# A Multi-Frame PCA-Based Stereo Audio Coding Method
## 3 Quantization Design Based On PVQ
### Design of the Low-Pass Filter  
The noise generated from quantization and matrix calculation is white noise. There are two ways to reduce it. The first way is introducing noise shaping to make noise more comfortable for human hearing, and the second way is introducing a filter in decoder.  
For most signals, the energy concentrates on low frequency domain, therefore the noise in low frequency domain does not sound obvious because of simultaneous masking. While in the high frequency part, if the original signal does not have high frequency components, the noise signal will not be masked and can be heard. So, a low-pass filter can be set to mask the high frequency noise signal, without affecting the original signal. The key point of the filter design is to determine the cut-off frequency.  
Figure 11: Bit allocation for rotation matrix (darker color means more bits needed; white color means no bits).  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Region** & **Bits Needed (bit/per Point)** \\ \hline The first region & 4 \\ The second region & 3 \\ The third region & 2 \\ The fourth region & 2 \\ The fifth region & 0.5 \\ The sixth region & 0.5 \\ The seventh region & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Quantization bits for rotation matrix.  
Given the original matrix \(X_{[\begin{array}{ccc}16&15\end{array}]}=\left[\begin{array}{ccc}a_{1}& \ldots&a_{225}\\ \vdots&\ddots&\vdots\\ a_{16}&\ldots&a_{240}\end{array}\right]\), there are 15 subbands in \(X\), in which the first subband is the first row, the second subband is the second row, and so on. When C = \(\frac{1}{m}X^{T}X\) is calculated in PCA, the first value \(e_{1}\) on the diagonal line \(e_{1},e_{2}\ldots e_{15}\) is calculated by  
\[\begin{array}{l}e_{1}=((a_{1}-\overline{a})*(a_{1}-\overline{a})+(a_{2}- \overline{a})*(a_{2}-\overline{a})+\ldots(a_{16}-\overline{a})*(a_{16}- \overline{a}))/16\\ \phantom{e_{1}=(a_{1}^{2}+a_{2}^{2}+\ldots a_{16}^{2}+16\overline{a}^{2}-2 \overline{a}(a_{1}+a_{2}+\ldots a_{16}))/16}\\ \phantom{e_{1}=(a_{1}^{2}+a_{2}^{2}+\ldots a_{16}^{2}-16\overline{a}^{2})/16 }\\ \end{array} \tag{4}\]  
in which \({a_{1}}^{2}+{a_{2}}^{2}+\ldots a_{16}^{2}\) is equal to the energy of the first subband \(E_{1}\), and \(\overline{a}\) is the average value of the first subband. Therefore, the relationship between \(E_{1}\) and \(e_{1}\) is  
\[E_{1}=16(e_{1}+\overline{a}^{2}) \tag{5}\]  
Actually, the value of \(\overline{a}^{2}\) is far less than \(e_{1}\), so \(E_{1}\) is equal to \(16e_{1}\), and the relationships between \(E_{2}\ldots E_{15}\) and \(e_{2}\ldots e_{15}\) can be gotten by analogy. Therefore, through PCA, the energy of each subband is calculated, and the filter can be determined by the energy of each band. Considering the proportion of energy accumulation, \(A_{k}\) is  
\[A_{k}=\frac{\sum_{i=1}^{k}e_{i}}{\sum_{i=1}^{15}e_{i}} \tag{6}\]  
According to some experiments, when \(A_{k}=99.6\%\), k is the proper cut-off band. When the signal passes through the filter, the noise signal will be filtered out, and the signal itself will not be too much damaged.  
Considering the frequency characteristics of the audio signal, the stop band setting is not low, and the signal with more than 20,000 Hz is often ignored by default, so each band of the above 15 bands will not be transmitted. Taken together, \(e_{1},e_{2},e_{3},e_{12},e_{13},e_{14},e_{15}\) will not be transmitted, and the index of the left 8 bands are quantized by 3 bits, so the bitrate for cut-off band is 75 bps.