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”
The Fourier transform underpins so much of our technological lives, in most cases probably without our realising it. The ability to mathematically split a waveform into its frequency components and vice versa underpins much of the field of digital signal processing, and DSP has become an essential part of many electronic devices we take for granted.
But while most of us will know what a Fourier transform is, fewer of us will know anything of how one works. They are a function called from a library rather than performed in themselves. Even when they are taught in schools or university courses they remain something that not all students “get”, and woe betide you if (as your scribe did) you have a sub-par maths lecturer.
The video below the break then is very much worth a look if Fourier transforms are a bit of a mystery to you. In it [Grant Sanderson] explains them through a series of simple graphical examples in a style that perhaps may chalk-and-talk mathematics teachers should emulate. You may still only use Foruier transforms through a library, but after watching this video perhaps some of their mysteries will be revealed.
Continue reading “All The Stuff You Wished You Knew About Fourier Transforms But Were Afraid To Ask”
Every machine has its own way of communicating with its operator. Some send status emails, some illuminate, but most of them vibrate and make noise. If it hums happily, that’s usually a good sign, but if it complains loudly, maintenance is overdue. [Ariel Quezada] wants to make sense of machine vibrations and draw conclusions about their overall mechanical condition from them. With his project, a 3-axis Open Source FFT Spectrum Analyzer he is not only entering the Hackaday Prize 2016 but also the highly contested field of acoustic defect recognition.
For the hardware side of the spectrum analyzer, [Ariel] equipped an Arduino Nano with an ADXL335 accelerometer, which is able to pick up vibrations within a frequency range of 0 to 1600 Hz on the X and Y axis. A film container, equipped with a strong magnet for easy installation, serves as an enclosure for the sensor. The firmware [Ariel] wrote is an efficient piece of code that samples the analog signals from the accelerometer in a free running loop at about 5000 Hz. It streams the digitized waveforms to a host computer over the serial port, where they are captured and stored by a Python script for further processing.
From there, another Python script filters the captured waveform, applies a window function, calculates the Fourier transform and plots the spectrum into a graph. With the analyzer up and running, [Ariel] went on testing the device on a large bearing of an arbitrary rotating machine he had access to. A series of tests that involved adding eccentric weights to the rotating shaft shows that the analyzer already makes it possible to discriminate between different grades of imbalance.