The DFT is the most important discrete transform, used to perform Fourier analysis in many practical applications. If the original sequence is one cycle of a periodic function, the DFT provides all the non-zero values of one DTFT cycle. If the original sequence spans all the non-zero values of a function, its DTFT is continuous (and periodic), and the DFT provides discrete samples of one cycle. The DFT is therefore said to be a frequency domain representation of the original input sequence. It has the same sample-values as the original input sequence. An inverse DFT (IDFT) is a Fourier series, using the DTFT samples as coefficients of complex sinusoids at the corresponding DTFT frequencies. The interval at which the DTFT is sampled is the reciprocal of the duration of the input sequence. In mathematics, the discrete Fourier transform ( DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency.

Its similarities to the original transform, S(f), and its relative computational ease are often the motivation for computing a DFT sequence. The respective formulas are (a) the Fourier series integral and (b) the DFT summation. The spectral sequences at (a) upper right and (b) lower right are respectively computed from (a) one cycle of the periodic summation of s(t) and (b) one cycle of the periodic summation of the s(nT) sequence. Fig 2: Depiction of a Fourier transform (upper left) and its periodic summation (DTFT) in the lower left corner.

Matlab convert 2d image to 3d