Transforms

This chapter discusses four transforms: the Fourier transform, the Laplace transform, the integral transform, and the Z-transform. The Fourier series is the sum of trigonometric or periodic functions. The Fourier series is used to solve various ordinary and partial differential equations. The Parseval's identity, essentially, is a result on the summability of the Fourier series of a function. Generally speaking, Fourier transform is a tool used in physical concepts such as frequency or energy. The chapter also discusses the discrete Fourier transform (DFT). In practice, to calculate the Fourier transform, such as in signal processing, a DFT algorithm, called fast Fourier transform or FFT, is used. The Laplace transform is used for the analysis of continuous-time systems. Fourier and Laplace transforms are examples of a broader class of transforms referred to as integral transform. The Z-transform has applications not only in mathematics but also in signal processing.