What is DFT 1

DFT - Discrete Fourier Transform

In the following article, the DFT (Discrete Fourier Transformation) explained and their location in the Fourier analysis set out. Furthermore, important properties of the DFT are shown as well as the IDFT (Inverse DFT) explained.

If that is too detailed for you, we have the most important information about the Discrete Fourier Transformation for you in one clear Videosummarized.

DFT, Fourier Transformation and Fourier Series

The DFT (Discrete Fourier Transformation) is a transformation from the field of Fourier analysis.

She is performing an operation to determine the Frequency spectrum of a finite discrete signal.

The DFT is related to the continuous Fourier transformation and can be based on considerations Fourier series derive.

Fourier transform

With the help of Fourier transformation can be non-periodic functions by a Linear combination from sine and cosine functions of various Frequencies represent. For this the Fourier transform assigns any frequency the amplitude with which the oscillation of this frequency occurs in the linear combination:

The non-periodic functions can then go through the Fourier integral - the so-called inverse Fourier transform - can be approximated:

This summarizes the Exponential function the sine and cosine oscillations together.

Fourier series

For periodic functionsf with the period T the frequencies involved in the linear combination are discrete. You are a n-fold of the base frequency and will be with designated. The integral thus becomes the Fourier series,

where the Amplitudes represent the analog of the Fourier transform. They can be calculated as follows:

,

By determining these amplitudes or the Fourier transform, the functions under consideration become so-called Frequency spectrum disassembled.

Definition: Discrete Fourier Transform

With the help of the Fourier transformation and the Fourier series, the frequency spectra of non-periodic and periodic functions can be determined. In practice, however, the signals to be examined are not available as functions, but as Set of discrete values. The Discrete Fourier Transform deals with the Spectral analysis such discrete finite signals which are periodically continued for investigation.

The will be examined complex values that represent the signal values ​​and in a vector can be summarized.

The Discrete Fourier Transform of Vector owns the Coefficients

For efficient computation of the discrete Fourier transformation there are FFT (Fast Fourier Transformation) an algorithm.

Relationship Between Discrete Fourier Transforms and Fourier Series

The DFT is closely related to the theory of the Fourier series or it can be derived from it. The individual values ,, of the to be examined discreet and finite signal as equidistant Function values ​​of a -periodic function viewed:

With

For such a -periodic function f a Fourier series can be developed and the Fourier coefficients can be calculated as already mentioned by the following integral:

,

The limits of the integral can also be shifted. The only important thing is that it has a complete Period duration is integrated. The Coefficients can also be calculated as follows:

,

Here are the frequencies a multiple of the basic frequency:

In the case of the DFT, however, not all continuous function values ​​are known, only those discrete readings. This means that this integral cannot be calculated. Hence the integral is replaced by a Riemann sum approximated: the function is at the support points evaluated, with the difference between two support points multiplied and this product is over the running index k summed up. The following results Riemann sum:

This corresponds to the calculation formula of the n-th coefficient the Discrete Fourier Transform of the vector .

Looking at the formula for calculating the coefficient so it is noticeable that this is the coefficient corresponds, which is why the discrete Fourier transform only between and is defined:

From the continuous to the discrete Fourier transformation

Also for continuous Fourier transform the DFT has great similarities. To make this clear, we use a non-periodic function considered those outside the interval disappears and on this in the distance each select an entry as a function value assumes:

, For

Now be the number of values ​​examined N significantly larger than the length of the selected interval , then the integral of Fourier transform the function f Approach sensibly by a discrete sum:

Will this sum only for calculated, it results:

That corresponds to the constant factor T the calculation formula of the n-th entry the Discrete Fourier Transform .

IDFT: Inverse Discrete Fourier Transformation

To get to the formula of Inverse DFT to get there can be the Fourier series same periodic function which has been described above.

In the case of the Discrete Fourier Transform, the coefficients through the entries of the Discrete Fourier Transform. Since this is only for are defined, the index only runs between these two limits. Accordingly, one calculates this series for the values and for the frequencies the following result is obtained:

Definition: IDFT

The coefficients are thus calculated the Inverse Discrete Fourier Transform of after renaming the indices in the following way:

To summarize: the coefficients the Discrete Fourier Transform of ring:

For the prefactor There are different conventions for the Discrete Fourier Transformation. Sometimes it is inserted into the inverse DFT instead of the DFT.

Overview of Fourier Analysis