Fourier TransformNote 1
: This gif is long ~ 44sec which might take time to load on slower connections.Note 2
: The gif is self-explanatory if you're familiar with the concept. If not the below article contains basic explanation and instruction on how to read/understand the gif. And also to the unfamiliar one, I really encourage you to research on the topic yourself because its very interesting. First of all, what is Fourier Transformation?
An intro:Virtually everything in the world can be described via a waveform - a function of time, space or some other variable. For instance, sound waves, electromagnetic fields, the elevation of a hill versus location, a plot of VSWR versus frequency, the price of your favourite stock versus time, etc.
The Fourier Transform gives us a unique and powerful way of viewing these waveforms. All waveforms, no matter what you scribble or observe in the universe, are actually just the sum of simple sinusoids of different frequencies.
More info at thefouriertransform.com/transform/fourier.phpStarting with Wikipedia’s definition on Fourier transform:The Fourier transform (English pronunciation: /ˈfɔərieɪ/), named after Joseph Fourier, is a mathematical transformation employed to transform signals between time(or spatial) domain and frequency domain, which has many applications in physics and engineering. It is reversible, being able to transform from either domain to the other. The term itself refers to both the transform operation and to the function it produces.In the case of a periodic function over time (for example, a continuous but not necessarily sinusoidal musical sound), the Fourier transform can be simplified to the calculation of a discrete set of complex amplitudes, called Fourier series coefficients. They represent the frequency spectrum of the original time-domain signal. Also, when a time-domain function is sampled to facilitate storage or computer-processing, it is still possible to recreate a version of the original Fourier transform according to the Poisson summation formula, also known as discrete-time Fourier transform.And a ‘simple’ and ‘not so number and equation filled’ explanation.
The Fourier Transform is a tool that breaks a waveform (a function or signal) into an alternate representation, characterized by sine and cosines. The term "Fourier transform" refers to both the transform operation and to the complex-valued function it produces.
In the case of a periodic function the Fourier transform can be simplified to the calculation of a discrete set of complex amplitudes, called Fourier series coefficients.
Moreover, the Fourier series is used to represent a periodic function by a discrete sum of complex exponentials, while the Fourier transform is then used to represent a general, non-periodic function by a continuous superposition or integral of complex exponentials. The Fourier transform can be viewed as the limit of the Fourier series of a function with the period approaches to infinity, so the limits of integration change from one period to (-infinity, infinity).
Sound is probably the easiest thing to think about when talking about Fourier transforms. If you could see sound, it would look like air molecules bouncing back and forth very quickly. But oddly enough, when you hear sound you’re not perceiving the air moving back and forth, instead you experience sound in terms of its frequencies. Just for “the more you know” purpose, Here’s the definition for *Fourier series
Fourier series is a way to REWRITE the signal, i.e. replicating it. You have to compute Fourier coefficients for that, using simple integral formulas, and after that you plug in those Fourier coefficients into the summation series
Fourier transform is the integral of a function using the formula f(t)exp(-jt) where j is the complex number. And just for “the more you know^2” purpose, here’s the definition forFourier analysisFourier analysis is the study of Fourier transform and Fourier series combined
The Fourier series is used to represent a periodic function by a discrete sum of complex exponentials, while the Fourier transform is then used to represent a general, non-periodic function by a continuous superposition or integral of complex exponentials. The Fourier transform can be viewed as the limit of the Fourier series of a function with the period approaches to infinity, so the limits of integration change from one period to (−∞, ∞).
In a classical approach it would not be possible to use the Fourier transform for a periodic function which cannot be in L1 (−∞, ∞).
The use of generalized functions, however, frees us of that restriction and makes it possible to look at the Fourier transform of a periodic function. It can be shown that the Fourier series coefficients of a periodic function are sampled values of the Fourier transform of one period of the function.Great article by 1ucasvb.tumblr.com and DETAILS about the gifThe continuous Fourier transform takes an input function f(x) in the domain and turns it into a new function, ƒ̂(x) in the frequency domain. (These can represent other things too, but that’s beside the point.)In the first animation, the Fourier transform (as usually defined in signal processing) is applied to the rectangular function, returning the normalized sinc function.In the second animation, 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. We say it is a 4-periodic automorphism.
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.
In this example, it also happens that there are no imaginary/sine components, so you’re looking at the real/cosine components only.
Shown at left, overlaid on the red time domain curve, you’ll notice 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, by integrating along the entire surface (this is continuous, remember?)
As we add more and more of the components, the approximation improves. In some special cases, it is exact. For the rect function, it isn’t, and you get some wavy artefacts in some places (the sudden jumps, aka discontinuities). These are due to Gibbs phenomenon, and are the main cause of ringing artifacts. As you’ll probably notice, the approximation is pretty much dead on for the sinc function, as shown in the second animation.The illustration shows the domains in the interval [-5, 5], but the Fourier transform extends infinitely to all directions, of course.The surface illustrated here isn’t too far off from the approach used in Fourier operators. If you consider the surfaces defined by z = cos (xy) and z = sin (xy), you get the cosine and sine Fourier operators. Having complex values lets you mix both into one thing.The surface you see in the first animation is just z = cos (2πxy) sinc (πy). The Fourier transform can be thought of as multiplying a function by these continuous operators, and integrating the result. This can be very neatly performed using matrix multiplication in the discrete cases. (New drinking game: take a shot every time linear algebra shows up in any mathematical discussion.)This also explains why the Fourier transform is cyclic after 4 iterations: rotating 90° four times gets you back to your original position. By using different rotation angles, you get fractional Fourier transforms. Awesome stuff.And finallyWhy is the Fourier Transform so important?http://dsp.stackexchange.com/questions/69/why-is-the-fourier-transform-so-important
Research material:http://www.thefouriertransform.com/http://betterexplained.com/articles/an-interactive-guide-to-the-fourier-transform/The continuous Fourier Transform of rect and sinc functions (animation) thefouriertransform.com/transform/fourier.php1ucasvb.tumblr.com