Number theoretic Hilbert transform

The number theoretic Hilbert transform is an extension* {{citation|last=Kak |first=Subhash |authorlink=Subhash Kak|title=Number theoretic Hilbert transform|journal=Circuits, Systems and Signal Processing |volume= 33 |issue=8 |year=2014|pages= 2539–2548|doi=10.1007/s00034-014-9759-8 |arxiv=1308.1688 |s2cid=253639606 }} 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 $\backslash mathbb\{Z\}\_m$, when the modulus $m$ is not prime, provided a principal root of order n exists.

The $n\backslash times\; n$ 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:$NHT^\backslash mathrm\{T\}\; NHT\; =\; NHT\; NHT^\backslash mathrm\{T\}\; =\; I\; \backslash bmod\backslash \; p,\; \backslash ,$ 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.{{citation|last=Kak |first=Subhash |authorlink=Subhash Kak|title=Orthogonal residue sequences|journal=Circuits, Systems and Signal Processing |volume= 34 |issue=3 |year=2015|pages= 1017–1025|doi=10.1007/s00034-014-9879-1 |s2cid=253636320 }} [https://link.springer.com/article/10.1007%2Fs00034-014-9879-1#page-1] Other ways to generate constrained orthogonal sequences also exist.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.