How To Draw Anything With A Fourier Series

Chapter 4But what is a Fourier series? From heat flow to circle drawings

Intro

Here, we await at the math backside an animation like this, what's known as a "complex Fourier series".

Each petty vector is rotating at some abiding integer frequency, and when yous add them all together, tip to tail, they depict out some shape over time. Past tweaking the initial size and bending of each vector, nosotros tin make it depict anything we desire, and hither you'll meet how.

Before diving in, take a moment to linger on just how hit this is. Go full screen for this if you tin can, the intricacy is worth it.

A single-line portrait of Joseph Fourier, as fatigued by a Fourier serial.

How many rotating arrows would you guess are used in the animation above to depict a picture of Joseph Fourier?

Recall about this, the action of each private arrow is possibly the simplest affair you lot could imagine: Rotation at a steady rate. Yet the collection of all added together is anything just simple. The mind-boggling complexity is put into fifty-fifty sharper focus the farther we zoom in, revealing the contributions of the littlest, quickest arrows.

Considering the chaotic frenzy you're looking at, and the clockwork rigidity of the underlying motions, it's bizarre how the swarm acts with a kind of coordination to trace out some very specific shape. Unlike much of the emergent complexity you notice elsewhere in nature, though, this is something we have the math to draw and to control completely. Just by tuning the starting conditions, nothing more, you can make this swarm conspire in all the right ways to draw anything you want, provided you have enough little arrows. What's even crazier, as you'll see, is the ultimate formula for all this is incredibly short.

$$c_n = \int_0^1 e^{-ii \pi i n \cdot t} f(t) dt$$

Nosotros'll explain what this expression ways and how to read it in just a bit.

Ofttimes, Fourier serial are described in terms of functions of real numbers being broken downwards equally a sum of sine waves. That turns out to be a special case of this more general rotating vector miracle that we'll build upward to, just it's where Fourier himself started, and there's good reason for us to start the story there as well.

Already, a coupld questions might come to mind. What does adding sine waves have to exercise with the circumvolve animations? And perhaps more than pertinantly, why would anyone care? What problems does this solve?

Who cares?

Technically, this is the tertiary lesson in a series about the rut equation, which Fourier was working on when he developed his big idea. I'd like to teach you nearly Fourier series in a style that doesn't depend on you coming from those chapters, only if you have at to the lowest degree a loftier-level idea of the problem from physics which originally motivated this piece of math, it gives some indication for how unexpectedly far-reaching Fourier serial are.

If y'all don't care well-nigh the historical and physics-based origins of this math, or if you're coming from the previous capacity, feel free to skip the side by side few sections. If you do care, or you lot want a quick recap, let'due south dive on in.

The oestrus equation (epitomize)

Suppose you lot have some object, which for simplicity we'll remember of every bit beingness ane-dimensional, like a rod, and y'all know the temperature at every point on this rod at a given snapshot in time, which we'll call $t = 0$ . We'll call this the initial temperature distribution on the rod.

Every bit time ticks forward, the hot points will tend to cool down, and the cold points will tend to warm up, with an all around trend for a more even temperature distribution. How quickly this happens is determined by a constant $\alpha$ , known equally the thermal diffusivity, which depends on the fabric. Merely how specifically the shape of this distribution changes over time is determined by what'southward known every bit the heat equation.

The heat equation is a partial differential equation, which the last two lessons describe in much more than detail. For now all you need to know is that it does non directly tell y'all what future distributions will wait like. Instead, together with an initial distribution and some condition on the boundary of the rod, it gives a constraint that this evolution of future distribution must satisfy. It is up to the mathematician to actually solve this equation if they want a specific formula for how the temperature is distributed at any time $t > 0$ .

Enter Fourier.

While it'south exceedingly challenging to direct to straight solve this equation for a given initial distribution, there's a simple solution if that initial function happens to look like a cosine wave with a frequency tuned to make it flat at each endpoint. Specifically, equally you graph what happens over time, these waves simply become scaled down exponentially, with college frequency waves decaying faster.

Which of the following functions could describe the temperature distribution irresolute over fourth dimension pictured to a higher place.

$T(x, t) = \cos(4\pi t / Fifty) e^{-\blastoff (4\pi / L)^2 10}$

$T(x, t) = \cos(4\pi t / L) + e^{-\alpha (four\pi / 50)^2 10}$

$T(ten, t) = \cos(4\pi ten / L) e^{-\alpha (4\pi / L)^2 t}$

$T(x, t) = \cos(4\pi ten / 50) + due east^{-\blastoff (4\pi / L)^2 t}$

The heat equation happens to be what'due south known in the concern as a "linear" equation, meaning if you lot know two solutions and yous add together them up, that sum is also a new solution. You tin fifty-fifty scale them each past some abiding, which gives you some dials to plough to construct a custom office solving the equation.

Then even though information technology would be cool to discover a rod whose initial temperature distribution is a perfect cosine wave, this means nosotros know how to solve the equation for a much bigger class of functions, anything which tin can be written as a scaled sum of waves. You lot solve the equation for each of those waves separately, so add them together.

Information technology's hard to overstate how powerful this kind of linearity is. It means we can accept our infinite family unit of solutions, these exponentially decaying cosine waves, scale a few of them past some custom constants of our choosing, and combine them to get a solution for a new tailor-made initial condition which is some combination of cosine waves. Linearity is the difference between having a scattered set up of haphazard solutions in isolated cases, and having a massive space of solutions with knobs and dials to adjust that melody this initial status to our liking.

Something important I desire you to notice about combining the waves like this is that because college frequency ones disuse faster, this sum which you construct will polish out over time as the high-frequency terms quickly go to zero, leaving only the low-frequency terms dominating. So in some sense, all the complexity in the evolution that the estrus equation implies is captured by this difference in decay rates for the dissimilar frequency components.

Fourier series

It'due south at this point that Fourier gains immortality. I call up virtually normal people at this phase would say "well, I can solve the oestrus equation when the initial temperature distribution happens to wait like a moving ridge, or a sum of waves, merely what a shame that most existent-world distributions don't at all look similar this!"

Certain, it works for waves...so what?

For example, let's say you lot brought together two rods, each at some uniform temperature, and you wanted to know what happens immediately after they come into contact. To make the numbers simple, let's say the temperature of the left rod is $1^{\circ}$ , and the correct rod is $-1^{\circ}$ , and that the total length L of the combined rod is 1.

Our initial temperature distribution is a stride role, which is so obviously different from sine waves and sums of sine waves, don't you recall? It'southward most entirely apartment, not wavy, and for god's sake, it's even discontinuous!

And yet, Fourier thought to inquire a question which seems absurd: How do you express this every bit a sum of sine waves? Even more boldly, how do you express any initial temperature distribution as a sum of sine waves?

And it's more constrained than just that! You have to restrict yourself to calculation waves which satisfy a certain boundary condition, which as we saw in the last lesson ways working only with these cosine functions whose frequencies are all some whole number multiple of a given base frequency.

Not-mathematical interlude

It's strange how often progress in math looks like asking a new question, rather than simply answering an erstwhile ane.

Fourier really does accept a kind of immortality, with his name essentially synonymous with the idea of breaking down functions and patterns every bit combinations of simple oscillations.

Fourier could never have imagined how significant and far-reaching this thought would turn out to be, ranging from the study of prime numbers, to quantum calculating, to point processing and much more. And withal, the origin of all this is in a piece of physics which upon first glance has goose egg to exercise with frequencies and oscillations.

Infinite sinusoids?

"Now hang on," I hear some of you saying, "none of these sums of sine waves beingness shown are really the pace function." It's true, any finite sum of sine waves will never be perfectly flat (except for a constant function), nor discontinuous. But Fourier thought more broadly, considering infinite sums. In the example of our step part, it turns out to exist equal to this space sum:

$$2 \cdot H(10 - 0.v) - i = \frac{4}{\pi} \left( \frac{\cos(one \pi ten)}{1} - \frac{\cos(3 \pi ten)}{three} + \frac{\cos(5 \pi x)}{5} - \frac{\cos(7 \pi x)}{7} + \cdots \right)$$

I'll explain where these numbers come from in a moment.

Before that, it's worth being clear about what we mean with a phrase like "space sum", which runs the take chances of existence vague. Consider the simpler context of adding numbers, say this alternating sum of fractions with odd denominators:

$$\frac{1}{ane} - \frac{i}{3} + \frac{1}{5} - \frac{one}{7} + \frac{1}{9} - \frac{1}{11} \cdots = \frac{\pi}{iv}$$

If you add these terms successively, ane-by-one, at all times what you have will exist rational. At no point volition your partial sum of $N$ terms equal the irrational $\frac{\pi}{iv}$ . But this sequence of partial sums approaches $\frac{\pi}{4}$ . The numbers you run into become arbitrarily shut to that value, and stay arbitrarily close to that value.

Referencing partial sums, and what that sequence of values approaches, is all a bit of mouthful, so instead we abridge and say the infinite sum "equals" $\frac{\pi}{4}$ .

With functions, you're doing the same thing but with many dissimilar values in parallel. Consider a specific input, like $x=0.2$ , and the value of all these scaled cosine functions for that input.

If that input is less than 0.5, as y'all add more than and more terms, the sum volition approach 1.

If that input is greater than 0.five, as y'all add together more than and more terms it would arroyo -1.

At the input 0.5 itself, all the cosines are 0, so the limit of the partial sums is 0.

Somewhat awkwardly, and then, for this space sum of functions to be strictly true, we do have to prescribe the value of the footstep part at the indicate of discontinuity to exist 0.

Coordinating to an infinite sum of rational numbers existence irrational, the infinite sum of wavy continuous functions can equal a discontinuous flat function. Limits allow for qualitative changes which finite sums lone never could.

There are multiple technical nuances I'thousand sweeping under the rug hither.

• Does the fact that we're forced into a certain value for the step part at its indicate of discontinuity brand whatever departure for the heat flow problem?
• For that matter, what does it really mean to solve a PDE with a discontinuous initial condition?
• Can we be sure the limit of solutions to the oestrus equation is besides a solution?
• Tin can all functions exist expressed as an infinite sum of waves similar this?

These are exactly the kind of questions real analysis is built to answer, but it falls a chip deeper in the weeds than I call up we should get here.

The issue is that when you take the estrus equation solutions associated with these cosine waves and add them all upward, all infinitely many of them, you do go an exact solution describing how the step function will evolve over time.

How do we compute the coefficients?

Generalize

The central challenge, of course, is to discover these coefficients, the terms which we scale each moving ridge by before calculation them up. And so far, nosotros've been thinking about functions with real number outputs, simply for the computations I'd like to show you something more general than what Fourier originally did, applying to functions whose output tin can be any circuitous number, which is where those rotating vectors from the opening come back into play.

Why the added complexity? Bated from being more than full general, in my view the computations become cleaner and it'southward easier to see why they work. More importantly, it sets a skilful foundation for ideas that will come again later on in the series, like the Laplace transform and the importance of exponential functions.

Waves and rotation

We'll withal think of functions whose input is some real number on a finite interval, say the one from 0 to i for simplicity. But whereas something similar a temperature role will have an output confined to the real number line, we'll broaden our view to outputs anywhere in the two-dimensional complex plane.

Y'all might recall of such a function as a drawing, with a pencil tip tracing along different points in the circuitous plane every bit the input ranges from 0 to i.

Instead of sine waves being the key edifice block, we'll focus on breaking these functions down as a sum of fiddling vectors, all rotating at some abiding integer frequency.

Functions with real number outputs are essentially really boring drawings; a ane-dimensional pencil sketch confined to the real number line. You might not exist used to thinking of them similar this, since usually we visualize such a part with a graph, just right at present the path being fatigued is only in the output space.

When we do the decomposition into rotating vectors for these irksome 1d drawings, what will happen is that all the vectors with frequency i and -1 will have the same length, and they'll exist horizontal reflections of each other. When you lot just look at the sum of these two as they rotate, that sum stays fixed on the real number line, and oscillates like a sine wave.

This might exist a weird mode to call back about a sine wave, since we're used to looking at its graph rather than the output alone wandering on the existent number line. Only in the broader context of functions with complex number outputs, this is what sine waves look similar. Similarly, the pair of rotating vectors with frequency ii, -2 volition add another sine wave component, so on, with the sine waves we were looking at earlier now respective to pairs of vectors rotating in opposite directions.

And then the context Fourier originally studied, breaking downwardly real-valued functions into sine moving ridge components, is a special case of the more general idea with 2d-drawings and rotating vectors.

I'chiliad lamentable, you said this was easier?

At this signal, peradventure you don't trust me that widening our view to circuitous functions makes things easier to understand, but bear with me. It actually is worth the added effort to see the fuller movie, and I think you'll be pleased by how clean the actual computation is in this broader context.

Complex numbers

You may likewise wonder why, if we're going to crash-land things upwardly to 2-dimensions, we don't we merely talk near second vectors; What does $\sqrt{-1}$ have to do with annihilation? Well, the heart and soul of Fourier series is the complex exponential, $e^{it}$ . As the value of $t$ ticks forward with time, this value walks effectually the unit circle at a rate of 1 unit per second.

In the side by side lesson, you'll run into a quick intuition for why exponentiating imaginary numbers walks in circles similar this from the perspective of differential equations, and beyond that, equally the series progresses I hope to give you some sense for why circuitous exponentials are important.

You see, in theory, you could describe all of this Fourier series stuff purely in terms of vectors and never breathe a word of i. The formulas would become more convoluted, but across that, leaving out the function $e^ten$ would somehow no longer authentically reflect why this idea turns out to be so useful for solving differential equations. The key feature of $e^x$ is that it's a role whose derivative equals itself, which makes it a useful edifice cake for describing functions whose derivatives depend on the function itself in more complicated ways.

For right now you can think of this $e^{i t}$ as a notational shorthand to describe a rotating vector, but just keep in the dorsum of your listen that it's more meaning than a mere shorthand. In fact, if the dorsum of your mind has a piddling extra space, you can recall almost how the 2nd derivative of this function is equal to $-i$ times itself, and how this might correspond to the negative sign showing up in the oestrus equation.

I'll be loose with language and utilise the words "vector" and "complex number" somewhat interchangeably, since thinking of complex numbers as little arrows makes the idea of adding many together clearer.

So, what are the constants?

Armed with the part $eastward^{it}$ , allow's write down a formula for each of these rotating vectors we're working with. For now, think of each of them as starting pointed i unit of measurement to right, at the number 1.

The easiest vector to draw is the constant i, which just stays at the number 1, never moving. Or, if you prefer, it'due south "rotating" at a frequency of 0.

And so at that place will be a vector rotating 1 bicycle every second which nosotros write as $e^{ii \pi i t}$ .

The $2 \pi$ is there considering as $t$ goes from $0$ to $1$ , it needs to cover a distance of $two \pi$ along the circumvolve.

We also have a vector rotating at one bicycle per second in the other direction, $due east^{-2 \pi i t}$ .

Similarly, the 1 going 2 rotations per 2nd is $e^{2 \cdot 2 \pi i t}$ , where that $2 \cdot ii \pi$ in the exponent describes how much distance is covered in 1 2nd. And we go on like this over all integers, both positive and negative, with a general formula of $e^{n \cdot 2 \pi i t}$ for each rotating vector.

Find, this makes information technology more consequent to write the constant vector is written every bit $e^{0 \cdot 2 \pi i t}$ , which feels like an awfully complicated to write the number i, but at least then it fits the pattern.

The control we take, the set of knobs and dials we get to turn, is the initial size and direction of each of these numbers. The way we command that is by multiplying each i by some complex number, which I'll telephone call $c_n$ .

For example, if we wanted that constant vector non to exist at the number one, simply to accept a length of 0.5, we'd calibration it by 0.5.

If we wanted the vector rotating at one cycle per second to start off at an angle of $45^{\circ}$ , what factor would we multiply $e^{2 \pi i}$ past?

$e^{\frac{\pi}{iv} i}$

$e^{45i}$

$e^{\frac{three \pi}{iv} i}$

$\frac{\pi}{4}$

Anybody in our infinite family of rotating vectors has some complex constant being multiplied into it which determines its initial angle and magnitude. Our goal is to express whatever arbitrary part $f(t)$ , say the one below cartoon an eighth annotation, as a sum of terms like this, so we need some manner to pick out these constants i-by-one given the data of the function.

The integration trick

The easiest one is the constant term. This term represents a sort of center of mass for the full drawing; if you were to sample a bunch of evenly spaced values for the input t as information technology ranges from 0 to 1, the boilerplate of all the outputs of the office for those samples will be the constant term $c_0$

Or more accurately, every bit you lot consider finer and finer samples, their boilerplate approaches $c_0$ in the limit.

What I'm describing, effectively and finer sums of f(t) for a sample of t from the input range, is an integral of f(t) from 0 to i.

$$c_0 = \int_{0}^{i} f(t) dt$$

Unremarkably, since the aim is to compute an average, you'd divide this integral by the length of the interval. But that length is 1, then information technology amounts to the same thing.

$$c_0 = \frac{ane}{1-0} \int_{0}^{i} f(t) dt$$

Integrating a circuitous office?

There'south a very nice way to think about why this integral would pull out $c_0$ . Since nosotros want to think of the role as a sum of these rotating vectors, consider this integral (this continuous average) as being applied to that sum.

$$\int_{0}^{1} f(t) dt =\int_{0}^{1}\left(\cdots+c_{-1} e^{-1 \cdot 2 \pi i t}+c_{0} east^{0 \cdot 2 \pi i t}+c_{ane} e^{1 \cdot 2 \pi i t}+c_{2} eastward^{2 \cdot two \pi i t}+\cdots\right) d t$$

This boilerplate of a sum is the aforementioned as a sum over the averages of each part.

$$=\cdots+\int_{0}^{1} c_{-1} eastward^{-ane \cdot ii \pi i t} d t+\int_{0}^{1} c_{0} eastward^{0 \cdot 2 \pi i t} d t+\int_{0}^{1} c_{1} e^{1 \cdot 2 \pi i t} d t+\int_{0}^{1} c_{2} e^{2 \cdot 2 \pi i t} d t+\cdots$$

You tin read this move as a shift in perspective. Rather than looking at the sum of all the vectors at each point in time, and taking the average value of the points they trace out, look at the average value for each individual vector as t goes from 0 to 1, and add upwards all these averages.

It's a subtle shift, simply think nearly what each of those inner integrals existence added now means. Each of these rotating vectors makes a whole number of rotations around 0 as $t$ ranges from $0$ to $1$ , so its average value will be 0. The only exception is that constant term; since information technology stays static and doesn't rotate, its average value is only whatever number it started on, which is $c_0$ . So doing this boilerplate over the whole function kills all the terms except $c_0$ .

Pull out any other term

But now allow'southward say you wanted to compute a different term, like $c_2$ in front of the vector rotating 2 cycles per second. The trick is to first multiply $f(t)$ by something which makes that vector hold nevertheless, the mathematical equivalent of giving a smartphone to an overactive child. Specifically, if you multiply the whole function by $e^{-2 \cdot 2 \pi i t}$ , think most what happens to each term.

Call back, we're bold you can write $f(t)$ as a sum which looks similar this:

$$f(t) = \left(\cdots c_{-2}eastward^{-two \cdot 2\pi i t} + c_{-1}e^{-1 \cdot 2\pi i t} + c_{0}e^{0 \cdot ii\pi i t} + c_{i}east^{i \cdot 2\pi i t} + c_{2}e^{2 \cdot two\pi i t} + \cdots\right)$$

For a role $f(t)$ with the grade above, what is the boilerplate value of $f(t)e^{-2 \cdot two\pi i t}$ as $t$ ranges from $0$ to $1$ ?

$c_{-two}$

$\frac{1}{2}c_{-ii}$

$c_0$

$\frac{ane}{two}c_{2}$

$c_2$

In short, multiplying by $e^{-ii \cdot 2\pi i t}$ is a fashion to make the rotating vectors associated with $c_2 e^{2 \cdot 2\pi i t}$ hold still while all the others movement around.

Of form, in that location's nothing special nearly two here. If nosotros replace it with whatever other $n$ , you accept a formula for any other term $c_n$ .

$$c_n = \int_{0}^{1} f(t) east^{-n \cdot 2 \pi i t} d t$$

Again, you can read this expression as modifying our office, our 2d drawing, then as to make the $n^{\text{th}}$ little vector hold still, and then performing an average so that all other vectors get canceled out. Isn't that crazy? All the complexity of this decomposition as a sum of many rotations is entirely captured in this expression.

Doing this in practice

To create the animations for this postal service, that formula is exactly what I'm having the computer evaluate. It treats a path like a complex function, and for a certain range of values for due north, information technology computes this integral to notice each coefficient $c_n$ . For those of you curious about where the data for the path itself comes from, I'm going the easy road having the plan read in an svg, which is a file format that defines the prototype in terms of mathematical curves rather than with pixel values. Skipping over some details of accordingly massaging the data of those curves, this essentially means the mapping $f(t)$ from a time parameter to points in space comes predefined.

The animation to a higher place uses 101 rotating vectors, computing values of n from -50 up to fifty. In practise, the integral is computed numerically, essentially meaning it chops up the unit interval into many small pieces of size $\Delta t$ and adds up this value $f(t)eastward^{-n \cdot two \pi i t} \Delta t$ for each one of them. There are fancier methods for more efficient numerical integration, merely that gives the basic idea.

After computing these 101 values, each one determines an initial position for the little vectors, and then you set up them all rotating, adding them all tip to tail, and the path drawn out past the concluding tip is some approximation of the original path. As the number of vectors used approaches infinity, it gets more and more accurate.

Relation to step part

To bring this all dorsum downward to earth, consider the example we were looking at before of a pace function, which was useful for modeling the heat dissipation betwixt two rods of unlike temperatures after coming into contact.

Since it's a real-valued function, a stride function is like a boring cartoon confined to one-dimension. But this 1 is an peculiarly dull cartoon, since for inputs between 0 and 0.five, the output merely stays static at the number one, and so it discontinuously jumps to -ane for inputs between 0.5 and i.

What does this mean for the Fourier series approximation? The vector sum stays really close to one for the showtime half of the cycle, so really quickly jumps to -1 for the 2d half. Call up, each pair of vectors rotating in opposite directions correspond to one of the cosine waves we were looking at earlier.

To discover the coefficients, you'd need to compute this integral. For the ambitious reader among you itching to work out some integrals past mitt, this is ane where you lot can exercise the calculus to get an exact answer, rather than just having a figurer do it numerically for you. I'll go out it as an exercise to piece of work this out, and to relate it back to the idea of cosine waves past pairing off the vectors rotating in contrary directions.

For the even more ambitious, I'll too leave some other exercises upward on how to relate this more general ciphering with what you might come across in a textbook describing Fourier series only in terms of real-valued functions with sines and cosines.

Determination

If you're looking for more than Fourier series content, I highly recommend the videos by Mathologer and The Coding Train on the topic, and the weblog post by Jezzamoon. Here's an interactive to describe your own path with the respective Fourier series.

On the ane mitt, this concludes our word of the heat equation, which was a little window into the report of partial differential equations.

On the other hand, this foray into Fourier series is a first glimpse at a deeper thought. Exponential functions, including their generalization into complex numbers and fifty-fifty matrices, play a very important role for differential equations, especially when it comes to linear equations. What yous simply saw, breaking down a function every bit a combination of these exponential functions, comes up again in different shapes and forms.

Thank you

Special thanks to those beneath for supporting the original video behind this mail service, and to current patrons for funding ongoing projects. If yous observe these lessons valuable, consider joining.

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