Fourier Transform of Periodic Signals

Abstract

While the Fourier transform was originally incepted as a way to deal with aperiodic signals, we can still use it to analyze period ones! The tackle method is twofold. First we know that any periodic signals can be decomposed in terms of a Fourier series which is a summation of complex exponentials. Second we know that each exponential has the Fourier transform as a delta function in the frequency domain. By simply using superposition, then we would expect that the Fourier transform of the periodic signal to be comprised of a sequence of delta functions in the frequency domain, uniformly spaced by the fundamental frequency of the periodic signal in the time domain. We apply this on a few examples, including the periodic pulse, periodic hat function, periodic parabola function, and finally the periodic impulse function. We also derive the relation between the Fourier transform of the aperiodic version and that of periodic on. In essence in going from an aperiodic version to one that is periodic, the frequency spectrum collapses from a continuous one to a discrete one, while still maintaining the overall shape. Similar to before strong emphasis is put on graphics and visual to illustrate the essence of both aperiodic and periodic spectrums.