Analysis of the output spectrum for direct digital frequency synthesizers in the presence of phase truncation and finite arithmetic precision

A simple algorithm is presented for the calculation of the output spectrum of direct digital frequency synthesizers (DDFS) in the presence of phase accumulator truncation and finite arithmetic precision. Using no approximations, a simple formula calculates the magnitudes of the spurious noise frequencies (spurs) due to phase accumulator truncation and due to finite precision implementation. Their spectral position is also determined by a simple expression. The derivation process itself provides strong insight into spur magnitude and spectral location, and it makes evident that the set of spurs due to phase word truncation and the set resulting from finite arithmetic precision are disjoint.