Fourier Analysis

The Fourier analysis includes both the Fourier transform (or Fourier integral) and the Fourier series. The Fourier transform is applicable to functions that are defined on the real line, while the Fourier series is used to analyze functions that are periodic. Fourier series and Fourier transform are often separately treated by mathematicians since they involve two different classes of functions. This chapter focuses only on the properties of Fourier analysis that are relevant to wavelet analysis. It discusses the Poisson’s sum whose derivation is made much simpler by using some properties of the Fourier transform. The partial sum of a Fourier series is a least square approximation to the original periodic function. The sampling theorem is fundamentally important to digital signal analysis. Fourier series and discrete-time Fourier transform are directly computable. Controlled Vocabulary Terms digital signal processing; discrete Fourier transforms; discrete time systems; Poisson equation