## 3 Adaptive Sinusoidal Modeling with eaQHM
### Parameter Interpolation across Frames  
The model parameters \(\hat{a}_{k}\), \(\hat{F}_{k}\), and \(\hat{\phi}_{k}\) from Equation (5) are estimated as samples at the frame rate \(\
icefrac{{1}}{{H}}\) of the amplitude- and frequency-modulation (AM-FM) functions \(\hat{a}_{k}\left(n\right)\) and \(\hat{\phi}_{k}\left(n\right)=2\
icefrac{{\pi}}{{f_{s}}}\hat{F}_{k}\left(n \right)+\hat{\phi}_{k}\), which describe, respectively, the long-term amplitude and frequency temporal variation of each sinusoid \(k\). For each frame \(m\), \(\hat{a}_{k}\left(\tau,m\right)\) and \(\hat{F}_{k}\left(\tau,m\right)\) are estimated using the sample index at the center of the frame \(n=\tau\) as reference. Resynthesis of \(\hat{x}\left(n,m\right)\) requires \(\hat{a}_{k}\left(n,m\right)\) and \(\hat{F}_{k}\left(n,m\right)\) at the signal sampling rate \(f_{s}\). Equation (5) uses constant values, resulting in locally stationary sinusoids with constant amplitudes and frequencies inside each frame \(m\).  
However, the parameter values might vary across frames, resulting in discontinuities such as \(\hat{a}_{k}\left(\tau,m\right)\
eq\hat{a}_{k}\left(\tau,m+1\right)\) due to temporal variations happening at the frame rate \(\
icefrac{{1}}{{H}}\). OLA resynthesis [52] uses the analysis window \(w\left(n\right)\) to taper discontinuities at the frame boundaries by resynthesizing \(\hat{x}\left(n,m\right)=\hat{x}\left(n\right)w\left(n\right)\) for each \(m\) similarly to Equation (2) and then overlap-adding \(\hat{x}\left(n,m\right)\) across \(m\) to obtain \(\hat{x}\left(n\right)\).  
Additive synthesis is an alternative to OLA that results in smoother temporal variation [20] by first interpolating \(\hat{a}_{k}\left(\tau,m\right)\) and \(\hat{\phi}_{k}\left(\tau,m\right)\) across \(m\) and then summing over \(k\). In this case, \(\hat{a}_{k}\left(n\right)\) is obtained by linear interpolation of \(\hat{a}_{k}\left(\tau,m\right)\) and \(\hat{a}_{k}\left(\tau,m+1\right)\). Recursive calculation across \(m\) results in a piece-wise linear approximation of \(\hat{a}_{k}\left(n\right)\). \(\hat{F}_{k}\left(n\right)\) is estimated via piece-wise polynomial interpolation of \(\hat{F}_{k}\left(\tau,m\right)\) across \(m\) with quadratic splines, and \(\hat{\phi}_{k}\left(n\right)\) is obtained integrating \(\hat{F}_{k}\left(n\right)\) in two steps because \(\hat{\phi}_{k}\left(\tau,m\right)\) is wrapped around \(2\pi\) across \(m\). First, \(\hat{\phi}_{k}\left(n\right)\) is calculated as  
\[\hat{\phi}_{k}\left(n\right)=\hat{\phi}_{k}\left(\tau,m\right)+\frac{2\pi}{f_{s }}\sum_{n=m}^{m+1}\hat{F}_{k}\left(u\right). \tag{6}\]The calculation of \(\hat{\phi}_{k}\left(n\right)\) using Equation (6) does not guarantee that \(\hat{\phi}_{k}\left(\tau,m+1\right)=\hat{\phi}_{k}\left(\tau,m+1\right)+2\pi P\), with \(P\) the closest integer to unwrap the phase (see details in [45]). Thus, \(\hat{\phi}_{k}\left(n\right)\) is calculated as  
\[\hat{\phi}_{k}\left(n\right)=\hat{\phi}_{k}\left(\tau,m\right)+\frac{2\pi}{f_{ s}}\sum_{u=m}^{m+1}\left[\hat{F}_{k}\left(u\right)+\gamma\sin\left(\frac{\pi \left(u-m\tau\right)}{\left(m+1\right)\tau-m\tau}\right)\right], \tag{7}\]  
where the term given by the sine function ensures continuity with \(\hat{\phi}_{k}\left(\tau,m+1\right)\) when \(\gamma\) is  
\[\gamma=\frac{\pi}{2}\left[\frac{\hat{\phi}_{k}\left(\tau,m+1\right)+P-\hat{ \phi}_{k}\left(\tau,m+1\right)}{\left(m+1\right)\tau-m\tau}\right], \tag{8}\]  
with \(P\) given by \(\left|\hat{\phi}_{k}\left(\tau,m+1\right)-\hat{\phi}_{k}\left(\tau,m+1\right) \right|\) (see [45]).