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A set of mathematical formulas used to convert a time function, such as a seismic trace, to a function in the frequency domain (Fourier analysis) and back (Fourier synthesis). The function is expressed as a convergent trigonometric series, similar to that first formulated by French mathematician Jean-Baptiste-Joseph, Baron Fourier (1768 to 1830). The Fourier transform is used extensively in signal processing to design filters and remove coherent noise. Many filtering operations are performed in the frequency domain. The Fourier transform has applications in image analysis and in pattern recognition in geological systems.