## 4 Real and Imaginary Parts of Interference Correlation
In [7], it was first established that for the single-input-single-output (SISO) channel, there is a correlation between the real part of \(J_{m_{0},n_{0}}\) and the imaginary part of \(J_{m_{0},n_{0}\pm t}\), where \(t\) is a nonzero time shift. Thus, knowing the imaginary part of the interference and the correlation between the real and imaginary parts, it becomes possible to estimate the real part. To do this, we define the unnormalized correlation matrix between the real part of \(\mathbf{J}_{m_{0},n_{0}}(\mathbf{J}_{m_{0},n_{0}}^{\text{Re}})\) and the imaginary part of \(\mathbf{J}_{m_{0},n_{0}}\) (\(\mathbf{J}_{m_{0},n_{0}}^{\text{Im}}\)) for the given channel \(\mathbf{H}(l)\) as:  
\[C(t,m_{0})=E\bigg{[}\Big{(}J_{m_{0},n_{0}}^{\text{Re}}\Big{)}^{T}J_{m_{0},n_{0 }-t}^{\text{Im}}\bigg{]}=\sigma_{a}^{2}\mathbf{I}_{\mathbf{N}_{T}\times N_{T}} \sum_{ \begin{array}{c}p,q\\ \
eq(0,0)\end{array} }(-1)^{pt}\bigg{[}\text{Re}\bigg{(}\overset{-p,q}{\mathbf{ H}}_{m_{0}}^{p,q}\mathbf{B}_{m_{0}}\bigg{)}\bigg{]}^{T}\text{Im}\bigg{(} \overset{-p,q-t}{\mathbf{H}}_{m_{0}}^{p,q-t}\mathbf{B}_{m_{0}}\bigg{)} \tag{14}\] \[\
eq(0,0)\] \[\
eq(0,t)\]  
Since the mathematical expectation of the real and imaginary components of internal noise is zero, \(M\Big{[}\overset{-0}{\mathbf{J}}_{m_{0},n_{0}}^{\text{Re}}\Big{]}=M\Big{[} \overset{-0}{\mathbf{J}}_{m_{0},n_{0}}^{\text{Im}}\Big{]}=\mathbf{0}\) (based on Equation (13) and equality to zero \(M[\mathbf{A}]\)), and their variances are, respectively, equal to \(D\Big{[}\overset{-0}{\mathbf{J}}_{m_{0},n_{0}}^{\text{Re}}\Big{]}=\sigma_{RPI}^ {2}\mathbf{I}_{R\times R}\) and \(D\Big{[}\mathbf{J}_{m_{0},n_{0}}^{\text{Im}}\Big{]}=\sigma_{IPI}^{2}\mathbf{I }_{R\times R}\), where \(\mathbf{I}_{R\times R}\) is an identity matrix of size \(R\times R\). The variances \(\sigma_{IPI}^{2}\) and \(\sigma_{RPI}^{2}\), considering that \(M\Big{[}\overset{-0}{\mathbf{J}}_{m_{0},n_{0}}^{\text{Im}}\Big{]}=M\Big{[} \overset{-0}{\mathbf{J}}_{m_{0},n_{0}}^{\text{Re}}\Big{]}=\mathbf{0}\), equal  
\[\sigma_{RPI}^{2}\mathbf{I}_{Tr\times Tr}=M\bigg{[}\Big{(}\mathbf{J}_{m_{0},n_{0 }}^{\text{Re}}\Big{)}^{2}\bigg{]}=\sigma_{a}^{2}\mathbf{I}_{Tr\times Tr} \sum_{ \begin{array}{c}p,q\\ \
eq(0,0)\end{array} }\bigg{[}\text{Re}\bigg{(}\overset{-p,q}{\mathbf{H}}_{m_{ 0}}^{p,q}\mathbf{B}_{m_{0}}\bigg{)}\bigg{]}^{T}\text{Re}\bigg{(}\overset{-p,q}{ \mathbf{H}}_{m_{0}}^{p,q}\mathbf{B}_{m_{0}}\bigg{)} \tag{15}\] \[\sigma_{IPI}^{2}\mathbf{I}_{Tr\times Tr}=M\bigg{[}\Big{(}\mathbf{J }_{m_{0},n_{0}}^{\text{Im}}\Big{)}^{2}\bigg{]}=\sigma_{a}^{2}\mathbf{I}_{Tr \times Tr} \sum_{ \begin{array}{c}p,q\\ \
eq(0,0)\end{array} }\bigg{[}\text{Im}\bigg{(}\overset{-0}{\mathbf{H}}_{m_{ 0}}^{p,q}\mathbf{B}_{m_{0}}\bigg{)}\bigg{]}^{T}\text{Im}\bigg{(}\overset{-0}{ \mathbf{H}}_{m_{0}}^{p,q}\mathbf{B}_{m_{0}}\bigg{)}\] (16) \[\
eq(0,t)\]Thus, the normalized correlation coefficient can be defined as:  
\[\rho(t,m_{0})=\frac{\mathbf{C}(t,m_{0})}{\sigma_{RP1}\sigma_{IP1}} \tag{17}\]  
Figure 4 shows the dependence of the normalized correlation coefficient \(\rho\) on \(t\) for the \(2\times 2\) MIMO scheme, showing the presence of a theoretical relationship between the nearest FBMC-OQAM symbols.