Fourier Geometry

Last time I opened with basic exponentiation and raised it to the idea of complex exponents (which may, or may not, have been surprising to you). I also began exploring the ubiquitous exp function, which enables the complex math needed to deal with such exponents.

The exp(x) function, which is the same as ex, appears widely throughout physics. The complex version, exp(ix), is especially common in wave-based physics (such as optics, sound, and quantum mechanics). It’s instrumental in the Fourier transform.

Which in turn is as instrumental to mathematicians and physicists as a hammer is to carpenters and pianos.

Warning: To make sense of this post, you need to have read the previous post (or have some background with the topics covered thus far). I’m going to jump in and pick up where I left off.

Which involved this pair of charts:

Figure 1. A waveform (upper) and its Fourier transform (lower).

Both show the same thing, a superposition of three sine waves of different frequencies (13, 27 & 42 cycles per second). The upper chart shows the energy of the waveform over time (for a period of two seconds). The lower chart shows the energy of the waveform over frequency (between 0-50).

The Fourier transform, along with its inverse, allow us to derive either from the other. Specifically, a Fourier transform generates frequency data from time data. An Inverse Fourier transform generates time data from frequency data.

[They use complex numbers, so the values have both a magnitude (or amplitude) and an angle (often called an argument). That angle carries the wave’s phase information. The magnitude is its energy.]

So a Fourier transform decomposes a signal (which is comprised of one or more frequencies) into a spectrum of those frequencies. It’s like how a prism transforms a beam of white light into a spectrum of rainbow colors.

Implicit in all this is that any waveform can be analyzed in terms of its component specific frequencies.

Any waveform is the sum of some set of sine waves.

§ §

Here’s the basic definition of the Fourier transform function:

Depending on one’s comfort level with math, it’s either a hugely intimidating expression or a familiar simple one. (Full disclosure, I’m a bit past intimidating, but a long ways from familiar.)

What it says is actually fairly simple, and I’ll explore it here, but below is a video with a very cool geometric way illustrate exactly how and why it works. (For me it turned the idea of a Fourier transform from Huh? to Duh!)

What the math above does is, on the left, define a function f-hat as a Fourier transform of the function f (which is inside the scary expression on the right). This function f-hat takes a parameter, κ (kappa) that represents a “winding” frequency (which is explained in the video below).

Notice that just to the right of the function f is our friend from last time, e-to-the-power-of-something-really-weird. I’ll take a closer look at that below.

So the inner part says multiply the values generated by function f times the values generated by the exponential expression.

Note that function f is the source of the waveform we’re analyzing. It gives us the energy of that waveform for any given time t. The function might read data from a table (some captured physical signal), or it might generate it mathematically.

The outer part of the right side, the giant curly -shape, and the dt on the far right, turn this into an integral. That means we measure the area beneath the curve created by the inner part. Negative area (below the axis) is subtracted from positive area (above the axis), so the integral is, in part, a measure of how offset from zero the inner equation is.

In this case, the integral’s value increases when the “winding” frequency matches, or is close to, one of the component frequencies of the f function. When the “winding” frequency isn’t close to a component frequency, the value of the integral is near zero.

Therefore we can find the component frequencies by changing κ and watching for the integral to increase.

§

Which is a fine and true explanation, but huh?

Grant Sanderson has a video on his 3Blue1Brown YouTube channel that illustrates exactly what’s happening in a very clear way:

I gotta say, this just blew me away. It makes how Fourier transforms work almost obvious.

Essentially it detects the constructive interference between the winding frequency and a matching component frequency. (I think it’s actually due more to resonance than interference, but the latter is perhaps an easier metaphor.)

The center-of-mass idea Grant uses here is the geometric idea I found so cool I had to share it. It turns the idea of an integral (area under a curve) into an even more physical idea — center-of-mass is very easy to understand.

The video is so good that I’m just going to let it stand. (That’s the genius of Grant’s presentations. He finds extremely evocative ways of illustrating math concepts. Hands down my favorite math channel.)

§

What I can try to add is a deeper look at certain parts of Fourier transforms and at Grant’s “winding machine” in more detail.

Here, again, for easy reference is the Fourier transform definition:

Let’s take a closer look at that exponential function:

We know that ei puts us on the complex unit circle. We also know that 2π is a full trip around that circle. That leaves the κ and t variables. Above I mentioned that κ is the “winding” frequency (from the video you know why it’s called that). I also mentioned that t is for time — it’s the ticking clock of our winding machine.

Think of ei2π as a static “wheel” of 360° — which, remember, is also 2π radians. The expression, e-to-the-power-of-i-2-pi, evaluates to the complex number +1+0i. (It’s a case of Euler’s Formula where a=2π.)

Note that ei2π×0 = e0 = +1+0i. This is both a case that anything to the power of zero is one and a case of zero rotation of the wheel.

In fact, for any ei2π×n, where n is a whole number, the resulting value is +1+0i — zero rotation of the wheel. In other words, the wheel makes a full rotation for every whole value of n.

Going back to the expression above, consider the t variable. It has some value we interpret as seconds. Every whole value of t (0,1, 2, 3, 4,…) is a multiple of 2π, so each “second” involves a full trip around the circle. After four seconds, we’ve gone around four times. (Fractions of a second go partway around.)

Lastly, the variable κ acts as a multiplier. If our winding frequency should be 3.14 cycles per second, κ=3.14, which makes our wheel go around 3.14 times per whole value of t (i.e. per second).

That’s all there is to it. The expression ei2πκt is a “wheel” we can make go around at some winding frequency κ. The output of the function is a series of complex numbers describing that circular motion around the unit circle.

Figure 2. Plot of ei2πκt for κ=3.0 and t=0.0 to 2π

Above is a plot of a “wheel” with a winding frequency of 3.0. We’re using complex numbers, so the result has a real part and an imaginary part.

The real part (blue) is the same as the cosine of t, and the imaginary part (red) is the same as the sine of t. Which is exactly what Euler’s Formula says:

eia = exp(ia) = cos(a) + i sin(a)

(I gotta be honest: plotting sine waves without actually using the cos and sin functions is kind of weird! It’s amazing we get that out of the exp function.)

Note that a wheel has a constant magnitude of 1.0 although its angle varies. A wave represents something with a constant energy, but with a rotating phase angle. This is important in quantum mechanics because it’s the phases that interfere.

§

The other part of the transform definition is the function f(t), which is the source of the signal we’re analyzing. What we need is the signal energy at any given time.

In many circumstances we’d read data from a table of {time, energy} pairs — that data presumably representing some real world signal. In other circumstances, such as solutions for the Schrödinger equation, we want to analyze a mathematical function.

The demos here are another case of using mathematical functions to create the waveform to be analyzed. In the previous post I showed you the math behind the waveform in the upper chart of Figure 1:

As you can now see, these are just “wheels” (three of them!) that crank out sine waves. These use x rather than t (which is just notation). The constants (13, 27 & 42) are the values for κ.

Their outputs are averaged (summed and divided by n=3) which produces a superposed waveform (the upper chart of Figure 1).

§ §

HUP

Heisenberg’s Principle

Last time I said that Fourier transforms underpin the Heisenberg Uncertainty Principle, which, as written, is a statement about position and momentum:

However, the actual principle underlying the Principle is more general and applies to all conjugate pairs in quantum mechanics.

The short form is that, in wave mechanics, position and momentum (and other such pairs) are mutually exclusive Fourier transforms of each other.

This video explains:

I like his pointing out, in German, the term refers to sharpness. That’s a good way to think about it and clearly Heisenberg’s intention.

I’ve always thought the sound analogy is the easiest to grasp, and asking about the pitch (frequency) of a clap highlights the point. (In many regards, the correct answer is: superposition of all frequencies.)

The shorter the sound (or the smaller the window used to measure sound), the greater the uncertainty about the sound’s pitch, but the more localized in time it is. We know exactly when it happened.

The longer the sound (or measurement window), the more we’re certain of its pitch, but the less we can say about when it happened (because it happens over a period of time). At the extreme, infinite time means perfect certainty about pitch and zero information about time.

This uncertainty relationship is a fundamental aspect of any wave mechanics.

§ §

I liked Grant’s idea about a winding machine so much I just had to write some Python code to see for myself. Here’s a sample of the output:

The first set show a winding frequency of 1, 2 & 3 cycles per second (cps). In all the examples here, the input wave form — the output of function f — is a superposition of two sine waves, one at 9 cps and the other at 14 cps.

In the leftmost image, the winding frequency of 1 wraps a single second in a circle, and the input waveform is somewhat apparent. Notice there are nine large lobes (from the 9 cps input) and fourteen lobes in total (due to the 14 cps input).

The next two show the winding frequency near and matching the inputs:

The winding frequencies here are 8, 9 & 10 cps. The middle image shows the winding frequency matching the 9 cps input frequency. Note how off-center the total shape is.

Here the winding frequencies are 13, 14 & 15 cps. The middle image again shows the match. The key is how off-center the shape is — the center-of-mass is shifted away from the center.

§

I had intended to dig a little deeper into the exp function, but this has gone on long enough. (Perhaps I’ll do a Sideband about it — it does get a bit technical.)

Stay transformed, my friends. Go forth and spread beauty and light.

About Wyrd Smythe

The canonical fool on the hill watching the sunset and the rotation of the planet and thinking what he imagines are large thoughts. View all posts by Wyrd Smythe

7 responses to “Fourier Geometry

  • SelfAwarePatterns

    I know this isn’t the same thing, but one thing the winding machine seemed to click in me is a hazy idea of how spin might actually manifest in a wave system.

    • Wyrd Smythe

      How so? What are you seeing with your mind’s eye?

      Pretty cool way to explain the Fourier transform, I thought. Did it click with you? (Same question to anyone else who reads this.)

      • SelfAwarePatterns

        Don’t know how true to reality this maps to, but I think about the original conception of spin, as a ball spinning about its axis. The problem, as I understand it, is that actual spin doesn’t work this way, and when thinking about waves, I wasn’t sure how they manifested. The video made me realize that maybe we’re talking about a repeating pattern that can be modeled as an abstract spin of some type.

        I’ve also read that Schrodinger had to incorporate spin into his equations to make them work, which means it should be accounted for in them.

        Which led me to this wiki, which quickly reminded me how out of my depth I am with this stuff.
        https://en.wikipedia.org/wiki/Spinor

        The video did sort of click, but I’m in a haze with much of this stuff. I suspect it would have clicked much better if I was more steeped in the prerequisites.

      • Wyrd Smythe

        As you may have experienced, one of the hardest parts of explaining technical stuff is judging what it’s like for people who aren’t as familiar with the material. We talked about the upward inflection in the knowledge curve — once one is far up that slope, and things seem so clear, it’s really hard to judge what it can be like for those who aren’t at that level.

        On some level, all I can do is say this is really cool, and if one likes this sort of thing, here’s a great doorway into it. Obviously it does require certain foundation knowledge. (I will say that, if one wanted to get deeper into QM, follow me — that’s where I’m headed. 🙂 )

        Quantum spin and the mathematical objects spinners are both on my TODO list. Spin is apparently a more complex topic in QM. That series I’m watching hasn’t touched on it in any detail (and from the titles of the remaining lectures, I’m not sure it will).

        In fact, in the first lecture, the teacher talked about two “magical” boxes that you send an electron in one side, and that electron either comes out the opposite side or is deflected out the top. Which way the electron exits depends on the box “magically” determining a given property of the electron. In one case, if the electron is “white” or “black” (it’ll always be one or the other). In the other case, if the electron is “hard” or “soft” (again, it’ll always be one or the other).

        Then he sets up a bunch of scenarios, different combinations of one box after another, to illustrate the weirdness of superposition in QM. An electron measured to be “white” will continue to be measured as “white” unless sent through a “hard/soft” box — then color information goes back into superposition. And he ends with scenarios that really bring home how weird that is.

        But he’s actually talking about vertical and horizontal spin. It’s just that spin is so weird that they don’t really get into it in a first year quantum mechanics class even at MIT! So definitely on my TODO list. 🙂

        They always say nothing is actually spinning, yet so many of its properties sure seem like something is spinning. One measures spin by looking at deflection when passing through a magnetic field, and some notions of angular momentum seem to apply. So I quite agree that something seems to be, in some sense, actually spinning.

        I do know that a full description of a particle system does require including spin (and some other properties). AIUI, those are all part of the operator applied to the wave-function, and they make the math much more complicated. (Hence leaving it out at first.)

        Spinners keep popping up. One more bit of mathematics to explore. (One thing that’s almost discouraging about this math path is that it’s endless. There are always more branches to explore.)

      • SelfAwarePatterns

        I’m definitely familiar with the problem of explaining technical stuff. Some readers have little grounding in it, and are quickly left behind. Others are thoroughly read in and impatient with simplified explanations. Sometimes you just can’t win and have to decide which audience you’re going to cater to.

        From the little I know about spin, it sounds like it might at least be a rotation of states. But unlike a true spin, which would always return to the initial state after one rotation, the fact that particles can be in spin 1/2, 1, 3/2, 2, 5/2, etc, before returning to their initial state, which along with all the other odd properties we’ve discussed, makes it seem very weird. Tegmark, I believe, said it could only be understood mathematically, but he said that in a book dedicated to arguing that it’s all math.

      • Wyrd Smythe

        “Sometimes you just can’t win and have to decide which audience you’re going to cater to.”

        Yep. And the internet is filled with people explaining things, which sometimes makes me wonder why I bother. A lot of these posts are as much for me and for readers. Trying to explain something is such a good way to get it clear in one’s own mind.

        Here’s a really cool way to illustrate an aspect of spin:

        Stick your right hand out in front, palm up, fingers pointing forward. Stick your thumb out to the right as an additional direction indicator. Now, focusing on keeping your palm up, swing your arm to the right and around over your head and back to sticking out in front of you. Remember to keep your palm always facing up.

        The end result should be your right hand in front, palm up, fingers pointing forward, but now your arm is twisted. In fact, it’s twisted 360° because your palm rotated 360° in a CW direction (as viewed by you).

        You can apply another 360° CW rotation, but this time you’ll have to swing your hand down under your arm and back to the starting position. This revolution untwists your arm and you’re now back where you started.

        So it’s an example of a system for which a single rotation of 360° puts the system in a different state, but a second rotation returns it to the first state. Likewise, an electron (a 1/2 spin particle) requires two “rotations” (of state) to return to the starting state.

        Tegmark pretty much echoes every other person I’ve heard talk about spin — it’s mathematical. At least our understanding of it is. My assumption is that there’s something physical going on. One of the things that initially drew me to string theory was it offers a possible physical account of spin — some aspect of the vibrating string would be spin.

        My guess is still that something along those lines is going on. Measuring one axis aligns whatever is spinning such that further readings along that axis return the same result, but readings along other axes only have probabilities. (Even a small rotation creates some chance of getting the other reading. Orthogonal rotation makes it a complete crapshoot.) So it’s spin is aligned, but there’s some “wobble”.

  • Wyrd Smythe

    FWIW, the intended takeaway here is the idea that, in the Heisenberg Uncertainty relationship, things like position and momentum are Fourier transforms of each other, which is why only one can be well localized at a time.

    A pure sine wave is infinite in time, but a sharp spike — well localized — in frequency. On the other hand, a sharp sound spike (like a clap),which is localized in time, has infinitely many component frequencies.

    Likewise, a well-localized particle position means an infinite number of possible momentum values, and a well-localized momentum means the particle “could be anywhere.”

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