# Joint Interference and Phase Alignment among Data Streams in Multicell MIMO Broadcasting
## 4 Bayesian Estimation of Delay Error CSI  
In a communication system that needs feedback CSI, due to limited hardware performance and feedback CSI propagation delay, the CSI of the sending end often has both time delay and error. This section will use a Bayesian algorithm to predict CSI. The CSI vectors \(\mathbf{L}_{k_{\mathrm{g}}\cdot\mathrm{j}\tau}\) and \(\mathbf{L}_{k_{\mathrm{g}}\cdot\mathrm{j}}\) of the true channel \(\mathbf{H}_{k_{\mathrm{g}}\cdot\mathrm{j}}\) and the delay channel \(\mathbf{H}_{k_{\mathrm{g}}\cdot\mathrm{j}\tau}\) are, respectively:  
\[\mathbf{L}_{k_{\mathrm{g}}\cdot\mathrm{j}\tau}=vec\Big{(}\mathbf{H}_{k_{\mathrm{g}} \cdot\mathrm{j}\tau}\Big{)} \tag{3}\]  
\[\mathbf{L}_{k_{\mathrm{g}}\cdot\mathrm{j}}=vec\Big{(}\mathbf{H}_{k_{\mathrm{g}}\cdot \mathrm{j}}\Big{)} \tag{4}\]  
Let the time-varying channel satisfy the [PERSON] model [19], then the covariance of \(\mathbf{L}_{k_{\mathrm{g}}\cdot\mathrm{j}\tau}\) and \(\mathbf{L}_{k_{\mathrm{g}}\cdot\mathrm{j}}\) satisfies the joint Gaussian distribution, namely \(E\Big{\{}\mathbf{L}_{k_{\mathrm{g}}\cdot\mathrm{j}}\mathbf{L}_{k_{\mathrm{g}}\cdot \mathrm{j}\tau}^{H}\Big{\}}=\rho\mathbf{I}_{T}\), where \(T=N\times M,\rho=\mathbf{J}_{0}(2\pi\mathbf{f}_{d}\tau)\) is the correlation coefficient, \(\tau\) is the delay, \(\mathbf{f}_{d}\) is the maximum Doppler shift, and \(\mathbf{J}_{0}(x)\) is the first zero-order Bessel function. At this point, the statistical model of the equivalent channel is:  
\[\mathbf{L}_{k_{\mathrm{g}}\cdot\mathrm{j}}=\rho\mathbf{L}_{k_{\mathrm{g}}\cdot\mathrm{ j}\tau}+\mathbf{\sigma}_{k_{\mathrm{g}}\cdot\mathrm{j}} \tag{5}\]  
where \(\mathbf{L}_{k_{\mathrm{g}}\cdot\mathrm{j}}\) and \(\mathbf{L}_{k_{\mathrm{g}}\cdot\mathrm{j}\tau}\) obey a complex Gaussian distribution with a mean of 0 and a variance of 1. \(\mathbf{\xi}_{k_{\mathrm{g}}\cdot\mathrm{j}}\) and \(\mathbf{L}_{k_{\mathrm{g}}\cdot\mathrm{j}\tau}\) are mutually independent, and \(\mathbf{\xi}_{k_{\mathrm{g}}\cdot\mathrm{j}}\sim CN(0,\ \varrho_{\xi}^{2})\); \(\varrho_{\xi}^{2}=(1-|\rho|^{2})\mathbf{I}_{T}\). Here, the receiver only knows the current time and the CSI of the estimation error exists (i.e., \(\hat{\mathbf{L}}_{k_{\mathrm{g}}\cdot\mathrm{j}}\)) and the sender only knows the CSI before the time \(\tau\) and there is an estimation error (i.e., \(\hat{\mathbf{L}}_{k_{\mathrm{g}}\cdot\mathrm{j}\tau}\)). Thus, when \(\mathbf{L}_{k_{\mathrm{g}}\cdot\mathrm{j}\tau}\) and \(\mathbf{L}_{k_{\mathrm{g}}\cdot\mathrm{j}}\) have estimated errors \(\mathbf{e}_{k_{\mathrm{g}}\cdot\mathrm{j}\tau}\) and \(\mathbf{e}_{k_{\mathrm{g}}\cdot\mathrm{j}}\), respectively, the corresponding expression should be:  
\[\hat{\mathbf{L}}_{k_{\mathrm{g}}\cdot\mathrm{j}\tau}=\bmHere, we use the Bayesian algorithm [20] to estimate \(\hat{L}_{k_{g}j\tau}\) (real-time CSI) from \(L_{k_{g}j}\) (time delay error CSI). First, calculate the conditional probability \(E\left(L_{k_{g}j}\middle|\hat{L}_{k_{g}j\tau}\right)\) of \(L_{k_{g}j}\), and the conditional covariance matrix \(D\left(L_{k_{g}j}\middle|\hat{L}_{k_{g}j\tau}\right)\) as:  
\[E\left(L_{k_{g}j}\middle|\hat{L}_{k_{g}j\tau}\right)=E\left(L_{k_{g}j}L_{k_{g}j \tau}^{H}\right)\times E\left(\hat{L}_{k_{g}j\tau}L_{k_{g}j\tau}^{H}\right)^{-1 }\hat{L}_{k_{g}j\tau}=\rho\left(diag\left(1+\delta_{e}^{2}\right)_{T\times T} \right)^{-1}\hat{L}_{k_{g}j\tau} \tag{11}\]  
\[D\left(L_{k_{g}j}\middle|\hat{L}_{k_{g}j\tau}\right)=E\left(L_{k_{g}j}L_{k_{g}j }^{H}\right)-E\left(L_{k_{g}j}\hat{L}_{k_{g}j\tau}^{H}\right)\times E\left( \hat{L}_{k_{g}j\tau}\hat{L}_{k_{g}j\tau}^{H}\right)^{-1}E\left(\hat{L}_{k_{g}j \tau}\hat{L}_{k_{g}j\tau}^{H}\right) \tag{12}\]  
For this reason, under the time delay error CSI, the actual channel \(\mathbf{H}_{k_{g}j}\).