Demystifying DFT-Based Harmonic Phase Estimation, Transformation, and Synthesis

Many natural signals exhibit quasi-periodic behaviors and are conveniently modeled as combinations of several harmonic sinusoids whose relative frequencies, magnitudes, and phases vary with time. The waveform shapes of those signals reflect important physical phenomena underlying their generation, requiring those parameters to be accurately estimated and modeled. In the literature, accurate phase estimation and modeling have received significantly less attention than frequency or magnitude estimation. This paper first addresses accurate DFT-based phase estimation of individual sinusoids across six scenarios involving two DFT-based filter banks and three different windows. It has been shown that bias in phase estimation is less than 0.001 radians when the SNR is equal to or larger than 2.5 dB. Using the Cramér–Rao lower bound as a reference, it has been demonstrated that one particular window offers performance of practical interest by better approximating the CRLB under favorable signal conditions and minimizing performance deviation under adverse conditions. This paper describes the development of a shift-invariant phase-related feature that characterizes the harmonic phase structure. This feature motivates a new signal processing paradigm that greatly simplifies the parametric modeling, transformation, and synthesis of harmonic signals. It also aids in understanding and reverse engineering the phasegram. The theory and results are discussed from a reproducible perspective, with dedicated experiments supported by code, allowing for the replication of figures and results presented in this paper and facilitating further research.