## 6 Experimental Results
### Amr-Nb  
The adaptive multirate (AMR) set of codecs is a widely installed, popular speech codec used in digital cellular and Voice over Internet Protocol (VoIP) applications [22; 26]. A wideband version (50 Hz to 7 kHz input bandwidth) is standardized, but a narrowband version is also included. The AMR-NB codec has rates of 12.2, 10.2, 7.95, 7.4, 6.7, 5.9, 5.15, and 4.75 Kbps. In Tables 5 and 6, we list the percentage of frames with the log likelihood ratio in several ranges for all of the AMR-NB bit rates and the two sentences \"We were away a year ago\" and \"A lathe is a big tool\", respectively. At a glance, it is seen that although the PESQ-MOS changes by over 0.5 for the sentence \"We were away a year ago\" and by more than 0.8 for the sentence \"A lathe is a big tool\" as the rates decrease, there are no frames above the perceptually significant threshold! This observation points out that the \"perceptually significant\" threshold is fairly arbitrary, and also that the data need further analysis. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Bit Rate (Kbps)** & \begin{tabular}{c} **\% Frames with** \\ **\(D<24\)** \\ \end{tabular} & \begin{tabular}{c} **\% Frames with** \\ **24 \(<D<72\)** \\ \end{tabular} & \
\begin{tabular}{c} **\% Frames with** \\ **\(D>72\)** \\ \end{tabular} &
\begin{tabular}{c} **\% Frames with** \\ **PESQ-MOS** \\ \end{tabular} \\ \hline
8 Kbps & 42.19 & 57.81 & 0 & 3.810 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of \(D\) and PESQ values for G.729 “A lathe is a big tool”. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Bit Rate (Kbps)** & \begin{tabular}{c} **\% Frames with** \\ **\(D<24\)** \\ \end{tabular}  & \begin{tabular}{c} **\% Frames with** \\ **24 \(<D<72\)** \\ \end{tabular}  & \ \begin{tabular}{c} **\% Frames with** \\ **\(D>72\)** \\ \end{tabular}  & \begin{tabular}{c} **\% Frames with** \\ **PESQ-MOS** \\ \end{tabular}  \\ \hline
12.2 & 82.67 & 17.33 & 0 & 4.113 \\
10.2 & 80 & 20 & 0 & 4.084 \\
7.95 & 41.33 & 58.67 & 0 & 3.836 \\
7.4 & 45.33 & 54.67 & 0 & 3.871 \\
6.7 & 44 & 56 & 0 & 3.803 \\
5.9 & 53.33 & 46.67 & 0 & 3.807 \\
5.15 & 30.37 & 69.33 & 0 &A more subtle observation is that, for the sentence \"We were away a year ago\", changes in PESQ-MOS of more than 0.1 align with a significant increase in the percentage of frames satisfying \(24<D<72\). The converse can be stated for the sentence \"A lathe is a big tool\", namely that when the percentage of frames satisfying \(24<D<72\) increases substantially, there is a change in PESQ-MOS of 0.1 or more. There are two changes in PESQ-MOS of about 0.1 for this sentence, which correspond to only slight increases in the number of frames greater than the statistically significant threshold. Analyses of the number of bits allocated to the different codec parameters allow further important interpretations of \(D\), particularly in terms of the new expressions involving differential entropy and mutual information. Table 7 shows the bits/20 ms frame for each AMR-NB bit rate for each of the major codec parameter categories, predictor coefficients, pitch delay, fixed codebook, and codebook gains. Without discussing the AMR-NB codec, we point out in summary that, first, the AMR-NB codec does not quantize and code the predictor coefficients directly, but quantizes and codes a one-to-one transformation of these parameters called line spectrum pairs [15; 23], but we use the label \"predictor coefficients\", since these are the model parameters discussed in this paper. Further, as shown in Figure 5, the synthesizers or decoders in these speech codecs (as well as G.729) are linear prediction filters with two excitations added together, the adaptive codebook, depending on the pitch delay, and the fixed codebook, which is intended to capture the elements of the residual error that are not predictable using the predictor coefficients and the long-term pitch predictor.