Number theoretic transforms with independent length and moduli

Number theoretic transforms introduced by Rader [1], and Agarwal and Burrus [2] have gained increased recognition. These transforms are used to perform convolutions and have the advantage of exact calculations. All operations are performed in a finite ring with the arithmetic carried out modulo an integer P. The disadvantage is this structure causes a rigid relationship between the transform length, the integer P, and the computer word size. This paper defines a generalized number theoretic transform which allows for independent length and the modulus P. This class of transforms retain the advantage of exact calculation but not all transforms possess the convolution property. The Walsh transform and the Haar transform are defined as number theoretic transforms and the constraints necessary to transform and inverse transform a sequence are presented.