# Demystifying DFT-Based Harmonic Phase Estimation, Transformation, and Synthesis
## 3 An Interpretable Time-Shift Invariant Harmonic Phase Model
### NRD Estimation Examples Using Natural Voice Signals  
An example of NRD-based analysis with real-world signals involves voiced sounds, i.e., those natural voice sounds that possess a periodic nature as a consequence of the vibration of the vocal folds in the larynx. Figure 23(a) illustrates an overlay of NRD vectors resulting from a sustained vowel (22,050 Hz sampling rate and 1 second long) that was uttered by a female speaker.  
It can be seen that NRD vectors are very consistent up to harmonic 22, beyond which, dispersion in the NRD vectors is apparent. This reflects the fact that phase estimation is disturbed after harmonic 22, which can be easily understood by observing one representative magnitude spectrum of the signal, as represented in Figure 23(b). In fact, it can be confirmed that harmonics 22 and 23, at around 5 kHz, have a very small magnitude, which makes them particularly vulnerable to the noise influence, and leakage effects, which obviously adversely affect phase estimation. In any case, the most important harmonics are correctly characterized in their phase structure, which means reliable NRD models can be built that can be used in parametric-oriented speech processing, as illustrated in Figure 2. Our recent research, which assumes this framework, has revealed that when simple NRD  
Figure 23: Overlay representation of magnitude spectra (**top**), and unwrapped NRD vectors (**bottom**) of the negative of a sawtooth derivative signal that is FM modulated (2.5 Hz deviation around the mean) and whose SNR is 10 dB.  
Figure 24: Example of a harmonic phase model (**a**) and harmonic magnitude model (**b**) pertaining to a female sustained vowel signal.  
modeling is used in the synthesis process, signal reconstruction with transparent audio quality can be achieved, even if the NRD region that exhibits dispersion is replaced by a simple line resulting from extrapolation based on the reliable NRD region (i.e., for which the unwrapped NRD coefficients are consistent) [28].  
A similar NRD analysis for a sustained vowel that was uttered by a male speaker is illustrated in Figure 25a.  
It can be seen that the NRD vectors are quite consistent up to harmonics 43-44, which are located around 5 kHz. Figure 25b depicts one representative magnitude spectrum of the vowel signal and it can be confirmed that, similarly to the previous case, the magnitudes of higher-order harmonics approach the noise floor, which makes phase estimation essentially unreliable.  
However, the region of the unwrapped NRD vectors that span the most important voice resonances can be reliably modeled. When this is combined with simple modeling of higher-order NRD coefficients through interpolation, very high-quality synthesis is achieved which sounds essentially indistinguishable from the original signal [28].  
Our recent research in parametric voice re-synthesis, wideband speech coding, and speaker identification, highlighted that NRD patterns are idiosyncratic, and that they reflect more strongly the influence of the glottal excitation (i.e., the glottal pulse) than the influence of the vocal tract filter [29, 23, 28]. The fact that \"the vocal tract phase during voicing is not important in achieving naturalness\" has been understood already in previous studies ([30], p. 208).