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.
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
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 nÂ·Ï‰0, where Ï‰0=2Ï€f0 is angular frequency of mains. Amplitudes of cosine coefficients can be calculated as
For DC component, Fourier cosine coefficient is
amplitudes of sine coefficients can be uniformly calculated for both n=0 and n>0
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:
Amplitudes of cosine DFT – Discrete Fourier Transform can be calculated as
DFT DC cosine coefficient is
Amplitudes of sine DFT – Discrete Fourier Transform can be calculated as
Discrete Fourier Transform coefficients can be used for signal compression, signal processing in frequency domain and signal interpolation.