# Digital Signal Processing of the Inharmonic Complex Tone
## 6 Conclusions  
In this paper, the analysis of single-tone signals from string musical instruments based on subchannel decomposition is presented. By this approach, the complex signal is divided into an array of less complex signals, and further analysis of each channel is performed simultaneously in both the time and frequency domains. The decomposition of the signal is performed by a non-uniform multichannel filter bank with additional phase correction, preserving time instances between channels. It is expected that each subchannel contains only one strong quasi-sinusoidal component. Sometimes the subchannel signal has an additional strong quasi-sinusoidal component, such as phantom partials. Due to the relatively short intervals of quasi-stationarity in the musical signal, it is challenging to detect and accurately verify the presence of those irregularities. The STFT is a powerful tool for analyzing non-stationary signals; however, due to the short window lengths, the close picks in frequency are non-separable. In the presented approach, the combination of time and frequency analysis of each channel and analysis of the overall signal improves the analysis. MuPI enables more accurate analysis of tone signals thanks to a specialized filter bank design that is not present in applications of similar purpose. Unlike other tools, it is focused on parameters like inharmonicity and reconstruction, which makes it suitable for the analysis of stringed instruments. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Signal**} & \multirow{2}{*}{**Tone**} & \multicolumn{2}{c}{**Number of**} & \multirow{2}{*}{**Fundamental**} & \multirow{2}{*}{**Inharmonicity**} \\  & & **Pominant** & & \\  & & **Partials** & & \\ \hline Harp\_As4\_recored & As4 & 15 & 416.9 & \(2.2\times 10^{-4}\) \\ Piano\_A3\_recorded & A3 & 25 & 220.6 & \(2.65\times 10^{-4}\) \\ Harpsichord\_F3\_recored & F3 & \(>\)70 & 174.4 & \(1.8\times 10^{-5}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Signal parameters used in the experiments. The presented application features a modular structure, enabling the implementation of various additional modules. In future work, the additional processing block can be developed that relies on the extrapolation of the signal as a tool for improving the resolution of the STFT [39]. The parameter M2M is calculated as a reasonable merit of the prominence of the partial. However, it is only provided as a value but not fully used in the current version of the application. In further work, it can be seen as a tool for excluding the non-prominent or completely \"missing\" partials from the evaluation of the signal parameters, including the inharmonicity coefficient \(B\). For the proposed structure of the filter bank, it is unexpected to detect two strong partials in the same band. The future work will also focus on finding a quick and simple tool for the detection of channels with more than one frequency peak. Conceptualization, D.S.P. ; methodology, M.B. and J.C.; software, J.C.; validation, T.M. and M.B. ; formal analysis, T.M. and J.C.; investigation, T.M. ; resources, D.S.P. ; data curation, J.C. and T.M. ; writing--original draft preparation, M.B., J.C. and T.M. ; writing--review and editing, J.C. and D.S.P. ; visualization, J.C. and M.B. ; supervision, D.S.P. ; project administration, T.M. ; funding acquisition, D.S.P. and J.C. All authors have read and agreed to the published version of the manuscript. This work was financially supported by the Ministry of Science, Technological Development and Innovation of the Republic of Serbia, under contract numbers 451-03-136/2025-03/200103, 451-03-137/2025-03/200103. Not applicable.