Fourierbased

Fourierbased, or Fourier-based, is an adjective used to describe methods, models, or analyses that rely on Fourier analysis and the transformation of data into the frequency domain. The term encompasses both continuous and discrete formulations, and is commonly used in engineering and applied mathematics.

Core idea: a signal is expressed as a sum or integral of sinusoids; this makes certain operations, like convolution, easier via the convolution theorem; often implemented via the discrete Fourier transform (DFT) and the fast Fourier transform (FFT).

Applications: in signal processing for denoising, filtering, and compression; in image processing for frequency-domain filtering and pattern recognition; in scientific computing for solving differential equations with spectral methods; in spectroscopy and communications; in medical imaging such as MRI.

Advantages and limitations: Fourierbased methods provide a global representation, offer linearity, and benefit from efficient algorithms; they assume or work best with stationary signals and periodic boundaries, and can produce edge artifacts or misrepresent localized features. For non-stationary data, time-frequency methods or wavelets are often used.

Historical note: the Fourier transform was developed by Jean-Baptiste Joseph Fourier in the early 19th century, with later refinements such as the discrete Fourier transform and the fast Fourier transform enabling practical computation on digital data.