Fourier Transform

Useful skill to build - or cheat sheet framework

Look at questions and decide what form of the Fourier transform you’d use! Exponential? Cosine form? Sine form? Think about what your reasoning is: even, odd, already has an exponential in it? etc.

A Fourier transform is where a non-periodic function is represented as the integral over trigonometric functions, usually in the form of complex exponentials.

$$\tilde{f}(\omega )=\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{\infty }f(t)e^{-i\omega t}dt,\text{and its inverse }f(t)=\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{\infty }\tilde{f}(\omega )e^{i\omega t}d\omega$$

Some books use different definitions for a Fourier transform - as long as they multiply to give the same pre-factor as Fourier’s inversion theorem, $\frac{1}{2\pi }$, then it’s fine.

Notation for Fourier transforms vary, too, writing the Fourier transform of $f(t)$ as $\tilde{f}(\omega )$ or $\mathcal{F}[f(t)]$.

You find the Fourier transform just by substituting your function into the formula and simplifying, often reducing the limits of integration by the definition of the function, too, and even using the exponential definitions of trigonometric functions. This then defines the level of frequency for the function across its values.

Sometimes finding inverse Fourier transforms can be simplified by using Euler’s formula to write it in terms of trigonometric functions to then analyse the parts of the function that are odd/even, as long as the integral is defined symmetrically.

Calculating Fourier transforms and then their inverses to find an equivalent integral definition of a function can be useful after equating them to find the solutions of very complicated integrals.

We can generalise the transforms for even and odd functions to the following:

• For an even function $f(t)=f(-t)$, the Fourier transform generalises to ${\tilde{f}}_c(\omega )=\sqrt{\frac{2}{\pi }}{\int }_0^{\infty }f(t)\cos \omega tdt$ and its inverse as $f(t)=\sqrt{\frac{2}{\pi }}{\int }_0^{\infty }{\tilde{f}}_c(\omega )\cos \omega td\omega$.
• For an odd function $-f(t)=f(-t)$, the Fourier transform generalises to ${\tilde{f}}_s(\omega )=\sqrt{\frac{2}{\pi }}{\int }_0^{\infty }f(t)\sin \omega tdt$ and its inverse as $f(t)=\sqrt{\frac{2}{\pi }}{\int }_0^{\infty }{\tilde{f}}_s(\omega )\sin \omega td\omega$.

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Example One

Calculate the Fourier transform of $f(t)=\{\begin{array}{ll}1& -1<t<1\\ 0& |t|\ge 1\end{array}$::$\sqrt{\frac{2}{\pi }}\frac{\sin (\omega )}{\omega }$

Example Two

Calculate the inverse Fourier transform of $\tilde{f}(w)=\sqrt{\frac{2}{\pi }}\frac{\sin (\omega )}{\omega }$::$\frac{2}{\pi }{\int }_0^{\infty }\frac{\sin (\omega )\cos (\omega t)}{\omega }d\omega$

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Properties of Fourier Transforms

Fourier transforms have four main properties: scaling, differentiation, translation, and exponential multiplication.

The scaling property of Fourier transforms is defined as::$\mathcal{F}[f(at)]=\frac{1}{|a|}\tilde{f}\left(\frac{\omega }{a}\right):a\ne 0$.

The differentiation property of Fourier transforms, often used for transforming differential equations into algebraic equations, is defined as:::$\mathcal{F}[f^{\prime }(t)]=i\omega \tilde{f}(\omega )$.

The translation property of Fourier transforms is defined as::$\mathcal{F}[f(t+a)]=e^{ia\omega }\tilde{f}(\omega )$.

The exponential multiplication properties of Fourier transforms is defined as::$\mathcal{F}[e^{\alpha t}f(t)]=\tilde{f}(\omega +i\alpha )$.