The Fourier Transform

(an interactive introduction)

What's a Fourier Transform?

The Fourier Transform is one of the most important mathematical concepts in signal processing.

In 1822, Joseph Fourier asserted that certain functions could be represented as infinite sums of other functions, which simplifies analysis considerably; this concept has been refined over decades and is now at the heart of what drives much of modern audio signal processing, image processing, video processing, wireless communication, control systems, financial signal processing, and dozens of other very important things.

Fourier series

A natural place to start thinking about the decomposition of functions into other functions is the Fourier series: a decomposition of a periodic function into a sum of sines and cosines.

Consider a square wave:

A square wave is a discontinuous function, but with a Fourier Series, we can approximate it by a set of continuous sinusoids, in a summation of the form:

$\frac{4}{\pi}\sum_{n=1, 3, 5, \dots}^{\infty} \frac{1}{n}\sin \Big( \frac{n \pi x}{L} \Big)$

Oberserve that, while any sinusoid bears little resemblance to a square wave in itself, when multiple sinusoids are added in linear combination we converge quickly to something that very nearly approximates our signal:

The square wave is a simple example, but Fourier series can also be constructed to approximate any periodic function: triangle waves, sawtooth waves, and any other sinusoud are all fair game. As the number of terms in our approximation goes to infinity, our approximation gets closer and closer to our function.

Visualizing the Frequency Domain

To proceed further in our understanding of the Fourier transform and the Fourier series, it's useful to visualize our functions in the frequency domain, instead of the time domain. In the frequency domain, we plot the frequencies of the sinusoidal components of the function, instead of time. The y-axis can represent phase shift or amplitude. For our purposes, let's make a plot of the frequency-domain representation of the square wave Fourier series that we just saw in the last section:

Each component at frequency n is represented by an impulse at n on the x-axis and amplitude equal to 1/n, its coefficient in the linear combination that makes up our signal.

Experiment with how the amplitudes of different frequency components affect the shape of our periodic pattern! In the widget below, drag the impulses up and down and observe the changes in the periodic signal.

A quick note: in many of the formulae relevant to signal processing, it is convenient to employ another way of conceptualizing sinusoids--not as simply sines and cosines, but as complex exponentials. The relationships between trigonometric functions and the complex exponential function is established by Euler's Formula:

$e^{ix} = \cos x + i \sin x$

In particular, the formula for the sine and cosine functions are

$\cos x = \frac{1}{2}(e^{ix} + e^{-ix})$

$\sin x = \frac{1}{2i}(e^{ix} - e^{-ix})$

Using these, our Fourier series approximation of our signal be compactly represented by the expression

$\sum_{k=-\infty}^{\infty}c_ke^{ikw_0t}$

A (countably) infinite sum of complex exponentials and their respective coefficients at integer multiples of some frequency $\omega_0$.

To Finity and Beyond

This is great--we've shown that we can represent arbitrary periodic signals with sums of sines and cosines. But we needn't stop there; Fourier analysis can be used on finite signals, as well.

You can't approximate arbitrary finite signals with finite sums of sinusoids. However, consider what would happen if we didn't limit ourselves to finite sums of sinusoids. A Fourier series consists of impulses in the frequency domain, but what happens with signals in the frequency space that are continuous, that can have any value of frequency?

The Fourier transform is just such a function; it can be thought of as a continuous analogue to the Fourier series, where instead of a sum of countably many complex exponentials, we have an integral over every possible angular frequency. Our infinite summation from our Fourier series representation of our signal now becomes an integral:

$$ \int _{-\infty}^{\infty} R(\omega)e^{i \omega t} d \omega$$

Where $R(\omega )$ is a continuous function in the frequency domain called the Fourier transform.

Some Fourier transforms of simple functions are shown below.