## 4 Real and Imaginary Parts of Interference Correlation  
For further analysis, we present the expression for the symbols at the demodulator input, taking into account the MIMO channel, by substituting Equation (3) into Equation (6):  
\[\overset{\mathbf{A}}{\mathbf{A}}_{m_{0},n_{0}}=\sum\limits_{l=0}^{L-1}\sum \limits_{m=0}^{M-1}\sum\limits_{n\in Z}\mathbf{A}_{m,n}\mathbf{H}(l)g(k-l-nN )e^{j\frac{2\pi}{M}m(k-l-\frac{Lg-1}{2})}e^{j\phi_{m,n}}+\mathbf{n}(k), \tag{7}\]  
where \(\mathbf{A}_{m_{0},n_{0}}=\left[a_{1,m_{0},n_{0}},a_{2,m_{0},n_{0}},\dots,a_{T,m_{0},n_{0}}\right]\) is a row vector containing the characters to be transmitted by PAM. By separating the useful and interference parts of Equation (7) based on Equation (3), the symbols at the demodulator output before taking the real part and without considering noise \(\mathbf{n}(k)\) can be represented as:  
\[\overset{\mathbf{A}}{\mathbf{A}}_{m_{0},n_{0}}=\mathbf{A}_{m_{0},n_{0}} \overset{\mathbf{-0},0}{\mathbf{H}}_{m_{0}}+\mathbf{I}_{m_{0},n_{0}}, \tag{8}\]  
where the component \(\mathbf{A}_{m_{0},n_{0}}\overset{\mathbf{-0},0}{\mathbf{H}}\) describes the useful part of the symbol evaluation, taking into account the influence of the channel and the signal passing through the prototype filter with a delay, and \(\mathbf{I}_{m_{0},n_{0}}\) specifies the interference component, determined by the influence of all surrounding symbols on the given one:  
\[\mathbf{I}_{m_{0},n_{0}}=\sum\limits_{p,q\
eq(0,0)}{(-1)^{n_{0}p}\mathbf{A}_ {m_{0}+p,n_{0}+q}\overset{\mathbf{-p},q}{\mathbf{H}}_{m_{0}}^{p,q}}, \tag{9}\]  
where the matrix \(\overset{\mathbf{-p},q}{\mathbf{H}}_{m_{0}}^{p,q}\) is the FBMC-OQAM system transmultiplexer response describing the effect of a symbol with a shift on \(p\) symbols and \(q\) subcarriers from the \(m_{0},n_{0}\) symbol appearing due to a non-ideal separation by a prototype filter for rays arriving with a delay:  
\[\widetilde{\mathbf{H}}_{m_{0}}^{p,q}= \begin{pmatrix}\sum\limits_{l=0}^{L-1}h_ {1,1}(l)A_{g}(-qN-l,p)e^{-j\frac{2\pi}{M}m_{0}l}&\dots&\sum\limits_{l=0}^{L-1} h_{N_{T},N_{R}}(l)A_{g}(-qN-l,p)e^{-j\frac{2\pi}{M}m_{0}l}\\ \vdots&\ddots&\vdots\\ \sum\limits_{l=0}^{L-1}h_{N_{T},1}(l)A_{g}(-qN-l,p)e^{-j\frac{2\pi}{M}m_{0}l}& \dots&\sum\limits_{l=0}^{L-1}h_{N_{T},N_{R}}(l)A_{g}(-qN-l,p)e^{-j\frac{2\pi} {M}m_{0}l}\end{pmatrix}  \tag{10}\]  
After removing the channel effect using an ideal single-tap equalizer, calculated by the zero-forcing (ZF) criterion as:  
\[\mathbf{B}_{m_{0}}= \begin{pmatrix}\left[\overset{\mathbf{-0},0}{\mathbf{H}}_ {m_{0}}\right]^{T}\overset{\mathbf{-0},0}{\mathbf{H}}_{m_{0}}\end{pmatrix} ^{ -1}\left[\overset{\mathbf{-0},0}{\mathbf{H}}_{m_{0}}\right]^{T}, \tag{11}\]the demodulated signal can be expressed as:  
\[\overset{\wedge}{\mathbf{Z}}_{m_{0},n_{0}}=\mathbf{A}_{m_{0},n_{0}}+\mathbf{J}_{m _{0},n_{0}} \tag{12}\]  
where \(\mathbf{J}_{m_{0},n_{0}}\), similarly to Equation (9), is equal to  
\[\mathbf{J}_{m_{0},n_{0}}= \sum_{pq} (-1)^{n_{0}p}\mathbf{A}_{m_{0}+p,n_{0}+q}\overset{-0,0}{\mathbf{ H}}_{m_{0}}^{p,q}\mathbf{B}_{m_{0}} \tag{13}\] \[\
eq(0,0)\]  
Based on Equation (13), it is possible, with known true symbol values \(\mathbf{A}\), to calculate and compensate for the intrinsic interference that occurs in the FBMC-OQAM modem. The pseudoinverse matrix calculation in Equation (11) requires \(\overset{-0,0}{\mathbf{H}}_{m_{0}}\) to be a full row-rank matrix. For further analysis, we assume that \(\overset{-0,0}{\mathbf{H}}_{m_{0}}\) meets these conditions. However, in practice, the true values of \(\mathbf{A}\) are unknown, which requires a different way to estimate the intrinsic interference. Obviously, when using the real PAM modulation, after eliminating the influence of the channel, the imaginary part of the symbol estimate \(\hat{z}_{m_{0},n_{0}}\) will consist only of the imaginary part of the intrinsic interference \(\mathbf{J}_{m_{0},n_{0}}\); therefore, it can be easily defined in the receiver as \(\Im[\varepsilon_{m_{0},n_{0}}]\).