Fourier transforms

Fourier transforms are a powerful approach to gaining insights into some physics problems such as solving differential equations and expanding solutions into their fundamental modes in field theory, to name only two examples. In terms of data analysis, spectral analysis and signal processing, which are based on the Fourier transform, are two common techniques used in several fields of physics. Fourier transforms thus form the topic of this chapter. We learn the concepts by first developing the discrete Fourier transform. Python library functions based on the fast-Fourier transform are used throughout the chapter. Some numerical considerations such as aliasing and spectral leakage, are discussed. Finally, differentiation using Fourier transforms is recalled which can sometimes be a more accurate approach than finite differences for representing differentiation numerically.