The Fourier series handles steady-state sinusoidal functions, but does not have the ability to model transients effectively. As we have seen, the Fourier transform provides us with the capability to model signals of finite duration, but (due to aliasing and other factors) the FFT does not converge for some functions. The Laplace transform, encompasses the the entire complex plane and (in some sense) is a superset of the Fourier transform. We will look at a few applications for the Laplace transform and related functions. To obtain the Laplace transform of a function of time f(t), we multiply f(t) by e -st and then integrate with respect to time from zero to infinity The Laplace Transform This is the one-sided Laplace transform. To use the one-sided Laplace transform we must assume that f(t)=0 for t<0. The inverse Laplace transform is given by,

The quantity s in the Laplace transform is a comlex number and the root of the characteristic equation s=  i  is imaginary and is called he radian or angular frequency. In time domain functions  is expressed as sin  t or as cos  t. The  is called the neper and represents a nondimensional logarithmic unit. They are combined as shown to produce the complex frequency. The s-plane is analogous to the complex plane with i  as the imaginary axis and  as the real axis. If  =0 in s=  i , then s is entirely dependent upon the value of  (the exponential). If  >0 then the exponential function will be increasing, and if  <0 it will be decreasing. s-plane If s=0  i ,e i  t is just a rotating unit vector. A counterclockwise (+) direction corresponds to cosine terms while a clockwise direction (-) corresponds to sine terms. In this form the Laplace transform is equivalent to the Fourier transform.

Laplace Transform Pairs While their derivations are beyond the scope of this course, it is important to be aware of the existence of Laplace transform pairs for many common time-domain functions. f(t) F(s)