Why are fast Fourier transforms so important
The Fourier transform is the method for determining the Fourier transform. This plays an essential role in breaking down a non-periodic output function in trigonometric functions With different frequencies. The Fourier transform describes the so-called Frequency spectrum, i.e. it assigns the appropriate amplitude to each frequency for the desired decomposition.
In this article we will show you using one example, as the Fourier transformation works and go on that application a. At the end we will show you a clear one again table to Fourier transformation. The main points are also in oursVideosummarized. Look inside!
Fourier transformation explained simply
In the Fourier analysis is the general goal of a given function as a linear combination of the periodic Sine- and Cosine functions to describe.
There should generally be two types of the given function A distinction is made between: periodic functions and non-periodic functions.
Periodic functions: Fourier series
One function is called periodic with period T. designated, if for all applies: . Such a T-periodic function lets itself be on the interval under certain conditions well through the following trigonometric series approximate:
The development coefficients are calculated and as follows:
Becomes referred to as the base frequency, so is represented by a sum of sine and cosine functions of different amplitudes, the frequencies of which are positive integer multiples the fundamental frequency are.
Are the trigonometric functions using Euler's formula by
replaced, the result is the following, somewhat clearer representation of this trigonometric series:
The coefficients are calculated as follows:
The series is called Fourier series the T-periodic function designated. If this function is continuously differentiable piece by piece, the Fourier series converges uniformly to . Is but only steadily in so the Fourier series only converges to the function in terms of the root mean square (Dirichlet condition or Dirichlet's theorem).
By determining the Fourier series, a so-called frequency spectrum is obtained, which corresponds to each frequency the Amplitude of the oscillation assigns. In acoustics, a noise that is made up of tones of different frequencies can be broken down into its individual tones with an indication of their frequency and amplitude.
However, the function was initially mentioned the important prerequisite that it must be periodic, which is why an analysis of the Frequency spectrum in this way is only possible for periodic signals. Since such signals rarely occur in practice, it is of interest to be able to determine the frequency spectrum for non-periodic functions as well.
Non-periodic functions: Fourier integral
The aim of the Fourier transformation is to be able to approximate non-periodic functions by linear combination of the trigonometric functions. For this, the knowledge about T-periodic functions is used and passed on. The basic idea here is that it is a non-periodic function is understood as a periodic function, the period of which, however, is infinitely large. This leads to the following phenomenon: During the distance two neighboring frequencies and for a finite period T exactly this distance disappears in the borderline case :
This means that through the transition from a finite period T to an infinitely large period a discrete frequency spectrum becomes a continuous frequency spectrum.
The more precise consequences of the transition should now be based on the Fourier series to be viewed as. This is done in the formula first the amplitude used and it results:
By establishing the relationship follows further:
Will now by replaced, the following expression is obtained:
This represents a Riemann sum with the decomposition if one now looks at the Border crossing, so applies as already shown and from the discrete values becomes a function:
It is also in the border crossing from the sum an integral and this is the result Fourier integral:
Fourier transformation formula
If one compares the Fourier integral derived above with the original formulation of the Fourier series, certain analogies become visible.
While for periodic functions to the discrete frequencies the amplitude heard, for non-periodic functions there is a Amplitude function depending on the continuous frequencies . The function is called Fourier transform of designated. It represents the frequency spectrum of the function and the Fourier transform is nothing other than the operation of its determination. An alternative notation for the Fourier transform is . The Reverse operation the Fourier transform is the so-called inverse Fourier transform and the Fourier integral is also known as the inverse Fourier transform.
Definition: Fourier transform and Fourier integral
A - Space is defined as follows:
For a function
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