# A Multi-Frame PCA-Based Stereo Audio Coding Method
## 3 Quantization Design Based On PVQ
### Quantization Design of the Rotation Matrix  
According to \(Y=XP\) in encoder and \(X_{restore}=YP^{T}\) in decoder, some properties of the rotation matrix can be found:  
1. The higher row in matrix \(P\) is used to restore the region of higher frequencies in the restored signal. 2. The first column in matrix \(P\) corresponds to the first principal component in the reduced-dimensional matrix. That means that the first column of the rotation matrix only multiplies with the first column (first principal component) of the reduced-dimensional matrix when calculating the restored signal in the decoder. The second column of the rotation matrix only \begin{table}
\begin{tabular}{l c} \hline \hline
**Principal Component** & **Bits Needed (bit/per Point)** \\ \hline First principal component & 3 \\ Second principal component & 3 \\ Third principal component & 2.5 \\ Fourth principal component & 1.5 \\ Fifth principal component & 0.45 \\ Sixth principal component & 0.45 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Quantization bit for reduced-dimensional matrix. Figure 9: Bits allocation for reduced-dimensional matrix (darker color means more bits needed). multiplies with the second column (second principal component) of the reduced-dimensional matrix, and so on. According to the above properties of the rotation matrix, the quantization distribution of the rotation matrix has been made clearer, that is, the larger the row number is, and the larger the column number is, the fewer allocation bits there are. In addition to the above two properties of the rotation matrix, there is another important property. Generally, the data in the first four rows around the diagonal are bigger than others. The thinking of this characteristic in this paper is as follows: common audio focuses more energy on low-band in frequency domain, and the method of matrix construction described in Section 2.4 can keep the coefficients of low-band stay in low-column. Thus, the first diagonal value that is calculated from the first column must be the largest one of overall values in rotation matrix or autocorrelation matrix. The second diagonal value could quite possibly be the second-largest value, and so on. That means these data are more important for decoder, so the quantization accuracy of these regions with larger absolute values can determine the error between the restored signal and the original signal. Therefore, the data around the diagonal need to be allocated with more bits. Figure 10 shows the \"average value\" rotation matrix of a piece of audio as an example to show this property more clearly. The rotation matrix also has the following quantization criterion:  
1. The first column of the rotation matrix needs to be precisely quantized, because the first principal component of the reduced-dimensional signal is only multiplied by the first column of \(P\) in decoder to restore signal. 2. Data in columns 2-6 in row 1 have little effect on the restored signal, so that few bits can be allocated for this region. 3. The higher row in matrix \(P\) is used to restore the region of higher frequencies in the restored signal. The data in lines 13, 14, and 15 correspond to the frequency that exceeds the range of frequencies perceptible to the human ear, so these data do not need to be quantized. According to the above quantization criteria, the rotation matrix that is divided into the following regions according to bit allocation is shown in Figure 11. The darker the color is, the more bits should be allocated. Figure 10: An example rotation matrix (“value” means the average value of cells in rotation matrices, “column” means the column of rotation matrix, and “row” means the row of rotation matrix). The same test method as the one for reduced-dimensional matrix was used to determine the number of bits needed in each region in rotation matrix. In Table 3, the first region corresponds to the region with the darkest color in Figure 11; the second corresponds to the area with the second-darkest color, and so on.