Fourier transform is widely used in physics, electronics, number theory, combinatorial mathematics, signal processing, probability theory, statistics, cryptography, acoustics, optics, oceanography, structural dynamics and other fields.

For example, in signal processing, the typical use of Fourier transform is to decompose the signal into frequency spectrum-the amplitude corresponding to the display frequency.

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The basic contents of signal processing include transformation, filtering, modulation, demodulation, detection, spectrum analysis and estimation. Such as Fourier transform, sine transform, cosine transform, Walsh transform, etc. Filtering includes Qualcomm filtering, low-pass filtering, band-pass filtering, Wiener filtering, Kalman filtering, linear filtering, nonlinear filtering and adaptive filtering.

Spectrum analysis includes deterministic signal analysis and random signal analysis. Usually, the most common research is random signal analysis, also known as statistical signal analysis or estimation, which is usually divided into linear spectrum estimation and nonlinear spectrum estimation.

Spectral estimation includes periodic graph estimation, maximum entropy spectral estimation and so on. Because of the complexity of signal types, when the analyzed signal cannot meet the Gaussian distribution and non-minimum phase conditions, there is a high-order spectral analysis method.

Higher-order spectral analysis can provide phase information, non-Gaussian information and nonlinear information of signals. Adaptive filtering and equalization are also important areas of application research. Adaptive filtering includes horizontal LMS adaptive filtering, lattice adaptive filtering, adaptive cancellation filtering and adaptive equalization filtering. In addition, there is array signal processing.