# Aidan's question via email about Fourier Transforms

• MHB
• Prove It
In summary, the inverse Fourier Transform of $\displaystyle \frac{1}{6 + 5\,\mathrm{i}\,\omega - \omega ^2}$ is $\displaystyle \mathrm{e}^{-2\,t}\,H\left( t \right) - \mathrm{e}^{-3\,t} \,H\left( t \right)$.
Prove It
Gold Member
MHB
Find the Inverse Fourier Transform of $\displaystyle \frac{1}{6 + 5\,\mathrm{i}\,\omega - \omega ^2}$.

In order to factorise this quadratic, we will need to recognise that $\displaystyle \mathrm{i}^2 = -1$, so we can rewrite this as

\displaystyle \begin{align*} \frac{1}{6 + 5\,\mathrm{i}\,\omega - \omega ^2 } &= \frac{1}{\mathrm{i}^2\,\omega ^2 + 5\,\mathrm{i}\,\omega + 6} \\ &= \frac{1}{ \left( \mathrm{i}\,\omega \right) ^2 + 5\,\mathrm{i}\,\omega + 6 } \\ &= \frac{1}{\left( \mathrm{i}\,\omega + 2 \right) \left( \mathrm{i}\,\omega + 3 \right) } \end{align*}

Now apply Partial Fractions:

\displaystyle \begin{align*} \frac{A}{\mathrm{i}\,\omega + 2 } + \frac{B}{\mathrm{i}\,\omega + 3 } &\equiv \frac{1}{\left( \mathrm{i}\,\omega + 2 \right) \left( \mathrm{i}\,\omega + 3 \right) } \\ A\left( \mathrm{i}\,\omega + 3 \right) + B \left( \mathrm{i}\,\omega + 2 \right) &\equiv 1 \end{align*}

Let $\displaystyle \mathrm{i}\,\omega = -2 \implies A = 1$.

Let $\displaystyle \mathrm{i}\,\omega = -3 \implies -B = 1 \implies B = -1$. Thus

\displaystyle \begin{align*} \mathcal{F}^{-1}\,\left\{ \frac{1}{6 + 5\,\mathrm{i}\,\omega - \omega ^2 } \right\} &= \mathcal{F}^{-1}\,\left\{ \frac{1}{2 + \mathrm{i}\,\omega} - \frac{1}{3 + \mathrm{i}\,\omega} \right\} \\ &= \mathrm{e}^{-2\,t}\,H\left( t \right) - \mathrm{e}^{-3\,t} \,H\left( t \right) \end{align*}

where $\displaystyle H\left( t \right) = \begin{cases} 1 & t \geq 0 \\ 0 & t < 0 \end{cases}$ is the Heaviside step function.

## What is a Fourier Transform?

A Fourier Transform is a mathematical tool used to decompose a complex signal into its individual frequency components. It essentially converts a signal from the time domain to the frequency domain.

## Why do we use Fourier Transforms?

Fourier Transforms are useful in many scientific and engineering fields, including signal processing, image processing, and quantum mechanics. They allow us to analyze and understand complex signals and systems by breaking them down into simpler components.

## How do you perform a Fourier Transform?

To perform a Fourier Transform, you need to have a mathematical function or signal and apply the Fourier Transform formula to it. This involves integrating the function over a specific range and using complex numbers to represent the amplitude and phase of each frequency component.

## What is the difference between a Fourier Transform and a Fourier Series?

A Fourier Transform is used for continuous signals, while a Fourier Series is used for periodic signals. Additionally, a Fourier Transform produces a continuous spectrum of frequencies, while a Fourier Series only produces discrete frequencies.

## What are some applications of Fourier Transforms?

Fourier Transforms have many applications, including audio and image compression, filtering and noise reduction, spectral analysis, and solving differential equations. They are also used in areas such as medical imaging, radar and sonar systems, and telecommunications.

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