Discrete Integer Fourier Transform in Real Space: Elliptic Fourier Transform

The concept of the N-point DFT is generalized, by considering it in the real space (not complex). The multiplication by twiddle coefficients is considered in matrix form; as the Givens transformation. Such block-wise representation of the matrix of the DFT is effective. The transformation which is called the T-generated N-block discrete transform, or N-block T-GDT is introduced. For each N-block T-GDT, the inner product is defined, with respect to which the rows (and columns) of the matrices X are orthogonal. By using different parameterized matrices T, we define metrics in the real space of vectors. The selection of the parameters can be done among only the integer numbers, which leads to integer-valued metric. We also propose a new representation of the discrete Fourier transform in the real space R(2N). This representation is not integer, and is based on the matrix C (2x2) which is not a rotation, but a root of the unit matrix. The point (1, 0) is not moving around the unite circle by the group of motion generated by C, but along the perimeter of an ellipse. The N-block C-GDT is therefore called the N-block elliptic FT (EFT). These orthogonal transformations are parameterized; their properties are described and examples are given.