# Enhancement of Conventional Beat Tracking System Using Teager-Kaiser Energy Operator
## 3 Results  
Table 3 presents results of the GT detection based on the first database for the first 30 analyzed tracks. The complete table is in Table 12. Average deviation from the reference tempo was 9.99 BPM (11.28%) in the case of TS and 18.19 BPM (17.74%) in the case of DS. The least accurate estimation was done on the recordings of a solo acoustic guitar. We also applied the \(t\)-Test (Paired Two Sample for Means) for each system (compared to the reference). P-value for the TS is 0.038 and 0.024 for the DS (\(\alpha=0.05\)). Next, Table 4 presents general results of the GT testing: median, mean, standard deviation, relative standard deviation and variance for each tested system and the deviations from the reference tempo. The mean value of the reference GT was 76.78 BPM, the average computed GT 83.75 BPM for the TS and 88.97 BPM for the DS. Table 5 shows the mean value and the median of the ABD testing for all analyzed tracks. The average difference between consecutive beat time positions of the reference and the TS was 2.30 s and 2.84 s for the DS. Table 6 shows the average of the arithmetic mean and the median of time \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Track** & **Beg** & **A** & **B** & **C** & **D** & **E** \\ \hline CD01 & 80.61 & 69.37 & 34.41 & 88.56 & 55.60 & 74.50 \\ \hline CD02 & 77.80 & 69.03 & 44.14 & 81.84 & 59.52 & 72.43 \\ \hline CD03 & 77.93 & 73.19 & 41.60 & 87.09 & 62.36 & 79.14 \\ \hline. &. &. &. &. &. \. &. &. &. &. &. \. &. &. &. &. &. \\ \hline CD33 & 76.92 & 63.24 & 42.05 & 74.62 & 56.26 & 68.31 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The EAT of all motifs of the second database. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Motif** & **Beginning** & **A** & **B** & **C** & **D** & **E** \\ \hline Bars & 1–70 & 71–110 & 111–118 & 119–164 & 165–225 & 226–262 \\ Quarter notes & 280 & 160 & 32 & 184 & 244 & 148 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Calculation of quarter notes in all motifs. difference values of the recordings in which the ABD were less than 1 s. This means 11 recordings for the TS (37% of the tested database) and 9 recordings for the DS (30%). The TS detected the right metric pulse in more recordings than the DS. Average deviations from the reference beat positions were 0.39 s and 0.29 s for the TS and 0.95 s and 0.36 s for the DS respectively. Table 2 presents the length of the first movement of each motif of the second database and the corresponding number of quartet notes. Then, the EAT was calculated. Table 1 contains results based on the EAT of all motifs of our second database--33 different interpretations of _String Quartet No. 1 e minor \"From My Life\"_. Finally, Table 7 shows the difference between the estimated GT and the EAT for both proposed systems. The complete table is in Table 10. The average deviation for the TS is 6.42 BPM and 6.59 BPM for the DS. Due to the nature of the results of the second dataset, no further statistical processing of the values was used. Figure 13 shows differences between the reference GT and calculated GT of the TS and DS of the first database. The TS generally follows the reference tempo more accurately mainly because it more often determined the correct metric pulse. The DS shows greater local deviations of the GT from the tested tracks. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multicolumn{1}{c}{\multirow{2}{*}{**Dev. \(<\) 1 s in the Average of TS**}} & \multicolumn{6}{c}{**\textless{1 s in the Average of DS**}} \\ \cline{2-11}  & **TS** & \multicolumn{2}{c}{**DS**} & \multicolumn{2}{c}{**TS**} & \multicolumn{2}{c}{**DS**} \\ \cline{2-11}  & **\%** & **Me** & \%** & **Me** & \% & **Me** & \% & **Me** & \% & **Me** \\ \hline Average & 0.39 & 0.38 & 0.95 & 0.70 & 0.29 & 0.12 & 0.36 & 0.22 \\ \hline \hline \end{tabular}  
* TS—System with the TKEO; DS—System without the TKEO; Dev.—deviation from the reference beat positions;  
* Mean—median; \(\bar{\kappa}\)—mean value.  
\end{table}