Properties of the Fourier Transforms

Abstract

In this chapter we cover various properties of the Fourier transform. In addition to getting a deeper understanding of the machinery of the Fourier transform, by understanding the properties of the Fourier transform we are better fit to deal with new problems and/or deal with older ones more efficiently and faster. The studied properties include linearity, time scaling, reciprocity, time shifting, frequency shifting, time differentiation, time integration, frequency differentiation, frequency integration, time convolution, and frequency convolution. For each of the properties we do a thorough derivation accompanied with one or more examples demonstrating an actual application of the property. We also demonstrate the property flow visually and simultaneously on a time-domain and frequency-domain plot. We single out the convolution property due to its importance and in anticipation of further use throughout the text.