When it comes to mathematics, the average person can probably get through most of life well enough with just basic algebra. Some simple statistical concepts would be helpful, and a little calculus couldn’t hurt. But that leaves out a lot of interesting mathematical concepts that really do have applications in everyday life and are just plain fascinating in their own right.
Chief among these concepts is the Fourier transform, which is the key to understanding everything from how JPEGs work to how we can stream audio and video over the Internet. To help get your mind around the concept, [Jez Swanson] has this interactive Fourier transform visualizer that really drives home the important points. This is high-level stuff; it just covers the basic concepts of a Fourier transform, how they work, and what they’re good for in everyday life. There are no equations, just engaging animations that show how any function can be decomposed into a set of sine waves. One shows the approximation of a square wave with a slider to control to vary the number of component sine waves; a button lets you hear the resulting sound getting harsher as it approaches a true square wave. There’s also a great bit on epicycles and SVGs, and one of the best introductions to encoding images as JPEGs that we’ve seen. The best part: all the code behind the demos is available on GitHub.
In terms of making Fourier transform concepts accessible, we’d put [Jez]’s work right up there with such devices as the original Michelson harmonic analyzer, or even its more recent plywood reproduction. Plus the interactive demos were a lot of fun to play with.
[via the Adafruit blog]
One of the nice things about living in the Internet age is that creating amazing simulations and animations is relatively simple today. [SmarterEveryDay] recently did a video that shows this off, discussing a blog post (which was in Turkish) to show how sine waves can add together to create arbitrary waveforms. You can see the English video, below.
We’ve seen similar things before, but if you haven’t you can really see how a point on a moving circle describes a sine wave. Through adding those waves, anything can then be done.
Continue reading “Explaining Fourier Again”
One of the things hard about engineering — electrical engineering, in particular — is that you can’t really visualize what’s important. Sure, you can see a resistor and an LED in your hands, but the real stuff that we care about — electron flow, space charge, and all that — is totally abstract. If you just tinker, you might avoid a lot of the inherent math (or maths for our UK friends), but if you decide to get serious, you’ll quickly find yourself in a numerical quicksand. The problem is, there’s mechanically understanding math, and intuitively understanding math. We recently came across a simple site that tries to help with the latter that deserves a look.
If you don’t know what we mean by that, consider a simple example. You can teach a kid that 5×3 is 15. But, hopefully, a teacher at some point in your academic career pointed out to you what the meaning of it was. That if you had five packages of three items, you have 15 items total. Or that if you have a room that is five feet on one side and three feet on the other, the square footage is 15 square feet.
Continue reading “Understanding Math vs Understanding Math”
When I say “siren” what do you think of? Ambulances? Air raids? Sigh. I was afraid you were going to say that. We’ve got work to do.
You see, the siren played an important role in physics and mathematics about 150 years ago. Through the first half of the 1900s, this fine apparatus was trivialized, used for its pure noise-making abilities. During the World Wars, the siren became associated with air raids and bomb shelters: a far cry from its romantic origins. In this article, we’re going to take the siren back for the Muses. I want you to see the siren in a new light: as a fundamental scientific experiment, a musical instrument, and in the end, as a great DIY project — this is Hackaday after all.
Continue reading “Drawn in by the Siren’s Song”
Here’s a really quick video which takes a different approach to understanding the Fourier Series than we’re used to. If you’re a regular reader we’re sure you’ve heard of the Fourier Series (often discussed as FFT or Fast Fourier Transform), but there’s a good chance you know little about it. The series allows you to break down complex signals (think audio waves) into combinations of simple sine or cosine equations which can be handled by a microcontroller.
We’ve had that base level of understanding for a long time. But when you start to dig deeper we find that it becomes a math exercise that isn’t all that intuitive. The video clip embedded after the break changes that. It starts off by showing a rotating vector. Mapping the tip of that vector horizontally will draw the waveform. The Fourier Series is then leveraged, adding spinning vectors for the harmonics to the tip of the last vector. The result of summing these harmonics produces the sine-based square wave approximation seen above.
That’s a mouthful, and we’re sure you’ll agree that the video demo is much easier to understand. But the three minute clip just scratches the surface. If you’re determined to master the Fourier Series give this mammoth Stanford lecture series on the topic a try.
Continue reading “Retrotechtacular: The Fourier Series”