The implementation of the generalized Lagrange FIR filter structure defined over finite fields or rings

This paper discusses the use of the Complex Number Theoretic z-transform in implementing a recursive FIR filter structure for frequency samples spaced around the unit circle in the complex number theoretic z-domain. The complex arithmetic operations have been implemented using the Quadratic Residue Number System (QRNS) and Modified Quadratic Residue Number System (MQRNS) for uniformly spaced frequency samples around the unit circle. We discuss the extension of this technique to non-uniformly spaced samples around the unit circle and the resulting filter structure has been termed the generalized number theoretic FIR filter structure. We demonstrate that the MQRNS is the only suitable tool in implementing this recursive FIR filter structure for non-uniformly spaced frequency samples.