# Audio Steganalysis Estimation with the Goertzel Algorithm
## 4 Statistical Coefficient Results
### Linear Prediction Coefficient Algorithm  
To confirm the assumption that the original audio was altered, we obtained a linear prediction and error from the linear prediction. The LPC algorithm is primarily used to encode voice data at low data rates using lossy or lossless compression techniques. The fundamental assumption in LPC is that the prediction of the \(n\)-th sample in a sequence of noisy speech samples is made by taking the weighted sum of the previous \(k\) segments/samples of the final signal as follows [21; 22; 23]:  
\[s[n]=\sum_{k=1}^{p}a_{k}s[n+k]+e[k] \tag{12}\]  
where \(a_{k}\) represents the \(p\)th linear prediction coefficient and \(e[k]\) is the error in the linear prediction. Moreover, the linear predictions of the frequency spectrums of both channels in each pair of audio vectors were used for comparison. To calculate the corresponding vector in the frequency spectrum by means of the linear prediction, a \(p=10\) was chosen; that is, 10 previous elements of the original frequency spectrum vector were used to estimate the following frequency spectrum. Afterwards, we compared the vectors to identify the range of alteration between the frequency spectrum of the original and the stegoaudios. Figure 16 depicts graphics comparing the corresponding vectors with the frequency spectrums of the left and right channels in the original audio in conjunction with the vectors of the linear predictions of the homologous channels of the stegoaudios. Figure 17 shows the graphics comparing the vectors of the frequency spectrum of the right and left channels of the stegoaudios with the vectors of the linear predictions of the homologous channels. In both figures, the vectors overlap, indicating no apparent difference exists between these vectors. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Test**} & \multicolumn{2}{c}{**Mean**} & \multicolumn{2}{c}{**Standard Deviation**} & \multicolumn{2}{c}{**Variance**} & \multicolumn{2}{c}{**Skewness**} & \multicolumn{2}{c}{**Kurtosis**} & \multicolumn{2}{c}{**Average Difference**} \\ \cline{2-13}  & **Left** & **Right** & **Left** & **Right** & **Left** & **Right** & **Left** & **Right** & **Left** & **Right** \\ \hline \multirow{2}{*}{8} & 0.1275 & 0.1313 & 0.1862 & 0.1766 & 0.0347 & 0.0312 & 2.1355 & 1.9504 & 7.4058 & 6.5890 & 1.9779 & 1.7757 \\  & 0.1332 & 0.1380 & 0.1827 & 0.1764 & 0.0334 & 0.0311 & 2.1572 & 2.0062 & 7.4645 & 6.8348 & 1.9942 & 1.8373 \\ \hline \multirow{2}{*}{9} & 0.1263 & 0.1205 & 0.1854 & 0.1764 & 0.0344 & 0.0311 & 2.0611 & 2.1085 & 6.8736 & 7.3121 & 1.8561 & 1.9497 \\  & 0.1272 & 0.1222 & 0.1854 & 0.1772 & 0.0344 & 0.0314 & 2.0689 & 2.1083 & 6.9027 & 7.3051 & 1.8637 & 1.9488 \\ \hline \multirow{2}{*}{10} & 0.1034 & 0.1415 & 0.1555 & 0.2102 & 0.0242 & 0.0442 & 2.2328 & 1.9909 & 8.1453 & 6.3936 & 2.1322 & 1.7560 \\  & 0.1076 & 0.1478 & 0.1539 & 0.2105 & 0.0237 & 0.0443 & 2.2941 & 2.0060 & 8.4384 & 6.4523 & 2.1035 & 1.7721 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._Figure 16: Comparison between the magnitude spectrum of the original audio and the magnitude spectrum recreated from the linear prediction. (**a**) Comparison between the magnitude spectrum of the left channel of the original audio and the spectrum recreated from the linear prediction in analysis. (**b**) Comparison between the magnitude spectrum of the right channel of the original audio and the spectrum recreated from the linear prediction in analysis. (**c**) Comparison between the magnitude spectrum of the left channel of the original audio and the spectrum recreated from the linear prediction in analysis. (**d**) Comparison between the magnitude spectrum of the right channel of the original audio and the spectrum recreated from the linear prediction in analysis. (**e**) Comparison between the magnitude spectrum of the left channel of the original audio and the spectrum recreated from the linear prediction in analysis.