The discrete cosine transform. (English) Zbl 0939.42021

In this paper the author derives the Discrete Cosine Transform (DCT) bases as eigenvectors of a symmetric second-difference matrix with certain boundary conditions. The type of boundary condition (Dirichlet or Neumann, centered at a meshpoint or a midpoint) determines the applications that are appropriate for each transform. The noteworthy observation is that all these “eigenvectors of cosines” come from simple and familiar matrices. More specific, each matrix contains a circular block between the second and before the last row, constructed with \((-1,2,-1)\) around the diagonal. The first and last row are different for each particular boundary condition.


42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
94A11 Application of orthogonal and other special functions
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
65T60 Numerical methods for wavelets
Full Text: DOI