Fourier Transform

Abstract

The Fourier transform is the evolution of the complex Fourier series with the adaptation to deal with aperiodic signals. This is accomplished by stretching the period to infinity and replacing the discrete Fourier series spectrum with that of a continuous one. Now we are able to deal with a whole new class of functions, including—but not limited to—the single pulse function, the unit step, the negative exponentials, the continuous sine/cosine functions, as well as the single-sided (causal) sine/cosine functions. For the first time we pick up the delta function δ(ω) as a Fourier transform of various singular functions, such as the DC, unit step, and signum functions. Also the shifted delta function is used to represent the spectrum of both the continuous and the causal sine/cosine functions. We make extensive use of spectrum plots (both real and imaginary) and do actual frequency integration to reproduce the time signal. We are to do so by limiting the frequency integration over that range where the spectrum is substantial, ensuring to always include both real and imaginary components, as well as negative and positive frequencies.