# Demystifying DFT-Based Harmonic Phase Estimation, Transformation, and Synthesis
## 3 An Interpretable Time-Shift Invariant Harmonic Phase Model  
In this section, we take advantage of phase estimation of individual sinusoids using the signal representation in a discrete-time and discrete-frequency Fourier domain, as it was discussed in the previous section, especially Section 2.4. The objective is to identify a vertical harmonic phase model that is time-shift invariant, and that can be approximated by a simple parametric model. In our development in this section, we use the deterministic part of (1) taking into consideration the harmonic relationship given by (2). Thus, in our analysis, we use the following:  
\[x(t) = \sum_{\ell=0}^{L-1}A_{\ell}\sin((\ell+1)\Omega_{0}t+\phi_{\ell}) \tag{27}\] \[= \sum_{\ell=0}^{L-1}A_{\ell}\sin((\ell+1)\Omega_{0}(t+t_{\ell}))\] \[= A_{0}\sin(\Omega_{0}(t+t_{0}))+v(t)\,\]  
where \(t_{\ell}\) represents the starting delay of harmonic \(\ell\), and \(v(t)\) represents all harmonics above the fundamental frequency. If \(\Omega_{0}=2\pi/T_{0}\), where \(T_{0}\) is the fundamental period, i.e., the reciprocal of the fundamental frequency (\(F_{0}\)), a convenient manipulation of \(v(t)\) leads to the following:  
\[v(t) = \sum_{\ell=1}^{L-1}A_{\ell}\sin((\ell+1)\Omega_{0}(t+t_{\ell})) \tag{28}\] \[= \sum_{\ell=1}^{L-1}A_{\ell}\sin((\ell+1)\Omega_{0}(t+t_{0}-t_{0} +t_{\ell}))\] \[= \sum_{\ell=1}^{L-1}A_{\ell}\sin((\ell+1)\Omega_{0}(t+t_{0})+2\pi \frac{t_{\ell}-t_{0}}{T_{0}/(\ell+1)})\] \[= \sum_{\ell=1}^{L-1}A_{\ell}\sin((\ell+1)\Omega_{0}(t+t_{0})+2\pi \text{NRD}_{\ell})\,\]  
where \(\text{NRD}_{\ell}\) denotes the normalized relative delay of harmonic \(\ell\), and is defined as follows:  
\[\text{NRD}_{\ell}=\frac{t_{\ell}-t_{0}}{T_{0}/(\ell+1)}=\frac{\phi_{\ell}-( \ell+1)\phi_{0}}{2\pi}. \tag{29}\]  
\(\text{NRD}_{\ell}\) expresses the difference between the starting delay of harmonic \(\ell\), and the starting delay (i.e., the onset) of the fundamental frequency, which is further normalized by the period of harmonic \(\ell\)[24]. Therefore, in its most intuitive interpretation, NRD can be taken modulo 1, which means that \(\text{NRD}_{\ell}\in[0.0,1.0]\). Equation (29) allows two other important conclusions. First, by definition, \(\text{NRD}_{0}=0\). This means that, by definition, the NRD phase-related feature is intrinsically time-shift invariant. Second, as the second part of (29) highlights, the NRD does not depend on the fundamental frequency. Therefore, we may rewrite (27) as follows:  
\[x(t)=\sum_{\ell=0}^{L-1}A_{\ell}\sin((\ell+1)\Omega_{0}(t+t_{0})+2\pi\text{NRD }_{\ell})\, \tag{30}\]which highlights the fact that, in terms of phase, all harmonics can be expressed as a part that depends on the time-varying phase of the fundamental frequency, and on its starting phase, and another part, the NRD, which is time-shift invariant. Thus, the NRD acts as a holistic harmonic phase model that is identified in Figure 2 as PM. Other harmonic phase descriptors that are similar to NRD were proposed by [PERSON] in 1996 (phase envelope [25] (p. 44)), [PERSON] in 1998 [26], and [PERSON] in 2009 (relative phase Shift-RPS [27]). An inspiring metaphor from nature, elucidating the meaning of NRD, is depicted in Figure 12.  
Each bird in the flock formation is like a harmonic in a harmonic structure, and the dynamics of the wings of the former are like the dynamics of the phase rotation of the latter. The NRD feature represents a shift-invariant harmonic phase structure, similar to the space-invariant bird formation structure of the flock that is represented by the line connecting the birds in the upper branch of the flock. Ultimately, the whole flock's flying dynamics are governed by the position and wing evolution of the leading bird in the flock, since the whole harmonic dynamics are governed by the phase evolution of the fundamental frequency in the harmonic structure. On the other hand, since the NRD inherits the properties of phase, then phase wrapping and phase unwrapping also apply to NRD feature vectors.