# A Multi-Frame PCA-Based Stereo Audio Coding Method
## 2 Multi-Frame PCA-Based Coding Method
### Principle of PCA  
The PCA's mathematical principle is as follows: after coordinate transformation, the original high-dimensional samples with certain relevance can be transferred to a new set of low-dimensional samples that are unrelated to each other. These new samples carry most information of the original data and can replace the original samples for follow-up analysis. There are several criteria for choosing new samples or selecting new bases in PCA. The typical method is to use the variance of new sample \(F_{1}\) (i.e., the variance of the original sample mapping on the new coordinates). The larger Var (\(Fi\)) is, the more information \(Fi\) contains. So, the first principal component should have the largest variance \(F_{1}\). If the first principal component \(F_{1}\) is not qualified to replace the original sample, then the second principal component \(F_{2}\) should be considered. \(F_{2}\) is the principal component with the largest variance except \(F_{1}\), and \(F_{2}\) is uncorrelated to \(F_{1}\), that is,  
Figure 1: Flowchart of mono encoder. (TF, Time-to-Frequency; PCA, Principle Component Analysis). Figure 2: Flowchart of mono decoder. (iPCA, inverse Principle Component Analysis; FT, Frequency-to-Time). \(\text{Cov}(F_{1},F_{2})=0\). This means that the base of \(F_{1}\) and the base of \(F_{2}\) are orthogonal to each other, which can reduce the data redundancy between new samples (or principal components) effectively. The third, fourth, and \(p\)-th principal component can be constructed similarly. The variance of these principal components is in descending order, and the corresponding base in new space is uncorrelated to other new base. If there are \(m\)\(n\)-dimensional data, the process of PCA is shown in Table 1.  
The contribution rate of the principal component reflects the proportion that each principal component accounts for the total amount of data after coordinate transformation, which can effectively solve the problem of dimension selection after dimensionality reduction. In PCA application, people often use the cumulative contribution rate as the basis for principal components selection. The cumulative contribution rate \(M_{k}\) of the first \(k\) principal components is  
\[M_{k}=\frac{\sum_{i=1}^{k}\lambda_{i}}{\sum_{i=1}^{n}\lambda_{i}} \tag{1}\]  
If the contribution rate of the first \(k\) principal components meets the specific requirements (the contribution rates are different according to different requirements), the first \(k\) principal components can be used to describe the original data to achieve the purpose of dimensionality reduction. PCA is a good transformation due to its properties, as follows:  
1. Each new base is orthogonal to the other new base;
2. Mean squared error of the data is the minimum after transformation;
3. Energy is more concentrated and more convenient for data processing. It is worthwhile noting that PCA does not simply delete the data of little importance. After PCA transformation, the dimension-reduced data can be transformed to restore most of the high-dimensional original data, which is a good character for data compression. In this paper, as is shown in Figure 3, the spectrum coefficients of the input signal are divided into multiple samples according to specific rules; then, these samples will be constructed to the original matrix \(X\). After the principal component analysis, matrix \(X\) is decomposed into reduced-dimensional matrix \(Y\) and rotation matrix \(P\); the process of calculating matrix \(Y\) and \(P\) is shown in Table 1. The matrix \(Y\) and \(P\) are transmitted to the decoder after quantization and coding. In decoder, the original matrix can be restored by multiplying reduced-dimensional matrix and transposed rotation matrix. There is some data loss during dimension reduction, but the loss is much less, so we can ignore it. For example, we can recover 99.97% information through a 6-dimension matrix, when the autocorrelation matrix has the 15 th dimension.