## 5 Results and Discussion
### Full-Band Quasi-Harmonic Analysis with AM-FM Sinusoids  
To simplify the comparison and reduce the information, we present the differences of SRER instead of absolute SRER values. For each sound, we subtract the absolute SRER values (in dB) for the SM and EDS from that of eaQHM to obtain the differences of SRER. The local value measures the fit for the attack and the global value measures the overall fit. Positive values indicate that eaQHM  
Figure 6: Comparison between local and global SRER as a function of the size of the frame for the three models (SM, EDS, and eaQHM). The bars around the mean are the standard deviation across different sounds from the family indicated. The distributions are not symmetrical as suggested by the bars.  
results in higher SRER than the other method for that particular sound, while a negative value means the opposite. The different SRER values are averaged across all musical instruments that belong to the family indicated. Table 2 shows the comparison of eaQHM against EDS and the SM with \(K=K_{\text{max}}\) and \(L=3T_{0}f_{s}\) clustered by instrumental family. The distributions are not symmetrical around the mean as suggested by the standard deviation.  
Thus, Table 2 summarizes the result of full-band quasi-harmonic analysis with adaptive AM-FM sinusoids from eaQHM comparing with the SM and EDS under the same conditions, namely the same number of sinusoids \(K=K_{\text{max}}\) and frame size \(L=3T_{0}f_{s}\). When eaQHM is compared to the SM, both local and global difference SRER are positive for all families. This means that full-band quasi-harmonic modeling with eaQHM results in a better fit for the analysis and synthesis of musical instrument sounds.  
When eaQHM is compared to EDS, all global difference SRER are positive and all local difference SRER are positive except for _Brass_ and _Bowed Percussion_. Thus, EDS can fit the attack of _Brass_ and _Bowed Percussion_ better than eaQHM. The exponential amplitude envelope of EDS is considered suitable to model percussive sounds with sharp attacks such as harps, pianos, and marimbas [36; 37]. The musical instrument families that contain percussive sounds are _Plucked strings_, _Struck percussion_, and _Keyboard_. Table 2 shows that eaQHM outperformed EDS locally and globally for all percussive sounds. The ability to adapt the amplitude of the sinusoidal partials to the local characteristics of the waveform makes eaQHM extremely flexible to fit both percussive and nonpercussive musical instrument sounds. On the other hand, both _Brass_ and _Bowed Percussion_ present slow attacks typically lasting longer than one frame \(L\). Note that \(\
icefrac{{L}}{{f_{s}}}=3T_{0}\approx 22\) ms for C3 (\(f_{0}\approx 131\) Hz) while _Bowed Percussion_ can have attacks longer than 100 ms. Therefore, one frame \(L=3T_{0}f_{s}\) does not measure the fit for the entire duration of the attack.  
Note that the local SRER is important because the global SRER measures the overall fit without indication of _where_ the differences lie in the waveform. For musical instrument sounds, differences in the attack impact the results differently than elsewhere because the attack is among the most important perceptual features in dissimilarity judgment [56; 57; 58]. Consequently, when comparing two models with the global SRER, it is only safe to say that a higher SRER indicates that resynthesis results in a waveform that is closer to the original recording.