Part 6 - Fourier Transform

Introduction

In this part we’ll define and understand Fourier transforms. In the last part we covered the concept of Fourier series.

Which was a way of describing periodic functions as a linear combination (sum) of simple trigonometric functions. The Fourier transform takes this one step further, it’s a way to extend the Fourier series of non-periodic functions.

Definition

Given a function in the time-domain, the equivalent function in the frequency domain is: $$ F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-j\omega t}\ dt $$

The other way around: $$ f(t) = \dfrac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{j\omega t}\ d\omega $$

We call these functions the Fourier pair. Meaning they are the same functions, in different domains. They represent the same signal.

$$ f(t) \iff F(\omega) $$

So the notation for the Fourier transform is: $$ F(\omega) = \mathcal{F}[f(t)] $$

$$ f(t) = \mathcal{F}^{-1}[F(\omega)] $$

Let’s do our first Fourier transform!

Example

Let’s do this on a simple unit gate function, let’s do it on a general unit gate function.

$$ f(t) = rect\left(\dfrac{t}{\tau}\right) $$

$$ F(\omega) = \int_{-\infty}^{\infty} rect\left(\dfrac{t}{\tau}\right) e^{-j\omega t}\ dt $$

Since the gate function is only defined at $t = -\dfrac{\tau}{2}$ and $t = \dfrac{\tau}{2}$, and in this interval it is constant 1. $$ F(\omega) = \int_{-\frac{\tau}{2}}^{\frac{\tau}{2}} e^{-j\omega t}\ dt $$

$$ F(\omega) = -\dfrac{1}{j\omega} e^{-j\omega t} \bigg\rvert_{\frac{\tau}{2}}^{-\frac{\tau}{2}} $$

$$ F(\omega) = -\dfrac{1}{j\omega} \left(e^{-j\omega \dfrac{\tau}{2}} - e^{j\omega t \dfrac{\tau}{2}}\right) $$

Using Euler’s formula: $$ F(\omega) = \dfrac{2 sin\left(\dfrac{\omega \tau}{2}\right)}{\omega} $$

Let’s rewrite it as: $$ F(\omega) = \tau \dfrac{sin\left(\dfrac{\omega \tau}{2}\right)}{\dfrac{\omega \tau}{2}} $$

We can rewrite using $sinc(x) = \dfrac{sin(\pi x)}{\pi x}$: $$ F(\omega) = \tau sinc\left(\dfrac{\omega \tau}{2}\right) $$

So, this means that: $$ rect\left(\dfrac{t}{\tau}\right) \iff \tau sinc\left(\dfrac{\omega \tau}{2}\right) $$

Now, we can do this manually each time, but that’s tedious, that’s why we’ll usually use a Fourier table.

Let’s cover some important Fourier transforms.

Important Fourier transforms

$$ cos(\omega_0 t) \iff \pi [\delta(\omega + \omega_0) + \delta(\omega - \omega_0)] $$

$$ sin(\omega_0 t) \iff j\pi [\delta(\omega + \omega_0) + \delta(\omega - \omega_0)] $$

We’ll use these often.

Fourier transform properties

Symmetry

After we have obtained one representation of the signal, it doesn’t stop us from just replacing $t$ or $\omega$ with the other.

For example: $$ rect\left(\dfrac{t}{\tau}\right) \iff \tau sinc\left(\dfrac{\omega \tau}{2}\right) $$

We can do: $$ F(t) = \tau sinc\left(\dfrac{t \tau}{2}\right) $$

We can utilize: $$ F(t) \iff 2\pi f(-\omega) $$

Now this means: $$ F(t) \iff 2\pi f(-\omega) = 2\pi \ rect\left(\dfrac{-\omega}{\tau}\right) $$

Since the unit gate function is an even function: $$ F(t) \iff 2\pi f(-\omega) = 2\pi \ rect\left(\dfrac{\omega}{\tau}\right) $$

Scaling

$$ f(at) \iff \dfrac{1}{|a|} F\left(\dfrac{\omega}{a}\right) $$

As we can see, an expansion in the time domain, means a compression in the frequency domain and vice-versa.

From this we can also gain the time and frequency inversion property: $$ f(-t) \iff F(-\omega) $$

Time and frequency shifting

$$ f(t - t_0) \iff F(\omega) e^{-j\omega t_0} $$

This also means: $$ f(t) e^{j\omega_0 t} \iff F(\omega - \omega_0) $$

Time differentiation and integration

$$ \dfrac{df}{dt} \iff j\omega F(\omega) $$

$$ \int_{-\infty}^{t} f(\tau)\ d\tau \iff \dfrac{F(\omega)}{j\omega} + \pi F(0)\delta(\omega) $$

Now, for the most powerful and beautiful property, which we’ll use.

Convolution

Given $f_1(t) \iff F_1(\omega)$ and $f_2(t) \iff F_2(\omega)$.

Time convolution: $$ f_1(t) * f_2(t) \iff F_1(\omega) F_2(\omega) $$

Frequency convolution $$ f_1(t) f_2(t) \iff \dfrac{1}{2\pi} F_1(\omega) * F_2(\omega) $$

Does the Fourier transform always exist?

The Fourier transform exist, if: $$ \int_{-\infty}^{\infty} |f(t)|\ dt < \infty $$