## 3 Adaptive Sinusoidal Modeling with eaQHM
### The Extended Adaptive Quasi-Harmonic Model (eaQHM)  
[PERSON] _et al._ [45] proposed adapting the phase of the sinusoids. The adaptive procedure applies LS, frequency correction, and frequency interpolation iteratively (see Figure 2), projecting \(x\left(n,m\right)\) onto \(\hat{x}\left(n,m\right)\). Figure 3 shows the first and second iterations to illustrate adaptation of one sinusoid. [PERSON] _et al._ [43] adapted both the instantaneous amplitude and the instantaneous phase of \(\hat{x}\left(n,m\right)\) with a similar iterative procedure in eaQHM. The analysis stage uses  
\[\hat{x}\left(n,m\right)=\sum_{k=-K}^{K}\left(\mathbf{a}_{k}+n\mathbf{b}_{k} \right)\hat{A}_{k}\left(n,m\right)e^{i\hat{\phi}_{k}\left(n,m\right)}, \tag{9}\]  
where \(\hat{A}_{k}\left(n,m\right)\) and \(\hat{\phi}_{k}\left(n,m\right)\) are functions of the time-varying instantaneous amplitude and phase of each sinusoid, respectively [43; 45], obtained from the parameter interpolation step and defined as  
\[\hat{A}_{k}\left(n,m\right) =\frac{\hat{a}_{k}\left(n\right)}{\hat{a}_{k}\left(\tau,m\right)}, \tag{10a}\] \[\hat{\phi}_{k}\left(n,m\right) =\hat{\phi}_{k}\left(n\right)-\hat{\phi}_{k}\left(\tau,m\right), \tag{10b}\]  
where \(\hat{a}_{k}\left(n\right)\) is the piece-wise linear amplitude and \(\hat{\phi}_{k}\left(n\right)\) is estimated using Equation (7). Finally, eaQHM models \(x\left(n\right)\) as a set of amplitude and frequency modulated nonstationary sinusoids given by  
\[\hat{x}_{i}\left(n\right)=\sum_{k=-K}^{K}\hat{a}_{k,i-1}\left(n\right)e^{i \hat{\phi}_{k,i-1}\left(n\right)}, \tag{11}\]  
where \(\hat{a}_{k,i-1}\left(n\right)\) and \(\hat{\phi}_{k,i-1}\left(n\right)\) are the instantaneous amplitude and phase from the previous iteration \(i-1\). Adaptation results from the iterative projection of \(x\left(n\right)\) onto \(\hat{x}\left(n\right)\) from \(i-1\) as the model \(\hat{x}\left(n\right)\) are used as nonstationary basis functions locally adapted to the time-varying behavior of \(x\left(n\right)\). Note that Equation (9) is simply Equation (3) with a nonstationary basis \(\hat{A}_{k}\left(n,m\right)e^{i\hat{\phi}_{k}\left(n,m\right)}\). In fact, Equation (9) represents the next parameter estimation step, which will be again followed by frequency correction as in Figure 2. The convergence criterion for eaQHM is either a maximum number of iterations \(i\) or an adaptation threshold \(\varepsilon\) calculated as  
\[\frac{\text{SRER}^{i-1}-\text{SRER}^{i}}{\text{SRER}^{i-1}}<\varepsilon, \tag{12}\]  
where the signal-to-reconstruction-error ratio (SRER) is calculated as  
\[\text{SRER}=20\log_{10}\frac{\text{RMS}\left(x\right)}{\text{RMS}\left(x- \hat{x}\right)}=20\log_{10}\frac{\text{RMS}\left(x\right)}{\text{RMS}\left( \varepsilon\right)}. \tag{13}\]The SRER measures the fit between the model \(\hat{x}\left(n\right)\) and the original recording \(x\left(n\right)\) by dividing the total energy in \(x\left(n\right)\) by the energy in the residual \(e\left(n\right)\). The higher the SRER, the better the fit. Note that \(\varepsilon\) stops adaptation whenever the fit does not improve from iteration \(i-1\) to \(i\) regardless of the absolute SRER value. Thus, even sounds from the same instruments can reach different SRER.