4 ms·

One thing I don't fathom is why FT is always explained in terms of circles? For me it was always confusing this way; the concept was much more graspable when vi

One thing I don't fathom is why FT is always explained in terms of circles? For me it was always confusing this way; the concept was much more graspable when visualized in the terms superposition of sinusoidal waves.

waldir 12y ago
Because those sinusoidal waves are merely projections (in the real and complex plane) of the true elemental component of the FT, the complex exponential, which is really a helix: http://m.eet.com/media/1068017/lyons_pt2_3.gif http://m.eet.com/media/1068017/lyons_pt2_3.gif A while ago I made some drafts for a series of diagrams to help visualize this (http://imgur.com/vEcnVdn http://imgur.com/vEcnVdn) but unfortunately never got around to finish it (https://en.wikipedia.org/wiki/Wikipedia:Graphics_Lab/Illustration_workshop/Archive/Sep_2013#RfC_-_Visualization_of_exponential_function https://en.wikipedia.org/wiki/Wikipedia:Graphics_Lab/Illustr...). Anyway, for those who find it easier to think of sinusoidal curves, the animation in the Wikipedia article (https://commons.wikimedia.org/wiki/File:Fourier_transform_time_and_frequency_domains_(small).gif https://commons.wikimedia.org/wiki/File:Fourier_transform_ti...) is a very good visualization (also, BetterExplained's rant on sines being explained as circles may resonate: http://betterexplained.com/articles/intuitive-understanding-of-sine-waves/ http://betterexplained.com/articles/intuitive-understanding-...)
smosher_ 12y ago
To be fair, the expression does contain (edit: ) Euler's Formula, which is basically drawing circles in the complex plane, so people are going to draw circles. I started thinking about sinusoids this way after seeing stuff like this: https://3.bp.blogspot.com/_6t_ZmJSkbL4/TJLIDoBQMzI/AAAAAAAACb8/WEbPI4KjoJQ/s1600/sinewave_spiral.gif https://3.bp.blogspot.com/_6t_ZmJSkbL4/TJLIDoBQMzI/AAAAAAAAC... — if you look down at the complex plane, you will see a circle.
bluecalm 12y ago
Sine an cosine are functions which tells you what your coordinates are when you travel around a circle (of diameter 1) so anything containing those (here hidden in Euler's formula) will provoke thinking in terms of circles. I find it very useful btw. Once you realize sin(x) represents how high you are on a circle and cos(x) how far to the right (if you start walking on it from [0,1] counter clockwise) then a lot of stuff start making intuitive sense.

waldir 12y ago

Because those sinusoidal waves are merely projections (in the real and complex plane) of the true elemental component of the FT, the complex exponential, which is really a helix: http://m.eet.com/media/1068017/lyons_pt2_3.gif http://m.eet.com/media/1068017/lyons_pt2_3.gif A while ago I made some drafts for a series of diagrams to help visualize this (http://imgur.com/vEcnVdn http://imgur.com/vEcnVdn) but unfortunately never got around to finish it (https://en.wikipedia.org/wiki/Wikipedia:Graphics_Lab/Illustration_workshop/Archive/Sep_2013#RfC_-_Visualization_of_exponential_function https://en.wikipedia.org/wiki/Wikipedia:Graphics_Lab/Illustr...). Anyway, for those who find it easier to think of sinusoidal curves, the animation in the Wikipedia article (https://commons.wikimedia.org/wiki/File:Fourier_transform_time_and_frequency_domains_(small).gif https://commons.wikimedia.org/wiki/File:Fourier_transform_ti...) is a very good visualization (also, BetterExplained's rant on sines being explained as circles may resonate: http://betterexplained.com/articles/intuitive-understanding-of-sine-waves/ http://betterexplained.com/articles/intuitive-understanding-...)

smosher_ 12y ago

To be fair, the expression does contain (edit: ) Euler's Formula, which is basically drawing circles in the complex plane, so people are going to draw circles. I started thinking about sinusoids this way after seeing stuff like this: https://3.bp.blogspot.com/_6t_ZmJSkbL4/TJLIDoBQMzI/AAAAAAAACb8/WEbPI4KjoJQ/s1600/sinewave_spiral.gif https://3.bp.blogspot.com/_6t_ZmJSkbL4/TJLIDoBQMzI/AAAAAAAAC... — if you look down at the complex plane, you will see a circle.

bluecalm 12y ago

Sine an cosine are functions which tells you what your coordinates are when you travel around a circle (of diameter 1) so anything containing those (here hidden in Euler's formula) will provoke thinking in terms of circles. I find it very useful btw. Once you realize sin(x) represents how high you are on a circle and cos(x) how far to the right (if you start walking on it from [0,1] counter clockwise) then a lot of stuff start making intuitive sense.