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In signal processing and data analysis, the DFT is widely used to convert a time-domain signal into its frequency-domain representation, which provides valuable information about the signal's frequency components. This Wikipedia page on the Discrete Fourier Transform provides an overview of the concept, its mathematical formulation, and properties. The page explains the historical context of the DFT and discusses its relation to the more general Fourier Transform. The mathematical definition of the DFT along with the inverse DFT is given, demonstrating how the transform can be applied to a given sequence of samples. The article also explores various properties of the DFT, such as linearity, time and frequency shifting, convolution, and correlation theorem. It describes the fast Fourier transform (FFT), an efficient algorithm that can compute the DFT in a much shorter time compared to the traditional direct approach. The FFT has become a popular technique in many applications due to its computational efficiency. Additionally, the Wikipedia page covers applications of the DFT in various fields, including signal processing, image processing, data compression, and noise analysis. Examples of DFT applications are provided, such as the analysis of periodic and non-periodic signals, frequency-domain filtering, and spectrum analysis. The Discrete Fourier Transform page also provides links to related topics such as the continuous Fourier transform, discrete cosine transform, and other mathematical concepts related to the DFT. Overall, this Wikipedia page serves as a comprehensive resource for understanding the Discrete Fourier Transform, its applications, and its role in signal and data analysis.