The Fourier Integral-A Basic Introduction

The object of this introduction is to give the reader an idea of the various kinds of problems in which the Fourier series or integral is useful, and to discuss not only the essential properties which make it useful, but also to point out the limitations and restrictions imposed by these properties. Collaterally it is significant to recognize the flexibility that is inherent in the Fourier representation, if we consider its mathematical form from a nonconventional point of view. Thus it is a common belief, that only the integral representation applies to aperiodic functions, and that the Fourier series is restricted to the representation of periodic time functions. If we are interested in sampled time functions, this attitude turns out to be incorrect, and we find that the series rather than the integral is at once the pertinent tool. Inherent properties of the partial sums, such as the Gibbs phenomenon and the mean-square-error criterion are discussed, as well as the fact that in dealing with stable systems for which the response must be zero before the advent of the excitation, it turns out that the real or imaginary parts of the system function are separately sufficient to characterize the response: and one may dispense with the need of having to deal with complex system functions, or to distinguish between such things as minimum-phase and nonminimum-phase transfer functions.