Why does fourier transform work

The Fourier Transform equation is essentially a measurement of the energy i. In practice, we can use this notion to sweep over a range of frequencies, and quantify how dominant each particular frequency is within the original signal.

Thus, based on the measure, we can actually identify which frequencies of oscillations were used to construct our original signal! While this might not make sense yet, if you bear with me to the end of the post, it will hopefully become more clear.

The first step to attaining the measure of energy at a particular frequency will be to redraw our signal from a linear framework into a circular one. As a starting point, you need to just trust me that this expression forms a circle with radius 1 when plotted in the complex plane — where the x-axis is real numbers and the y-axis is imaginary.

The underlying math, for why this works, is somewhat beyond this post, but if you are interested here is a great video explanation. Below we see the final result of plotting the expression in the complex plane.

As we increase t , the original signal forms a circle rotating in the clockwise direction if we removed the negative sign it would move counter clockwise. The value of f will be used to control the amount of time it takes to complete a rotation around the circle. We would thus say 0. This can be illustrated by the grey dotted lines appearing every full rotation. Thus by scaling our frequency term up and down, we can speed up and slow down the amount of time it takes to travel around the circle.

Rather than focusing on the cosine expression, think of the value of the function g t as the height of the orange bar in the animation.

By introducing this new term, the rotating complex number is scaled up and down according to the value of g t. When g t is large, the rotating complex number is far from the origin. When g t is small, the rotating complex number is close to the origin. Essentially, you can think of the rotating vector, with its changing length, as drawing the function wound up to a particular frequency the cycling frequency f. While this not only makes for a super beautiful spiral, it also is an elegant way to define our signal in such a way that new properties emerge that can be utilized to define an energy that is associated with the particular cycling frequency.

As stated earlier, our goal is to define a metric that can be used to help assess the prevalence of a particular frequency in our signal. Fourier Transform Applications What good is theory if it isn't applied to something practical? Mathematical Background This section gives some mathematical background helpful for understanding the Fourier Transform. This website is intended to be a source of knowledge for learning about and understanding the Fourier Transform.

The goal is to present a comprehensive tutorial on the Fourier transform and related topics. In the spirit of Einstein: "Everything should be made as simple as possible, but not simpler. Check out our partner site, antenna-theory. Introduction to the Fourier Transform Virtually everything in the world can be described via a waveform - a function of time, space or some other variable.

For instance, sound waves, electromagnetic fields, the elevation of a hill versus location, a plot of VSWR versus frequency, the price of your favorite stock versus time, etc. The Fourier Transform gives us a unique and powerful way of viewing these waveforms. The purpose of this entire website is to explain the following fundamental fact: One of the Fundamental Secrets of the Universe All waveforms, no matter what you scribble or observe in the universe, are actually just the sum of simple sinusoids of different frequencies.

Figure 1. First fundamental frequency left and original waveform right compared. The second frequency will have a period half as long as the first twice the frequency. The second component is shown on the left in Figure 2, along with the sum of the first two frequencies compared to the original waveform. Figure 2. Second fundamental frequency left and original waveform compared with the first two frequency components.

We see that the sum of the first two frequencies is starting to look like the original waveform. The third frequency component is 3 times the frequency as the first. Claiming "planets move around in epicycles" is mathematically equivalent to saying "planets move around in two dimensions". Well, that's not saying nothing, but it's not saying much, either!

A simple mathematical way to represent "moving around in a circle" is to say that positions in a plane are represented by complex numbers, so a point moving in the plane is represented by a complex function of time.

We can then imagine three, four, or infinitely-many such circles being added. If we allow the circles to have every possible angular frequency, we can now write. If you start by tracing any time-dependent path you want through two-dimensions, your path can be perfectly-emulated by infinitely many circles of different frequencies, all added up, and the radii of those circles is the Fourier transform of your path.

Caveat: we must allow the circles to have complex radii. This isn't weird, though. It's the same thing as saying the circles have real radii, but they do not all have to start at the same place. At time zero, you can start however far you want around each circle. If your path closes on itself, as it does in the video, the Fourier transform turns out to simplify to a Fourier series. Most frequencies are no longer necessary, and we can write. The only circles we need are the slowest circle, then one twice as fast as that, then one three times as fast as the slowest one, etc.

There are still infinitely-many circles if you want to reproduce a repeating path perfectly, but they are countably-infinite now. If you take the first twenty or so and drop the rest, you should get close to your desired answer. In this way, you can use Fourier analysis to create your own epicycle video of your favorite cartoon character. That's what Fourier analysis says. The questions that remain are how to do it, what it's for, and why it works.

I think I will mostly leave those alone. Why it works is a rather deep question. It's a consequence of the spectral theorem.

What it's for has a huge range. It's useful in analyzing the response of linear physical systems to an external input, such as an electrical circuit responding to the signal it picks up with an antenna or a mass on a spring responding to being pushed. It's useful in optics; the interference pattern from light scattering from a diffraction grating is the Fourier transform of the grating, and the image of a source at the focus of a lens is its Fourier transform.

It's useful in spectroscopy, and in the analysis of any sort of wave phenomena. It converts between position and momentum representations of a wavefunction in quantum mechanics. Check out this question on physics. Fourier techniques are useful in signal analysis, image processing, and other digital applications. Finally, they are of course useful mathematically, as many other posts here describe. It took me quite a while to understand what exactly is meant by Fourier transform since it can refer to various algorithms, operations and results.

Though I'm quite new in this topic, I'll try to give a short but hopefully intuitive overview on what I came up with feel free to correct me :. Take for example the red function from here.

And that's what the continuous Fourier transform does. Often though, problems can be solved much easier in this other representation which is like finding the appropriate coordinate system. Most importantly, the Fourier transform has many nice mathematical properties i. It's often much easier to work with the Fourier transforms than with the function itself. So we transform, have an easy job with filtering, transforming and manipulating sine waves and transform back after all.

Let's say we want to do some noise reduction on a digital image. Let me partially steal from the accepted answer on MO, and illustrate it with examples I understand: The Fourier transform is a different representation that makes convolutions easy.

Or, to quote directly from there: "the Fourier transform is a unitary change of basis for functions or distributions that diagonalizes all convolution operators. This is the use of the discrete Fourier transform I'm most familiar with. Suppose you want to multiply two polynomials of degree n, given by their coefficients a 0 , …, a n and b 0 , …, b n. This is a convolution, and doing it naively would take O n 2 time. Instead, suppose we represent the polynomials by their values at 2n points.

Then the value of the product polynomial the one we want at any point is simply the product of the values of our original two polynomials. Here's the position of each cycle at every instant:. When our cycle is 4 units long, cycle speeds a half-cycle apart 2 units will either be lined up difference of 0, 4, 8… or on opposite sides difference of 2, 6, 10….

When every cycle has equal power and 0 phase, we start aligned and cancel afterwards. I don't have a nice proof yet -- any takers? In my head, I label these signals as "time spikes": they have a value for a single instant, and are zero otherwise the fancy name is a delta function.

Here's where phase comes in. Imagine a race with 4 runners. Normal races have everyone lined up at the starting line, the 4 0 0 0 time pattern. What if we want everyone to finish at the same time?

Just move people forward or backwards by the appropriate distance. Maybe granny can start 2 feet in front of the finish line, Usain Bolt can start m back, and they can cross the tape holding hands. Phase shifts, the starting angle, are delays in the cycle universe.

Here's how we adjust the starting position to delay every cycle 1 second:. If time points 4 0 0 0 are made from cycles [1 1 1 1] , then time points 0 4 0 0 are made from [1 ]. Note: I'm using "1Hz", but I mean "1 cycle over the entire time period". Test your intuition: Can you make 0 0 4 0 , i. The big insight: our signal is just a bunch of time spikes!

If we merge the recipes for each time spike, we should get the recipe for the full signal. This was my most challenging article yet. I was constantly bumping into the edge of my knowledge.

But there's always simple analogies out there -- I refuse to think otherwise. The analogy is flawed, and that's ok: it's a raft to use, and leave behind once we cross the river.

I realized how feeble my own understanding was when I couldn't work out the transform of 1 0 0 0 in my head. Why not? Shouldn't we have an intuition for the simplest of operations? That discomfort led me around the web to build my intuition.

In addition to the references in the article, I'd like to thank:. Today's goal was to experience the Fourier Transform. We'll save the advanced analysis for next time. Stuart Riffle has a great interpretation of the Fourier Transform:. Imagine spinning your signal in a centrifuge and checking for a bias. You already know why: we need a phase delay so spikes appear in the future.

Lucas Vieira , author of excellent Wikipedia animations , was inspired to make this interactive animation:. Fourier Toy - Click to download, requires flash.