Number theoretic Hilbert transform (original) (raw)

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

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

N H T = [ 0 a m … 0 a 1 a 1 0 a m 0 ⋮ a 1 0 ⋱ ⋮ 0 ⋱ ⋱ a m a m 0 … a 1 0 ] . {\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}}.} {\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}}.}

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: N H T T N H T = N H T N H T T = I mod p , {\displaystyle NHT^{\mathrm {T} }NHT=NHTNHT^{\mathrm {T} }=I{\bmod {\ }}p,\,} {\displaystyle 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]

  1. ^ * 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
  2. ^ 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]
  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.