# Intrinsic Interference Cancellation Scheme for FBMC-OQAM Systems Based on Power Multiplexing
## 2 System Model
### Coded Auxiliary Pilots (CAP) Method for Intrinsic Interference Cancellation  
In order to make correct channel estimations, many methods are proposed to eliminate intrinsic interference to pilot symbols, among which the CAP method in literature [19] is a good solution. However, it eliminates intrinsic interference completely with sacrificing data symbols, resulting in a decrease of spectral efficiency. Based on CAP method, we propose a new method which has a higher spectral efficiency, whereas an introduction of CAP is necessary before our study. Details of CAP method are as follows.  
Denote OQAM data symbols around the pilot by \(\mathbf{a}_{CAP}=\left[a_{1}\,a_{2}\,\cdots\,a_{N-1}\right]^{T}\) and represent the coded symbols as \(\mathbf{s}_{CAP}=\left[s_{1}\,s_{2}\,\cdots\,s_{N}\right]^{T}\), which is obtained by \(\mathbf{s}_{CAP}=\left[s_{1}\,s_{2}\,\cdots\,s_{N}\right]^{T}\). The \(\mathbf{s}_{CAP}\) is the \(\(\mathbf{Ca}_{CAP}+\delta_{CAP}\), where \(\mathbf{C}\) is a coding matrix of size \(N\times(N-1)\) and \(\delta_{CAP}\) is a vector waiting to be solved. Denote coefficients of intrinsic interference in Table 1 by \(\gamma=[\gamma_{1}\;\gamma_{2}\;\cdots\;\gamma_{L}]^{T}\), and the interference generated by the surrounding data symbols to the pilot can be expressed as  
\[\theta_{CAP}=\boldsymbol{\gamma}^{T}\begin{bmatrix}\mathbf{s}_{CAP}\\ \mathbf{a}_{N+1,\;L}\end{bmatrix}=\boldsymbol{\gamma}_{1,\;N}^{T}\mathbf{s}_ {CAP}+\theta^{\prime}_{CAP}, \tag{6}\]  
where \(\mathbf{a}_{N+1,\;L}=[a_{N+1}\;a_{N+2}\;\cdots\;a_{L}]^{T}\), \(\boldsymbol{\gamma}_{1,\;N}=[\gamma_{1}\;\gamma_{2}\;\cdots\;\gamma_{N}]^{T}\) and \(\theta^{\prime}_{CAP}=\boldsymbol{\gamma}_{N+1,\;L}^{T}\mathbf{a}_{N+1,\;L}\). Supposing that the coding matrix \(\mathbf{C}\) satisfies the equation \(\boldsymbol{\gamma}_{1,\;N}^{T}\mathbf{C}=0\), (6) is then simplified as \(\theta_{CAP}=\boldsymbol{\gamma}_{1,\;N}^{T}\delta_{CAP}+\theta^{\prime}_{ CAP}\), where \(\theta_{CAP}\) must be equal to zero in order to eliminate intrinsic interference. Under these equality constraints, the vector \(\delta_{CAP}\) can be calculated as  
\[\delta_{CAP}=-\frac{\theta^{\prime}_{CAP}}{\|\boldsymbol{\gamma}_{1,\;N}\|^{2 }}\boldsymbol{\gamma}_{1,\;N}. \tag{7}\]  
Pilot symbols are no longer subjected to intrinsic interference after the coding operations described above, and the receiver can estimate channels correctly based on these pilot symbols. It is assumed that the channel is perfectly estimated, and vector \(\mathbf{\hat{s}}_{CAP}\) is an estimate of \(\mathbf{s}_{CAP}\) after equalization in the receiver. Supposing the coding matrix \(\mathbf{C}\) satisfies the equation \(\mathbf{C}^{T}\mathbf{C}=\mathbf{I}\), then the estimate of \(\mathbf{a}_{CAP}\) can be calculated by \(\mathbf{\hat{a}}_{CAP}=\mathbf{C}^{T}\mathbf{\hat{s}}_{CAP}\) at the receiver. Note that the coding matrix \(\mathbf{C}\) satisfying both \(\boldsymbol{\gamma}_{1,\;N}^{T}\mathbf{C}=0\) and \(\mathbf{C}^{T}\mathbf{C}=\mathbf{I}\) can be constructed based on [19], and the case for \(N=4\) used in this article is given directly as follows:  
\[\mathbf{C}=\frac{1}{\sqrt{2}}\begin{bmatrix}1&0&0.3903\\ 1&0&-0.3903\\ 0&1&0.9207\\ 0&1&-0.9207\end{bmatrix}. \tag{8}\]