Recursive realization of finite impulse filters using finite field arithmetic

Recursive filter design techniques are described and developed for finite impulse filters using finite field arithmetic. The finite fields considered have the form
GF(q^{2})
, the Galois field of
q^{2}
elements, and are analogous to the field of complex numbers when
q
is a prime such that
(-1)
is not a quadratic residue. These filters can be designed to yield either a desired finite impulse or finite frequency response function. This filtering technique has other possible applications, including the encoding or decoding of information and signal design. Infinite signal trains can be decomposed naturally into orthogonal sequences which may be useful in the encoding and decoding process and may provide another approach to convolutional coding. Since the recursive filters developed here do not have the accumulation of round-off or truncation error that one might expect in recursive computations, such filters are noise-free transducers in the sense of Shannon.