Number theoretic Hilbert transform explained

The number theoretic Hilbert transform is an extension of the discrete Hilbert transform to integers modulo a prime

p . The transformation operator is a circulant matrix.

Z _{m} , when the modulus m is not prime, provided a principal root of order n exists. The n x n NHT matrix, where n =2 m , has the formNHT = \begin{bmatrix} 0 & a _{m} & ... & 0 & a _{1} \\ a _{1} & 0 & a _{m} & & 0 \\ \vdots & a _{1} & 0 & \ddots & \vdots \\ 0 & & \ddots & \ddots & a _{m} \\ a _{m} & 0 & ... & 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 ^{T} NHT = NHT NHT ^{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.^[1] Other ways to generate constrained orthogonal sequences also exist.^{[2] [3]}

See also

• Number theoretic transform

Notes and References

1. https://link.springer.com/article/10.1007%2Fs00034-014-9879-1#page-1
2. Donelan, H. (1999). Method for generating sets of orthogonal sequences. Electronics Letters 35: 1537-1538.
3. 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.