# A Multi-Frame PCA-Based Stereo Audio Coding Method
## 4 PCA-Based Parametric Stereo
### Procession of Stereo Signal  
Since the signals in two channels of the stereo tend to have high correlation. The signal of the left and right channels can be constructed into one original matrix. Firstly, the coefficients from left channel and right channel construct original matrices \(X_{I[m\ n]}\) and \(X_{I[m\ n]}\) respectively. Then, matrices \(X_{I[m\ n]}\). and \(X_{r[m\ n]}\) are used to form a new matrix \(X\), in which \(=\left[\begin{array}{c}X_{I[m\ n]}\\ X_{r[m\ n]}\end{array}\right]\). Matrix \(X\) is used to obtain one rotation matrix \(P_{[n\ k]}\) by PCA, and \(P_{[n\ k]}\) can handle both left and right channel signals. That is,  
\[Y_{I[m\ k]} = X_{I[m\ n]}\times P_{[n\ k]} \tag{7}\] \[Y_{r[m\ k]} = X_{r[m\ n]}\times P_{[n\ k]} \tag{8}\]  
If the first six principal components are preserved, most mono audio signals can be well restored. At this time, we keep the first six bases in principal component matrix and obtain rotation matrix \(P_{[}\ \ 15\ \ \ 6\ \ \ ]\). The reduced-dimensional matrices of each sub-frame are \(Y_{1[156]},Y_{2[156]},\ldots,Y_{8[156]}\). Experiments were done to verify the design for stereo signals: 10 normal audio files and 5 artificial synthesized audio files (the left channel and right channel have less correlation) were chosen as the test materials. Results of the subjective listening experiments are shown in Figures 14 and 15. We can consider that for most stereo signals, in which two channels have high relevance with each other, the proposed method for stereo signals perform as well as for mono signals.  
Figure 12: Flowchart of stereo encoder. (MDCT, Modified Discrete Cosine Transform; PCA, Principle Component Analysis; IC, Interaural Coherence; ILD, Interaural Level Difference).  
Figure 13: Flowchart of stereo decoder. (MDCT, Modified Discrete Cosine Transform; iPCA, inverse Principle Component Analysis; IC, Interaural Coherence; ILD, Interaural Level Difference).  
in which \(Y(k)\) is the spectrum coefficients, \(x(n)\) is the input signal in time domain, and \(w(n)\) is the window function. Then, a new transform MDFT is introduced, \(Z(k)=X(k)+jY(k)\), in which \(X(k)\) is the MDCT spectral coefficients, \(Y(k)\) is the MDST spectral coefficients, and IPD can be calculated by  
\[\text{IPD[b]}=\angle\left(\sum\limits_{k=A_{b-1}}^{A_{b}-1}Z_{l}(k){Z_{r}}^{*} (k)\right) \tag{12}\]  
Traditional decoder uses these parameters and a downmix signal to restore left channel's signal and right channel's signal. Compared with formula (4, 9, 10), when the method described in Section 4.1 is used to deal with stereo signals, \(\sum\limits_{k=A_{b-1}}^{A_{b}-1}X_{l}(k)X_{l}(k)\) and \(\sum\limits_{k=A_{b-1}}^{A_{b}-1}X_{r}(k)X_{r}(k)\) can be calculated in the processing of PCA; therefore, parametric stereo and PCA have high associativity. After PCA, we can get ILD and IC only by calculating \(\langle X_{bl}(k),X_{br}(k)\rangle\). In addition, we also need to calculate IPD by Formula (12); however, introducing MDST will bring computational complexity, and ITD or IPD mainly works on signals below 1.6 kHz that play smaller roles in high frequency domain. Thus, some improvements can be made to the parametric stereo according to the nature of the PCA.