Fourier transform and DFT discrete Fourier transform

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DFT – Discrete Fourier Transform calculator

Purpose of Fourier Transform as well as DFT – Discrete Fourier Transform is to transform input signal (analog or digital) to sum of sine and cosine coefficients. Schematically it is presented in figure bellow.

DFT-Transform

Fourier Transform decompose analog signal into sum of sine and cosine signals on frequencies that are multiplies of main frequency. Formal definition of Fourier Transform for integrable function x(t) is given in complex form

dft_eq01

In engineering practice, analog signal that is transformed with Fourier Transform has to be periodic. If signal is not periodic, but limited to certain time interval, signal can be repeated, so formally can be treated as periodic one. In that case cosine and sine functions in the sum that represents input signals, has frequencies of n·ω0, where ω0=2πf0 is angular frequency of mains. Amplitudes of cosine coefficients can be calculated as

dft_eq02

For DC component, Fourier cosine coefficient is

dft_eq03

Since

dft_eq04

amplitudes of sine coefficients can be uniformly calculated for both n=0 and n>0

dft_eq05

DFT i.e. Discrete Fourier Transform is Fourier Transform adjusted to digital (sampled signal). In case of digital input signal, analog integral is replaced by finite sum:

dft_eq06

Amplitudes of cosine DFT – Discrete Fourier Transform can be calculated as

dft_eq07

DFT DC cosine coefficient is

dft_eq08

Amplitudes of sine DFT – Discrete Fourier Transform can be calculated as

dft_eq09

Discrete Fourier Transform coefficients can be used for signal compression, signal processing in frequency domain and signal interpolation.

DFT – Discrete Fourier Transform calculator

External links:

Fourier Transform on Wikipedia
Fourier Transform on Mathworld
Fourier Transform on Thefouriertransform
DFT – Discrete Fourier Transform on Wikipedia
DFT – Discrete Fourier Transform on Mathworld

 

 

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