Relatives of Fourier Transforms

There are various ways of introducing a student to different forms of transforms. We chose the approximation of a function by using Fourier series first and then came up with the Fourier transforms. Fourier cosine and sine series were considered using the Fourier series. The next step is to study some of the other transforms that are related to the Fourier transforms. These include cosine, sine, Laplace, discrete, and fast Fourier transforms. Discrete and fast Fourier transforms will be included in Chapters 8 and 9. In many of the undergraduate engineering curricula, Laplace transforms are introduced first and then the Fourier transforms. Fourier transforms are considered more theoretical. The development of the sine and cosine transforms parallel to the Fourier cosine and sine series discussed in Chapter 3. For a good review on many of these topics, see the handbook by Poularikas (2000). We can consider the Laplace transform as an independent transform or a modified version of the Fourier transform. One problem with Fourier transforms is that the signal under consideration must be absolutely integrable. (the periodic functions are exceptions). Therefore, the transformation to the Fourier domain is limited to energy signals or to finite power signals that are convergent in the limit. Fourier and Laplace transforms have been widely used in engineering; Fourier transforms in the signal and communications area and the Laplace transform in the circuits, systems, and control area. Neither one is a generalization of the other. Both transforms have their own merits.