Kosinustransform

Kosinustransform is a generalized cosine-based transform used in signal processing and data analysis to represent a function as a weighted sum of cosine basis functions. It is inspired by classical transforms such as the discrete cosine transform and the Fourier transform, but is defined with additional flexibility through parameters that control sampling, phase, and weighting. In practice, kosinustransform variants aim to adapt the basis to nonuniform data or to emphasize particular spectral components.

In a discrete form, the kosinustransform coefficients K[k] can be written as K[k] = sum over n of w[n] f[n] cos(ω_k (n + α) + β_k), where f[n] is the input sequence, w[n] is a nonnegative weight sequence, ω_k are chosen angular frequencies, α is a shift parameter, and β_k is a phase parameter. A continuous form exists as K(ω) = ∫ w(t) f(t) cos(ω t + φ) dt. The corresponding basis functions are cos(ω_k (n + α) + β_k).

Invertibility and reconstruction depend on the chosen parameters: if the basis is orthogonal with respect to the weight w[n], the original signal can be recovered by a parallel sum of the coefficients times the basis functions, with appropriate normalization constants. This makes kosinustransform useful for exact or near-exact reconstruction under suitable conditions.

Variants of the kosinustransform include setting α, β_k, and w[n] to recover known transforms. For example, with w[n] = 1, α = 0, β_k = 0, and ω_k = π k / N, the kosinustransform reduces to a discrete cosine transform. Nonuniform sampling, adaptive weighting, or phase adjustments yield generalized forms suited to specific data characteristics.

Applications encompass data compression, denoising, feature extraction, and spectral methods for solving differential equations. The term appears in several research discussions as a flexible, generalized cosine-based transform, though there is no universally standardized definition.