Comparing the Fourier and Laplace Transform

It is worthwhile to take some moments to compare the Laplace and Fourier transforms. The following similarities and differences should be noted:

• The Laplace transform defined in Chapter 21 ( Frequency Response) is one-sided in that the integral is over $ 0 < t < \infty $, making it only useful for positive-time functions, $ f(t), t > 0 $. The Fourier transform is applicable to functions defined for all time.
• For a function $ f(t) $ that is nonzero for positive time only (i.e., $ f(t)=0, \quad t < 0) $ and $ \int_{0}^{\infty}|f(t)| d t < \infty $, the two transforms are related by $$F(\omega)=\left.F(s)\right|_{s=j \omega} \tag{1}$$ This equation also shows that the Fourier transform can be regarded as a special case of the Laplace transform with $ s=j \omega $. Recall that $ s=\sigma+j \omega $. Therefore, Eq. (1) shows that the Laplace transform is related to the entire $ s $ plane, whereas the Fourier transform is restricted to the $ j \omega $ axis.
• The Laplace transform is applicable to a wider range of functions than the Fourier transform. For example, the function $ t u(t) $ has a Laplace transform but no Fourier transform. But Fourier transforms exist for signals that are not physically realizable and have no Laplace transforms.
• The Laplace transform is better suited for the analysis of transient problems involving initial conditions, since it permits the inclusion of the initial conditions, whereas the Fourier transform does not. The Fourier transform is especially useful for problems in the steady state.
• The Fourier transform provides greater insight into the frequency characteristics of signals than does the Laplace transform.