# Demystifying DFT-Based Harmonic Phase Estimation, Transformation, and Synthesis
## 4 Conclusions  
This paper focused on a harmonic signal analysis, modeling, and processing paradigm that eases significantly the representation, transformation, and synthesis of harmonic signals, especially from the point of view of the phase information.  
In the first part of the paper, practical DFT-based approaches that build on a filterbank perspective were discussed for estimating the starting phases of individual sinusoids. Their performance was characterized by considering the CRLB for the variance of an unbiased phase estimator. In particular, it was shown that contrary to harmonic frequency and magnitude estimation, accurate phase estimation depends only on 'coarse search' and not on 'fine search', which makes the estimation more robust. Six phase estimation alternatives were studied by combining two DFT-based filter banks and three different window functions. Results were explained in a reproducible manner.  
In the second part of the paper, it was shown that the starting phases of individual sinusoids that are harmonically related may be converted into a phase-related feature (NRD) that expresses the holistic phase structure of a harmonic signal, has the advantage of being time-shift-invariant, helps to explain the waveform shape of a quasi-periodic signal, and helps to provide insight into the physical process that generates it.  
Finally, it was shown that the unique information that exists in a phasegram resulting from a stationary harmonic signal consists of the starting phase of the fundamental frequency and the NRD feature vector.  
Matlab code ([[https://github.com/[PERSON]/demo_AccPhaseEst](https://github.com/[PERSON]/demo_AccPhaseEst)]([https://github.com/[PERSON]/demo_AccPhaseEst](https://github.com/[PERSON]/demo_AccPhaseEst)), accessed on 1 January 2024) is provided that illustrates the most relevant concepts and results that are discussed in the paper.  
Many application scenarios may benefit from the results in this paper paving the way for new research results, namely speech coding, pitch, and time-scale modification of speech, singing, audio, and music; speech enhancement; whispered-speech to voiced-speech conversion and voice rehabilitation; audio forensics; physiological signal analysis and diagnosis (e.g., using ECG signals); and monitoring of the operation of mechanical systems using sound.  
Conceptualization, A.F.; methodology, M.O., V.S. and A.S.; software, M.O., V.S. and A.F.; validation, M.O., V.S., A.S. and A.F.; formal analysis, V.S. and A.F.; investigation, M.O., V.S. and A.S.; writing--original draft preparation, A.F.; writing--review and editing, M.O., V.S., A.S. and A.F.; visualization, M.O., V.S. and A.F.; supervision, A.F.; project administration, A.F.; funding acquisition, A.F. All authors have read and agreed to the published version of the manuscript.  
This work was developed within the context of Research Project COMPETE2030-FEDER-00923600 supported by the Portuguese Foundation for Science and Technology (FCT). Support was also received from the FCT-POCH Program under the SFRH/BSAB/150372/2019 Sabbatical Leave Fellowship.  
Data Availability Statement:The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.  
**Acknowledgments:** Parts of this paper were written while the corresponding author ([PERSON]) was with the Digital Signal Processing Group at MIT during a sabbatical leave. The corresponding author would like to thank Prof. [PERSON] for allowing magic to happen. The four co-authors would like to thank the anonymous reviewers for their comments and suggestions, which helped to improve the quality of the manuscript.  
The authors declare no conflicts of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.