Further enhancements to the analysis of spectral purity in the application of practical direct digital synthesis

In general, the output of a DDS (direct digital synthesis) system will inherently include spurious signals due to two effects: (1) phase truncation errors; and (2) phase to amplitude conversion errors. Phase truncation errors have traditionally been analyzed separately from the fundamental DDS output signal making the derivations awkward and hard to follow. We have recognized that by: (1) only dealing with the active portion of the phase accumulator; (2) carefully reformulating the instantaneous phase; and (3) including all possible initial conditions, the spectral analyses of the DDS output signal can be performed directly, exactly and in a way that is much easier to follow. Our analysis approach results in a simple formulation of the frequency locations and spacing of the phase truncation spurs. We have also recognized that, as long as the phase-to-amplitude conversion errors are "static" in nature, all possible DDS output spectra are inter-related. This is true without any a priori knowledge of the errors. The equations for this interrelationship have been derived in this paper. They show that if a specific DDS spectrum is known, all other possible spectra can be derived from it. We referred to this specific spectrum as the mother spectrum. Also from the properties of these inter-relationships, a different insight has been given to the interpretation of the phase truncation spurs. Based on our derivations, it has also been shown that observations of the DDS spectrum may be affected by using a phase-locked loop phase detector.