## 5 Results and Discussion
### Full-Band Modeling and Quasi-Harmonicity  
Time-frequency transforms such as the STFT represent \(L\) samples in a frame with \(N\) DFT coefficients provided that \(N\geq L\). Note that \(N\in\mathbb{C}\), corresponding to \(p=2N\) real numbers. There is signal expansion whenever the representation uses \(p\) parameters to represent \(L\) samples and \(p>L\). Sinusoidal models represent \(L\) samples in a frame with \(K\) sinusoids. In turn, each sinusoid is described \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**SRER (eaQHM-EDS)**} & \multicolumn{2}{c}{**SRER (eaQHM-SM)**} \\ \hline
**Family** & **Local (dB)** & **Global (dB)** & **Local (dB)** & **Global (dB)** \\ \hline
**Brass** & \(-9.4\pm 7.0\) & 12.5 \(\pm\) 6.8 & 27.3 \(\pm\) 5.8 & 31.9 \(\pm\) 4.0 \\
**Woodwinds** & 7.8 \(\pm\) 3.9 & 22.0 \(\pm\) 5.9 & 30.9 \(\pm\) 7.5 & 36.1 \(\pm\) 4.7 \\
**Bowed Strings** & 12.2 \(\pm\) 4.2 & 24.1 \(\pm\) 6.7 & 35.0 \(\pm\) 4.7 & 40.0 \(\pm\) 4.7 \\
**Plucked Strings** & 8.3 \(\pm\) 5.0 & 4.7 \(\pm\) 3.4 & 49.5 \(\pm\) 4.3 & 46.6 \(\pm\) 5.1 \\
**Bowed Percussion** & \(-2.7\pm\) 2.5 & 16.3 \(\pm\) 2.2 & 12.7 \(\pm\) 2.6 & 37.6 \(\pm\) 3.6 \\
**Struck Percussion** & 10.5 \(\pm\) 4.8 & 10.1 \(\pm\) 2.6 & 28.6 \(\pm\) 13.3 & 26.0 \(\pm\) 11.3 \\
**Popular** & 6.3 \(\pm\) 3.3 & 11.9 \(\pm\) 7.0 & 26.5 \(\pm\) 10.8 & 27.5 \(\pm\) 11.6 \\
**Keyboard** & 5.7 \(\pm\) 3.4 & 5.4 \(\pm\) 4.3 & 37.0 \(\pm\) 8.0 & 34.6 \(\pm\) 2.0 \\
**Total** & 5.3 \(\pm\) 2.4 & 13.2 \(\pm\) 3.3 & 31.0 \(\pm\) 7.1 & 35.0 \(\pm\) 5.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Local and global difference of signal-to-reconstruction-error ratio (SRER) comparing eaQHM with exponentially damped sinusoids (EDS) and eaQHM with the standard sinusoidal model (SM) for the frame size \(L=3T_{0}f_{s}\) and number of partials \(K=K_{\text{max}}\). The three C2 sounds are not included. by \(p\) parameters, requiring \(pK\) parameters to represent \(L\) samples. Therefore, there is a maximum number of sinusoids to represent a frame without signal expansion. For example, white noise has a flat spectrum across that would take a large number of sinusoids close together in frequency resulting in signal expansion. The \(pK\) parameters to represent \(L\) samples can be interpreted as the degrees of freedom of the fit. As a general rule, more parameters mean greater flexibility of representation (hence potentially a better fit), but with the risk of over-fitting. Table 3 shows a comparison of the number of real parameters \(p\) (per sinusoid \(k\) per frame \(m\)) for the analysis and synthesis stages of the SM, EDS, and eaQHM. Note that eaQHM and EDS require more parameters than the SM at the _analysis_ stage, but eaQHM and the SM require fewer parameters than EDS for the _synthesis_ stage. The difference is due to the resynthesis strategy used by each algorithm. EDS uses OLA resynthesis, which requires all analysis parameters for resynthesis, while both eaQHM and the SM use additive resynthesis. Harmonicity of the partials guarantees that there are no signal expansions in full-band modeling with sinusoids. Consider \(L=qT_{0}f_{s}\) with \(q\) an integer and \(T_{0}=1/f_{0}\). Using \(K_{\max}\approx f_{s}/2f_{0}\) quasi-harmonic partials and \(p\) parameters per partial, it takes at most \(pK_{\max}=\
icefrac{{(pf_{s})}}{{2}}/f_{0}\) numbers to represent \(L=qT_{0}f_{s}=\
icefrac{{(qf_{s})}}{{f_{0}}}/f_{0}\) samples, which gives the ratio \(r=\
icefrac{{(pK_{\max})}}{{L}}=\
icefrac{{p}}{{2q}}\). Table 3 shows that analysis with eaQHM requires \(p=4\) real parameters. Thus, a frame size with \(q>2\) is enough to guarantee no signal expansion. This result is due to the full-band paradigm using \(K_{\max}\) harmonically related partials, not a particular model. The advantage of full-band modeling results from the use of one single component instead of decomposition. Table 4 compares the complexity of SM, EDS, and eaQHM in Big-O notation.