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Binomial transform; Discrete Fourier transform, DFT Fast Fourier transform, a popular implementation of the DFT; Discrete cosine transform. Modified discrete cosine transform; Discrete Hartley transform; Discrete sine transform; Discrete wavelet transform; Hadamard transform (or, Walsh–Hadamard transform) Fast wavelet transform
A discrete cosine transform (DCT) expresses a finite sequence of data points in terms of a sum of cosine functions oscillating at different frequencies. The DCT, first proposed by Nasir Ahmed in 1972, is a widely used transformation technique in signal processing and data compression .
Clenshaw–Curtis quadrature is essentially a change of variables to cast an arbitrary integral in terms of integrals of periodic functions where the Euler–Maclaurin approach is very accurate (in that particular case the Euler–Maclaurin formula takes the form of a discrete cosine transform). This technique is known as a periodizing ...
In addition to spectral analysis of signals, discrete transforms play important role in data compression, signal detection, digital filtering and correlation analysis. [2] The discrete cosine transform (DCT) is the most widely used transform coding compression algorithm in digital media, followed by the discrete wavelet transform (DWT).
The sine-only expansion for equally spaced points, corresponding to odd symmetry, was solved by Joseph Louis Lagrange in 1762, for which the solution is a discrete sine transform. The full cosine and sine interpolating polynomial, which gives rise to the DFT, was solved by Carl Friedrich Gauss in unpublished work around 1805, at which point he ...
Discrete sine and cosine transforms: When the input sequence has odd or even symmetry around the origin, the DTFT reduces to a discrete sine transform (DST) or discrete cosine transform (DCT). Regressive discrete Fourier series, in which the period is determined by the data rather than fixed in advance.
By applying Euler's formula (= + ), it can be shown (for real-valued functions) that the Fourier transform's real component is the cosine transform (representing the even component of the original function) and the Fourier transform's imaginary component is the negative of the sine transform (representing the odd component of the ...
The Fourier series is an example of a trigonometric series. [2] By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Joseph Fourier to find solutions to the heat equation. This ...