12. Discrete Fourier Transform

By Sadasivan Puthusserypady, Technical University of Denmark, Denmark

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Published: 31 Mar 2021

© 2021 Sadasivan Puthusserypady


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