Fourier Transforms

A.3 Fourier Transforms

This appendix explores the mathematics of the Fourier transform, some of the results of which are employed throughout the course in the derivation of various equations and explanation of physical results. Therefore, a full grasp of what a Fourier transform is, whilst not essential, may aid with the comprehension of the course material.

A.3.1 Representing a function as a series

A.3.1.1 Taylor series

A.3.1.2 Periodic functions and Fourier Series

Figure A.1: Periodic function f(x)

Figure A.2: Periodic function f(x) fitted with sin(kx)

Figure A.3: Three sin functions sin(kx), sin(2kx) and sin(3kx)

This expansion is known as a Fourier series and is commonly used break down components of periodic functions. A example of including more terms in the expansion is shown in Fig. A.4.

Figure A.4: Fourier expansion up to the term (a) sin(kx) (b) sin(3kx), (c) sin(6kx) and (d) sin(9kx)

This expansion can also be used for non-periodic functions if they are evaluated over a limited region. If we are interested in a function between bounds x = 0 and x = L we can use a Fourier expansion for the region 0 < x < L if we assume the function is ‘pseudo-periodic’ over this region, i.e. f(x) = f(x + L). Note that one key difference between Taylor and Fourier expansions is that Taylor expansions are carried out around a given point x₀. Hence, Taylor expansions give a good estimate around this specific location, and deviate more as we move further from x₀. By contrast a Fourier series should given an approximation that is equally good across the x range.

A.3.1.3 Determining Fourier coefficients

This expansion can also be used for non-periodic functions if they are evaluated over a limited region. If we are interested in a function between bounds x = 0 and x = L we can use a Fourier expansion for the region 0 < x < L if we assume the function is ‘pseudo-periodic’ over this region, i.e. f(x) = f(x + L). Note that one key difference between Taylor and Fourier expansions is that Taylor expansions are carried out around a given point x₀. Hence, Taylor expansions give a good estimate around this specific location, and deviate more as we move further from x₀. By contrast a Fourier series should given an approximation that is equally good across the x range.