Number theoretic Hilbert transform

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

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

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: $NHT^{\mathrm {T} }NHT=NHTNHT^{\mathrm {T} }=I{\bmod {\ }}p,\,$ 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

^ * Kak, Subhash (2014), "Number theoretic Hilbert transform", Circuits, Systems and Signal Processing, 33 (8): 2539–2548, arXiv:1308.1688, doi:10.1007/s00034-014-9759-8, S2CID 253639606
^ Kak, Subhash (2015), "Orthogonal residue sequences", Circuits, Systems and Signal Processing, 34 (3): 1017–1025, doi:10.1007/s00034-014-9879-1, S2CID 253636320 [1]
^ Donelan, H. (1999). Method for generating sets of orthogonal sequences. Electronics Letters 35: 1537-1538.
^ 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.

References

See also