Exact analysis of DDS spurs and SNR due to phase truncation and arbitrary phase-to-amplitude errors

This paper presents the theoretical basis for an algorithm that performs an exact analysis of the output spectrum of direct digital frequency synthesizers (DDS or DDFS) in the presence of phase accumulator truncation, finite arithmetic precision and arbitrary approximations and errors in the sine/cosine mapping function (SCMF). The derivation provides strong insight into spurious frequency (spur) magnitude and spectral location, and makes evident that the set of spurs due to phase-word truncation and the set resulting from SCMF imprecision and errors are effectively disjoint. Phase-truncation spurs are shown to have distinct magnitudes and their spectral locations, ordered from largest to smallest in magnitude, are easily ascertained. Without generating all spur magnitudes, the algorithm supported by this theory is capable of exactly computing the signal-to-noise ratio (SNR) and spurious-free dynamic range (SFDR) and exactly computing the magnitudes and locations of the TV worst (i.e., largest-magnitude) spurs or all spurs with magnitudes larger than epsiv, due to the combined effect of phase truncation and arbitrary SCMF imprecision (where N and epsiv are user-specified parameters)