The Fourier transform takes an input function f (in red) in the “time domain” and converts it into a new function f-hat (in blue) in the “frequency domain”.
In other words, the original function can be thought of as being “amplitude given time”, and the Fourier transform of the function is “amplitude given frequency”.
Shown here, a simple 6-component approximation of the square wave is decomposed (exactly, for simplicity) into 6 sine waves. These component frequencies show as very sharp peaks in the frequency domain of the function, shown as the blue graph. In practice, these peaks are never that sharp. That would require infinite precision.
I’m not too happy with this one yet. It’s a bit of a mixture of Fourier series and Fourier transform. The animation could also be a bit smoother in some steps. I may tweak it later.
Different modes of oscillation for a pendulum
The period of a simple pendulum is not a trivial thing, and it depends on the initial conditions.
Shown here are ten different modes of oscillation for the same pendulum. The only difference is the total amount of mechanical energy in the system.
As a result, each one has a completely different period of oscillation, unlike what the small-angle approximation (as taught in high-school) would suggest. They can’t be in sync. You may see some really interesting patterns based on the delay between them in your browser.
The red graph above each pendulum represents the phase portrait for the respective mode of oscillation, with the current state marked as a blue dot. The horizontal axis represents angle (hence why it wraps around the sides) while the vertical axis represents angular velocity.
Pendulums are very interesting dynamical systems, as they are relatively simple to understand but can produce surprisingly complex results in certain cases, such as the chaotic behavior of double pendulums and the odd behavior displayed by coupled pendulums.
This guy’s tumblr is really great. I hope to be making original content like his soon. This post fits in perfectly with my current ece course. Enjoy!
The continuous Fourier transform takes an input function f(x) in the time domain and turns it into a new function, ƒ̂(x) in the frequency domain. (These can represent other things too, but that’s besides the point.)
(Tumblr kept rejecting the proper sized GIFs, so they may look blurry, pixelated or compressed to you. There’s also HD video.)
In the second animation, the transform is reapplied to the normalized sinc function, and we get our original rect function back.
It takes four iterations of the Fourier transform to get back to the original function. We say it is a 4-periodic automorphism.
However, in this particular example, and with this particular definition of the Fourier transform, the rect function and the sinc function are exact inverses of each other. Using other definitions would require four applications, as we would get a distorted rect and sinc function in the intermediate steps.
For simplicity, I opted for this so I don’t have very tall and very wide intermediate functions, or the need for a very long animation. It doesn’t really work visually, and the details can be easily extrapolated once the main idea gets across, I think.
In this example, it also happens that there are no imaginary/sine components, so you’re looking at the real/cosine components only.
Shown at left, overlaid on the red time domain curve, you’ll notice a changing yellow curve. This is the approximation using the components extracted from the frequency domain “found” so far (the blue cosines sweeping the surface). The approximation is calculated by adding all the components, by integrating along the entire surface (this is continuous, remember?)
As we add more and more of the components, the approximation improves. In some special cases, it is exact. For the rect function, it isn’t, and you get some wavy artifacts in some places (the sudden jumps, aka discontinuities). These are due to Gibbs phenomenon, and are the main cause of ringing artifacts. As you’ll probably notice, the approximation is pretty much dead on for the sinc function, as shown in the second animation.
The illustration shows the domains in the interval [-5,5], but the Fourier transform extends infinitely to all directions, of course.
The surface illustrated here isn’t too far off from the approach used in Fourier operators. If you consider the surfaces defined by z = cos(xy) and z = sin(xy), you get the cosine and sine Fourier operators. Having complex values lets you mix both into one thing.
The surface you see in the first animation is just z = cos(2πxy)sinc(πy). The Fourier transform can be thought of as multiplying a function by these continuous operators, and integrating the result. This can be very neatly performed using matrix multiplication in the discrete cases. (New drinking game: take a shot every time linear algebra shows up in any mathematical discussion.)
This also explains why the Fourier transform is cyclic after 4 iterations: rotating 90° four times gets you back to your original position. By using different rotation angles, you get fractional Fourier transforms. Awesome stuff.
NOTE: This animation is a follow-up to the previous one on time/frequency domains, showing discrete frequency components. Check that one out too, as it may help with understanding this one.
Sadly, I had to reduce the images to 400 pixels wide instead of 500. Tumblr wouldn’t accept it otherwise. However, a HD video is also available:
This animation would probably look better with a different way of rendering that surface. Sorry, I don’t have anything better available at the moment, but I’ll work on it. If I do come up with something, I’ll post an update.
New Software Reveals “Hidden Information” In Video Footage
Leave it up to those brainiacs at MIT to come up with a technology like this. With a nifty video processing program they’re able to see changes impossible to see with the naked eye. Blood flow across a person’s face, vibrations of individual guitar strings, or the breathing patterns of a tiny infant are all made visible by their software which allows us to see subtle changes in the world that would otherwise go undetected.
The process, called Eulerian Video Magnification, extracts the “hidden information” in video footage by dividing the image up spatially, tracking the frequency at which the color in those smaller regions change, then amplifying changes that occur only at certain frequencies. By amplifying signals that change, say, between 50 and 70 times per minute – a probable range for heart rates – they can actually see blood flow across a person’s face and thus measure his or her pulse rate. (via New Software Reveals “Hidden Information” In Video Footage | Singularity Hub)
Line integral of a scalar field.
The line integral of a two-dimensional scalar field can be visualized by the area under the path over the surface defined by the scalar field. This is shown in this animation as a blue “curtain”.
This connects the idea of line integral with the familiar integral. The idea extends to more dimensions without much effort, except one cannot visualize the values along the field so easily.
This is a slight variation of the animation currently on Wikipedia, which was a finalist of the 2012 Wikimedia Commons Picture of the Year competition.
Below, you see 8 components corresponding to the integer frequencies from 1 to 8. The blue circle represents the amplitude and phase of each component. You can change these by clicking and dragging around each box. You can also set the amplitude manually by typing in a value from 0 to 1.
Holding shift will lock the phase in 45° multiples, and holding control will lock the amplitude in place.
The right-click context menu has a few wave presets.
Space bar: reset everything to zero
Up / down: speed up and slow down the animation
Left / right: rotate the phases of all components at once
Ctrl + 1-8: set the associated component to zero
T: toggle the blue trace
Ctrl+backspace: clear graph