The number theoretic transform is meaningful in the
ring, when the modulus is not prime, provided a principal root of order n exists.
The NHT matrix, where , has the form
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: 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]
^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.