Fourier Series Representation of Continuous Time Periodic Signals

Summary

This chapter introduces the Fourier series representation of continuous time periodic functions. A periodic function can be represented as a vector in an infinite‐dimensional space. Harmonically related complex exponential functions form the basis of this space. They are periodic functions with angular frequencies. They are orthogonal to each other. Thus, they span an infinite‐dimensional function space, called Hilbert space, where each periodic function is uniquely represented by a set of coordinates, called the spectral coefficients, provided that the function satisfies the Dirichlet conditions. The representation of a signal in Hilbert space is called the Fourier series, named after J. B. Fourier, a French mathematician. Fourier series representation enables us to decompose a periodic signal into its harmonically related frequency components. Since a periodic function is represented in terms of its frequency content, we refer to this specific Hilbert space the frequency domain.