View source for Number theoretic Hilbert transform

The '''number theoretic Hilbert transform''' is an extension<ref>* {{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 }}</ref> of the discrete [[Hilbert transform]] to [[modular arithmetic|integers modulo a prime <math>p</math>]].  The transformation operator is a [[circulant matrix]].

The number theoretic transform is meaningful in the [[ring (mathematics)|ring]] <math>\mathbb{Z}_m</math>, when the modulus <math>m</math> is  not prime, provided a principal root of order ''n'' exists. 
The <math>n\times n</math> NHT matrix, where <math>n =2m</math>, has the form

:<math>
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}.
</math>

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:<math>NHT^\mathrm{T} NHT = NHT NHT^\mathrm{T} = I \bmod\ p, \,</math> 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]].<ref>{{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]</ref> Other ways to generate constrained orthogonal sequences also exist.<ref>Donelan, H. (1999). Method for generating sets of orthogonal sequences. Electronics Letters 35: 1537-1538.</ref><ref>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.</ref>

==References==
{{reflist|}}

== See also ==
* [[Number theoretic transform]]

[[Category:Signal processing]]