t) is said to be the inverse Laplace transform of X( s). It can be shown that ^[1]

where c is a constant chosen to ensure the convergence of the integral in Eq. (4.1), as explained later.

This pair of equations is known as the bilateral Laplace transform pair, where X( s) is the direct Laplace transform of x( t) and x( t) is the inverse Laplace transform of X( s). Symbolically,

Note that

It is also common practice to use a bidirectional arrow to indicate a Laplace transform pair, as follows:

The Laplace transform, defined in this way, can handle signals existing over the entire time interval from ? ? to ? (causal and noncausal signals). For this reason it is called the bilateral (or two-sided) Laplace transform. Later we shall consider a special case-the unilateral or one-sided Laplace transform-which can handle only causal signals.

LINEARITY OF THE LAPLACE TRANSFORM

We now prove that the Laplace transform is a linear operator by showing that the principle of superposition holds, implying that if

then

The proof is simple. By definition

This result can be extended to any finite sum.