Number theoretic Hilbert transform

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

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

[math]\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}. }[/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]\displaystyle{ 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.[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, 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 [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

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