Digital oscillators over finite fields

This paper shows how the error-free computation properties of finite fields can be used to eliminate aperiodicity and SNR degradation problems in digital oscillators. Galois fields are used to eliminate representation and truncation errors in the values of sinusoid samples. Theorems are presented that characterize the fields which admit all operands necessary for a given phase resolution. Two oscillator architectures (exponential feedback and direct form) are presented and various design issues associated with each are discussed. It is also shown how field arithmetic can be replaced by arithmetic in the direct product ring associated with a set of specially chosen small fields. This is important in high-speed applications where the loop delay must be minimized. We also present analytical estimates of clocking frequency, latency, phase and frequency resolution, along with an example design in an Altera 7256 CPLD.