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| View Original: | Continuous Fourier transform of rect and sinc functions.gif (500x375) | |||
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| Keywords: Continuous Fourier transform of rect and sinc functions.gif Fourier transform of the rect function and sinc functions 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 In the first part of the animation the Fourier transform as usually defined in signal processing is applied to the rectangular function returning the normalized sinc function In the second part 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 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 In this example it also happens that there are no imaginary/sine components so only the real/cosine components are displayed Shown at left overlaid on the red time domain curve there's 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 integrating along the entire surface with the appropriate amplitude correction due to the specific Fourier transform and ranges used own 2013-03-03 Lucas V Barbosa HD video and a variant with square wave with discrete frequencies File Fourier_transform_time_and_frequency_domains gif Fourier analysis Fourier transformation Mathematical animations Sinc function | ||||
