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]

References

1. • Kak, Subhash (2014), "Number theoretic Hilbert transform", Circuits Systems Signal Processing, 33 (8): 2539–2548, arXiv:1308.1688, doi:10.1007/s00034-014-9759-8
2. Kak, Subhash (2015), "Orthogonal residue sequences", Circuits Systems Signal Processing, 34 (3): 1017–1025, doi:10.1007/s00034-014-9879-1
3. Donelan, H. (1999). Method for generating sets of orthogonal sequences. Electronics Letters 35: 1537-1538.
4. Appuswamy, R., Chaturvedi, A.K. (2006). A new framework for constructing mutually orthogonal complementary sets and ZCZ sequences. IEEE Trans. Inf. Theory 52: 3817-3826.

See also

• Number theoretic transform