Visualizing the Discrete Fourier Transform (opens in new tab)

Rather than funny analogies about signals spinning on poles, I think it's relatively easy to understand what the DFT is with just two things that many people already know: 1) knowledge of the complex exponential definition of Fourier series [1], and 2) how to approximate integrals with the left rectangle rule [2].

Take your continuous signal and represent it with a Fourier series. Since the Fourier series is a linear decomposition into integer frequency sinusoids, the coefficients of the series tell you the amount of each frequency contained in the signal. The DFT gives you an approximation of these.

The coefficients of the Fourier series of a function are integrals. Approximate these integrals with a left Riemann sum. Integrals turn into sums ... sums turn into a linear system ... the linear system turns into a matrix ... bingobango there's the DFT matrix [3].

[1]: http://users.wpi.edu/~goulet/Matlab/overlap/efs.html

[2]: https://en.wikipedia.org/wiki/Riemann_sum#Left_Riemann_Sum

[3]: https://en.wikipedia.org/wiki/DFT_matrix

Why not link to the original site? http://www.billconnelly.net/?p=276

On a similar note, The Engineer Guy's walkthrough of Albert Michelson's Harmonic Analyzer gave me a much better understanding of Fourier analysis in general.

https://www.youtube.com/playlist?list=PL0INsTTU1k2UYO9Mck-i5...

I found this video to be an incredibly useful explanation of the Fourier Transform: https://www.youtube.com/watch?v=FjmwwDHT98c

I love the way they used color and plain English to describe that function. It would be awesome if a general app for this sort of color coded simple translation were available to help kids learn about mathematics. It'd have to be more inclusive for people with dichromacy and anamalous trichromacy, but there could be settings in the app to compensate for that.