Number theoretic Hilbert transform

The number theoretic Hilbert transform is an extension^[1] of the discrete Hilbert transform to integers modulo a prime ${\displaystyle p}$. The transformation operator is a circulant matrix.

The number theoretic transform is meaningful in the ring ${\displaystyle \mathbb {Z} _{m}}$, when the modulus ${\displaystyle m}$ is not prime, provided a principal root of order n exists. The ${\displaystyle n\times n}$ NHT matrix, where ${\displaystyle n=2m}$, has the form

${\displaystyle NHT={\begin{bmatrix}0&a_{m}&\dots &0&a_{1}\\a_{1}&0&a_{m}&&0\\\vdots &a_{1}&0&\ddots &\vdots \\0&&\ddots &\ddots &a_{m}\\a_{m}&0&\dots &a_{1}&0\\\end{bmatrix}}.}$

The rows are the cyclic permutations of the first row, or the columns may be seen as the cyclic permutations of the first column. The NHT is its own inverse:${\displaystyle NHT^{\mathrm {T} }NHT=NHTNHT^{\mathrm {T} }=I{\bmod {\ }}p,\,}$ where I is the identity matrix.

The number theoretic Hilbert transform can be used to generate sets of orthogonal discrete sequences that have applications in signal processing, wireless systems, and cryptography.^[2] Other ways to generate constrained orthogonal sequences also exist.^[3][4]